A GAME THEORETIC APPROACH TO

ANALYZING CONTAINER TRANSSHIPMENT PORT

COMPETITION

BAE MIN JU

NATIONAL UNIVERSITY OF SINGAPORE

2013

A GAME THEORETIC APPROACH TO ANALYZING CONTAINER TRANSSHIPMENT PORT

COMPETITION

BAE MIN JU

(M.Eng., Korea Maritime University, South Korea)

A THESIS SUBMITTED

FOR THE DEGREE OF MASTER OF ENGINEERING

DEPARTMENT OF INDUSTRIAL AND SYSTEMS ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2013

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Declaration

I hereby declare that this thesis is my original work and it has been written by me in its

entirety. I have duly acknowledged all the sources of information which have been used in

the thesis.

This thesis has also not been submitted for any degree in any university previously.

Bae Min Ju

22 April 2013

  

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Acknowledgements

First and foremost I offer my sincerest gratitude to my supervisors, Associate Professor Ek

Peng Chew and Loo Hay Lee who supported me with kind guidance and great

encouragement. This thesis would not have been possible without their help, support and

patience. I am also grateful to all the other faculty members and staffs in the Department of

Industrial and Systems Engineering, especially Ms Lai Chun Ow and Ms Celine Neo for their

heartwarming support. My sincere thanks go to examiners of my thesis, Associate Professor

Qiang Meng, from Department of Civil and Environmental Engineering, and Kien Ming Ng,

for their thorough review, critical comments and constructive advices. I would like to

acknowledge Professor Anming Zhang, from Sauder School of Business in University of

British Columbia, for his kindness and invaluable comments during the preparation of the

conference paper.

My special thanks to fellow postgraduate students in the Department of Industrial

and Systems Engineering, Liqin Chen, Yinghui Fu, Juxin Li, Qiang Wang, Qi Zhou, Aoran

Mu, Yi Luo and Nugroho Artadi Pujowidianto for their kindness and cherished friendship.

I am deeply indebted to my parents for their unconditional love and support in all

my pursuits and my parents-in-law for their great help. Finally, I would express my deepest

appreciation to my loving, supportive, encouraging and patient husband Ming Guang and my

adorable daughter Shanyi. They have been the greatest source of strength and support in my

work and in my life.

Min Ju Bae

National University of Singapore

April 2013

  

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Contents

Declaration................................................................................................................................. i 

Acknowledgements .................................................................................................................. ii 

Summary ................................................................................................................................... v 

List of Tables ........................................................................................................................... vi 

List of Figures ......................................................................................................................... vii 

List of Symbols ......................................................................................................................... x 

List of Abbreviation ................................................................................................................ xi 

Chapter 1 Introduction............................................................................................................ 1 1.1  Background .............................................................................................................................. 1 1.2  Motivation ............................................................................................................................... 3 1.3  Objective.................................................................................................................................. 5 1.4  Overview ................................................................................................................................. 6 

Chapter 2 Literature Review .................................................................................................. 8 2.1  Port Selection ........................................................................................................................... 8 2.2  Port Competition ................................................................................................................... 10 

2.2.1  Empirical Research .................................................................................................... 10 2.2.2  Game Theory ............................................................................................................. 11 

2.3  Game Theory in Aviation Industry ........................................................................................ 13 

Chapter 3 Base Model Development .................................................................................... 15 3.1  Overview ............................................................................................................................... 15 3.2  The Model ............................................................................................................................. 16 3.3  A Non-Cooperative Two-stage Game ................................................................................... 19 

3.3.1  Stage two: Shipping Lines’ Port Call Decision.......................................................... 20 

3.3.2  Stage one: Port Pricing Strategies ............................................................................. 23 3.4  Ports Collusion and Social Optimum .................................................................................... 23 

3.4.1  Ports Collusion Model ............................................................................................... 23 3.4.2  Social Optimum Model ............................................................................................. 24 

3.5  Numerical Results ................................................................................................................. 25 3.7  Conclusions ........................................................................................................................... 33 

  

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Chapter 4 Nonlinear Transshipment Demand and Multiple Nash Equilibria ................ 35 4.1  Overview ............................................................................................................................... 35 4.2  The model .............................................................................................................................. 35 

4.3  Non-cooperative two-stage game .......................................................................................... 37 

4.4  Numerical Analysis ............................................................................................................... 38 

4.4.1  Shipping lines’ port call decision ............................................................................... 38 

4.4.2  Port Pricing Strategies ............................................................................................... 64 4.5  Conclusions ........................................................................................................................... 65 

Chapter 5 Analysis on Asymmetric Shipping lines ............................................................. 67 5.1  Overview ............................................................................................................................... 67 5.2  The model .............................................................................................................................. 67 5.3  Numerical analysis ................................................................................................................ 69 

5.3.1  Asymmetric Shipping lines’ Port Call Decision ........................................................ 69 

5.4  Conclusion ............................................................................................................................. 79 

Chapter 6 Conclusions ........................................................................................................... 81 6.1  Conclusions of the Study ....................................................................................................... 81 6.2  Future Research ..................................................................................................................... 82 

Bibliography ........................................................................................................................... 84 

Appendix A ............................................................................................................................. 89 

Appendix B ............................................................................................................................. 90 

Appendix C ............................................................................................................................. 94 

Appendix D ............................................................................................................................. 96 

Appendix E ............................................................................................................................. 98 

Appendix F ........................................................................................................................... 100 

Appendix G ........................................................................................................................... 102 

Appendix H ........................................................................................................................... 104 

Appendix I ............................................................................................................................ 107 

  

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Summary

This thesis investigates the competition between transshipment container ports. The main

purpose of this thesis is to develop the model that can capture the trends in transshipment

container port competition and behaviours of key players, ports and shipping lines, in their

decision making.

Studying industrial events and relevant literatures led us to choose the key factors

that play an important role in port competition problem. The key determinants such as port

capacity, price, congestion and transshipment level are taken into account model

development. Subsequently, multiple functions are created to examine the interdependency

among shipping lines when determining port demand.

With a linear transshipment container demand, it is able to achieve analytical

properties through a two-stage game approach. Considering a nonlinear transshipment

container demand in the model, it results in Multiple Nash equilibria in shipping lines’ port

call decision. Asymmetric shipping lines are also studied in the model for investigating their

demand driving forces.

Overall, this thesis addresses the characteristics of transshipment container

demand and shipping lines’ port call decision towards transshipment benefit and congestion-

associated effect. Most of all, this thesis can help advance the analysis on ports’

transshipment capabilities and enable the ports to uncover and balance its demand and

capacity levels so as to strategize for the long term. It can also provide better visibility on the

shipping lines’ criteria when choosing a transshipment port.

  

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List of Tables

Table 4.1 Symmetric port capacity levels……….…………………………...……………43

Table 4.2 Summary of port call strategy under various symmetric capacities……….………48

Table 4.3 Summary of port call strategy under changes in symmetric prices……….……….52

Table 4.4 Summary of port call strategy under changes in symmetric TS levels……….…53

Table 4.5 Asymmetric port capacity levels……….………………………………………….56

Table 4.6 Summary of port call strategy under asymmetric capacities……….……………58

Table 4.7 Summary of port call strategy under asymmetric prices……….…………………60

Table 4.8 Summary of port call strategy under asymmetric TS levels ……….…………63

Table 5.1 Symmetric port capacity levels for ASLs……….………………………………...70

Table 5.2 Experimental cases for proposed pricing strategy to ASLs……….………………77

  

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List of Figures

Figure 1.1 World container port traffic 1990-2011 ……….…………………………………..1

Figure 1.2 Trends and challenges in container shipping market……….……………………...2

Figure 2.1 Port selection criteria found in port selection literatures……….….………………9

Figure 3.1 Market structure and key variables……….………………………………………16

Figure 3.2 Effect of price differences and capacity levels on SL’s port call decision……….26

Figure 3.3 Effect of transshipment and capacity levels on SL’s port call decision……….....27

Figure 3.4 Equilibrium port prices while varying capacity differences……….……..............28

Figure 3.5 Equilibrium port prices while varying transshipment level differences……….…29

Figure 3.6 Comparison between Non-cooperative and Social optimum model….…..............29

Figure 4.1 SLs’ BRCs with a Unique Nash Equilibrium……..…….………………………..40

Figure 4.2 Multiple Nash equilibria and Stability……..…..……..…………………………..41

Figure 4.3 Multiple Nash equilibria and Robustness……..………………….……………....41

Figure 4.4 SLs’ BRCs when symmetric capacities at CU=100%……..…..….………….......43

Figure 4.5 Trends of SL 2’s TS benefits, congestion cost and total profit when 11q =1 …….44

Figure 4.6 SLs’ downward sloping BRCs under various levels of symmetric capacities …...45

  

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Figure 4.7 SLs’ upward sloping BRCs under various levels of symmetric capacities…........46

Figure 4.8 SLs’ BRCs under different transshipment effects…..............................................49

Figure 4.9 Summary port call strategy under different transshipment effects….....................49

Figure 4.10 Comparison base case: CU=80%, 0.3g , 0.3 …..........................................50

Figure 4.11 Comparison of base case with changes in symmetric capacities and prices….....51

Figure 4.12 Comparison of base case with changes in symmetric capacities and TS levels...53

Figure 4.13 SLs’ BRCs when 1 2K > K …................................................................................57

Figure 4.14 SLs’ BRCs when 1 2 ……………………………………………………….59

Figure 4.15 SLs’ BRCs with different levels of price difference at CU=20%.........................60

Figure 4.16 SLs’ BRCs when 1 2g g ……………………………………………………..…62

Figure 4.17 SLs’ BRCs with different levels of TS differences at CU=20%……………..…63

Figure 4.18 Unique Nash equilibrium in port pricing………………………………………..64

Figure 5.1 ASLs’ BRCs under various levels of symmetric capacities……………………...71

Figure 5.2 Comparison base case: CU=80%, g =0.3, = 0.3………………………………72

Figure 5.3 Comparison of base case with changes in symmetric capacities and prices…...…73

Figure 5.4 Comparison of base case with changes in symmetric capacities and TS levels.…74

Figure 5.5 ASLs’ BRCs when 1 2K K …………………………………………………....…75

  

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Figure 5.6 ASLs’ BRCs when 1 2K K at CU=20%…………………………………….……75

Figure 5.7 ASLs’ BRCs when 1 2g g ………………………………………………….……76

Figure 5.8 ASLs’ BRCs when 1 2g g at CU=20%…………………………………….….…77

Figure 5.9 ASLs’ BRCs under proposed pricing strategies at CU=90%…………………….78

Figure 5.10 ASLs’ BRCs under proposed pricing strategies at CU=30%…………………...78

  

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List of Symbols

f : coefficient of loaded and unloaded gateway container demand

rg : coefficient of loaded and unloaded transshipment container demand at port r

N : number of shipping lines

irq : a fraction of transshipment port calls made by shipping line i at port r

ip : price per container charged by shipping line i

ic : shipping line i ’s constant per container cost

in : shipping line i ’s total port call size

irF : total loaded and unloaded containers of shipping line i ’s at port r

i : shipping line i ’s profit

r : port r ’s price per container

ir : port r ’s price per container charged to shipping line i

rO : port r ’s operation cost per container

rm : port r ’s capacity marginal cost

rK : capacity of port r

  

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r : profit of port r

1

N

r iriQ q

: total port calls at port r

1

N

r iriF F

: total container volume at port r

,r r rD F K : congestion delay cost per container at port r

List of Abbreviation

SWOT : Strengths Weaknesses Opportunities and Threats

LTCD : Linear Transshipment Container Demand

NTCD : Nonlinear Transshipment Container Demand

BR : Best Response

BRC : Best Response Curve

NE : Nash Equilibrium

CU : Capacity Utilization

SL : Shipping Line

ASL : Asymmetric Shipping Line

TS : Trans-Shipment

KKT : Karash Kuhn-Tucker

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Chapter 1 Introduction

Introduction

1.1 Background

Container transshipment market, as shown in Figure 1.1, has been expanding

significantly during last two decades. Growth in demand for transshipment containers

has been led by several changes in container shipping market. These changes, as an

attempt to be a price competitive and improve services, aim towards cost savings

through the emergence of the global and total logistics.

Figure 1.1 World container port traffic 1990-20111

                                                            1 Drewry Shipping Consultants Ltd.

  

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Figure 1.2 Trends and challenges in container shipping market2

In particular, shipping lines, the key player in container transport market, develop the

larger form of strategic arrangements such as alliances. Globalization of shipping lines

gave a strong market power to shipping alliances as international shipping lines have

more choices in calling at ports. As a result, alliances and other cooperative

arrangements have power to control significant cargo flows on the major global trade

routes. With much appreciation of advanced technology and importance of economies

of scale in ship size, currently 8,000-10,000 TEU ships dominated major Asian trade

routes. Since 2002, ships above 6,000 TEU have come into operation on Asian routes

and some of the shipping lines have been deploying even larger ships (16,000 TEU is

the largest ship size by 2012). The implication of increase in ship size has been given

great attention to the hub and spoke system, where the biggest ships will call at only a

limited number of efficient ports on the main trade routes, with other ports being

connected by extended feeder networks. As illustrated in Figure 1.2, the strategic

alliances of global shipping lines and ever growing ship size, together with hub and

                                                            2 Ocean Shipping Consultants (2006). Report. “East Asian Container port markets to 2020”.  

  

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spoke system resulted in expansion of transhipment market. Major container ports are

competing to be a regional hub to capture this significant amount of container demand.

The competition status of transshipment hub ports especially in Asia region, where the

dominant ports are located in, is getting intense. Moreover, current dominant

transshipment hub ports are being challenged by the emergence of other fast-

developing ports that are located within proximity of the region.

1.2 Motivation

In 2000, Maersk Sealand relocated its major transshipment operations from the Port of

Singapore (PSA) to the Port of Tanjung Pelepas (PTP) in Malaysia. The impact of this

relocation on the regional transshipment market structure was significant. Maersk

Sealand was then the largest shipping operator in Singapore. Its shift to PTP resulted

in a decline of approximately 11% in PSA’s overall business. In 2001, PSA’s total

container throughput fell from 17.09 million TEUs to 15.52 million TEUs (Tongzon

2006), marking a year-on-year drop of 8.9 %. In the same period, PTP’s container

throughput had increased nearly 5 folds, from 0.42 million to 2.05 million TEUs.3The

shipping industry in Singapore and the region grew concerned about Maersk Sealand’s

relocation and the potential ripple effect on other shipping lines’ decisions and related

business activities (Allison 2000; Kleywegt et al. 2002). As shipping lines form

strategic alliances to achieve economies of scale, the interdependency among alliance

members and small- and medium-size shipping lines heightens. Consequently, Maersk

Sealand’s decision on changing its transshipment port-of-call could well induce

similar decisions among affiliating carriers. In 2002, Evergreen and its subsidiary

Uniglory also shifted most of their container operations, amounting to 1-1.2 million

                                                            3 The container throughput data is taken from the official website of Port of Tanjung Pelepas, http://www.ptp.com.my/history-2000.html and http://www.ptp.com.my/history-2001.html.

  

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TEUs of annual throughput, from PSA to PTP. Since then, other shipping lines have

also started to provide direct services to PTP. APL, for example, had chosen PTP for

its West Asia Express service between Asia and the Middle East (Kleywegt et al.

2002).

In the case of competition between PTP and PSA, the acquisition of

transshipment cargo is critical. Both ports are subjected to stringent growth limitations

as gateway ports but possess excellent locations along the Strait of Malacca.

Transshipment presents a good opportunity for these ports to expand beyond the

demands of their respective catchment economies and more importantly, tap into the

international cargo flows to enjoy superior profits. Beyond the potential spike in the

number of cargo handling jobs and value-added activities, a transshipment port would

also gain access to profitable feeder line networks which serve to transport containers

to/from tributary ports4. These networks give transshipment ports good connectivity,

which in turn strengthens itself through the ripple effect. As the importance of

achieving dominance in the market becomes apparent, it is foreseeable for regional

ports to compete for transshipment container traffic.

There are many possible attributions to the above mentioned relocations

(mainly from the cost and operational perspectives), of which port pricing emerges as

one of the probable causes. As global customers exert increasing pressure on shipping

lines to lower their prices, the competition to reduce costs among shipping lines

inevitably intensifies. Shipping lines are forced to explore options which give the most

cost-saving. With these drivers in mind, the attractiveness of PTP’s port price, which

was some 30% lower than that of PSA’s at that time, becomes apparent. In fact,

                                                            4 The feeder line networks between transshipment port and tributary port have been mentioned in the recent study of Low and Tang (2012) as an example of indirect network effect.

  

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Evergreen had estimated that their shift to PTP would save them between US$ 5.7

million and US$ 30 million per annum (Kleywegt et al. 2002).

A similar event took place in February 2006 between two major

transshipment ports in South Korea, Busan and Gwangyang port. Maersk, after

merging with P&O Nedlloyd, relocated 50-60% of P&O’s transshipment operation

that was originally handled in Busan port, to Gwangyang port. As a result, Busan port,

after 6-month later, lost about 250,000 TEU from its transshipment business with P&O

Nedlloyd. Two major reasons behind this event were found, i.e., port charge and

congestion. The port charge in Busan port at that time was almost double compared to

Gwangyang port’s charge. Moreover, Gwangyang port provided a dedicated berth,

thus there is no congestion issue to shipping line, while Busan port had high

congestion problems due to its general usage of the berth.5

Considering the abovementioned industrial events, this thesis attempts to

study a regional hub port competition for transshipment containers with respect to port

price and congestion associated issues.

1.3 Objective

This study tries to understand the complexity of the situation in port competition,

especially for competition over the transshipment cargos. Author believes that to

understand the real world problems, the modeling a simple, yet representative version

of the real situation must take precedence, so as to appreciate the broad characteristics.

Therefore, this study seeks to simplify the underlying complexities in exchange for a

more mathematical approach to the port competition problem. In particular, this thesis

                                                            5 KT Press(2006) and Gynet (2006)

  

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aims to provide the managerial insight to explain the trend and behaviour of ports and

shipping lines in transshipment container market. This thesis is a fundamental study to

develop a model that provides insights on basic market behaviors with potential

possibility to extend the model uncovering real situation of port competition problem

in the future.

1.4 Overview

The remaining parts of this thesis are organized as follows:

In Chapter 2, studies relevant to port competition are reviewed. The key factors

considered in this study are found in port selection literatures, where it supports

our choice of critical determinants. Port competition studies that adopted an

empirical method and game theory suggest a way to approach the port

competition problem for this study.

In Chapter 3, the base model for analysing transshipment port competition is

developed with a linear transshipment container demand function. It presents

the findings obtained from both analytical works and numerical experiments.

In Chapter 4, our base model is advanced to a nonlinear transshipment

container demand. In this case, Multiple Nash equilibria occur in shipping lines’

port call decision. The intensive numerical analyses are conducted in order to

explore the combined effect among key factors.

In Chapter 5, some pricing strategies are proposed to study the preferences of

asymmetric shipping lines. It is also investigated a demand driving force in

asymmetric shipping lines.

  

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Finally, Chapter 6 discusses about limitations and contributions of this thesis, and

provides comments on possible future research.

  

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Chapter 2 Literature Review

Literature Review

2.1 Port Selection

Port selection6 studies usually aim to find the key factors that affect the port users’

decision for selecting a port. In other words, it approaches port competition problem

from the port users’ perspective. There is an issue on decision maker for selecting a

port; shippers (Slack 1985; Bird and Bland 1988; Murphy and Daley 1994; Tongzon

2002; Nir et al. 2003; Tiwary et al. 2004) or shipping lines7. Although Slack (1985)

conducted his study from the shipper’s perspective, he mentioned in his analysis of

survey results that shipping lines are the key actors in the port selection process. Also,

D’Este et al.(1992a,1992b) suggested that as shipping lines increased their scale of

operations and shippers began soliciting prices for door-to-door service rather than

individual segments, the port selection shifted from the shipper to the shipping lines.

Tongzon and Sawant (2007) have stated that globalization, mergers and acquisitions

among shipping lines enable to form the greater volumes that are controlled by a

single line or alliance. This implies that the capacity of shipping line to seriously affect

the business of a port is much greater than it has been in the past. This warrants a need

for the port operators to understand the underlying factors of port competitiveness

from the shipping lines’ perspective. Therefore, more recent studies approach port

                                                            6 “Port choice” is often used as similar term. 7 In the literatures, carriers or ocean carriers often used to represent shipping lines.

  

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selection problem from the shipping lines’ perspective (D’Este et al., 1992a; 1992b;

Tongzon and Sawant 2007; Chang et al. 2008). Based on this perspective, this thesis

considers shipping lines as main port users.

Figure 2.1 summarizes the port selection criteria found in the existing port

selection literatures. In particular, with increasing importance of the port function as a

transshipment facility, recent port selection studies have paid attention to

transshipment port selection problem. Lirn et al.(2003; 2004) found in their studies

that the transshipment port selection problem is depending mainly on port

competitiveness and efficiency, as represented by the cost and the container loading

and discharging rates, respectively. Similarly, Chou (2007) suggested port manager

that if they want to become a transshipment hub port, it will be the most efficient way

to attract ocean carriers by increasing the volume of import/export/transshipment

containers and decreasing port charge. Chang et al.(2008) considered port selection

problem from the trunk liners’ and feeder service providers’ perspective. Authors

concluded that local cargo volume, terminal handling charge, berth availability, port

location, transshipment volume and feeder network are the most important factors.

Figure 2.1 Port selection criteria found in port selection literatures

  

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Our consideration of key factors in transshipment port competition problem, i.e. port

price, port capacity, transshipment level and congestion cost, seems valid as these

factors have been consistently selected in the existing literatures. Therefore, we will

take into account these factors to our model development.

2.2 Port Competition

The methodologies adopted in port competition studies are summarized mainly into

the categories of empirical research and game theory.

2.2.1 Empirical Research

Existing empirical researches have used various models to analyse actual data to draw

the insights of competition between or among ports. Chou et al.(2003) used SWOT

analysis to analyze the competitiveness of major container ports in Asia eastern region,

where the major ports are including Hong Kong, Singapore, Busan, Kaohsiung and

Shanghai. Yang and Yang (2005) attempted to develop an index system to evaluate

container port competition ability. For this, authors combined the empirical survey

method and AHP(analytical hierarchy process), and optimal path-searching algorithm.

An AHP analysis has been used by Yuen et al.(2012) to study port competitiveness.

Authors approached the problem from the users’ perspective and studied the major

container ports in China and neighboring countries. Adopting the branch of regression,

Veldman and Bückmann(2003) used a logit model for routing options and derived the

demand function to be used for port traffic forecasting and for the economic and

financial evaluation of container port project. The authors calibrated logit models in

the framework of the evaluation of the Massvlakte-2 container port expansion project

in the port of Rotterdam. Fung(2001) had set up the vector error correction

  

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model(VECM) with structural identification to capture the trade-interdependency and

oligopolistic relationship in the East and Southeast Asian market for container

handling services. Similarly, Yap and Lam(2006) employed unit root test,

cointegration test and error correction model to analyze given ports data.

In general, port competition studies approached by empirical method can

build theories and generate the hindsight wisdom as it is based on real data. However,

it is found that the results drawn from particular data may be limited to specific cases,

thus the method is more restrictive in usage.

2.2.2 Game Theory

Zan (1999) was one of the earliest authors who had attempted to use the game theory

to investigate the behaviour of port users (carriers and shippers) in transshipment port

management policy. He used a bi-level Stackelberg game to capture the flow of

foreign trade containers. Lam and Yap (2006) approached regional port competition

problem with Cournot simultaneous quantity setting model. Their model is used to

derive the overall costs of using the terminal, and applied to competition between

container terminal operators in Singapore, Port Klang and Tanjung Pelepas. Anderson

et al.(2008) developed a game-theoretic best response frame work for understanding

how competitor ports will respond to development at a focus port, and whether the

focus port will be able to capture or defend market share by building additional

capacity and applied their model to investment and competition currently occurring

between the ports of Busan and Shanghai. Saeed and Larsen (2010) studied intra-port

competition and examined the possible combinations of coalitions among container

terminals. The two-stage game has been employed to analyse possible coalitions: in

the first stage, three container terminals at Karachi Port decide whether to act

  

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individually or to join a coalition; and in the second stage, the resulting coalition plays

a non-cooperative game against non-members. De Borger et al. (2008) used a two-

stage game to analyse the interaction between the pricing behaviour of competing

ports and the optimal investment policies in the ports and hinterland capacity. Similar

form of analysis has been used in recent port competition problem that is further

investigated as part of rivalry between two alternative intermodal transportation chains;

hence, recent studies have taken into account hinterland access and road congestion in

order to observe their impact on ports and port competition (Zhang, 2008; Yuen et al.,

2008; Wan et al., 2012; Wan and Zhang, 2013). As noticed, it is only recently that

scholars and industry start to pay attention to applying game theory to competition

problem in maritime industry; hence, relatively not many studies have tackled the port

competition problem by game theory approaches, especially for transshipment port

competition.

Game theory provides a framework for analysing any competitive situation

game. We can identify the market players, their possible actions and reactions to the

actions of rivals, and the payoffs or rewards implicit in the game. Game theory models

reduce the world in which businesses operate from a highly complex one to one that is

simpler, while retaining some original important characteristics. By capturing and

clarifying the most significant aspects of competition and interdependence, game

theory models make it possible to extract a complex competitive situation into its key

components so as to analyse the complex dynamics between players. The main

assumption behind the game theory is that the players in a game are behaving

rationally, hence choosing their actions optimally; that is, players are choosing their

actions in the aim of maximising their payoff and that the other players are doing

likewise. This is especially valuable because it helps companies choose the right

  

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business strategies when confronted with a complex strategic situation. Moreover, the

concept of Nash equilibrium solution in game theory gives a strategy profile in the

competition. As we intend to approach port competition problem from the

optimization perspective, it would be pertinent to applying game theory to the ports

and shipping lines within the port competition arena.

2.3 Game Theory in Aviation Industry

As defined by De Borger and Van Dender (2006), congestible facilities are facilities

which are prone to congestion when the volume of simultaneous users increases amid

constant capacity. Examples of such facilities include seaports, airports, Internet

access providers and roads. Studies dealt with congestible facilities often choose a

game theory as their research methodology. For instance, Baake and Mitusch (2007)

analysed competition between two network providers. Authors analysed Bertrand

competition and Cournot competition in order to compare the equilibrium solution for

the competitive price and capacity expansion.

In the context of airports competition, a game theory has been intensively

used in this industry. In particular, Basso and Zhang (2007) developed a model for

congestible facility rivalry in vertical structures which explains the relationship among

the congestible facility and its intermediate user (airline) and final users (passengers).

Zhang and Zhang (2006) studied about airport capacity and congestion when carriers

have market power. Their study investigated the impact of carriers’ self-internalized

congestion on the airport. Similar to Basso and Zhang (2007), the model of airline and

airport behaviour is based on the two stage game, where in the first stage, the airport

decides on the airport charge and capacity, whereas in the second stage, each carrier

chooses its output in terms of the number of flights to maximize profit. It is observed

  

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that the market structure of aviation industry seems largely similar to the structure of

maritime industry. Furthermore, not only the competition problem but also market

issues such as privatization (Zhang and Zhang 2003), strategic alliances (Zhang and

Zhang 2006) and carriers’ market power are important issues in maritime industry as

well. Since the aviation industry has been successfully applied game theory to

analysing market structure and addressing management issue, it seems to stand

reasonable using game theory for port competition problem.

  

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Chapter 3 Base Model Development

Base Model Development

3.1 Overview

The main purpose of this chapter is to develop a base model for transshipment port

competition and deliver the theoretical properties to explain the trends and behaviours

of the shipping lines and ports. In particular, it is introduced a LTCD function, a

quadratic congestion delay cost function and profit functions for both shipping lines

and ports. Based on the model, analytical works are obtained and the findings are

demonstrated by numerical experiments. This chapter has been published in the

flagship journal of international shipping and port research, Maritime Policy &

Management.8

This chapter is organized as follows: Section 3.2 shows our model

formulation with a LTCD function. We then, in the section 3.3, proceed to apply the

non-cooperative two-stage game to our problem. In section 3.4, the port collusion

model and social optimum model which may reflect the current business models of

shipping lines are briefly analysed. Results from our numerical simulations are then

shown in section 3.5 to further explain the findings. Some of special cases, enabling to

                                                            8Min Ju Bae, Ek Peng Chew, Loo Hay Lee and Anming Zhang. 2013. “Container transshipment and port competition”. Maritime Policy & Management. DOI:10.1080/03088839.2013.797120.

  

16

show analytical derivatives for port pricing stage, are presented in section 3.6, and

finally section 3.7 concludes this chapter.

3.2 The Model

Consider two container transshipment ports, r =1, 2, which provide homogenous

container handling services to their customers within a stipulated period of time. The

customers are identical shipping lines, i =1,…, N. The market structure and key

variables are shown in Figure 3.1 for better understanding.

Figure 3.1 Market structure and key variables

Our LTCD function given in (3.1) denotes total number of containers which shipping

line i loads and unloads at port r.

ir r rF f g Q for 1, 2r (3.1)

where f represents the loaded and unloaded gateway container demand, rg is a

nonnegative coefficient that represents transshipment level at port r, and rQ represents

  

17

total transshipment port calls at port r, 1

N

r iriQ q

where 0 1irq and

1 for is irq q r s . It has to be noted that irq is the decision variable which

indicates a fraction of transshipment port calls that shipping line i makes at port r. Fir

is made up of two components: the gateway container and the transshipment container.

Since this study focuses on transshipment container demand, we assume that the

gateway container demand f is constant. This assumption helps simplify analytical

work. On the other hand, at a conceptual level, gr can be used to represent a port’s

“connectivity,” which can be defined as the port’s network connection to other

transport modes that extend to other destinations (e.g., feeder services, hinterland

connection, etc.). In logistics, a transshipment port is akin to a transit facility. As such,

shipping lines which adopt the hub-and-spoke transportation system are likely to

prefer a transshipment port that has an extensive and strong network connection.

Meanwhile, it has to be noted that our LTCD function generates an equal

amount of transshipment volume for all shipping lines calling at same port, regardless

of the number of port calls that each shipping line has made. For our transshipment

focus, it is assumed that the transshipment container demand depends only on ports’

handling capability and aggregate contribution of shipping lines’ port calls in the

stipulated period of time. From the container handling demand function (3.1), each

port’s demand function (3.2) can be derived:

1 1 1 1 2 2 2 21 1

,N N

i ii i

F F N f g Q F F N f g Q

(3.2)

where 2 2 1 11 11

N N

i ii iQ q q N Q

. And then, we can derive the following

properties by differentiating (3.2) with respect to 1iq :

1 2 1 21 2 1 2

1 1 1 1

, , , i i

i i i i

F F F Fg g Ng Ng

q q q q

(3.3)

  

18

Properties (3.3) illustrate that an increase in the expected number of port calls made at

port 1 would lead to a decrease in the expected number of port calls made at port 2.

This result is expected since we assume 2 11i iq q . Furthermore, the magnitude of

changes in the expected number of port calls depends on each port’s transshipment

level rg . We now consider the congestion delay cost function rD as shown in (3.4),

where this function possesses a quadratic form. Since this study only considers port

capacity and port demand as a measurement of port congestion, this quadratic

congestion function simply and efficiently captures the trend of congestion at port. We

further assume that / 1r rF K , with rK being 85% utilization of port r’s maximum

capacity maxrK .

2

, for 1,2rr r r r

r

FD F K a r

K

(3.4)

where ra is a positive parameter and max0.85 r rK K . Thus, the congestion delay cost

function rD is increasing with the number of port calls made to port r while

decreasing with port’s capacity rK . The following properties are derived:

2

2 2 2

12 0, 2 0r r r

r rr r r r

D F Da a

F K F K

(3.5)

2 2 2 2

3 2 4 32 0, 6 0, 4 0r r r r r r

r r rr r rr r r r

D F D F D Fa a a

K F KK K K K

(3.6)

Property (3.5) depicts that the congestion externality is convex in the port’s demand. It

is also intuitive that the congestion externality would decrease with port handling

capacity as shown in (3.6). The delay cost parameter ra is further assumed to be the

same across the two ports, and will be denoted a in further derivations.

  

19

3.3 A Non-Cooperative Two-stage Game

We now study a non-cooperative two-stage game with duopolistic transshipment ports

and a continuum of identical shipping lines. We first develop the profit functions for

shipping lines and ports. Shipping lines’ profit function and constraints are given by:

2

1

max for 1,...,

subject to

i i i r r irr

p c D F i N

2

1

0 1 ; 1, 2

1

; 1, 2

ir

irr

r r

q r

q

F K r

(3.7)

where ip is shipping line’s container price, ic is unit operating cost and r is port

price. Among shipping lines, both pricing and cost incurrence are assumed to be

identical, hence denoting them with p and c respectively. In the shipping line’s profit

function, the congestion delay cost function rD is captured as a cost component. This

is to model the shipping line’s preference for a less congested port, considering that

port congestion often leads to delays, which in turn translate to additional costs to the

shipping lines. The first constraint in (3.7) shows that irq is normalized between 0 and

1. The second constraint indicates that the shipping line’s total number of port call is

fixed to 1 so as to facilitate our analysis of shipping lines’ allocation decision in

response to the ports’ capacities, prices and transshipment levels. The third constraint

is, as mentioned previously, a capacity constraint. The ports’ objective is also to

maximize profits. Port’s profit function r is given by

max for 1,2r r r r r rO F m K r (3.8)

  

20

where rO is ports’ operation cost per unit, and rm is unit capacity cost. The ports’

operation and capacity costs are, for simplicity, assumed to be separable and constant.9

The unit capacity cost is the additional investment (capital) required to

increase one unit of capacity. In practice, the unit capacity cost is computed by

dividing the total development costs of the port by its total design capacity. It can be

measured on per berth basis (if the berths are largely homogenous) or on a per

container basis. In this case, we use the latter due to its ease in fitting into our model.

The cost of capacity is assumed to be a linear function of capacity. The operation cost

is the direct cost (variable cost) relating to running the port, which we expressed as the

cost of handling a unit of container. There are many ways to estimate port capacity in

the real world. One of the most popular methods is to estimate it based on the number

of berths, as the berth is commonly the bottleneck of the port. The unit capacity cost is

usually used for the port expansion analysis. There is an intention to expand this study

to capacity expansion analysis in the future as well as observe the effects of unit

capacity cost with respect to differing capacity levels.

Based on these functions, we now specify our two-stage game: in the first

stage, each port maximizes its profit by choosing its port price, and in the second stage,

each shipping line makes its port call decision to maximize profit, while observing the

ports’ capacities, prices and transshipment levels.

3.3.1 Stage two: Shipping Lines’ Port Call Decision

To examine the sub-game perfect Nash equilibrium, the game is to be solved using

backward induction, starting with the second-stage game. Given port capacities, prices,

                                                            9 The separable and constant port operation and capacity marginal cost has been assumed in, e.g., Basso and Zhang (2007) and Wan and Zhang (2013).

  

21

and transshipment levels, shipping lines simultaneously assign their port calls to both

ports. We assume Cournot behaviour in shipping lines competition10 leading to the

following first-order conditions:

1 1 1 2 2 21 1 1 2 2 2

1 1 1 1 1 2 1

0i i ii i

i i i i i

F D F F D Fp c D F p c D F

q q F q q F q

(3.9)

The assumption of symmetry among the shipping lines’ price ip p and cost ic c

for all i , and partial derivatives from (3.3) imply that the best response function of

shipping line i is identical for all i , i.e. ...ir Nrq q . Applying our earlier analysis on

the shipping lines’ profit function, we obtain:

Lemma 1 . Shipping line i’s profit i is concave in 1iq .

Proof. See Appendix A.

Lemma 1 shows that i has a maximum in 1iq . Hence, there exists a unique Nash

equilibrium in shipping lines’ port call decision. We solve (3.9) to obtain the best

response function of port call decision made by shipping line i at port 1:

2 2 2 2 3 22 1 1 2 2 1

1 3 2 3 22 1 1 2

3 2 3 22 1 1 2 2 2 1 11 2 1 2 1 2

1 2 23 2 3 2 22 1 1 2 1 2 1 23

Gi

f g K g K g NKq

N g K g K

g K g K g p c g p cK K f g g g g Ng g

N g K g K aN f g g g g N

(3.10)

where superscript G stands for the generalized case. A similar expression holds for

port 2. We focus on the solution range of greater than or equal to 0 and less than or

equal to 1. If the solution falls outside this range, the solution will be at the boundary

of the constraints.                                                             10 Cournot behavior by congestible facility users, such as airlines and shipping lines, has been assumed in, e.g., Lam and Yap(2006) and Zhang and Zhang (2006).

  

22

We now conduct the comparative statics analysis to see the changes in shipping lines’

equilibrium with respect to the changes in parameters such as port capacity, price and

transshipment level. Since the best response functions of shipping lines’ port call

decision are identical across the shipping lines and depend only on the aggregate port

calls at each port, we investigate the comparative statics of the shipping lines’

aggregate output at port 1 with respect to parameter set 1 2 1 2 1 2, , , , ,X K K g g .

The results are shown below and the details of derivations are given in Appendix B.

1 1 1 1

1 2 1 2

0, 0, 0, 0Q Q Q Q

K K

(3.11)

As shown in (3.11), the shipping lines’ aggregate output at port 1 decreases with port

1’s price and port 2’s capacity, and increases with port 1’s capacity and port 2’s price.

Similar results are obtained for 2Q . We further obtain the responses of aggregate

shipping lines’ output at port 1 to transshipment level:

2 21 1 1 21 1 1 1 1 1 2

1 1 21

3 2 3Q D D D

p c D g Q N D N g gF F Fg

(3.12)

2 21 2 1 22 2 2 2 2 1 2

2 1 22

3 2 3Q D D D

p c D g Q N D N g gF F Fg

(3.13)

The positive rate of return in (3.12) is contingent on the shipping lines’ marginal profit

at port 1 due to the increasing transshipment level of port 1. Similarly, the negative

rate of return in (3.12) is dependent on the shipping lines’ marginal cost at port 1 that

results from congestion due to increasing transshipment level of port 1. On the other

hand, the positive rate of return in (3.13) is contingent on the shipping lines’ marginal

cost at port 2 due to worsening congestion that results from increasing transshipment

at port 2, while the negative rate of return in (3.13) is pegged to the shipping lines’

marginal profit at port 2 that results from increasing transshipment in port 2.

  

23

3.3.2 Stage one: Port Pricing Strategies

The port pricing strategies are analyzed in the first stage of the game when both ports

maximize their profits (3.8) by choosing port price r . The first-order condition is

shown in (3.14),

0r rr r r

r r

FF O

(3.14)

Based on shipping lines’ decision behavior obtained from stage two and analyzing

ports’ profit functions, we can obtain Nash equilibrium port prices. However, the best-

response functions may not yield a closed form solution in this model. In this case we

will, in the following sections, explore the results of the port pricing stage using

numerical experiments.

3.4 Ports Collusion and Social Optimum

Thus far in this study, we had focused mainly on the non-cooperative game where

every port and shipping line makes independent decisions to maximize own profit. We

now consider the case of two ports cooperating on price, and of all market players, two

ports and N-shipping line, cooperating to maximize the market profit.

3.4.1 Ports Collusion Model

Consider the case in which two ports decide their prices concurrently. The profit

function of ports collusion model is shown in (3.15).

1 2 1 1 1 1 1 2 2 2 2 2maxr

O F m K O F m K

(3.15)

The first-order derivative for each port price from (3.15) is:

  

24

1 2 1 21 1 1 2 2

1 1 1

F FF O O

1 2 1 22 1 1 2 2

2 2 2

F FF O O

(3.16)

It is found that the partial derivatives are positive. The port price would approach

infinity if no boundary for maximum port price is set. Thus we can observe that the

port collusion model yields higher port price than that of the non-cooperative model.

The result stems from the monopolistic power that the ports now possess, rendering

them a free hand in escalating their prices to maximize their profits. Moving forward,

this analysis would help better explain the competitive landscape and strategies among

regionally bounded terminal operators.

3.4.2 Social Optimum Model

We now consider the social optimum model that reflects cooperation among all

players in the game to maximize their combined profits. This is captured in the social-

welfare function (SW) given below:

2 2 2

, 1 1 1 1 1

maxr ir

N N

r i r r r r r r r irq

r i r i r

SW O F m K p c D F

(3.17)

A reduced form of (3.17) is presented by

1 2 1 1 1 1 1 2 2 2 2 2SW p c F F O D F m K O D F m K (3.18)

It is important to note that the port price disappears from (3.17) to give (3.18). In this

case, the port price is internalized due to the equality between total port revenue and

sum of shipping lines’ port cost. Therefore, there is no pricing stage in this model, but

the port call decision game among shipping lines is considered. The shipping lines’

  

25

port call decision in social optimum model is characterized by the first order condition

below.

1 1 1 21 1 1 2 2

1 1 1 1 1

2 2 1 22

2 1 1 1

0

i i i i

i i i

F D F FSWO D F O D

q q F q q

D F F FF p c

F q q q

(3.19)

We solve (3.19) to achieve the best response of shipping lines’ port call decision

below in (3.20). It depicts that in social optimum model, the shipping lines’ port call

decision takes into account port operation cost in order to maximize overall profits. To

better appreciate the model, we can perceive the stipulated conditions as a vertical

expansion of shipping lines, where they extend their core businesses from liner

shipping to terminal operation. In this case, social optimum model explains the

structure of these shipping lines’ profit, where ports’ prices are internalized and ports’

demands are affected by port operation cost, capacity and transshipment volume.

2 2 2 2 3 22 1 1 2 2 1

1 3 2 3 22 1 1 2

3 2 3 22 1 1 2 2 2 1 11 2 1 2 1 2

1 2 23 2 3 2 22 1 1 2 1 2 1 23

SWi

f g K g K g NKq

N g K g K

g K g K g p c O g p c OK K f g g g g Ng g

N g K g K aN f g g g g N

(3.20)

3.5 Numerical Results

In this section, the shipping lines’ port call decisions and port pricing strategies are

further explored through numerical experiments. First, we explain the effect on

shipping lines’ port call decision, driven by differing port prices between two ports

while applying various levels of port capacities. Second, we show the effect of

  

26

transshipment level on shipping lines’ port call decision while applying various levels

of port capacities. Third, we explore the changes in equilibrium port price in

accordance with changes in capacities and transshipment levels. Finally, we compare

the results of the social optimum model and non-cooperative model in terms of

shipping lines’ port call decision and market profits.

In addition, we assume that the transshipment levels and capacities of both

ports are same in this case. In particular, we define “capacity utilization (CU)” for

experiments. The capacity utilization is calculated based on possible maximum market

demand, in which all shipping lines call at one port with their maximum port call.

Then, the capacity utilization is computed by given (3.21), a ratio of maximum market

demand to sum of two ports capacities. The higher CU signifies small capacity and the

lower CU indicates bigger capacity.

1 2

max(%) 100

DCU

K K

(3.21)

Figure 3.2 shows shipping lines’ port call decision at port 1 subjected to differing

prices between two equally-sized ports.

Figure 3.2 Effect of price differences and capacity levels on SLs’ port call decision

  

27

As expected, similar port prices yield an equal portion of port calls to both ports. The

portion of port calls to port 1 decreases as port 1’s price increases. This implies that

shipping lines are easily attracted by a cheaper port price. Another important finding

from Figure 3.2 is the combined effect of capacity and price level. Different slope

gradients were obtained when different port capacities were subjected to similar price

differentials (see CU=100% and CU=20%). While the capacity CU=100% carried a

gentle slope, the capacity CU=20% resulted in the steep slope. These results showed

that when both ports’ capacities are large, a marginal difference between two port

prices is sufficient to drive significant demand to the cheaper port. In contrast, when

both ports’ capacities are small, the shift of demand becomes inelastic to the difference

in port prices. Therefore, it is reasonable to assume that the congestion effect

associated with a small port capacity offsets the price difference between ports, and

that a large port capacity offsets the congestion effect and hence amplifies the effect of

price difference.

Figure 3.3 Effect of transshipment and capacity levels on SLs’ port call decision

Since gr is a factor that contributes to shipping lines’ transshipment demand, shipping

lines are inclined to make more port calls at the port that provides a higher gr.

  

28

However, this only holds true when the port capacity is sufficient to handle the

additional transshipment demand and offset the congestion effect, which is shown at

the intersection point between the curves in Figure 3.3. In other words, when the port

capacity is small, the congestion effect takes precedence over the transshipment effect,

hence resulting in the lower portion of port calls despite a higher gr.

Figure 3.4 Equilibrium port prices while varying capacity differences

We examine the equilibrium port price of port 1 and port 2 in the context of varying

difference between two ports’ capacities. This perspective translates into port

expansion in practice. As shown in figure 3.4, capacity expansion in either port will

decrease the equilibrium port price. This finding can be interpreted as such: a larger

port can aggressively lower its price to attract more demand as they are more likely to

have spare capacity and hence less congestion. In response, the rival port has to lower

its price as a countermeasure since its capacity level cannot retain its market share.

Figure 3.5 shows the equilibrium port price subjected to differing

transshipment level between two equally-sized ports. The Figure 3.5(a) and 3.5(b)

show the equilibrium port price of port 1 and port 2 under given congested (CU=100%)

and excessive capacity level (CU=20%), respectively.

  

29

(a) CU=100% (b) CU=20%

Figure 3.5 Equilibrium port prices while varying transshipment level differences

When both ports are congested, a high transshipment level is not preferred due to the

higher congestion cost to shipping lines. Thus port 1 lowers its price to capture the

port calls. On the other hand, when both ports are excessively large, the port that

provides more transshipment is preferred. The excessive capacity surpasses the

congestion effect, thus the port receives more port calls and in response, it increase its

price. Meanwhile, the other port, in response of this situation, will lower its price to

attract the shipping lines.

(a) Port call decision (b) Profit

Figure 3.6 Comparison between Non-cooperative and Social optimum model

  

30

In the previous analysis on the social optimum model, we found that the shipping lines’

port call decision depends on port capacity, transshipment level and port operation cost

when port prices are internalized in the objective function. We now take our analysis

further by comparing the non-cooperative model and the social optimum model. In this

analysis, we give both models identical parameters and compare the resulting shipping

lines’ port call decisions and total market profits. The conditions for this experiment

are given as follow: 1 2 1 2 1 2 , and K K O O g g . As shown in Figure 3.6(a), shipping

lines make more port call at port 1 in the social optimum model than in the non-

cooperative model. The intuition behind this phenomenon is that under the social

optimum condition, the shipping lines’ port call decision is bounded by the port

operation cost; whereas in the non-cooperative model, the same decision is bounded

by the port price. As assumed that the port price is always higher than the port

operation cost, and considering the inverse relationship between the port call decision

and the cost factors(port price and port operation cost), the shipping lines will thus

make more port calls at port 1 under the social optimum condition. As seen in Figure

3.6(b), the market profit of the social optimum model is greater than that of the non-

cooperative model. It is explained in shipping lines’ port call decision, where the port

call diverted to port 1 in social optimum model is shown to be comparatively higher.

3.6 Special Cases

As our model is limited to numerical results for port pricing stage, in this section, we

consider two cases that enable analytical work to achieve the best response function of

port pricing stage. We found that the results obtained in these special cases are

consistent with our numerical results.

  

31

3.6.1 Same capacities and transshipment levels for both ports

Thus far in this study, we have focused on the generalized case where the ports have

different prices, capacities and transshipment levels. This case is complicated by the

existence of possible combined effects among key factors. For this special case, we

reduce the complexities by considering same capacities and transshipment levels for

both ports, i.e. 1 2:K K K and 1 2:g g g , thus the ports are only differentiated

from their prices. The best response function of port call decision made by shipping

line i at port 1 is then:

22 1

1 3

1

2 6 2Si

Kq

agN f gN

(3.22)

where superscript S represents same capacities and transshipment levels for both ports.

Expression (3.22) shows that allocating port call decision by shipping lines initially

begins at the point 0.5, or half of maximum port call, as we have set 1 as the upper

bound of port call. Thereafter, depending on the difference between port prices, the

port call decision moves downward or upward. Based on (3.22) and the first order

condition of port profit function (3.14), we can obtain the analytical results of the

existence of Nash equilibrium in port prices and its uniqueness as shown in

Propositions 1 and 2.

Proposition 1. Port r’s profit function r is strictly concave in r .

Proof. See Appendix C.

Proposition 2. Port competition has a unique equilibrium in port price * * *1 2, .

Proof. See Appendix C.

The optimal port prices can be obtained as given in (3.23).

  

32

2

2 2 1 21 2

3 24

3S aN O O

f f gN g NK

2

2 2 1 22 2

3 24

3S aN O O

f f gN g NK

(3.23)

3.6.2 Linear Congestion Delay Cost

De Borger and Van Dender (2006) applied the linear volume-capacity ratio congestion

costs function in their study. As they focused on interior solutions, their linear

congestion function allows them to keep the analysis tractable while guaranteeing

firms produce positive output in the duopoly case, and avoid the use of exogenous

rationing rules. Similarly, we consider the linear congestion delay cost in this section

as one of special cases. The linear congestion delay cost function is given below.

rr

r

FD a

K (3.24)

We apply (3.24) to two-stage game instead of our quadratic congestion delay cost

function (3.4). Solving (3.9) yields the best response functions of shipping lines’ port

call decision at port 1:

22 1 1 2 2 1 1 2 2 2 1 1

1 2 2 2 2 22 1 1 2 2 1 1 22

Li

f g K g K g NK K K g p c g p cq

g K g K N a g K g K N

(3.25)

where superscript L stands for linear congestion delay cost. Similar to same capacities

and transshipment levels, based on (3.25) and the first order condition of port profit

function (3.14), we found that this case also satisfies the Proposition 1 and 2, although

the contract condition in Proposition 2 is bound to the condition, 1 22g g (See

Appendix D). Hence, this case also guarantees that there exists the unique Nash

equilibrium in port prices as long as port 2’s transshipment level is lower than twice of

  

33

port 1’s transshipment level. The best response function of port 1 can be obtained by

substituting (3.25) into port 1’s first order condition (3.14), and solving for its price 1 .

2 21 2 2 1 1 2 1 2 1 1 2 2

1 21 2 1 2 1

2 2 2

3 3L

aN g K g K f g g g g N g p c O g p c O

g g K K g

(3.26)

A similar expression holds for 2L . We apply comparative static analysis on (3.26)

with respect to the port capacities. The effects of port capacities are shown in (3.27)

and (3.28).

1 2 1 212

1 2 1

20

3

L aN f g g g g N

K g K

(3.27)

2 1 2 1 212 2

2 1 2

40

3

L ag N f g g g g N

K g K

(3.28)

Expression (3.27) and (3.28) show similar results obtained from the numerical

experiments that the equilibrium port price decreases with the port capacities of own

and competitor. Explaining from the long term strategic perspective, port expansion in

either port will decrease the equilibrium port prices.

3.7 Conclusions

This paper developed the two-stage duopoly model of container port competition

focusing on transshipment demand. The linear container demand function, among

others, was derived to facilitate a two-stage game analysis. In the main analysis, it was

shown that shipping lines have a tendency to assign more port calls to the port that

offers a cheaper price and a larger capacity. The port collusion model provided

evidence that a monopolistic port market will result in an infinite port price, whereby

in the social optimum model, the shipping lines’ decision in number of port calls is

contingent on the ports’ operation costs and congestion delay costs. Numerical

  

34

experiments were conducted to further our observation of the Nash equilibrium

solution. Several key discoveries were made. First, the level of port capacity has a

significant effect on other factors. The impact of price difference between two large

ports is further heightened as congestion delay costs become insignificant. Shipping

lines prefer the port that provides a higher transshipment level, only if its capacity is

sufficient to eliminate the accompanying congestion effect. Next, from the port

perspective, the port that has excessive capacity can cut the price to invite demand as

its spare capacity can balance the congestion effect, and in response, the rival port

needs to lower its price in order to maintain its competitiveness in the market. When

both ports are congested, a high transshipment port lowers the price to retain its

demand as its transshimment level results in high congestion cost to shipping lines.

When comparing the non-cooperative model with the social optimum model, we

showed that the internalized port price enables the social optimum model to achieve

more port calls, and also higher market profit. Finally, some special cases were

explored to increase the visibility on the best response functions and the existence of a

unique Nash equilibrium in port prices. These special cases reduced the complexities

in computation by considering (i) identical capacities and transshipment levels, and (ii)

linear congestion delay cost function. Both cases guaranteed the existence of a unique

Nash equilibrium in port prices.

  

35

Chapter 4 Nonlinear Transshipment Demand and Multiple Nash Equilibria

Nonlinear Transshipment Demand and

Multiple Nash Equilibria

4.1 Overview

This chapter extends our base model to a NTCD (nonlinear transshipment container

demand). The numerical analysis shows that shipping lines may have a unique Nash

equilibrium or multiple Nash equilibria in their port call decision. The conditions for

either outcome are complicated with interaction among nonlinearity, port price, port

capacity and transshipment level. Experimental results of Nash equilibrium port price

show that a unique Nash equilibrium for port price exists only when shipping lines’

port call decision has a unique Nash equilibrium.

This chapter is organized as follows: section 4.2 presents a NTCD function.

The solution concept of the non-cooperative two-stage game is briefly explained in

section 4.3. The intensive numerical analyses and findings are shown in section 4.4,

and finally the conclusion is given in section 4.5.

4.2 The model

Consider two container transshipment ports, r =1, 2, which provide homogenous

container handling services to their customers within a stipulated period of time. The

  

36

customers are two identical shipping lines, i =1, 2. Our NTCD function represents a

number of transshipment containers which shipping line i loads and unloads at port r .

We give our focus on transshipment containers, thus our NTCD function does not take

into account the gateway containers.

1,2; 1,2bir r ir rF g q Q i r (4.1)

where 0 1irq , 1is irq q for r s , rg is nonnegative coefficient and b is

nonlinearity index. As defined in the previous chapter, irq is a decision variable which

indicates a fraction of transshipment port calls that shipping line i makes at port r, and

is determined by the capacity of port r, its price and transshipment level during the

stipulated period of time. The nonlinearity index b can adjust transshipment effects,

where ‘b = 0’ indicates no transshipment benefits from other shipping lines, whereas

‘ 0 1b ’ denotes the moderate transshipment volume, whereas ‘ 1b ’ connotes the

exaggerated transshipment volume. It is noted that when the nonlinearity index b

becomes 1, rQ gives a linear trend. Our interest in this part of study lies on brQ as a

nonlinear trend, thus the moderate and exaggerated transshipment volumes are

considered.

We continue this study with the quadratic congestion delay cost function,

which has been introduced in chapter 3, expression (3.5). The shipping lines’ profit

function and constraints are given by:

  

37

2

1

2

1

max for 1, 2

subject to

0 1 ; 1, 2

1

; 1, 2

0 ; 1, 2

i i i r r irr

ir

irr

r r

i i r r

p c D F i

q r

q

F K r

p c D r

(4.2)

The first constraint in (4.2) shows that irq is normalized between 0 and 1. The second

constraint shows that the given shipping lines’ maximum call size is 1. This will help

to facilitate our analysis of the shipping lines’ allocation decision in response to the

ports’ capacities, prices and transshipment levels. The third constraint is a capacity

constraint and the fourth constraint is a nonnegative constraint that shipping lines only

consider a positive profit. We continue our study with port objective function that

introduced in the previous chapter, expression (3.9).

4.3 Non-cooperative two-stage game

Taking into account a NTCD function and the modified shipping lines’ profit function,

we advance our analysis via two-stage game. As defined the stages earlier, ports, in the

first stage, choose the price that maximize their profit and in the second stage,

shipping lines make port call decision to maximize their profit, observing ports’

capacities, prices and transshipment levels.

Due to the NTCD function and the quadratic congestion delay cost

function, it is not able to achieve the closed form solutions through the analytical work.

Thus, we will explore shipping lines’ port call decision and ports’ optimal pricing

through numerical experiments in the following section. In addition, our problem can

be defined as optimization with inequality and non-negativity constraints. The Kuhn-

  

38

Tucker formulation incorporates inequality and non-negative constraints by imposing

additional restrictions on the first-order conditions for the problem, thus we consider

the solution of Nash equilibrium problems by concatenating the KKT conditions of

each shipping line’s optimization problem to evaluate the optimal size of port calls.

Transforming shipping lines’ profit function to Lagrangian form and its conditions are

shown in Appendix E.

4.4 Numerical Analysis

The numerical experiments are conducted by Matlab software program, where its

optimization toolbox is developed based on KKT condition. The numerical

experiments are structured on the basis of two-stage game; hence, the shipping lines’

port call decision will be explored first with given port conditions, and then the

optimal port price will be found by searching method based on shipping lines’ decision

behaviour.

4.4.1 Shipping lines’ port call decision

The numerical experiments are conducted for examining two symmetric shipping lines’

port call decision based on given port conditions. In particular, BRC (best response

curve) method is adopted in this part of study since it is effective method to show the

strategy which produces the most favourable outcome for a player, taking opponent’s

strategy as given (Fudenberg and Tirole 1991). Before exploring numerical results, we

will explain the details of how to interpret the BRCs and two characteristics of

Multiple Nash equilibria.

  

39

1) Best Response Curves

BRCs can analyze and explain simultaneous decisions of competing players, and

indicate the best decision based on the decision that the player expects its rival will

make. In the graphical expression of BRCs, Nash equilibrium is found where players’

best-response curves intersect. That is a stable state of a situation in which no player

can gain by a change of strategy as long as the other player remains unchanged. Figure

4.1 shows the graphical illustration of two shipping lines’ port call decision BRCs at

port 1. In particular, the x-axis represents shipping line 1’s port call strategies at port 1,

while the y-axis represents shipping line 2’s port call strategies at port 1. Based on

shipping line 2’s strategies as given information, shipping lines 1’s BRC is presented

with solid line. Meanwhile, the shipping line 2’s BRC is presented with dotted line

based on given strategies of shipping line 1. Then, the intersection of two curves

indicates the Nash equilibrium. Specifically, the BRC can be studied as follow; if

shipping line 1 gives its strategy ① (see Figure 4.1), shipping line 2’s BR is found on

dotted line by following a vertical arrow. Then, shipping line 1 is taking this shipping

line 2’s BR as a given opponent strategy and produces its BR on solid line (follow the

horizontal arrow). By repeating this procedure, it eventually reaches to intersection of

two BRCs, where it is called the Nash equilibrium. It shows the similar moves for

shipping line 2 when it starts its move from ② and follows the arrows.

2) Multiple Nash equilibria

We now discuss about the Multiple Nash equilibria such as given in Figure 4.2.

Shipping lines’ BRCs, as in this case, result in three Nash equilibria, two at the corner

(0, 1), (1, 0) and one in the middle (0.5, 0.5). Such games with Multiple Nash

equilibria in which players choose the same or corresponding strategies are regarded

as coordination game (Cooper, 1998).

  

40

Figure 4.1 SLs’ BRCs with a Unique Nash Equilibrium

From a mathematical point of view, selecting the “Best” Nash equilibrium is an

important issue because it is approaching the coordination game as a problem that has

to be solved by some restriction of the assumptions that would rule out the multiple

equilibria. On the other hand, from a social scientific point of view, Multiple Nash

equilibria may provide us a possible “explanation” of coordination problems which

would be an important positive finding (McCain, 2010). With respect to these

perspectives, we will study Multiple Nash equilibria from a mathematical point of

view in the future. Instead, in this thesis, we will approach Multiple Nash equilibria

from the latter point of view mentioned above. We first discuss about two interesting

characteristics of Multiple Nash equilibria, i.e., stability and robustness.

Stability

Multiple Nash equilibria can be classified into a stable or an unstable equilibrium. The

unstable equilibrium means that a slight displacement from equilibrium departs further

from the original equilibrium. On the contrary, a stable equilibrium suggests a slight

disturbance from equilibrium returns to its original equilibrium position.

  

41

Figure 4.2 Multiple Nash equilibria and Stability

This can be observed in Figure 4.2; shipping line 1 decides to move from current

equilibrium in the middle 0.5 to ①, and then shipping line 2 takes ① as a known

opponent strategy and produces its BR on dotted line (the move follows vertical solid

arrow). By repeating this procedure, it is noticed that this move rapidly leads to one of

corner equilibria, (0, 1). For shipping line 2, it shows a similar result when it shifts its

port call from 0.5 to ②, reaching to the other corner equilibrium (1, 0). This suggests

that the two corner equilibria are stable while the centre equilibrium is unstable in this

case.

Figure 4.3 Multiple Nash equilibria and Robustness

  

42

Robustness

Robustness of Nash equilibrium is relevant to the probable region that would convert

an initial strategy to the certain Nash equilibrium. Figure 4.3 shows three regions of

initial strategies for shipping lines’ port call decision. For instance, any choice within

the blue region will be directed to the equilibrium (0, 1) while being directed to the

equilibrium (1, 0) within pink region. The centre equilibrium (0.5, 0.5) will be selected

by the choice made at the boundary between blue and pink region. Therefore, the

wider region gets in initial strategies the more chance to be selected.

4.4.1.1 Symmetric ports conditions

Given the existence of Multiple Nash equilibria and its characteristics, we first

examine the shipping lines’ port call decision under the symmetric ports conditions,

where the given ports’ capacities, prices and transshipment levels are 1 2:K K K ,

1 2: and 1 2:g g g . By analysing symmetric port conditions first, we can

clearly observe the impact of each port factor on shipping lines’ port call decision.

1) Analysis on capacity K

The port capacity is an important factor that has a direct influence on port congestion.

The numerical experiments begin with given parameters of 0.5, 0.3, 0.3b g .

The various levels of port capacity are given based on CU (referring to expression

(3.21)). The given ranges of CU cover between 20% and 100%. In order to calculate

the CU, a possible maximum market demand has to be estimated. It can be obtained

when both shipping lines call at the same port with their maximum port-of-call, i.e. 1.

The maximum market demand is calculated as follow.

  

43

21

1

max 0.3 2 2 0.8486b bir r r

i

D gq Q gQ

Assuming two ports capacities are identical in this case, the symmetric port capacity

level is determined in Table 4.1 in accordance with CU. In particular, a higher CU

signifies small capacity, representing congested capacity level, while a lower CU

indicates large capacity, representing excessive capacity level.

Table 4.1 Symmetric port capacity levels

CU(%) Port capacity K

100% 90% 80% 70% 60% 50% 40% 30% 20%

0.4243 0.4714 0.5303 0.6061 0.7071 0.8486 1.0607 1.4142 2.1213

Shipping lines’ BRCs when given CU=100% is presented in Figure 4.4. In this case, a

unique Nash equilibrium is found in the middle where the two BRCs are met.

Figure 4.4 SLs’ BRCs when symmetric capacities at CU=100%

  

44

Figure 4.5 Trends of SL 2’s TS benefits, congestion cost and total profit when 11q =1

Taking our analysis further, the effect of the congestion delay cost has been examined

and given in Figure 4.5. In particular, we observe the trends of shipping line 2’

transshipment benefits, congestion delay cost and total profit while varying its port call

decision at port 1. It is assumed that shipping line 1 only calls at port 1. The x-axis in

Figure 4.5 indicates the port call strategies of shipping line 2 made at port 1, and the y-

axis signifies the value of shipping line 2’s transshipment benefits, congestion delay

cost and total profit. The graph suggests that the shipping line 2 would achieve more

transshipment demand by assigning more calls to port 1(see solid line). However, the

congestion delay cost (dotted line) takes precedence over transshipment benefits; thus,

the optimal port call that the shipping line 2 assigns to port 1 is around 0.04 where its

total profit (dash-dotted line) is maximized. It is implied that a highly congested port

has limited handling capacity; hence forcing shipping lines to allocate their port calls

to both ports by matching the demand to capacity. This explains a trend of downward

sloping in both BRCs.

  

45

(a) CU=90% (b) CU=80%

(c) CU=70% (d) CU=60%

Figure 4.6 SLs’ downward sloping BRCs under various levels of symmetric capacities

Increased port capacities result in Multiple Nash equilibria in shipping lines’ port call

decision as shown in Figure 4.6. The BRCs in Figure 4.6(a)-(c) give the two corner

equilibria (0, 1), (1, 0) and the mid-point equilibrium (0.5, 0.5). The mid-point

equilibrium is the decision that shipping lines found a balance between transshipment

benefits and congestion delay cost by assigning equal amount of port calls to both

ports, while two corner equilibria are the decisions that shipping lines seek for

minimum congestion delay cost by calling either port. It is implied that shipping lines

avoid each other in order to reduce the congestion delay cost as it is predominant issue

in this capacity level; therefore, the BRCs are downward sloping. Different gradients

are obtained for the BRCs in Figure 4.6. It suggests that the larger capacity gets the

faster convergence rate to the two corner equilibria, hence implying that the corner

equilibria are stable. It is noticed that the five Nash equilibria are perceived in Figure

  

46

4.6(d), where the two new Nash equilibria appear between each corner and the mid-

point. The mid-point equilibrium in this case is considered stable while the two new

Nash equilibria are unstable. It is due to the coefficient between BRCs. This will be

explained further in the marginal analysis.

(a) CU=50% (b) CU=40%

(c) CU=30% (d) CU=20%

Figure 4.7 SLs’ upward sloping BRCs under various levels of symmetric capacities

Continuing our experiment on port capacity, further increased port capacity still

delivered Multiple Nash equilibria in shipping lines’ port call decision. As observed in

Figure 4.7, there are three Nash equilibria, the two corner equilibria (0, 0), (1, 1) and

the mid-point equilibrium (0.5, 0.5). It is noticed that both shipping lines’ BRCs are

becoming upward sloping. The trend of corner equilibria and upward sloping suggest

that shipping lines now opt for transshipment volume by calling at the same port as an

increase of excess capacity compensates the congestion effect. As studied previously

in Figure 4.6, a similar trend is observed regarding the gradient of BRCs; the larger

capacity gets the quicker convergence rate to the corner equilibria. Thus, two corner

  

47

equilibria are stable and the mid-point is unstable equilibrium. Another observation

made from Figure 4.7; there are five Nash equilibria in Figure 4.7 (a) and (b). Aside

two corner equilibria and mid-point equilibrium, two more equilibria occur between

each corner and mid-point. The mid-point equilibrium in both cases is stable while two

new Nash equilibria are unstable.

Figure 4.6(d) and, Figure 4.7(a) and (b) have presented a rare five Multiple

Nash equilibria. Such an occurrence is speculated that these cases represent a

transition stage of BRCs’ relationship from negative to positive. It is suspected that

there might be a discrete movement on the shipping lines’ port call decision. These

cases need to be further explored for deeper insights. However, as our interests lie on

capturing main trends in this study, we will consider further examination on this

phenomenon in the future study.

Based on our observation through the numerical analyses, we define three

types of Nash equilibrium solutions.

Unique Strategy

: There is only one Nash equilibrium solution for shipping lines’ port call

decision.

Exclusive Strategy

: Shipping lines assign considerable amount of port call at different port in

order to reduce the congestion delay cost. As congestion delay cost is

predominant issue in this capacity level, transshipment benefit in this strategy

is not as significant as congestion delay cost.

  

48

Inclusive Strategy

: Shipping lines opt for same port to achieve the transshipment benefit as much

as possible. In this case, the transshipment benefit is more significant than

congestion delay cost.

Table 4.2 Summary of port call strategy under various symmetric capacities

CU(%) Capacity

Level SLs’

port call decision No.of

Nash Equilibrium

100% Very congested Unique strategy 1

90% 80% 70% 60%

Congested Exclusive strategy 3

5 50% 40% 30% 20%

Excess capacity Inclusive strategy

3

Based on the definition given above, the numerical results of symmetric port capacity

are summarized in Table 4.2. It is found that a unique Nash equilibrium solution exists

when both ports are very congested. However, as port capacity increases, shipping

lines have Multiple Nash equilibria in their port call decision. ‘Congested capacity’

discourages shipping lines’ transshipment-prone behaviour as congestion delay cost is

more significant than transshipment benefits. On the other hand, ‘Excess capacity’

encourages shipping lines’ transshipment-prone behaviour since congestion delay cost

is less of their concerns due to the spare capacity.

2) Analysis on nonlinearity b

Our numerical experiment runs based on the moderate transshipment effect, i.e.

0.5b . Nonetheless, we assess the case of 2b , where the transshipment benefit is

obtained by squaring rQ , representing an exaggerated case of transshipment effect. In

order to compare the different transshipment effect by nonlinearity, we set a similar

  

49

level of capacity utilization for both ports, i.e. CU=70%, for the consistency of

experiments. As shown in Figure 4.8, the transshipment effect of b=0.5 is not as

significant as of b=2. The transshipment benefit and congestion delay cost of b=2 are

computed with one of stable Nash equilibrium (0.8, 0.8); the values are obtained 0.18

and 0.05, respectively. If we enforce shipping lines to make the same decision with

b=0.5, the transshipment benefit and the congestion delay cost for shipping lines are

computed respectively 0.09 and 0.1, where the transshipment benefit is less than

congestion delay cost.

(a) b=0.5

(b) b=2

Figure 4.8 SLs’ BRCs under different transshipment effects

 

Figure 4.9 Summary of port call strategy under different transshipment effects

  

50

Figure 4.9 summarizes shipping lines’ Nash equilibrium strategies under different

transshipment effect and various capacity levels. The Inclusive Strategy mostly occurs

when b=2, where the transhipment effect is amplified exponentially. It is implied that

shipping lines are willing to crowd at the same port due to the high profit made by

additional transshipment volume, in spite of the high congestion cost.

3) Analysis on port price μ

Considering Figure 4.10 as the comparison base case, where the given input

parameters are CU=80%, 0.3g and 0.3 , increased port price by 30% resulted in

Figure 4.11(b). It is noticed that this result is similar to Figure 4.11(a), where the port

capacity is reduced to CU=100%. On the other hand, decreased port price by 30%

yields Figure 4.11(d). It is also noticed that this result is similar to Figure 4.11(c),

where the port capacity is increased to CU=70%. These results suggest that lowering

port price has a similar effect as increasing port capacity. In other words, increasing

port price has a same effect as decreasing port capacity. From the shipping lines’

perspective, their interest is in not only maximizing own profit but also minimizing

cost components. Therefore, shipping lines’ cost including port price and congestion

delay cost can be reduced by selecting a cheaper and larger port.

Figure 4.10 Comparison base case: CU=80%, 0.3g = , µ=0.3

  

51

(a) CU=100% (b) µ increased by 30%

(c) CU=70% (d) µ decreased by 30%

Figure 4.11 Comparison of base case with changes in symmetric capacities and prices

The results of this experiment are summarized in Table 4.3. Increased port price

extends the range of capacity level to yield a Unique Strategy, while decreased port

price gives more capacity levels to opt for Inclusive Strategy. It is presumed that

shipping lines are willing to tolerate the congestion delay cost due to the cheaper port

price.

1) Analysis on transshipment level g

Continuing the experiment with comparison base case (Figure 4.10), we now explore

the effect of transshipment level to shipping lines’ port call decision. Increased

transshipment level by 10% gives Figure 4.12(b) while decreased transshipment level

by 10% results in Figure 4.12(d).

  

52

Table 4.3 Summary of port call strategy under changes in symmetric prices

CU(%)

Port call strategy

µ Increased by 30%

Base case µ

Decreased by 30%

100% Unique Strategy

Unique Strategy

Exclusive Strategy 90%

Exclusive Strategy80%

Exclusive Strategy 70%

60%

Inclusive Strategy

50%

Inclusive Strategy 40%

Inclusive Strategy 30%

20%

It is noticed that Figure 4.12(b) is similar to Figure 4.12(a), where the port capacity is

reduced to CU=90% and Figure 4.12(d) is similar to Figure 4.12(b), where the port

capacity is increased to CU=70%. Based on our NTCD function, a higher

transshipment level leads to a higher transshipment volume, thus contributing to an

increase of port congestion. This implies that a higher transshipment level gives a

similar effect of decreasing port capacity. The overall results of this experiment are

summarized in Table 4.4. Increased transshipment level generates more demand,

hence tightening port capacity and resulting in the Unique Strategy. Decreased

transshipment level gives spare space to loosen the congestion. Therefore, shipping

lines opt for Inclusive Strategy as the congestion delay cost is insignificant compared

to transshipment benefits.

To conclude the numerical analysis of shipping lines’ port call decision

under symmetric port conditions, the cheaper port price and the lower transshipment

level contribute to cost-saving for shipping lines, hence encouraging shipping lines to

aim for achieving transshipment benefits by choosing Inclusive Strategy.

  

53

(a) CU=90% (b) g increased by 10%

(c) CU=70% (d) g decreased by 10%

Figure 4.12 Comparison of base case with changes in symmetric capacities and TS levels

Table 4.4 Summary of port call strategy under changes in symmetric TS levels

CU(%) Port call strategy

g Increased by 10%

Base case g

Decreased by 10% 100%

Unique Strategy Unique Strategy

Exclusive Strategy 90%

Exclusive Strategy 80%

Exclusive Strategy 70% 60%

Inclusive Strategy 50%

Inclusive Strategy 40%

Inclusive Strategy 30% 20%

2) Marginal Analysis

Some theoretical works have been achieved based on marginal analysis in this section.

Having seen in numerical results in this section, regardless of number of Nash

  

54

equilibrium, the mid-point equilibrium (0.5, 0.5) is always obtained. Therefore, we can

build Theorem 1.

Theorem 1. The mid-point (0.5, 0.5) is always a Nash Equilibrium under the

symmetric port conditions.

Theorem 1 can be easily proved by solving one of shipping line’s optimal port call

decision with given rival’s port call strategy, *1 0.5 0.5iq . A solution of Multiple

Nash equilibria in this section shows mainly two trends, i.e. downward and upward

sloping. These two trends of Multiple Nash equilibria represent an Exclusive and

Inclusive port call strategy. Based on different two corner equilibria, we develop the

below lemmas.

Lemma 2. (1, 0) is not a Nash Equilibrium for shipping lines’ port call decision if

2 27K ag p c .

Proof. See Appendix F.

Lemma 3. (1, 1) is not a Nash Equilibrium for shipping lines’ port call decision if

2 288 5K ag p c .

Proof. See Appendix G.

Lemma 2 shows the lower bound of capacity condition for Exclusive Strategy, while

Lemma 3 gives the lower bound of capacity condition for Inclusive Strategy. It is

inferred that the lower bound of capacity condition for Exclusive Strategy is the upper

bound of capacity condition for Unique Strategy, whilst the lower bound of capacity

condition for Inclusive Strategy is the upper bound of capacity condition for Exclusive

Strategy. Thus, Lemma 2 and Lemma 3 suggest that there exist the capacity boundary

  

55

conditions for port call decision being led to different Nash equilibrium solutions and

this can be described in Theorem 2.

Theorem 2. There exist the threshold capacity levels that lead shipping lines’ port call

decision to the different Nash equilibrium solutions.

Finally, we study a stability of Nash equilibrium by marginal analysis. There exists the

interaction between two shipping lines’ move and this provides the criteria to evaluate

the equilibrium stability. This is summarized in Theorem 3.

Theorem 3 (Stability). A Nash Equilibrium (0.5, 0.5) is stable if the coefficient

between shipping lines’ move on BRCs is less than 1, and is unstable if the coefficient

between shipping lines’ move on BRCs is greater than 1.

Proof. See Appendix H.

Based on Theorem 3, we are now able to explain the phenomenon observed in Figure

4.6(d), 4.7(a) and 4.7(b), where the 5 Nash equilibria and the stable mid-point

equilibrium were found.

4.4.1.2 Asymmetric ports conditions

This section studies shipping lines’ port call decision when the given two port

capacities, prices and transshipment levels are different.

1) Analysis on capacity 1 2K > K

Considering the given parameters 0.5, 0.3, 0.3b g , we assume 1 2K K . Let

1 21 2 1 K KK K , where

1 2K K indicates the capacity difference between two ports.

CU is used for defining an experiment range of capacity level. In particular, 1K and

  

56

2K are calculated with relation to 20% difference for every range of CU. Testing

capacity levels for both ports are given in Table 4.5.

Table 4.5 Asymmetric port capacity levels

CU(%) 1 2K K =20%

1K 2K

100% 90% 80% 70% 60% 50% 40% 30% 20%

0.4628 0.5143 0.5785 0.6612 0.7714 0.9257 1.1571 1.5428 2.3141

0.3857 0.4286 0.4821 0.5510 0.6428 0.7714 0.9642 1.2856 1.9285

Figure 4.13 shows shipping lines’ BRCs under asymmetric ports capacities with

various levels of CU. Figure 4.13(a) shows a unique Nash equilibrium for shipping

lines’ port call decision, where it found the Nash equilibrium at (0.54, 0.54). It is

expected that the port 1 receives more port calls as its capacity is 20% larger than port

2’s. Figure 4.13(b)~(d) give the downward sloping Multiple Nash equilibria, where

these yield the Exclusive Strategy. Two corner equilibria which are stable in these

BRCs suggest that the congestion issue is more concerned than transshipment benefits.

Even though the mid-point equilibrium is unstable, it gives an increasing trend of

equilibrium point as the level of CU increases. At CU=50%, shipping lines reached the

capacity level that they can immediately opt for port 1 with their maximum port of

calls. Figure 4.13(g) and (h) show the upward sloping Multiple Nash equilibria. The

Inclusive Strategy suggests that the congestion effect is insignificant in this level of

capacity.

  

57

(a) CU=100%

(b) CU=90%

(c) CU=80%

(d) CU=70%

(e) CU=60%

(f) CU=50%

(g) CU=40%

(h) CU=30%

Figure 4.13 SLs’ BRCs when 1 2K > K

  

58

Another observation from Figure 4.13 (g) shows that the region to be converted to (1,

1) is wider than to (0, 0) due to the 20% larger capacity of port1. However, further

increased capacity led to a similar size of region for two corner equilibria to be

converted. This implies that when both ports’ capacities are large, capacity difference

does not have much impact on shipping lines’ decision. The overall experiment results

are summarized in Table 4.6. It is noticed that the transition stage between Exclusive

Strategy and Inclusive Strategy is identified at CU=50%. This transition stage of

capacity level is resulted in Unique Strategy as given ports capacities are different,

hence shipping lines’ preference is apparent.

Table 4.6 Summary of port call strategy under asymmetric capacities

CU(%) Port call strategy

1 220%K K

100% Unique Strategy 90%

Exclusive Strategy 80% 70% 60% 50% Unique Strategy 40%

Inclusive Strategy 30% 20%

2) Analysis on price 1 2μ < μ

Based on given parameters of 1 20.5, 0.3, :b g K K K , we assume 1 2 . It is

further assumed that port 2’s price is constant while varying port 1’s price. It defines

1 21 2 (1 ) , where 1 2 indicates price difference between two ports. Figure

4.14 shows the BRCs when the price difference between two ports is 20%. When both

ports are very congested, CU=100%, the shipping lines find a unique Nash equilibrium

at (0.57, 0.57) in Figure 4.14(a). It is expected that port 1 receives more calls as its

price is cheaper than port 2.

  

59

(a) CU=100%

(b) CU=90%

(c) CU=80%

(d) CU=70%

(e) CU=60%

(f) CU=50%

(g) CU=40%

(h) CU=30%

Figure 4.14 SLs’ BRCs when 1 2μ < μ

  

60

(a) 1 2 =20% (b)

1 2 =40%

Figure 4.15 SLs’ BRCs with different levels of price difference at CU=20%

Observing trends of BRCs in Figure 4.14, the results are apparent that as port capacity

increases, the effect of price difference becomes amplified. Figure 4.15, where it

compared 20% and 40% price difference when CU=20%, shows that the increase of

price difference immediately drives the shipping lines’ port call decision to port 1 with

their maximum port of call (1, 1). The numerical results are summarized in Table 4.7.

Table 4.7 Summary of port call strategy under asymmetric prices

CU(%) Port call strategy

Price difference 20%

Price difference 40%

100% Unique Strategy

Exclusive Strategy 90%

Exclusive Strategy 80% 70% 60% 50%

Unique Strategy Unique Strategy

40% 30%

Inclusive Strategy 20%

To conclude, the results obtained in this section suggest that when both ports’

capacities are small, the shift of demand becomes inelastic to the difference in port

prices. In contrast, when the both ports’ capacities are excessively large, a marginal

difference between two port prices is sufficient to drive significant demand to the

cheaper port. Therefore, it is reasonable to assume that the congestion effect associated

  

61

with a small port capacity offsets the price difference between ports, and that a large

port capacity offsets the congestion effect and hence amplifies the effect of price

differences.

3) Analysis on transshipment level 1 2g > g

Having observed in the symmetric condition, rg has an immediate influence on port

demand level, therefore contributing to congestion level. Conducting numerical runs

based on given parameters of 1 20.5, 0.3, :b g K K K , let 1 2g g and define

1 21 2 1 g gg g where

1 2g g indicates transshipment difference between two ports.

The numerical results shown in Figure 4.16 is perceived the trend of BRCs

while 1 2g g with respect to various levels of CU. The main trends observed in this

case are two-fold: first, when the given port capacity is small, the congestion effect

takes precedence over the transshipment effect, hence receiving less port calls. The

reason behind this is a higher transshipment level increases congestion delay costs for

shipping lines. Second, as given port capacity becomes larger, shipping lines are

inclined to make more port calls at the port that provides a higher gr . It is because the

congestion delay cost is less than the profit generated through transshipment level. The

BRCs in this experiment show the smiliar result found in different capacities and

prices. The higher transshipment level with the smaller capacities associated with

congestion effect leads to a unique Nash equilibrium, while with the larger capacities,

therefore less congestion, results in Multiple Nash equilibria.

  

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(a) CU=100%

(b) CU=90%

(c) CU=80%

(d) CU=70%

(e) CU=60%

(f) CU=50%

(g) CU=40%

(h) CU=30%

Figure 4.16 SLs’ BRCs when 1 2g g

  

63

(a) 1 2g g =20% (b)

1 2g g =40%

Figure 4.17 SLs’ BRCs with different levels of TS differences at CU=20%

The difference of transshipment level between ports gets larger when CU=20%, as

given in Figure 4.17, it immediately drives all shipping lines’ port call decision to port

1 that has a higher transshipment level. Table 4.8 summarizes port call strategies and

specific equiligrium points in this numerical runs.

Table 4.8 Summary of port call strategy under asymmetric TS levels

CU(%) 1 220% 1 2

40%

Port call strategy NE points Port call strategy NE points 100%

Unique Strategy (0.4866, 0.4866)

Unique Strategy

(0.4662, 0.4662) 90% (0.4964, 0.4964) (0.4798, 0.4798)

80%

Exclusive Strategy

(0.5144, 0.5144) (0,1) (1,0)

(0.4997, 0.4997)

70% (0.5542, 0.5542)

(0,1) (1,0)

Exclusive Strateegy

(0.5345, 0.5345) (0.05,1) (1, 0.051)

60% (0.6531, 0.6531)

(0,1)(1,0) (0.6113, 0.6113) (0.1, 1) (1, 0.1)

50% (0.8626, 0.8626)

(0.5515,1) (1, 0.5515)

(0.7475, 0.7475) (0.45,1) (1, 0.45)

40% Unique Strategy (1,1) (0.9, 0.9545)

(0.85, 1) (1, 0.85)

30% Inclusive Strategy

(0.1116, 0.1116) (0,0) (1,1)

Unique Strategy (1,1)

20% (0.1673, 0.1673)

(0,0) (1,1) (1,1)

  

64

4.4.2 Port Pricing Strategies

So far, we have conducted umpteen numerical experiments for shipping lines’ port call

decision. As defined in our two-stage game approach, a port pricing stage is solved

with obtaining shipping lines’ port call decision. However, we have witnessed the

existence of Multiple Nash equilibria in shipping lines’ port call decision, which it

makes difficult to study port pricing stage. The selection criteria or method needs to be

developed for Multiple Nash equilibria; however it is not in our scope of research.

Therefore, we only study the port pricing stage with the case in which unique Nash

equilibrium in shipping lines’ port call decision exists. Numerical analysis for port

price is conducted by Matlab program. Its code for searching port price and the

searching logic are given in Appendix I. We refer to given parameters from Figure 4.4,

0.5, 0.3, 0.3b g and 0.4243K , and behaviour of shipping lines’ port call

decision for this experiment.

Figure 4.18 Unique Nash equilibrium in port pricing

  

65

Figure 4.18 shows the BRCs of both ports’ prices. Due to the limitation of the

numerical runs, the BRCs are not smooth. However, it is still able to present the

unique Nash equilibrium in port pricing, where it is placed at (0.16, 0.16). Further

experiments have shown that when either port operation cost or capacity marginal cost

is raised in both ports, it results in the higher Nash equilibrium price. By cause of the

limited numerical runs, the Nash equilibrium of port prices is searched only for the

symmetric port conditions.

4.5 Conclusions

This chapter has extended the demand function to nonlinear, while accounting for the

market structure of two ports and two shipping lines. The objective of applying the

NTCD function was to capture the shipping lines’ preference towards transshipment

volume that was not observed in LTCD function. Intensive numerical analyses were

performed to investigate this behaviour due to the limitation of analytical feasibility

(the absence of a closed form solution). In particular, the BRC method has effectively

captured the results of unique or Multiple Nash equilibria in shipping lines’ port call

decision. Using the characteristics of equilibria, we were able to define the three types

of port call strategies: Unique Strategy, Exclusive Strategy and Inclusive Strategy. The

behaviour of shipping lines towards key factors, i.e. port capacity, price and

transshipment level, is similar to the findings from linear demand case. However, there

are a few more interesting points found in this study.

1 The shipping lines’ concern on transshipment volume is magnified with

excessively large capacities. In this case, the shipping lines’ interest is in

maximizing container demand by crowding at same port.

  

66

2 In the event of excessive capacity, small differences in prices or transshipment

levels between competing ports are sufficient to incentivize shipping lines to

switch from one port to the other.

3 In the event of limited capacity where congestion occurs, shipping lines would

prefer to ignore transhipment benefits and spread their container load across

different ports so as to achieve better services.

  

67

Chapter 5 Analysis on Asymmetric Shipping lines

Analysis on Asymmetric Shipping lines

5.1 Overview

Continued with our model and NTCD function, this chapter attempts to explore the

behaviour of asymmetric shipping lines. It is assumed that there is one large and one

small shipping line in the market in terms of their maximum call size. A different

pricing strategy to asymmetric shipping lines is proposed in order to observe the

demand driving force. Due to the complexity of the problem, studying the port pricing

stage is very limited. Therefore, the numerical analysis focuses on shipping lines’ port

call decision and regards port price as given.

The remaining parts of this chapter are organized as follows: section 5.2

presents modified profit functions for asymmetric shipping lines and ports. Section 5.3

demonstrates the results of numerical analyses and section 5.4 concludes this chapter.

5.2 The model

A port in the real operation offers different prices to different shipping lines. The main

reason behind this is due to the different demand driving force from different shipping

lines (or alliances). Those who have a strong demand driving force are the vital

customer for port since these customers enable the port to secure the certain level of

demand. Recalling the industrial event between PSA and PTP, one of key reasons that

  

68

Maersk line (it is the world largest shipping line that has an absolute demand driving

force) shifted to PTP was 40% lower port price offered by PTP. Thus, in this study, we

assume that ports offer the different prices to different shipping lines, and investigate

the behaviour of shipping lines towards the prices and transshipment benefits. The port

price is now defined by ir for i =1, 2 and r =1, 2, where i and r indicating the

shipping lines and the ports, respectively. Accompanying with proposed port price, the

profit function for shipping lines can be modified to (5.1).

2

1

max 1,2ii i i r r ir

r

p c D F i

2

1

subject to

0 1, 2

1, 2

0 1, 2

ir i

ir ir

r r

ii i r r

q n r

q n

F K r

p c D r

(5.1)

It is assumed that each shipping line has a different container charge ip , operation cost

ic and total port call size in . As size of total port call increases, it is expected

economies of scale in shipping lines’ container charge and operation cost. Thus, it is

reasonable to assume that a large shipping line has relatively lower profit margin but

also lower operation cost. The port profit function with different pricing for each

shipping line is updated to (5.2).

2

1

1,2ir r r ir r r

r

O F m K i

(5.2)

  

69

5.3 Numerical analysis

As witnessed in the chapter 4, the complexity that generated by nonlinearity of

functions makes our model unable to obtain any analytical properties. Therefore,

similar to previous chapter, we will continue our exploration through numerical

analyses. So far in this thesis, a non-cooperative two-stage game has been employed as

a solution method. However, due to the existence of Multiple Nash equilibria in

shipping lines’ port call decision, studying port pricing stage is very limited; hence we

will not examine the stage for Nash equilibrium port price in this chapter. Our

numerical analysis totally focuses on capturing the behaviour of shipping lines’ port

call decision.

5.3.1 Asymmetric Shipping lines’ Port Call Decision

The assumption is made that shipping line 1 and 2 represents a large shipping line and

a small shipping line, respectively. A large shipping line’s total port call size is given

to be double compared to small shipping line’s total port call size, i.e. 1 22, 1n n .

Furthermore, existence of economies of scale allows a large shipping line to lower its

container charge and operation cost; therefore, 1 2p p and 1 2c c . The BRCs present

the shipping lines’ port call decision at port1.

5.3.1.1 Symmetric port conditions

We first examine shipping lines’ port call decisions when both ports offer same port

capacities, prices and transshipment levels, where 1 2:K K K , 1 2: and

1 2:g g g .

  

70

1) Analysis on capacity K

With given parameters of 0.5, 0.3, 0.3b g , a similar approach from previous

chapter, expression (3.21), is used to calculate the various levels of port capacity. The

maximum market demand is estimated below and the symmetric port capacity levels

are defined in Table 5.1 in accordance with CU.

2

1

max 0.3 2 2 1 0.3 1 2 1 1.5588bir r

i

D gq Q

Table 5.1 Symmetric port capacity levels for ASLs

CU(%) Port capacity K

100% 90% 80% 70% 60% 50% 40% 30% 20%

0.7794 0.8660 0.9743 1.1135 1.2990 1.5588 1.9486 2.5981 3.8971

Asymmetric shipping lines’ BRCs with respect to given CU are shown in Figure 5.1.

From very congested to excessive capacity, the BRCs show the major trend of Unique

Strategy (a), Exclusive Strategy (b)-(d) and Inclusive Strategy (g)-(h). It seems Figure

5.1(e) and Figure 5.1(f) depicts a transition stage of BRCs converting from downward

sloping to upward sloping. The five Nash equilibria are found in Figure 5.1(c)-(e),

where the mid-point equilibrium becomes stable. Figure 5.1(g) and (h) suggest that

when the capacity is large enough to handle the both shipping lines’ demand, shipping

lines prefer to call at same port in order to maximize the transshipment benefit through

each other.

  

71

(a) CU=100%

(b) CU=90%

(c) CU=80%

(d) CU=70%

(e) CU=60%

(f) CU=50%

(g) CU=40%

(h) CU=30%

Figure 5.1 ASLs’ BRCs under various levels of symmetric capacities

  

72

2) Analysis on price

Figure 5.2 shows the shipping lines’ BRCs when the given parameters are CU=80%,

0.3g and 0.3 . It is resulted in downward sloping Multiple Nash equilibria, hence

perceiving Exclusive strategy. Based on this result, we now test the impact of different

levels of port price on asymmetric shipping lines’ decision.

Figure 5.2 Comparison base case: CU=80%, g =0.3, = 0.3

The outcome of increased port price from base case is shown in Figure 5.3(b). This is

greatly similar to Figure 5.3 (a), which is a result of decreasing port capacity from base

case. On the other hands, a decrease of port price from the base case generates a

similar BRCs to Figure 5.3(c), where the port capacity is increased to CU=70%. To

sum up, a rise of port capacity and a reduction of port price attribute similar trends to

shipping lines’ port call decision.

  

73

(a) CU=100% (b) Price increased by 40%

(c) CU=70% (d) Price decreased by 10%

Figure 5.3 Comparison of base case with changes in symmetric capacities and prices

3) Analysis on transshipment level g

Continued the numerical experiment with Figure 5.2, we now investigate the impact of

different transshipment levels on asymmetric shipping lines’ port call decision. As

given in Figure 5.3 and Figure 5.4, the BRCs are largely the same. An increase of

transshipment level by 30% from the base case led to similar BRCs in Figure 5.4(a),

where the port capacity is decreased to CU=100%. On the other hand, decreased

transshipment level yielded the result shown in Figure 5.4(d). This is similar to Figure

5.4(c), where the port capacity is increased to CU=70%. It can be concluded same as

price analysis, i.e. an expansion of port capacity and a reduction of transshipment level

draws similar behaviour of shipping lines’ port call decision.

  

74

(a) CU=100% (b) TS increased by 30%

(c) CU=70% (d) TS decreased by 10%

Figure 5.4 Comparison of base case with changes in capacities and TS levels

5.3.1.2 Asymmetric port conditions

In this section, asymmetric shipping lines’ behaviour is explored with asymmetric port

conditions, i.e. 1 2K K , 1 2g g and ir .

1) Analysis on 1 2K > K

Port 1’s capacity is set to be 20% higher than port 2’s in this experiment. It is observed

that a larger capacity difference between two ports results in the faster convergence

rate to the equilibrium point. Figure 5.5 shows the trend of shipping lines’ decision

behaviour towards larger capacities. Shipping lines’ preference on port 1 is apparent as

congestion is eased by larger capacity. However, it is given in Figure 5.6 that when

both ports are large, their capacity difference seems not so much influence on shipping

lines’ decision.

  

75

Figure 5.6 ASLs’ BRCs when 1 2K > K at CU=20%

(a) CU=100% (b) CU=80%

(c) CU=60% (d) CU=40%

Figure 5.5 ASLs’ BRCs when 1 2K > K

  

76

2) Analysis on 1 2g > g

Port 1’s transshipment level is set to be 20% higher than that of port 2 in this

numerical run. A highly congested port capacity yields a unique Nash equilibrium as

given in Figure 5.7(a). It is expected that the port 1 receives less port calls due to its

high transshipment level which contributes to congestion at port. Figure 5.7(b) and (c)

suggest the Exclusive Strategy while Figure 5.7(d) perceives that now the capacity is

large enough to offset the congestion effect caused by high transshipment level. Thus,

shipping lines opt for port 1 and enjoy the transshipment benefits by calling at the

same port. The further increased capacity such as CU=20% (see Figure 5.8) gives a

result similar to Figure 5.7(d). The positive effects of transshipment become more

apparent in this case.

(a) CU=100% (b) CU=80%

(c) CU=60% (d) CU= 40%

Figure 5.7 ASLs’ BRCs when 1 2g > g

  

77

Figure 5.8 ASLs’ BRCs when 1 2g > g at CU=20%

3) Analysis on differentiated prices, ir

We now study the proposed pricing strategy, where two ports are offering different

prices to asymmetric shipping lines. Amongst possible combinations, our interest lies

on exploring the cases appeared in Table 5.2. It is expected to observe the demand

driving force leading by each shipping line.

Table 5.2 Experimental cases for proposed pricing strategy to ASLs

Pricing A Pricing B 1 11 2

2 21 2

1 11 2

2 21 2

There is an attempt to direct shipping lines to either port by proposed price setting.

Next, it is believed that there exist the transshipment benefits that could lead shipping

lines to make a choice over the cheaper price. Therefore, hereby we can define the

demand driving force in our study as a pulling power that one shipping line

successfully convinces the other shipping line to choose the port over the price

advantage. Shipping lines’ decision will be made by comparing the marginal profit

from cheaper port price and the marginal profit from a balance between transshipment

benefit and congestion delay cost. An additional assumption is made for experiment

that both ports’ transshipment levels and capacities are identical.

  

78

(a) Pricing A (b) Pricing B

Figure 5.9 ASLs’ BRCs under proposed pricing strategies at CU=90%

(a) Pricing A (b) Pricing B

Figure 5.10 ASLs’ BRCs under proposed pricing strategies at CU=30%

Figure 5.9 represents asymmetric shipping lines’ BRCs with given port capacity

CU=90%, which is very congested. Hence, shipping lines call for either port because

the marginal profit from cheaper port price is higher than the marginal profit from a

balance between transshipment benefits and congestion delay cost. For instance, in

Figure 5.9(a), when the large shipping line opts for port 1 due to the cheaper price, this

could be a good opportunity for small shipping line to benefit from the large shipping

line’s transshipment demand. However, in this capacity level, the congestion takes

precedence over the transshipment benefits if small shipping line calls at port 1; hence

in the end small shipping line calls at port 2. A similar behaviour is observed in Figure

5.9(b).

  

79

Figure 5.10 shows a more desirable situation to observe the transshipment-prone

behaviour in shipping lines’ port call decision. As congestion is less inhibiting factor

in this level of capacity, i.e. CU=30%, shipping lines now have an option to compare

the price advantages and the transshipment benefits. Particularly, in Figure 5.10(a),

small shipping line has an opportunity to compare the profit margin obtained from

cheaper price (port2) and that from transshipment benefit (port1). The result shows

that small shipping line chooses port 1 in spite of cheaper price in port 2. A similar

decision process is made by large shipping line in Figure 5.10(b). A large shipping line

compares the profit margin obtained from cheaper price (port 2) and that from

transshipment benefit (port 1). However, the result is different from 5.10(a) that the

large shipping line’s profit margin from both cheaper price and transshipment benefit

is fairly compatible. This means a pulling power by small shipping line is not as strong

as large shipping line’s. Therefore, we can conclude that the large shipping line has an

absolute demand driving force than that of small shipping line.

To sum up, the findings from this analysis imply that if the port

successfully lures large shipping line, its demand driving force automatically pulls

small shipping line to the same port.

5.4 Conclusion

In this chapter, we have examined asymmetric shipping lines’ port call decision. The

asymmetric shipping lines’ behaviours towards port capacity, price and transshipment

level have shown trends that are largely similar to what we have found in the previous

chapter. However, we have uncovered some interesting results in the scenario where

the port charges different prices to asymmetric shipping lines. The key findings are:

  

80

1 Large shipping lines possess a stronger demand driving force than that of small

shipping lines.

2 Small shipping lines prefer the opportunity to obtain more transhipment

demand over a cheaper port price. This preference stems from the increased

probability to achieve transhipment benefits from larger shipping lines.

Based on the above findings, we can propose the following port strategies with respect

to the preferences of asymmetric shipping lines.

1 Port operators can effectively retain the market share for transhipment

containers by attracting large shipping lines. The presence of large shipping

lines would in turn draw smaller shipping lines to the port.

2 Provide differentiated port prices to large shipping lines to cater for their high

sensitivity towards port pricing.

3 The assurance of excessive operational capacity or prioritized service to

prevent congestion delay would be a good competitive strategy to attract

shipping lines.

  

81

Chapter 6 Conclusions

Conclusions

6.1 Conclusions of the Study

This thesis investigates the competition between transshipment container ports. To

approach this problem, an analytical model that focuses on transshipment container

demand is developed. In particular, multiple functions are created to examine the

interdependency among shipping lines when determining port demand. These

functions provide enhanced visibility on the decision making of shipping lines when

selecting a transshipment port-of-call. Due to the complexity of the model, the

analytical study of this thesis is limited to LTCD case. The study based on NTCD

function relies much on numerical experiments. Practical implementation, moreover,

is not applicable in this thesis due to the insufficient real data and also difficulty in

measuring qualitative factor such as transshipment level which represent connectivity

of feeder networks. Therefore, the numerical experiments are conducted based on

simplified data sets. We leverage on a two-stage game model to generate additional

insights to the responses of shipping lines amid various scenarios. These findings will

enable transshipment ports to strategize accordingly for port demand optimization.

Beyond these, we vary the symmetry of the shipping lines calling on transshipment

ports and uncovered the different behaviours of large and small shipping lines in

predetermined scenarios. These findings will aid transshipment ports in designing their

  

82

pricing and benefits strategies so as to entrench shipping lines of different sizes.

Overall, the contributions of our study can be summarized as follows.

1 Developed the model which characterized the transshipment container

demand and showed that the model is a valid predictor of the various

decisions undertaken by the shipping lines and ports within a transshipment

port competition setting.

2 Showed the Multiple Nash Equilibria in shipping lines’ port call decision

that explains the decision making amid dynamic market situations.

3 Provided ports the possible strategies to deal with asymmetric shipping

lines based on the information of different interests in asymmetric shipping

lines.

6.2 Future Research

This study is a first step towards the decoding of the competitive landscape of regional

competing transshipment ports. Beyond the factors of port size, efficiency and pricing

that were addressed in this thesis; future research can include key determinants such as

government policies, port operating models, hinterland compositions, local market

synergies, and regional feeder networks. A more in-depth understanding on how these

determinants interact to derive the market share and profitability can help

transshipment ports better understand their competitive landscape and, more

importantly, formulate the right strategies to maximize profits in face of intense

regional competition. Hence, the practical benefits of the research extended to existing

ports located in transshipment cargo catchment regions.

  

83

From the viewpoint of methodological analysis, Multiple Nash equilibria found in

shipping lines’ port call decision can be studied further in terms of its selection criteria

or evaluating equilibrium point so as to advance the port pricing stage in our two-stage

game.

  

84

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Appendix A

Proof of Lemma 1

From (3.2),(3.3),(3.4),(3.5), we obtain

2 22 2 21 21 1 1 1 2 2 2 2

1 22 2 21 1 1 1 1 1 2 1 1 2 1

2 22 22 21 1 2 2

1 1 1 2 2 22 21 1 2 2

22 21 11 12

1 1

2 2

2 2

2

i i ii i

i i i i i i i

i i

i

F FD F D F D F D FF F

q F q q F q F q q F q

D D D DNg Ng F Ng Ng F

F F F F

D Dg N N F g

F F

22 22 22 22

2 2

2 0i

D DN N F

F F

(A.1)

Therefore, i is concave in 1iq .

  

90

Appendix B

Derivations of comparative statics for shipping lines’ port call decision

Let 1 2 1 2 1 2, , , , ,X K K g g

thus, *1 1i iq q X

Using (3.9), we can express

*1

1

; 0ii

i

q Xq

Applying the total differential implicit function and chain rule, differentiate (3.9) with

respect to X. Then, we obtain

2 2 211

21 1 1 1

0ji i i i

i ji i j i

qq

q X q q X q X

(A.2)

2 21 11 1 1 1 1 1 1

121 2 1 1 1 1 1 1 1 1 1

22 22 2 2 2 2 2 2

222 1 1 2 1 1 2 1 2

0

i i ii

i j j i j i i j

i ii

j i j i i j

F FD F F D F D FF

q q F q q F q q F q q

F FD F F D F D FF

F q q F q q F q q

(A.3)

Substituting (A.1) and (A.3) into (A.2),

  

91

2 22 21 2 11 1 1 1 2 2 2 2

1 22 21 1 1 1 1 2 1 2 1 1

21 11 1 1 1 1 1 1

121 1 1 1 1 1 1 1 1

22 2

22

2 2i i ii i

i i i i i i

i ii

j i j i i j

j

F F qD F D F D F D FF F

F q F q q F q F q q X

F FD F F D F D FF

F q q F q q F q q

D F

F q

21

12 22 2 2 2 22

1 1 2 1 1 2 1 2

0j ii j

ii ii

i j i i j

q

X q XF FF D F D FF

q F q q F q q

(A.4)

From the properties of demand in (3.2) and (3.3), expression (A.4) can be written as:

22 21 1

1 1 121 1 1

222 2

2 2 2 222 2

22 21 1

1 1 1 12 21 1 1

212 22 2

2 2 2 222 2

2

2

2

0

2

i

i

i

ij i

i j ii

D DNg F Ng

F F q

XD DNg F Ng g

F F

D DN g F Ng g

F F q

X q XD DN g F Ng g

F F

22 2 21 1

1 1 121 1 1

22 22 2

2 2 2 222 2

22 21 1 1

1 1 1 1 1 121 1 1 1

22 22 2 2

1 2 2 2 2 2 222 2 2

21

2

2

i

i

i

i

iN

i i

D DN g F Ng

F F q

XD DN g F Ng g

F F

D D DN g F Ng g Ng g

F F F q

XD D DN g F Ng g Ng g

F F F

D

2 2 11 1 1 12 2

1 1 1

212 22 2

2 2 2 222 2

2

0

2

i

i i

ii

DN g F Ng g

F F q

X q XD DN g F Ng g

F F

Finally we have,

  

92

22 21 1

1 1 1 12 21 1 1

212 22 2

2 2 2 222 2

2

0

2

i

i

ii

D DN g F Ng g

F F Q

X q XD DN g F Ng g

F F

(A.5)

Summing up i in (A.5), then

2

1 11

2 22 2 2 21 2 1 21 1 2 2 1 22 2

1 2 1 2

1

2

Ni

i iN q XQ

X D D D DN g F g F g g

F F F F

(A.6)

Using (3.9) to derive 21i iq X , we obtain:

1 1

2 21 1 21 2

1 2

0

3

Q g

D DN g g

F F

(A.7)

1 2

2 22 1 21 2

1 2

0

3

Q g

D DN g g

F F

(A.8)

11

1 1

2 21 1 21 2

1 2

60

3

Dg

Q K

K D DN g g

F F

(A.9)

22

1 2

2 22 1 21 2

1 2

60

3

Dg

Q K

K D DN g g

F F

(A.10)

  

93

11 1 1 1 1

11

2 21 1 21 2

1 2

3 2

3

Dp c D g Q N D

FQ

g D DN g g

F F

(A.11)

22 2 2 2 2

21

2 22 1 21 2

1 2

3 2

3

Dp c D g Q N D

FQ

g D DN g g

F F

(A.12)

  

94

Appendix C

Same capacities and transshipment levels for both ports

Proof of Proposition 1

The first order derivate of (3.22) can be obtained

2

1

1

06 2

F K

aN f gN

(A.13)

Hence, the second order derivate of (3.22) is

2121

0F

The second order derivate for port 1’ profit with respect to its price is given below.

2 2

1 1 11 12 2

1 1 1

2F F

O

(A.14)

(A.13) and second order derivate of (3.22) give

21 1

21 1

2 0F

(A.15)

Therefore, 1 is concave in 1 . Similar result can be obtained for port 2.

Proof of Proposition 2

To prove uniqueness of port price, the contraction condition (Milgrom and Roberts,

1990) is assessed. The second order partial derivate for port 1 is

  

95

2 2

1 1 11 1

1 2 2 1 2

F FO

(A.16)

We can get the second order derivate for port 1’s demand from (A.13) and the first

order derivate for port 1’s demand with respect to port 2’s price from (3.22).

21

1 2

0F

, 1 1

2 1

F F

The obtained values give,

2 2

1 1 1 121 1 2 1 2

2 0F F

Similar result is obtained for port 2. As it satisfies the contract condition, the Nash

equilibrium port price, shown in (3.23), is unique.

  

96

Appendix D

Linear Congestion Delay Cost function Proof of Proposition 1

The first order derivate of (3.25) can be obtained

2

1 1 1 22 2

1 2 1 1 2

02

F g K K

a g K g K

(A.17)

Hence, the second order derivate of (3.25) is

2121

0F

The second order derivate for port 1’ profit with respect to its price is given (A.14).

(A.17) and second order derivate of (3.29) give

21 1

21 1

2 0F

(A.18)

Therefore, 1 is concave in 1 . Similar result can be obtained for port 2.

Proof of Proposition 2

To solve (A.16), we can get the second order derivate for port 1’s demand from (A.17)

and the first order derivate for port 1’s demand with respect to port 2’s price from

(3.25).

21

1 2

0F

, 1 1 2 1 2

2 22 2 1 1 22

F g g K K

a g K g K

  

97

The obtained values justify,

2 21 1

21 1 2

21 1 2 1 21 1 1 1 2 1 2 1 2

2 2 2 2 2 21 2 2 1 1 2 2 1 1 2 2 1 1 2

22

2 2

g K K g gF F g K K g g K K

a g K g K a g K g K a g K g K

The contract condition is satisfied only when 1 22g g . If port 2’ transshipment level is

more than double of port 1’s, the Nash equilibrium port price may not be unique.

Similar result is obtained for port 2.

  

98

Appendix E

Kuhn-Tucker condition

We conduct the numerical simulation based on Kuhn-Tucker condition. First, shipping lines’ profit function can be re-written in the canonical form.

max 1, 2

subject to

0 1, 2

( ) 0 1, 2

0, 1, 2

1 0 1, 2

i

r r

r r

ir

ir

i

K F r

p c D r

q r

q r

Transform the shipping line 1’s form to Lagrangian,

1 11 1 2 3 4 5 6; , , , , ,L q

1 1 1 1 2 2 2K F K F

3 1 1 4 2 2 5 11 6 111p c D p c D q q (A.19)

The first order conditions are as follows:

1)

1 1 1 2 1 1 2 21 2 3 4 5 6

11 11 11 11 1 11 2 11

0 L F F D F D F

q q q q F q F q

11 1111

0, 0L

q qq

2)

11 1

1

0L

K F

; 1 0 ;

11 1 1 1

1

0L

K F

3) 12 2

2

0L

K F

; 2 0 ; 12 2 2 2

2

0L

K F

4) 11 1

3

0L

p c D

; 3 0 ; 13 3 1 1

3

0L

p c D

5)

12 2

4

0L

p c D

; 4 0 ; 14 4 2 2

4

0L

p c D

  

99

6)

111

5

0L

q

; 5 0 ; 15 5 11

5

0L

q

7) 111

6

1 0L

q

; 6 0 ; 16 6 11

6

(1 ) 0L

q

  

100

Appendix F

Proof of Lemma 2

Let be a move made by shipping line 1 from current Nash Equilibrium point (1, 0).

Using Taylor series expansion, evaluate the shipping line 1’s shift, 1 , while

shipping line 2 remains at current Nash Equilibrium (1, 0). Taylor series expansion

gives, (1 ) 1 1 and solve for 1 gives

0

1 (1 )1 lim

The profit for shipping line 1 at Nash Equilibrium (1, 0) is:

11 1q ; 12 0q ; 21 0q ; 22 1q ; 1 1Q ; 2 1Q

11 1 11 1 1F g q Q g ; 12 2 12 2 0F g q Q

21 1 21 1 0F g q Q ; 22 2 22 2 2F g q Q g

1 11 21 1F F F g ; 2 12 22 2F F F g

2 21 1

1 21 1

F gD a a

K K

;

2 22 2

2 22 2

F gD a a

K K

21

1 121

1g

p c a gK

(A.20)

The profit for shipping line 1 at 1 , 0 is:

11 1q ; 12q ; 21 0q ; 22 1q ; 1 1Q ; 2 1Q

3

211 1 1F g ; 12 2 1F g ; 21 0F ; 22 2 1F g

3

21 1 1F g ;

3

22 2 1F g

  

101

321

1 21

1gD a

K

;

322

2 22

1gD a

K

32 31 2

1 121

322

2 222

11 1

11

gp c a g

K

gp c a g

K

(A.21)

Since it is a symmetric port case, assuming 1 2 1 2 1 2, , K K g g

0

1 1

2 2

3 7 7 501 2 2 22

312

1 (1 )1 lim

3 11 1 1

2 2lim

9 71 1 1

2 2

1 7

2 2

g p c

ag

K

agg p c

K

As assumed (1, 0) is not a Nash equilibrium,

312

1 70

2 2

agg p c

K

2

27 0

agp c

K

2

2 7agK

p c

Finally, the threshold capacity level is found. It gives that (1, 0) is not Nash

Equilibrium when the port capacity falls in above condition.

  

102

Appendix G

Proof of Lemma 3

Let be a move made by shipping line 1 from the Nash Equilibrium point (1, 1).

Using Taylor series expansion, evaluate the shipping line 1’s shift, 1 , while

shipping line 2 remains at current Nash Equilibrium (1, 1).

Taylor series expansion gives,

(1 ) 1 1

Solving for 1 gives

0

1 (1 )1 lim

.

The profit for shipping line 1 at Nash Equilibrium (1, 1) is :

11 1q ; 12 0q ; 21 1q ; 22 0q ; 1 2Q ; 2 0Q

11 1 11 1 1 2F g q Q g ; 12 2 12 2 0F g q Q

21 1 21 1 1 2F g q Q g ; 22 2 22 2 0F g q Q

1 12 2F g ; 2 0F

2 21 1

1 21 1

8F gD a a

K K

;

2

22

2

0F

D aK

21

1 121

81

gp c a g

K

(A.22)

The profit for shipping line 1 at 1 , 1 is:

11 1q ; 12q ; 21 1q ; 22 0q ; 1 2Q ; 2Q

  

103

11 1 1 2F g ; 12 2F g ; 21 1 2F g ; 22 0 F

1 1 2 2F g ; 2 2F g

321

1 21

2gD a

K

;

2 32

2 22

gD a

K

32

11 12

1

2 32

2 222

21 1 2

gp c a g

K

gp c a g

K

(A.23)

Since it is symmetric port case, assuming

0

1 11

2 22

1 1

2 22 3

0

3 4

3

2

1 (1 )1 lim

1 2 1 32 1 1 1

2 4 2 2

1 1 1lim 2 2 1 5 4 2 2 1 1

2 4 2

1 14

2

5 222 2

4

g p c

agg p c

K

As assumed (1, 1) is not the Nash equilibrium, then we have

3

2

5 222 2 0

4

agg p c

K

22 88

5( )

agK

p c

The threshold capacity level is found. It gives that (1, 1) is not Nash Equilibrium when

the port capacity falls in above condition.

1 2 1 2 1 2, , K K g g

  

104

Appendix H

Proof of Theorem 3

Let 1 be a move made by shipping line 1 from a Nash equilibrium point (1

2,

1

2) and

2 be a move made by shipping line 2 in order to respond to 1 . The purpose of this

marginal analysis is to find the coefficient between shipping lines’ move on BRCs.

Using Taylor series expansion, we first study the partial derivative of:

1 2 1

2

1 1 1 1, ,

2 2 2 2

We have

1 2 1

1 1 1 1, ,

2 2 2 2

321 2

1 1 22

1 11

2

gp c a g

K

32

1 21 1 22

1 11

2

gp c a g

K

321

1 12

1 11

2

gp c a g

K

321

1 12

1 11

2

gp c a g

K

1 1 2 1 1 2

1 1 1 1

1 11 1

2 2

1 11 1

2 2

g p c

  

105

3 3

1 2 1 1 2 1 2 1 1 23

23 3

1 1 1 1 1 1

1 11 1 1 1

2 2

1 11 1 1 1

2 2

ga

K

Let 1 11 , =1- , we have

2 21 2 1

1

2 21 2 1

1

7 77 7 72 2

2 22 2 21 2 1

1

7 77 7 72 2

2 22 2 21 2 1

1

1 1 1 11

1 1 1 11

1 1 1 11

1 1 1 11

Binomial Taylor series give:

1

22 2

1

22 2

77 72

2 22 2

77 72

2 22 2

1 12

1 12

71 1

2

71 1

2

The simplified partial derivative is obtained below,

1 2 1

2

7 732 2

1 1 1 12

1 1 5 532 2 2 2 2

1 1 1 12

1 1 1 1, ,

2 2 2 2

1 1 1 1

2 2 2 2

72 1 2 1 2 1 2 1

4

agg p c

K

agg p c

K

Using Taylor series expansion with respect to 1 , we get

  

106

7 732 2

1 1 1 12

1 1 5 532 2 2 2 2

1 1 1 12

1 1 1

1 1 1 1

2 2 2 2

72 1 2 1 2 1 2 1

4

1 1 11

2 2 2

agg p c

K

agg p c

K

g p c

1

7 732 2

1 1 1 12

1 1

2 21 1 1 1

2

5 532 2

1 12

11

2

1 7 1 71 1

2 2 2 2

1 12 2 1 1 2 2 1 1

2 2

4 7 52 2 1 1 2

2

ag

K

g p c

ag

K

1 1

52 1 1

2

7 73 32 2 2

1 2 2

1 1 5 53 32 2 2 2 2

1 2 2

3 32 1 2

1 2 2

71 1

21 73 2 2 2 2

4

37 21

4

ag agg p c g p c

K K

ag agg p c g p c

K K

ag agg p c g p c

K K

7 732 2

2

1 1 5 532 22 2 2 2

2

Remains

1 1

7

2 2

agg p c

K

agg p c

K

Then, we have

3 3

21 1 22 2

7 3 210

4

ag agg p c g p c R

K K

The coefficient between moves of shipping lines is obtained:

2

22

21

2

4 7

3 21

agp c

K

agp c

K

  

107

Appendix I

Searching equilibrium port price starts with port 1’s given pricing strategy. While port

1 fixes its price at a certain value, port 2 gives all possible prices of own and search for

the price that returns maximum profit. Referring to Table A.1, port 1’s price is given

as an arbitrary value while port 2 is varying the whole range of price values. Once the

best of profits is obtained, it is stored in representative variable with the information of

profit and the value of price.

Table A.1 An example of port price searching logic

µ1 µ2 Port 1’s profit

Port 2 ‘s profit

0.24 0.11 -0.75 -1.582730.24 0.12 -0.75 -1.374540.24 0.13 -0.75 -1.166360.24 0.14 -0.75 -0.958180.24 0.15 -0.75 -0.750.24 0.16 -0.75 -0.541820.24 0.17 -0.5544 -0.403860.24 0.18 -0.4786 -0.265750.24 0.19 -0.3986 -0.150870.24 0.2 -0.3133 -0.059920.24 0.21 -0.2227 0.006730.24 0.22 -0.1273 0.04950.24 0.23 -0.0289 0.06990.24 0.24 0.0706 0.070590.24 0.25 0.1688 0.0550.24 0.26 0.2639 0.026660.24 0.27 0.3546 -0.011290.24 0.28 0.4401 -0.056340.24 0.29 0.5203 -0.106670.24 0.3 0.5955 -0.161010.24 0.31 0.666 -0.218440.24 0.32 0.7322 -0.278390.24 0.33 0.7947 -0.340470.24 0.34 0.854 -0.404470.24 0.35 0.9107 -0.47042

Port 1’s given price 

Searching the best response price of port 

Maximum profit

  

108

Matlab Code for searching Nash equilibrium of port price %fix u1, vary u2% for u1=u1_lp:0.005:u1_up j=j+1; P_max=0; for u2=u2_lp:0.005:u2_up i=i+1; x0 = [0,0]; % Make a starting guess at the solution options = optimset('LargeScale','off'); options=optimset(options,'MaxFunEvals',10000000000000000000); options=optimset(options, 'MaxIter',10000000000000000000); x = fmincon(@(x)objfun(x,p,c,N,b,u1,u2,K,a,g,O,m),x0,[],[],[],[],[],[],... @(x)confuneq(x,p,c,N,b,u1,u2,K,a,g,O,m), options) [f,Q,F_11,F_12,F_21,F_22,F,D]= objfun(x,p,c,N,b,u1,u2,K,a,g,O,m); [ce, ceq] = confuneq(x,p,c,N,b,u1,u2,K,a,g,O,m); % Check the constraint values at x SL1_Profit=(p-c-u1-D(1))*F_11+(p-c-u2-D(2))*F_12; SL2_Profit=(p-c-u1-D(1))*F_21+(p-c-u2-D(2))*F_22; P_1=(u1-O(1))*F(1)-m(1)*K(1); P_2=(u2-O(2))*F(2)-m(2)*K(2); array1(i)=u1;array2(i)=u2;array3(i)=x(1);array4(i)=x(2); array5(i)=F(1);array6(i)=F(2); array7(i)=D(1);array8(i)=D(2); array9(i)=P_1;array10(i)=P_2; if (P_2 > P_max) P_max=P_2; best_r(j)=array2(i) end end end % fix u2, vary u1% k=j; l=i; for u2=u2_lp:0.005:u2_up k=k+1; P1_max=0; for u1=u1_lp:0.005:u1_up l=l+1; x0 = [0,0]; % Make a starting guess at the solution options = optimset('LargeScale','off');

  

109

options=optimset(options,'MaxFunEvals',1000000000000000000000); options=optimset(options, 'MaxIter',1000000000000000000000); x = fmincon(@(x)objfun(x,p,c,N,b,u1,u2,K,a,g,O,m),x0,[],[],[],[],[],[],... @(x)confuneq(x,p,c,N,b,u1,u2,K,a,g,O,m), options) [f,Q,F_11,F_12,F_21,F_22,F,D]= objfun(x,p,c,N,b,u1,u2,K,a,g,O,m); [ce, ceq] = confuneq(x,p,c,N,b,u1,u2,K,a,g,O,m); % Check the constraint values at x SL1_Profit=(p-c-u1-D(1))*F_11+(p-c-u2-D(2))*F_12; SL2_Profit=(p-c-u1-D(1))*F_21+(p-c-u2-D(2))*F_22; P_1=(u1-O(1))*F(1)-m(1)*K(1); P_2=(u2-O(2))*F(2)-m(2)*K(2); array1(l)=u1;array2(l)=u2;array3(l)=x(1);array4(l)=x(2); array5(l)=F(1);array6(l)=F(2); array7(l)=D(1);array8(l)=D(2); array9(l)=P_1;array10(l)=P_2; if (P_1 > P1_max) P1_max=P_1; best_br(k)=array1(l); end end end % best response table% t=0; for gp=u1_lp:0.005:u1_up t=t+1; array11(t)=gp; array12(t)=best_br(k-j+t); end