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Contests with Ambiguity

David Kelsey Tigran MelkonyanDepartment of Economics, Behavioural Science Group,University of Exeter. Warwick University.

University of Exeter.

August 2016

David Kelsey (University of Exeter.) Contests with Ambiguity August 2016 1 / 20

Introduction

In a contest there is a single indivisible prize.

Agents compete to win this prize by expending money or e¤ort.

Contests have been used to model the following interactions:

R&D and patent racesmilitary con�ictpolitical competitionlitigationrent-seeking, beauty contests and in�uence activitiessporting contests

Rent Dissipation

Tullock argues that the entire value of the prize will be expended during thecontest.In practice it seems that rent dissipation is signi�cantly less than 100%.

Introduction

Rent Dissipation

Introduction

Rent Dissipation

Introduction

R&D and patent races

military con�ictpolitical competitionlitigationrent-seeking, beauty contests and in�uence activitiessporting contests

Rent Dissipation

Introduction

R&D and patent racesmilitary con�ict

political competitionlitigationrent-seeking, beauty contests and in�uence activitiessporting contests

Rent Dissipation

Introduction

R&D and patent racesmilitary con�ictpolitical competition

litigationrent-seeking, beauty contests and in�uence activitiessporting contests

Rent Dissipation

Introduction

R&D and patent racesmilitary con�ictpolitical competitionlitigation

rent-seeking, beauty contests and in�uence activitiessporting contests

Rent Dissipation

Introduction

R&D and patent racesmilitary con�ictpolitical competitionlitigationrent-seeking, beauty contests and in�uence activities

sporting contests

Rent Dissipation

Introduction

Rent Dissipation

Introduction

Rent Dissipation

Introduction

Rent Dissipation

Tullock argues that the entire value of the prize will be expended during thecontest.

In practice it seems that rent dissipation is signi�cantly less than 100%.

Introduction

Rent Dissipation

Contest Model

There is a contest between 2 players, individual A and individual B.

The prize is worth V to both players.

Each contestant i = A,B chooses an expenditure or e¤ort level,xi 2 Xi = [κV ,λV ] , where κ < 1

4 and λ > 14 .

The probability that individual A will win the contest is given by:

pA (xA , xB ) =xA

xA + xB.

Contestant A�s utility function:

uA (xA , xB ) =xA

xA + xBV � xA .

Contest Model

4 and λ > 14 .

pA (xA , xB ) =xA

xA + xB.

uA (xA , xB ) =xA

xA + xBV � xA .

Contest Model

4 and λ > 14 .

pA (xA , xB ) =xA

xA + xB.

uA (xA , xB ) =xA

xA + xBV � xA .

Contest Model

4 and λ > 14 .

pA (xA , xB ) =xA

xA + xB.

uA (xA , xB ) =xA

xA + xBV � xA .

Contest Model

4 and λ > 14 .

pA (xA , xB ) =xA

xA + xB.

uA (xA , xB ) =xA

xA + xBV � xA .

Nash Equilibrium

Player A�s utility is given by

uA (xA , xB ) =xA

xA + xBV � xA .

One can derive A�s best response function which is:

xA =pVxB � xB .

The best response function has the following properties:

it is inverse U-shaped, (single peaked);

the peak occurs where it crosses the 45o line;

it is above (resp. below) the 45o line before (resp. after) the peak;

There is a unique Nash equilibrium where x�A = x�B =

half of the rent is dissipated in Nash equilibrium.

..............................................................................................

RAr.........................................V4

Figure: Reaction function

..............................................................................................

RAr.........................................V4

Many contests are unique events, thus one cannot base subjectiveprobabilities on relative frequencies.

World War I did not help to predict the outcome of World War II.Beauty contests happen often, but the past tells us little about the probabilityof success of a given contestant

Many contests depend on complex systems and/or new technologies, e.g.war, patent races.

The outcome of any contest depends on the behaviour of other people. Thisis intrinsically di¢ cult to predict.

Players may have ambiguous beliefs about what others will do.

Ambiguity is represented by assigning a set of probabilities to an event, e.g.the probability of winning the war is between 0.5 and 0.7.

World War I did not help to predict the outcome of World War II.

Beauty contests happen often, but the past tells us little about the probabilityof success of a given contestant

The Neo-additive Model of Ambiguity

We use the neo-additive model of ambiguity, axiomatised by Chateauneuf,Eichberger, and Grant (2007). They represent preferences by:

αδM (a) + δ (1� α)m (a) + (1� δ)Eπu (a) , (1)

M (a) denotes the maximum utility of act a,

m (a) denotes the minimum utility of act a,

Eπu (a) denotes the expected utility of act a.

This is a special case of the Choquet expected utility model, Schmeidler(1989), which represents beliefs as capacities.

δ is a measure of perceived ambiguity;

α measures ambiguity-attitude, α = 1 (resp. α = 0) corresponding to pureoptimism (resp. pessimism).

Only 2 additional parameters needed compared to expected utility.

αδM (a) + δ (1� α)m (a) + (1� δ)Eπu (a) , (1)

Ambiguity in Contests

There is a negative externality. The more your opponent contributes thelower are your chances of winning.

An optimist places a relatively large decision-weight on the event that theopponent will choose low e¤ort.

This reduces his/her marginal bene�t of e¤ort, since (s)he perceives that(s)he is likely to win easily.

A pessimist places over-weights the possibility that his/her opponent willchoose high e¤ort.

This reduces his/her marginal bene�t of e¤ort, since (s)he perceives that(s)he is likely to lose regardless of his/her own e¤ort.

Since ambiguity increases both optimism and pessimism it reduces themarginal bene�t of e¤ort.

The other major in�uence on behaviour is the intensity of competition. Ifone�s opponents are providing similar e¤ort levels competition is intense,which increases the incentive to provide e¤ort.

Symmetric Equilibrium

We start by considering a symmetric contest. The prize has the same value,V , for both players.

Both players perceive the same degree of ambiguity and have the sameambiguity-attitude, δA = δB = δ̂ and αA = αB = α̂.

Equilibrium e¤ort is a decreasing function of the degree of ambiguity.

It takes the value xA = xB =V4 when there is no ambiguity and

xA = xB =�p

λ� λ�V when there is maximal ambiguity.

Proposition

Assume δA = δB = δ̂ > 0 and αA = αB = α̂. Then:

1 a symmetric equilibrium exists and is unique;2 there is less e¤ort than in Nash equilibrium;3 the equilibrium e¤ort level, x̂ , is a strictly decreasing function of δ̂.

xA = xB =�p

Proposition

xA = xB =�p

Proposition

xA = xB =�p

Proposition

xA = xB =�p

Proposition

xA = xB =�p

Proposition

..............................................................................................

RAr.........................................V4

r...............................

Increasing ambiguity

Ambiguity-Attitude

The e¤ect of a change in ambiguity-attitude is summarised by the following result.

Proposition

Consider the symmetric case where δA = δB = δ̂ and αA = αB = α̂ then anincrease in ambiguity-aversion α̂ will reduce equilibrium e¤ort providedλκV 2 � x̂2 > 0.

Remark

Suppose that κ > 116λ , then λκV 2 � x̂2 > V 2

16 � x̂2 > 0 since xA <V4 by

Proposition 4.1. Henceforth we shall assume XA = XB = [κV ,λV ] , whereλ > 1

4 ,14 > κ > 1

16λ .

Thus an increase in optimism (ambiguity-loving) usually leads lead to higher e¤ort.

An decrease in α shifts decision weight from the worst outcome to the bestoutcome.

The inequality κ > 116λ implies that the best case is not too good.

Asymmetric Perceptions of Ambiguity

Continue to assume the prize has the same value for both players.

However we allow for asymmetric perceptions of ambiguity δA 6= δB anddi¤erent ambiguity-attitudes αA 6= αB .

With ambiguity both players provide less than the Nash equilibrium level ofe¤ort.

This is a possible explanation of why rent dissipation is not complete.

PropositionAssume that both players perceive ambiguity, 1 > δA > 0, 1 > δB > 0. Then inequilibrium both will make less than the Nash equilibrium level of contributions.

This result is not true if the value of winning is di¤erent for the two players,VA 6= VB .

Comparative Statics of Ambiguity I

Assume the players are initially in a symmetric equilibrium and there is anincrease in the ambiguity perceived by Player B.

Then in equilibrium both players will provide less e¤ort.More ambiguity causes Player B to put more weight on the possibility thathis opponent will play a high strategy. This decreases B�s perceived marginalbene�t.Player A responds by reducing her e¤ort, since the competition from B hasbecome less intense.

Proposition

Let x̃A = x̃B = x̃ denote the equilibrium e¤ort levels when δA = δB = δ̃,αA = αB = α̃. If

x 0A , x

�denotes the equilibrium e¤ort levels when

δA = δ̃ < δB = δ̂, then: