Forward-looking Network Formation

1st Workshop on Interactions between Math and Soc Sci 2014

Forward-looking Network Formation

Noemı Navarro

FAE I, EHU/UPV and IKERBASQUE, Basque Foundation for Science

BCAM Bilbao, September 26, 2014

BCAM 2014 Forward-Looking NetForm 1 of 14

Introduction

• Many social, economical and political interactions take

the form of a network of bilateral relationships

• Trading relationships, political alliances, pharma

marketing agreements

• The structure of the network can have a profound impact

on the welfare of the involved parties

• The first question is: which networks are likely to form?

• Once we have that, we can work out other questions like:

when is such a decentralized network a desirable

outcome?


Introduction

• Features of these interactions are the interdependence of

gains and the strategic decision-making

• The natural analytical tool is then game theory

• In particular, we define or formalize the idea of a “stable”

network by means of solution concepts

• A very well known solution concept for noncooperative

games is called Nash equilibrium


Introduction

• If we know the procedure we can use “over-the-counter”

noncooperative game theory

• If we do not know the procedure, or we want a solution

that does not depend on the particular procedure, a very

well known concept is pairwise stability

• A network is pairwise stable if no agent has an incentive

to unilaterally sever any of his/her connections in the

network and if no pair of agents have an incentive to

connect directly when their connection is absent from the

network


Introduction

Drawbacks of this definition:

• Pairwise stable networks may fail to exist

• It is a myopic notion: agents do not anticipate that

creating or severing a connection in the network might

make subsequent changes in the network

• Let us see this with an example


Example

Player 2 gains double than players 1 or 3

s ss

s ss

s ss

s ss

s ss

s ss

s ss

s ss

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AAAA �

��AAAA �

��

AAAA �

��

12

24

12 13

26

13 13

26

13 13

26

13

15

0

15 0

10

5 5

10

0 0

0

0

g1 g2 g3 g4

g5 g6 g7 g8


Example

P =

q1 q2 q3 q4 0 0 0 0

0 q5 0 0 q6 0 0 0

0 0 1 0 0 0 0 0

0 0 0 q7 q8 0 0 0

0 0 0 0 1 0 0 0

0 q9 q10 0 0 q11 0 0

0 0 q12 q13 0 0 q14 0

0 0 0 0 q15 q16 q17 q18

qi ∈ [0, 1]

q1 + q2 + q3 + q4 = 1

q5 + q6 = 1

q7 + q8 = 1

q9 + q10 + q11 = 1

q12 + q13 + q14 = 1

q15 + q16 + q17 + q18 = 1


Example

Myopic agents

ss s0

0 0

ss s0

15 15

ss s10

0 5

ss s10

5 0

ss s26

13 13

ss s26

13 13

ss s26

13 13

ss s24

12 12

@@ ��

@@ �� @@ ��

�� @@

?= s

+U

? ?6

��W

�

K

q

6 6

+

1

1


Example

Forward-looking agents

ss s0

0 0

ss s0

15 15

ss s10

0 5

ss s10

5 0

ss s26

13 13

ss s26

13 13

ss s26

13 13

ss s24

12 12

@@ ��

@@ �� @@ ��

�� @@

?= s

+U

6

?6

*

��W

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q

6 6

+

1


Overview of the model

• Time is discrete t = 0, 1, ...

• Network formation is represented by means of a

stationary transition probability matrix

• Agents obtain stationary gains from each network and

discount the future

• Adapt the notion of pairwise stability in discounted,

expected terms


Overview of the model

The transition probability matrix is a forward-looking network

formation scheme if it satisfies the following two conditions:

1. The probability of severing a direct connection is zero

whenever no participating agent gains from such a

decision (in discounted, expected terms)

2. The probability of creating a direct connection is zero

whenever it is not true that both participating agents

gain from such a decision (in discounted, expected terms)


Overview of the results

Results:

• A forward-looking network formation scheme exists such

that

– Discount factors can differ across agents

– Probabilities can differ across agents

– But all heterogeneous dynamics are related to a

homogeneous-normalized one

• If reallocations of (discounted, expected) gains are

possible, the solution concept with side payment exists if

the binding agreements are governed by a linear rule


What comes next?

• Are there conditions that guarantee the existence of

forward-looking pairwise stable networks?

• Are there reallocations of gains that satisfy notions as

individual rationality?

• How do these two sets of conditions relate?

• How does all this relate to desirable properties of final

networks, like efficiency?


The End

Thank you


Introduction: Related Literature

• Pairwise Stability: Jackson and Wolinsky JET 1996

• Network formation dynamics based on p.s.: Jackson and

Watts JET 2002

• Farsighted stability notions: Dutta, Ghosal and Ray JET

2005, Page, Wooders and Kamat JET 2005, Herings,

Mauleon and Vannetelbosch GEB 2009

• Network formation and allocation of worth: Currarini

and Morelli REcDesign 2000, Slikker and van den

Nouweland GEB 2001, Navarro SCW 2014, Carayol,

Delille and Vannetelbosch 2014

• Jackson GEB 2005, Bloch and Jackson JET 2007


Network Dynamics

• t = 0, 1, 2, ... with transitions gt → gt+1

• Player i receives yi(g) at t if gt = g and discounts the

future δi ∈ [0, 1)

• Transition probability matrix Pi = [Pi(g′|g)]g,g′∈G

• xi(y, Pi, δi) is player i’s discounted, expected payoff vector

xi(y, Pi, δi) = yi + δiPiyi + δ2i P2i yi + ... = (I − δiPi)

−1yi


Forward-looking Pairwise Stability

• Given (y, (P1, ..., Pn), (δ1, ..., δn)), g is pairwise stable in

discounted, expected terms if

1. For ij ∈ g:

xi(.)(g)−xi(.)(g\ij) ≥ 0 and xj(.)(g)−xj(.)(g\ij) ≥ 0

2. For ij /∈ g:

if xi(.)(g)− xi(.)(g ∪ ij) < 0

then xj(.)(g)− xj(.)(g ∪ ij) > 0


Forward-looking Network Formation Scheme

F = (y, (P1, ..., Pn), (δ1, ..., δn)) is a forward-looking network

formation scheme if Pi(g′|g) = 0 for any i ∈ N whenever g′

and g are not adjacent graphs and:

1. For ij ∈ g and k 6= i, j, Pk(g\ij|g) = 0 if

xi(.)(g)− xi(.)(g\ij) ≥ 0 and xj(.)(g)− xj(.)(g\ij) ≥ 0

2. For ij /∈ g, P (g ∪ ij|g) = 0

if whenever xi(.)(g)− xi(.)(g ∪ ij) < 0

we have that xj(.)(g)− xj(.)(g ∪ ij) > 0,

or if xi(.)(g)− xi(.)(g ∪ ij) ≥ 0 and

xj(.)(g)− xj(.)(g ∪ ij) ≥ 0


Existence

Proposition 1 Fix any vector y ∈ <n|G|. Then, for any

δ ∈ [0, 1)n there exist distinct transition probability matrices

(P1, ..., Pn) such that F = (y, (P1, ..., Pn), (δ1, ..., δn)) is a

forward-looking network formation scheme.

Corollary 1 The (distinct) transition probability matrices

(P1, ..., Pn) found above are increasing in the corresponding

difference of discounted, expected payoffs.


Existence

Proposition 2 For any FLNFS

F = (y, (P1, ..., Pn), (δ1, ..., δn)) there exists a δ∗ and a

homogenized/normalized matrix P ∗ such that

• P ∗(g′|g) = 1/|gN | iff Pk(g′|g) > 0 for any k, g, and g′,

and

• F = (y, (P ∗, ..., P ∗), (δ∗, ..., δ∗)) is also a FLNFS.


Introducing Side Payments

• Assume yi(g) are perfectly transferable and binding

agreements are possible

• If the agreement results a linear redistribution rule then

we can find a matrix of size n|G|, denoted R, such that

reallocation after an initial vector of gains yi(g) is given

by Ry in the myopic case


Introducing Side Payments: Two-player Examples

Two agents decide to share equally if connected:

Ry =

1 0 0 0

0 0.5 0 0.5

0 0 1 0

0 0.5 0 0.5

y1∅

y1(12)

y2∅

y2(12)

=

y1∅

0.5y1(12) + 0.5y2(12)

y2∅

0.5y1(12) + 0.5y2(12)


Introducing Side Payments: Two-player Examples

Two agents decide “fairly” if connected:

Ry =

1 0 0 0

0.5 0.5 −0.5 0.5

0 0 1 0

−0.5 0.5 0.5 0.5

y1∅

y1(12)

y2∅

y2(12)

=

=

y1∅

0.5[y1∅ + y1(12) − y2∅ + y2(12)

]y2∅

0.5[−y1∅ + y1(12) + y2∅ + y2(12)

]



Given F , a FLNFS, yr will yield the desired reallocation in

expected, discounted terms

yr = (I −∆P)R(I −∆P)−1y,

where ∆P is block diagonal, with δiPi being each of the

blocks in the main diagonal. Note that

• x(yr) = R(I −∆P)−1y, the reallocation of x(y)

• yr = Ry if R(I −∆P)−1 are commutable



Proposition 3 Fix any vector y ∈ <n|G| and any linear

reallocation rule R. Then, for any δ ∈ [0, 1)n there exist

distinct transition probability matrices (P1, ..., Pn) such that

F = (yr, (P1, ..., Pn), (δ1, ..., δn)) is a forward-looking network

formation scheme.

Corollary 2 The (distinct) transition probability matrices

(P1, ..., Pn) found above are increasing in the corresponding

difference of discounted, expected payoffs.

Note that the reallocation is as much forward-looking as

pairwise stability is (in discounted-expected terms)


Concluding remarks

• Purpose: Contribute to the development of foundational

theoretical models that can serve to analyse decentralized

network formation

• Defined and showed existence of a new solution concept:

forward-looking network formation scheme

• This solution concept:

– can represent subjective (increasing) probabilities but

are related to a normalized network formation process

– also exists when players are redistributing expected

discounted terms linearly


Example

Let us consider 3 players and the following gains from each

possible network structure

• Isolated players gain a profit of 0

• Two players directly connected gain a profit of 2

• Three players connected by two links gain a profit of 13/4

• Three players connected by three links gain a profit of 3

What if players distribute the gains proportionally to their

number of links?


Example

A proportional stage-wise allocation rule

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s ss

s ss

s ss

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1

1

1 13/16

13/16

13/8 13/16

13/8

13/16 13/8

13/16

13/16

1

0

1 0

1

1 1

1

0 0

0

0

g1 g2 g3 g4

g5 g6 g7 g8


Second Type Existence: An Example

s ss

s ss

s ss

s ss

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� � �1

v1 v2

w4

w3 0

0

0

v1 > 3w4, v2 > 2w4 +w32

, w3 > 0

s ss

s ss

s ss

s ss

��AAAA �

��AAAA

- � �1

v1 v2

w4

w3 0

0

0

v2 > 2max{w4,v13} + w3

2, w3 > 0


Second Type Existence: An Example

s ss

s ss

s ss

s ss

��AAAA �

��AAAA

- - �1

v1 v2

w4

w3 0

0

0

w3 > 0

s ss

s ss

s ss

s ss

��AAAA �

��AAAA

- - -1

v1 v2

w4

w3 0

0

0

w3 < 0


Efficiency

• If w is component additive and link monotonic or

• If w is strong critical-link monotonic

THEN

• gN is the strongly efficient graph

• ∃ y∗ monotonic w.r.t. own links

• (y∗, P ) is a forward-looking network formation scheme ∀ δwhere players never dissolve links and always create them


Efficiency: Discussion

s ss

s ss

s ss

s ss

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3 2

1

3/4 0

0

0

w link-monotonic, @ (P, y) monotonic w.r.t. own linksRemember! v1 > 3w4, v2 > 2w4 +

w32

, w3 > 0

s ss

s ss

s ss

s ss

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2.75 2.5

0

2.05 1

1

1

w link-monotonic and empty network strongly efficient


Examples

The component-wise egalitarian allocation rule

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s ss

s ss

s ss

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1

1

1 13/12

13/12

13/12 13/12

13/12

13/12 13/12

13/12

13/12

1

0

1 0

1

1 1

1

0 0

0

0

g1 g2 g3 g4

g5 g6 g7 g8


Examples

P =

p1 p2 p3 p4 0 0 0 0

0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 p5 0 p6 p7 0 0 0

0 p8 p9 0 0 p10 0 0

0 0 p11 p12 0 0 p13 0

0 0 0 0 p14 p15 p16 p17

pi ∈ [0, 1]

p1 + p2 + p3 + p4 = 1

p5 + p6 + p7 = 1

p8 + p9 + p10 = 1

p11 + p12 + p13 = 1

p14 + p15 + p16 + p17 = 1


Efficiency: Discussion

w link-montonic, ∃ (P, y) monotonic w.r.t. own links δ small

s ss

s ss

s ss

s ss

s ss

s ss

s ss

s ss

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3/4- 1/2+

3/4-

1/2+

1/2+

3 2 2 2

3/4− ε

1 + ε 3/4− ε

1 + ε 1 + ε

3/4− ε 0

0

0

g1 g2 g3 g4

g5 g6 g7 g8


Second Type Existence: Discussion

same w, δ > 910 , p(ij) = 1

3 created

s ss

s ss

s ss

s ss

s ss

s ss

s ss

s ss

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5/6 1/3

5/6

11/24

11/24

3 2 2 2

5/6

11/12 5/6

11/12 11/12

5/6 0

0

0

g1 g2 g3 g4

g5 g6 g7 g8


Second Type Existence: Discussion

same w, δ ∼ 1, p(ij) = 13 created

s ss

s ss

s ss

s ss

s ss

s ss

s ss

s ss

��AAAA

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6/7- 2/7+

6/7-

25/56

25/56

3 2 2 2

6/7

25/28 6/7

25/28 25/28

6/7 0

0

0

g1 g2 g3 g4

g5 g6 g7 g8


Forward-looking Network Formation

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