Equilibria in Network Games: At the Edge of Analytics and Complexity

Rachel Kranton Duke University Research Issues at the Interface of Computer Science and Economics, Cornell University September 2009

Introduction

• Growing research on games played on networks • Payoffs depend on players’ actions and graph structure: Π

i

(

x i

,

x

-i

;

G

) • Goal – characterize equilibrium play – as a function of G • Economics - emphasis on analytical solutions • unique vs. multiple equilibria • shape of equilibria – distribution of play • aggregates in different equilibria – e.g., total effort

Introduction

• Analytical solutions can give rise to well-defined computational problems – complexity. • Ripe area for economics-computer science collaboration • Today – give three precise examples for general class of games played on networks • Examples from research with Yann Bramoullé, Martin D’Amours • Game Class: linear best-reply functions • E.g., Ballester, Calvó-Armengol & Zenou (2006), Bramoullé & Kranton (2007) • Generalization – supercede results in previous work • Many other games in economics – IO, Macro – fit into this class • Opens door to network treatment of these games

The Model

•

n

individuals, set

N,

• vector of actions,

x

simultaneously choose = (

x i

,

x

-i

)

x i ≥

0 •

G

,

n

x

n

•

g ij

=

g ji

• matrix, otherwise

g ij

=

g ji g ij

either 0 or 1 = 1 iff

i

impacts

j’s

payoffs directly = 0, assume

g ij i

and

j linked

,

i = g ji ,

normalize

g ii

= 0.

and

j neighbors

• Payoff interaction parameter δ, 0 ≤ δ ≤1.

• Payoffs:

U i

(

x i

,

x

-i

; δ,

G

) •

x i

* ≡ optimal action in autarky • • maximizes

U i

let

x i

* =

x

*  for δ = 0, 1 for all

i g ij

= 0 for all – base case.

j

.

Class of Games - Examples

•

U i

(

x i

,

x

-i

; δ,

G

) has linear best replies • Give two examples of games previously studied in network literature, then precisely specify best replies • Note for 0 ≤ δ ≤1, strategic substitutes games.

• Public goods in networks generalization of Bramoullé & Kranton, 2007, (BK)

Ŭ i

(

x i

,

x

-i

; δ,

G

)

= b

(

x i +

δ

∑ j g ij x j

)

- cx i

with

b

increasing, strictly concave,

b'

(0)

< c < b'

(

+∞

).

Class of Games - Examples

• Negative externalities/quadratic payoffs, Ballester, Calvó-Armengol & Zenou, 2006 (BCZ)

Ũ i

(

x i

,

x

-i

; δ,

G

)

= x i

− ½

x i ² −

δ

∑ j g ij x i x j

• Other examples • investment games with quadratic payoffs • Cournot oligopoly, with linear demand, constant MC,

Linear Best Replies

• Each game yields exactly this best reply:

x i

(

x

-i

)

x i

(

x

-i

) = 1 − δ

∑ j g ij x j

if δ

∑ j g ij x j <

1 = 0 if δ

∑ j g ij x j ≥

1

Nash Equilibria

• Nash equilibrium • existence guaranteed, standard fixed point argument • Characterize the equilibrium set?

• Unique, finite, interior, corner etc…..?

• We solve for equilibria for any

G

• and any δ  [0, 1].

see interplay network structure, shape of equilibrium • • • Show two features of

G

key to equil. character. • Bonacich centrality & lowest eigenvalue of

G

(just flash today) links between analytics, computation, complexity

Nash Equilibrium: Matrix Formulation

• For a vector

x,

• • • •

x

S

G

S

G

N-S,S

consider its

support

set of “active” agents;

S

, such that

S

= vector of actions of agents in

S

= links between active agents = {

i

= links between active agents and inactive agents :

x i

> 0} •

Proposition

For any

δ  [0, 1] and any graph G, x is a Nash

equilibrium iff

(

I

+ δ

G

S

)x δ

G

N-S,S

x

S ≥

0

S

=

1

and

(1) Nash Equilibrium: Algorithm

• Simple algorithm to determine all (finite) equilibria.

• • • For any subset Q, invert ( Compute

x

Q =

(

I I

+ δ

G

Q

) if possible. + δ

G

Q

) −1

1

.

Equilibrium if

x

Q

≥ 0 and δ

G

N-Q,Q

x

Q

• Repeat for all subsets of

N

. ≥ 1 • Yields all finite equilibria.

• For any graph G, the number of equilibria is finite

for almost every

δ.

• Follows from when (

I

• If (

I

+ δ

G

Q

+ δ

G

Q

) invertible ) is not invertible, then equilibrium where

S = Q

is a continuum • Algorithm runs in exponential time – but number of equilibria can be exponential

Nash Equilibrium & Centrality

• Equilibria, in general, involve centrality of agents: • For any graph

G

, for almost every δ, for an equilibrium

x

:

x

S

= (

I

+ δ

G

Q

) -1

1

=

1

– δ

c

(−δ,

G

Q

) where

c

(

a

,

G

) = [I − aG ] −1

G1

(original Bonacich Centrality)

(2) Nash Equilibrium: Uniqueness

• Reformulate equilibrium conditions as a max problem • A

potential function φ

for a game with payoffs

V i φ

(

x i

,

x

-i

) −

φ

(

x i

 ,

x

-i

) =

V i

(

x i

,

x

-i

) −

V i

(

x i

 ,

x

-i

) for all

x i

,

x i

 and for all

i

. [Monderer & Shapley (1996)] • Game with quadratic payoffs,

Ũ i ,

has an exact potential:

φ

(

x

)

= ∑ i

[(

x i

− ½

x i ²

) − ½δ

∑ i,j g ij x i x j

] =

x

T

1

− ½

x

T

(

I

+ δ

G

)

x

Nash Equilibrium: Potential Function

•

Proposition

x is a Nash equilib of any game with best response x

i

(

x

-i

)

iff

x satisfies the Kuhn-Tucker conditions of the problem max

φ

(

x

)

s.t x i ≥

0 

I

•

x:

FOC, SOC satisfied for each agent

i

’s choice in game with

Ũ i

• equilibria are same for all games with best response

x i

(

x

-i

) • Thus, the set of equilibria for these games is the solution to a

quadratic programming

problem.

Equil & Quadratic Programming

• Key to solution is matrix (

I

+ δ

G

) •

Proposition

For any graph G ,

if

δ < −1/λ min (

G

)

,



a unique equil

.

• • (I+δ

G

) is positive definite iff δ < −1/λ min (

G

)

φ

(

x

) is strictly concave • unique global max, K-T conditions necessary and sufficient • Best known suff condition for uniqueness applicable to any

G

• Necessary and sufficient for many graphs (e.g. regular graphs)

Non-Convex Problem: Equilibria

• For δ

>

−1/λ min (

G

) • problem is NP-hard • non-convex optimization, • multiple equilibria possible.

• But we show this does not imply “anything goes.” • Obtain sharp results on shape of equilibria, depending on λ min (

G

) • And results on stability of equilibria, depending on λ min (

G S

)

(3) Aggregates?

• Among the equilibria for a given

δ

and

G

, which yield highest aggregate play?

• For

δ

= 1, in public goods model, we can now identify set of active agents to achieve this goal: • agents in largest maximal independent set of G • Proposition: If δ = 1

, equilibria with highest aggregate effort include those where agents in the largest maximal independent sets set x i =1,and all others set x j =0.

• Of course, here we again have an NP-hard problem.

Conclusion – Summary

• Analyze wide class of games – linear best response • Find and characterize Nash and stable equilibria for any graph, full range of payoff impacts • Results are at edge of analytics, computation, complexity.

• Obvious challenge for economics and computer science: how to find/compute/approximate/select equilibria?