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

Rachel KrantonDuke University

Research Issues at the Interface of Computer Science and Economics, Cornell UniversitySeptember 2009

Introduction

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

Πi(xi,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, simultaneously choose xi ≥ 0•vector of actions, x = (xi, x-i)

• G, nxn matrix, gij either 0 or 1• gij = gji = 1 iff i impacts j’s payoffs directly - i and j linked, i and j neighbors• otherwise gij = gji = 0, assume gij = gji, normalize gii = 0.

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

• Payoffs: Ui(xi,x-i; δ, G)

• xi* ≡ optimal action in autarky

•maximizes Ui for δ = 0, gij = 0 for all j.• let xi

* = x* 1 for all i – base case.

Class of Games - Examples

• Ui(xi,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)

Ŭi(xi,x-i; δ, G) = b(xi + δ∑jgijxj) - cxi

with b increasing, strictly concave, b'(0) < c < b'(+∞).

Class of Games - Examples

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

Ũi(xi,x-i; δ, G) = xi − ½xi² − δ ∑jgijxixj

• Other examples• investment games with quadratic payoffs

• Cournot oligopoly, with linear demand, constant MC,

Linear Best Replies

• Each game yields exactly this best reply: xi (x-i)

xi (x-i) = 1 − δ ∑jgijxj if δ ∑jgijxj < 1

= 0 if δ ∑jgijxj ≥ 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, consider its support

• set of “active” agents; S, such that S = {i: xi > 0}• xS = vector of actions of agents in S• GS = links between active agents• GN-S,S = links between active agents and inactive agents

• Proposition

For any δ [0, 1] and any graph G, x is a Nash equilibrium iff

(I + δGS)xS = 1 and

δ GN-S,SxS ≥ 0

(1) Nash Equilibrium: Algorithm

• Simple algorithm to determine all (finite) equilibria.•For any subset Q, invert (I + δGQ) if possible. •Compute xQ = (I + δGQ)−11.•Equilibrium if xQ ≥ 0 and δ GN-Q,QxQ ≥ 1•Repeat for all subsets of N. •Yields all finite equilibria.

• For any graph G, the number of equilibria is finite for almost every δ.

• Follows from when (I + δGQ) invertible• If (I + δGQ) 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:

xS = (I + δGQ)-11 = 1 – δc(−δ, GQ)

where c(a,G) = [I − aG] −1G1 (original Bonacich Centrality)

(2) Nash Equilibrium: Uniqueness

• Reformulate equilibrium conditions as a max problem

• A potential function φ for a game with payoffs Vi

φ(xi,x-i) − φ(xi,x-i) = Vi(xi,x-i) − Vi(xi,x-i)

for all xi, xi and for all i. [Monderer & Shapley (1996)]

• Game with quadratic payoffs, Ũi, has an exact potential:

φ(x) = ∑i[(xi − ½xi²) − ½δ∑i,jgijxixj]

= xT 1 − ½xT (I + δG)x

Nash Equilibrium: Potential Function

• Proposition

x is a Nash equilib of any game with best response xi(x-i) iff x satisfies the Kuhn-Tucker conditions of the problem

max φ(x) s.t xi ≥ 0 I

•x: FOC, SOC satisfied for each agent i’s choice in game with Ũi

•equilibria are same for all games with best response xi (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 + δG)

• Proposition For any graph G, if δ < −1/λmin(G), a unique equil.

• (I+δG) is positive definite iff δ < −1/λmin(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/λmin(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 λmin(G)

• And results on stability of equilibria, depending on λmin(GS)

(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 xi =1,and all others set xj =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?