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, simultaneously choose x i ≥ 0 vector of actions, x = (x i, x -i ) G, nxn matrix, g ij either 0 or 1 g ij = g ji = 1 iff i impacts j’s payoffs directly - i and j linked, i and j neighbors otherwise g ij = g ji = 0, assume g ij = g ji, normalize g ii = 0. 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 for δ = 0, g ij = 0 for all j. let x i * = x *  1 for all i – base case.

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 − ½x 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, consider its support set of “active” agents; S, such that S = {i: x i > 0} x S = vector of actions of agents in S G S = links between active agents G N-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 + δG S )x S = 1and δ G N-S,S x S ≥ 0

(1) Nash Equilibrium: Algorithm Simple algorithm to determine all (finite) equilibria. For any subset Q, invert (I + δG Q ) if possible. Compute x Q = (I + δG Q ) −1 1. Equilibrium if x Q ≥ 0 and δ G N-Q,Q x Q ≥ 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 + δG Q ) invertible If (I + δG Q ) 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 + δG Q ) -1 1 = 1 – δc(−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 − ½x i ²) − ½δ∑ i,j g ij x i x j ] = x T 1 − ½x T (I + δG)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 + δG) Proposition For any graph G, if δ < −1/λ min (G),  a unique equil. (I+δG) 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?