Ásbjörn H Kristbjörnsson1 The complexity of Finding Nash Equilibria Ásbjörn H Kristbjörnsson Algorithms, Logic and Complexity

Ásbjörn H Kristbjörnsson2 Game theory Game theory: A branch of applied mathematics, often used in the context of economics. Concerns the study of interactions between agents. Agent: An actor in a model that solves an optimization problem (generally). An equivalent term would be “player”. Strategic games: Agents choose strategies, given the strategic choices of the other agents, to maximize their returns. Strategy: Complete plan of action for w/e situation might arise. Implicit listing of all moves and counter moves for every possible situation. Complete algorithm for playing the game.

Ásbjörn H Kristbjörnsson3 Game theory (cont.) Strategy profile: A set of strategies for each player (one and only one strategy per player) Normal form / strategic form: Usually represented by a matrix showing players, strategies, and payoffs. Pure strategy: Complete definition on how a player plays a game. All choices available to the player are listed along with what options the player picks. Mixed strategy: Assignment of probability to each pure strategy.

Ásbjörn H Kristbjörnsson4 Nash Equilibrium Simply put: “A game is in Nash equilibrium if each agent is making the best decisions it can, taking the actions of the other agents into account” This is not necessarily the best cumulative payoff for the agents collectively. If competing agents would collectively agree on a different strategy they might increase their payoff.

Ásbjörn H Kristbjörnsson5 Nash equilibrium (more formal definition) Given a strategy profile (x 1 *,..., x n *) where x i * is an element from S i (S i is the strategy set of agent i) f is the set of payoff profiles σ -i is the strategy profile of all agents except for agent i i = 1,...,n where n is the number of players Then a strategy profile is a Nash equilibrium if the following holds:

Ásbjörn H Kristbjörnsson6 Finding Nash equilibrium, NP-complete? Nash’s theorem guarantees the existence of a set of mixed strategies for finite, noncooperative games. Article argues that this fact makes NP- completeness inappropriate TSP does always have a solution but this solution is hard to verify, so they argue that the TSP is not a good comparison to NASH

Ásbjörn H Kristbjörnsson7 PPAD Complexity class Polynomial Parity Arguments on Directed graphs Subset of TFNP (Total Function Nondeterministic Polynomial) TFNP is a subset of FNP (Function problem extension of NP)

Ásbjörn H Kristbjörnsson8 PPAD Complexity class (cont)

Ásbjörn H Kristbjörnsson9 PPAD Complexity class (cont) A directed graph is defined on a finite but exponentially large set of vertices Each vertex has an indegree and an outdegree at most 1 Given a string, it’s a computationally easy problem to (a) tell if it’s indeed a vertex of the graph (b) find its neighbours (one or two) (c) tell which one is a predecessor and/or which one is the successor There is one known source (“standard source”) Any sink of the graph or any source other than the standard one is a solution of the problem

Ásbjörn H Kristbjörnsson10 PPAD Complexity class (cont) TFNP: Consider a polynomial-time computable predicate P(x,y) where for every x there is at least one y such that P(x,y) is true. A TFNP function is the problem of given an input x, finding y such that P(x,y) TFNP is the class of all total NP-search problems PPA is the class of all NP search problems where the existence of a solution is guarenteed by the fact that in every finite graph which has vertices that are of at most degree 2, the number of leaves will be even.

Ásbjörn H Kristbjörnsson11 NASH is PPAD complete PPAD is a the directed version of PPA where the basic search problem is as follows: in a directed graph having vertices with in-degree and out-degree of at most 1, given a source find another source or sink. Article outlines recent proof that NASH is PPAD complete, even for 2 player games

Ásbjörn H Kristbjörnsson12 References Christos H. Papadimitriou, “The Complexity of Finding Nash Equilibria”, Algorithmic Game Theory Various articles about game theory on Wikipedia ( Various articles about game theory on Wolfram MathWorld (mathworld.wolfram.com)mathworld.wolfram.com Computation Complexity blog, Locally 2-dimensional Sperner Problems Complete for the Polynomial Parity Argument Classes,