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The complexity of Finding Nash Equilibria Ásbjörn H Kristbjörnsson

The complexity of Finding Nash Equilibria Ásbjörn H Kristbjörnsson. 08.73.11 Algorithms, Logic and Complexity. Game theory. Game theory: A branch of applied mathematics, often used in the context of economics. Concerns the study of interactions between agents.

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The complexity of Finding Nash Equilibria Ásbjörn H Kristbjörnsson

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  1. The complexity of Finding Nash EquilibriaÁsbjörn H Kristbjörnsson 08.73.11 Algorithms, Logic and Complexity Ásbjörn H Kristbjörnsson

  2. 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 Kristbjörnsson

  3. 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 Kristbjörnsson

  4. 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 Kristbjörnsson

  5. Nash equilibrium (more formal definition) • Given a strategy profile (x1*, ... , xn*) where xi* is an element from Si (Si 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 Kristbjörnsson

  6. 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 Kristbjörnsson

  7. 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 Kristbjörnsson

  8. PPAD Complexity class (cont) Ásbjörn H Kristbjörnsson

  9. 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 Kristbjörnsson

  10. 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 Kristbjörnsson

  11. 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 Kristbjörnsson

  12. References • Christos H. Papadimitriou, “The Complexity of Finding Nash Equilibria”, Algorithmic Game Theory • Various articles about game theory on Wikipedia (www.wikipedia.org) • Various articles about game theory on Wolfram MathWorld (mathworld.wolfram.com) • Computation Complexity blog, http://weblog.fortnow.com/2005/12/what-is-ppad.html • Locally 2-dimensional Sperner Problems Complete for the Polynomial Parity Argument Classes, http://www.lri.fr/~santha/Papers/fisv06.pdf Ásbjörn H Kristbjörnsson

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