Download The Complexity of Finding Nash Equlibria

The complexity of Finding Nash Equilibria
Ásbjörn H Kristbjörnsson
08.73.11 Algorithms, Logic and Complexity
30.10.2007
Ásbjörn H Kristbjörnsson
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Game theory
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Game theory:
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Agent:
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Strategic games:
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Strategy:
A branch of applied mathematics, often used in the context of
economics. Concerns the study of interactions between agents.
An actor in a model that solves an optimization problem (generally).
An equivalent term would be “player”.
Agents choose strategies, given the strategic choices of the other
agents, to maximize their returns.
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.
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Game theory (cont.)
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Strategy profile:
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Normal form / strategic form:
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Pure strategy:
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Mixed strategy:
A set of strategies for each player (one and only one strategy per
player)
Usually represented by a matrix showing players, strategies, and
payoffs.
Complete definition on how a player plays a game. All choices available
to the player are listed along with what options the player picks.
Assignment of probability to each pure strategy.
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Nash Equilibrium
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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.
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Nash equilibrium (more formal definition)
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Given a strategy profile (x1*, ... , xn*) where xi* is an element
from Si (Si is the strategy set of agent i)
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f is the set of payoff profiles
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σ-i is the strategy profile of all agents except for agent i
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i = 1,...,n where n is the number of players
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Then a strategy profile is a Nash equilibrium if the following
holds:
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Finding Nash equilibrium, NP-complete?
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Nash’s theorem guarantees the existence of a
set of mixed strategies for finite,
noncooperative games.
Article argues that this fact makes NPcompleteness 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
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PPAD Complexity class
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Polynomial Parity Arguments on
Directed graphs
Subset of TFNP (Total Function
Nondeterministic Polynomial)
TFNP is a subset of FNP (Function
problem extension of NP)
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PPAD Complexity class (cont)
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PPAD Complexity class (cont)
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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
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PPAD Complexity class (cont)
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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.
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NASH is PPAD complete
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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
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References
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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
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