Computation of Nash Equilibrium

Jugal Garg Georgios Piliouras

Recap – Two player game

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Nash Equilibrium

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Best Response Polyhedron

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Complementarity Conditions (CC)

Characterization

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Quadratic Programming Formulation

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Zero-sum Game

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Zero-sum Game

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Support Enumeration Algorithm

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Support Enumeration Algorithm

• How many possible support?

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Support Enumeration Algorithm

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Support Enumeration Algorithm

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Corollaries

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Lemke-Howson Algorithm (1964)

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Preliminaries

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Best Response Polyhedron (Recall)

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Linear Complementarity Problem (LCP) Formulation

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Complementarity Conditions (CC)

LCP Characterization

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Preliminaries

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Fully-labeled Points

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Fully-labeled Characterization

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Lemke-Howson Algorithm

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(0,0)

 

 

  

 

This path is called k-almost fully-labeled – only label k is missing

Convergence

• No cycling• Cannot comes back to starting vertex (0,0)• Has to terminate in finite time

• Exponential in worst case (Savani, von Stengel’04)

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Structural Results

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Break

Q. How we get existence and oddness of the number of equilibria from

Lemke-Howson algorithm?

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PTIME for Special Cases

• Zero-sum games - LP formulation• Both rank(A) and rank(B) are constant (Lipton,

Markakis, Mehta’04)

• Either rank(A) or rank(B) is constant (G., Jiang, Mehta’11)

– Proof on board• Rank-based hierarchy (rank(A+B)) – Zero-sum games = rank-0 games– Rank-1 games (Adsul, G., Mehta, Sohoni’ 11)

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Rank-1 Games

• Number of Nash equilibrium can be exponential (von Stengel’12)

• For any c, rank-1 games can have c number of connected components (Kannan-Theobold’07)

• Rank-1 games are strictly general than zero-sum games and LP (Dantzig’47)

• In general rank-1 QP is NP-hard to solve – surprisingly those arising from rank-1 games are

polynomial time solvable.27

Approximate Nash Equilibrium

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Approximate Nash Equilibrium

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FPTAS for Constant Rank Games

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Quasi-polynomial Time Algorithm (Lipton, Markakis, Mehta’03)

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Complexity Considerations

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Nash

YouArehere

Standard Complexity Classes

• P: The set of decision problems for which some algorithm can provide an answer in polynomial time

• NP: set of all decision problems for which for the instances where the answer is "yes“, we can verify in polynomial time that the answer is indeed yes.

• coNP: Same as above with yes->no

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Equilibrium Computation

• The goal is to find a function that maps objects (games) to the mixed strategy profiles.

• This is NOT a decision problem (yes/no).

• Examples:• Add two numbers and find the outcome• Is the sum of two numbers odd?

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Function Complexity Classes

• FP: The set of function problems for which some algorithm can provide an answer in polynomial time

• FNP: set of all function problems for which the validity of an (input, output) pair can be verified in polynomial time by some algorithm.

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Function Complexity Classes

• FP: A binary relation P(x,y) is in FP if and only if there is a deterministic polynomial time algorithm that, given x, can find some y such that P(x,y) holds.

• FNP: A binary P(x,y), where y is at most polynomially longer than x, is in FNP if and only if there is a deterministic polynomial time algorithm that can determine whether P(x,y) holds given both x and y.

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Function Complexity Classes

• FP: The set of function problems for which some algorithm can provide an answer in polynomial time

• FNP: set of all function problems for which the validity of an (input, output) pair can be verified in polynomial time by some algorithm.

• TFNP: The subclass of FNP for which existence of solution is guaranteed for every input!

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Non-constructive arguments• Local Search: Every directed acyclic graph must

have a sink.• Pigeonhole Principle: If a function maps n

elements to n-1 elements, then there is a collision.

• Handshaking lemma: If a graph has a node of odd degree, then it must have another.

• End of line: If a directed path has an unbalanced node, then it must have another.

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From non-constructive arguments to complexity classes

• PLS: All problems in TFNP whose existence proof is implied by Local Search arg.

• PPP: All problems in TFNP whose existence proof is implied by the Pigeonhole Principle.

• PPA: All problems in TFNP whose existence proof is implied by the Handshaking lemma.

• PPAD: All problems in TFNP whose existence proof is implied by the End-of-line argument.

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From non-constructive arguments to complexity classes

PPA

FNP

PPAD

PPPPLS

P 40

PPAD [Papadimitriou 1994]Suppose that an exponentially large graph with vertex set {0,1}n is defined by two circuits:

P

N

node id

node id

node id

node id

END OF THE LINE:Given P and N: If 0n is an unbalanced node, find another unbalanced node. Otherwise say “yes”.

PPAD = { Search problems in FNP reducible to END OF THE LINE}

possible previous

possible next

P(v)=u and N(u)=v

u

v Finding Nash equilibria, even in 2-person games is PPAD –complete. [Daskalakis, Goldberg, Papadimitriou 05], [Chen, Deng 06’]

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