computation of nash equilibrium jugal garg georgios piliouras
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Computation of Nash Equilibrium
Jugal Garg Georgios Piliouras
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Recap – Two player game
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Nash Equilibrium
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Best Response Polyhedron
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Complementarity Conditions (CC)
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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)
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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
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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
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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
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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
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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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