Approximate Nash Equilibrium Computation Paul W. Goldberg 1 1 Department of Computer Science University of Oxford, U. K. Invited talk, WINE conference 12th Dec. 2015 Goldberg Approximate Nash Equilibrium Computation
The computational challenge Input: payoff matrices R , C of bimatrix game G Output: a Nash equilibrium of G Centralised. No “strategic data”. I don’t care about social welfare. Goldberg Approximate Nash Equilibrium Computation
The computational challenge Input: payoff matrices R , C of bimatrix game G Output: a Nash equilibrium of G Centralised. No “strategic data”. I don’t care about social welfare. Two key (annoying?) features PPAD -complete pseudo-polytime (but not polytime) algorithm known for approximate version Goldberg Approximate Nash Equilibrium Computation
The computational challenge Input: payoff matrices R , C of bimatrix game G Output: a Nash equilibrium of G Centralised. No “strategic data”. I don’t care about social welfare. Two key (annoying?) features PPAD -complete pseudo-polytime (but not polytime) algorithm known for approximate version Unusual answers for both exact and approx versions... Coincidence?? Goldberg Approximate Nash Equilibrium Computation
The computational challenge Input: payoff matrices R , C of bimatrix game G Output: a Nash equilibrium of G Centralised. No “strategic data”. I don’t care about social welfare. Two key (annoying?) features PPAD -complete pseudo-polytime (but not polytime) algorithm known for approximate version Unusual answers for both exact and approx versions... Coincidence?? similar results for other classes of games. Goldberg Approximate Nash Equilibrium Computation
PPAD -completeness Goldberg Approximate Nash Equilibrium Computation
PPAD -completeness PPA ... what? — Papadimitriou Goldberg Approximate Nash Equilibrium Computation
PPAD -completeness PPA ... what? — Papadimitriou Pathways to Equilibria: Pretty Pictures And Diagrams (PPAD) — von Stengel Goldberg Approximate Nash Equilibrium Computation
PPAD -completeness PPA ... what? — Papadimitriou Pathways to Equilibria: Pretty Pictures And Diagrams (PPAD) — von Stengel My attempt at a talk title Goldberg Approximate Nash Equilibrium Computation
PPAD -completeness PPA ... what? — Papadimitriou Pathways to Equilibria: Pretty Pictures And Diagrams (PPAD) — von Stengel My attempt at a talk title P rospects for P rogress in A pproximate Goldberg Approximate Nash Equilibrium Computation
PPAD -completeness PPA ... what? — Papadimitriou Pathways to Equilibria: Pretty Pictures And Diagrams (PPAD) — von Stengel My attempt at a talk title P rospects for P rogress in A pproximate D uo.../ D ouble... (can’t think of a good word) Goldberg Approximate Nash Equilibrium Computation
PPAD -completeness PPA ... what? — Papadimitriou Pathways to Equilibria: Pretty Pictures And Diagrams (PPAD) — von Stengel My attempt at a talk title P rospects for P rogress in A pproximate D uo.../ D ouble... (can’t think of a good word) In fact, something like “ P olynomial P arity A rgument on a graph, D irected version” Papadimitriou, C.H.: On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence, J. Comput. Syst. Sci. (1994) Goldberg Approximate Nash Equilibrium Computation
What’s wrong with NP -completeness? NP -complete problems have yes-instances and no-instances... Goldberg Approximate Nash Equilibrium Computation
What’s wrong with NP -completeness? NP -complete problems have yes-instances and no-instances... In searching for a Nash equilibrium, every instance (game) is a yes-instance! Every game has one (Nash 1951), and a suggested NE is easy to check. Goldberg Approximate Nash Equilibrium Computation
What’s wrong with NP -completeness? NP -complete problems have yes-instances and no-instances... In searching for a Nash equilibrium, every instance (game) is a yes-instance! Every game has one (Nash 1951), and a suggested NE is easy to check. reduce from (say) SAT to NASH: what should happen to the no-instances? It’s conceivable some other “NASH is as hard as NP ” proof could exist... Goldberg Approximate Nash Equilibrium Computation
TFNP : total function computation in NP NASH ∈ TFNP : “TF”: every game has an outcome “NP”: a “transparent”, easily-checkable outcome Best-known “hard” TFNP problem: FACTORING — given a number, output its prime factorisation; hardness needed for much crypto Goldberg Approximate Nash Equilibrium Computation
TFNP : total function computation in NP NASH ∈ TFNP : “TF”: every game has an outcome “NP”: a “transparent”, easily-checkable outcome Best-known “hard” TFNP problem: FACTORING — given a number, output its prime factorisation; hardness needed for much crypto Hard TFNP problems: an unhappy family Happy families are all alike; every unhappy family is unhappy in its own way. — Leo Tolstoy For our purposes: NP -complete problems are all alike; every hard TFNP problem is hard in its own way. — don’t quote me Work in progress on this... Goldberg Approximate Nash Equilibrium Computation
PPAD : a happy subfamily of TFNP END OF THE LINE Circuits Succ and Pred; n inputs, n outputs; graph on 2 n vertices with arc from u to v iff Succ( u )= v , Pred( v )= u Given 0 has successor but no predecessor, find another vertex of degree 1. Goldberg Approximate Nash Equilibrium Computation
PPAD : a happy subfamily of TFNP PPA : same sort of thing, but undirected graph As it happens, FACTORING belongs to PPA , related to PPAD ... Emil Jeˇ r´ abek: Integer factoring and modular square roots J. Comput. Syst. Sci. , to appear suggestive —but only suggestive— that PPAD is hard Digression: oracle model of PPAD assumes query access to functions Succ and Pred:2 n → 2 n . Query complexity of search for a solution is poly in the circuit model but not in the oracle model. There are oracle separation results for PPAD and other subclasses of TFNP Goldberg Approximate Nash Equilibrium Computation
From NASH to ǫ -NASH: Bounded rationality fixes irrationality With 3 players, NE may have irrational values (Nash ’51); ....even for 3-player, 2-strategy anonymous games G and Turchetta: Query Complexity of Approximate Equilibria in Anonymous Games, these proceedings and in general, for any k > 2 players, n strategies, algebraic degree of values may be exponential in n ... also for graphical games ǫ -Nash equilibrium No incentive —————– ≤ ǫ incentive to deviate — solution can have values that are multiples of ǫ/ kn ∈ Q . To be meaningful, assume payoffs in some bounded range, usually [0 , 1]. Negative (hardness) results carry over to exact NE (useful for first PPAD -hardness results) Goldberg Approximate Nash Equilibrium Computation
ǫ -NASH versus ǫ - Well-Supported NASH ǫ -NASH: average payoff is worse than best-response by at most ǫ — but player may do much worse, with low probability ǫ -WSNE ( stronger! ): anything a player does with positive probability, pays at most ǫ less than best-response. The support of a probability distribution is the set of events that get non-zero probability — for a mixed strategy, all the pure strategies that may get chosen. i.e. anything in the support of a player’s mixed strategy, is within ǫ of best Goldberg Approximate Nash Equilibrium Computation
A good start: ǫ = 1 2 in poly time Daskalakis, Mehta, and Papadimitriou: A note on approximate Nash equilibria. WINE ’07; TCS 2009 0.2 0.9 0.2 1 0 0.1 0.2 2 0.2 0.1 0.2 0.3 0.4 0.5 0.2 0.2 0.8 0.6 0.7 0.8 1 Player 1 chooses arbitrary strategy i ; gives it probability 1 2 . Goldberg Approximate Nash Equilibrium Computation
A good start: ǫ = 1 2 in poly time Daskalakis, Mehta, and Papadimitriou: A note on approximate Nash equilibria. WINE ’07; TCS 2009 1 0.2 0.9 0.2 1 0 0.1 0.2 2 0.2 0.1 0.2 0.3 0.4 0.5 0.2 0.2 0.8 0.6 0.7 0.8 1 Player 1 chooses arbitrary strategy i ; gives it probability 1 2 . 2 Player 2 chooses best response j ; gives it probability 1. Goldberg Approximate Nash Equilibrium Computation
A good start: ǫ = 1 2 in poly time Daskalakis, Mehta, and Papadimitriou: A note on approximate Nash equilibria. WINE ’07; TCS 2009 1 0.2 0.9 0.2 1 0 0.1 0.2 2 0.2 0.1 0.2 0.3 0.4 0.5 0.2 0.2 0.8 1 0.6 0.7 0.8 2 1 Player 1 chooses arbitrary strategy i ; gives it probability 1 2 . 2 Player 2 chooses best response j ; gives it probability 1. 3 Player 1 finds best response k to j ; gives it probability 1 2 . They also find 5 6 -WSNE in poly-time... Goldberg Approximate Nash Equilibrium Computation
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