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Playing large games using simple strategies (2003)
| Venue: | IN: PROC. OF THE 4TH ACM CONF. ON EL. COMMERCE (EC ’03). ASSOC. OF COMP. MACH |
| Citations: | 122 - 4 self |
BibTeX
@INPROCEEDINGS{Lipton03playinglarge,
author = {Richard J. Lipton and Evangelos Markakis and Aranyak Mehta},
title = { Playing large games using simple strategies},
booktitle = {IN: PROC. OF THE 4TH ACM CONF. ON EL. COMMERCE (EC ’03). ASSOC. OF COMP. MACH},
year = {2003},
pages = {36--41},
publisher = {ACM Press}
}
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Abstract
We prove the existence of -Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payoffs to all players in any (exact) Nash equilibrium can be -approximated by the payoffs to the players in some such logarithmic support -Nash equilibrium. These strategies are also uniform on a multiset of logarithmic size and therefore this leads to a quasi-polynomial algorithm for computing an -Nash equilibrium. To our knowledge this is the rst subexponential algorithm for finding an -Nash equilibrium. Our results hold for any multiple-player game as long as the number of players is a constant (i.e., it is independent of the number of pure strategies). A similar argument also proves that for a xed number of players m, the payos to all players in any m-tuple of mixed strategies can be -approximated by the payos in some m-tuple of constant support strategies. We also prove that if the payoff matrices of a two person game have low rank then the game has an exact Nash equilibrium with small support. This implies that if the payoff matrices can be well approximated by low rank matrices, the game has an -equilibrium with small support. It also implies that if the payo matrices have constant rank we can compute an exact Nash equilibrium in polynomial time.
Keyphrases
large game payo matrix nash equilibrium simple strategy small support pure strategy xed number exact nash equilibrium polynomial time constant support strategy logarithmic support nash equilib-rium multiple-player game exact nash equi-librium rst subexponential algorithm low rank matrix support logarithmic mixed strategy similar argument loga-rithmic size nash equilibrium strategy constant rank person game low rank quasi-polynomial al-gorithm
