Abstract
The concept of a pure Nash equilibrium (NE) for a noncooperative game is simpler than that of a mixed NE, which always exists. However, pure NEs probably have more practical significance even though such a game may not have a pure NE. An efficient algorithm is presented here to determine whether an n-person game in normal form has a pure NE and, if so, to obtain all NEs. This algorithm uses the notion of regret, and the payoff matrix (PM) is transformed into a regret matrix (RM)—a loss matrix with an intuitive interpretation. The RM has the property that an action profile of the PM is a pure NE if and only if (0,· · ·,0) is the corresponding element of the RM. The computational complexity of the algorithm is O(N) in the number of individual utilities N in the PM, and so it is substantially faster than a total enumeration.
Highlights
In a Nash equilibrium (NE) for an n-person game, every player has a strategy that maximizes his payoff for the other n − 1 players’ strategies
An efficient algorithm is presented here to determine whether an n-person game in normal form has a pure NE and, if so, to obtain all NEs
We present here an efficient algorithm for computing a pure NE for a normal form game
Summary
In a Nash equilibrium (NE) for an n-person game, every player has a strategy that maximizes his payoff for the other n − 1 players’ strategies. Most work has focused on the computational complexity for finding mixed NEs, whose existence is guaranteed by Nash’s existence theorem (Nash, 1951). In addition to the computational difficulties associated with mixed strategies, their interpretation is controversial Both von Neumann & Morgenstern (1944) and later Nash (1953) considered a randomizing process an essential part of a mixed strategy, a requirement that Aumann (1985) and Rubinstein (1991) considered problematic since the reasons and methods for the players randomizing their decisions were not specified. We present here an efficient algorithm for computing a pure NE for a normal form game.
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