Lemke–Howson algorithm
The Lemke–Howson algorithm is an algorithm that computes a Nash equilibrium of a bimatrix game, named after its inventors, Carlton E. Lemke and J. T. Howson. It is said to be "the best known among the combinatorial algorithms for finding a Nash equilibrium", although more recently the Porter-Nudelman-Shoham algorithm has outperformed on a number of benchmarks.
Description
The input to the algorithm is a 2-player game. Here, is represented by two game matrices and, containing the payoffs for players 1 and 2 respectively, who have and pure strategies respectively. In the following, one assumes that all payoffs are positive.has two corresponding polytopes and, in dimensions and dimensions respectively, defined as follows:
- is in ; let denote the coordinates. is defined by inequalities, for all, and a further inequalities for all.
- is in ; let denote the coordinates. is defined by inequalities, for all, and a further inequalities for all.
pure strategies, such that player 2's expected payoff is at most 1. The first constraints require the probabilities to be non-negative, and the other constraints require each of the pure strategies of player 2 to have an expected payoff of at most 1. has a similar meaning, reversing the roles of the players.
Each vertex of is associated with a set of labels from the set
as follows. For vertex gets the label if at vertex.
For, vertex gets the label if
Assuming that is nondegenerate, each vertex is incident to facets of and has labels. Note that the origin, which is a vertex of, has the labels.
Each vertex of is associated with a set of labels from the set
as follows. For, vertex gets the label if at vertex. For, vertex gets the label if
Assuming that is nondegenerate, each vertex is incident to facets of and has labels. Note that the origin, which is a vertex of, has the labels.
Consider pairs of vertices,,. The pairs of vertices is said to be completely labeled if the sets associated with and contain all labels. Note that if and are the origins of and respectively, then is completely labeled. The pairs of vertices is said to be almost completely labeled if the sets associated with and contain all labels in other than. Note that in this case, there will be a duplicate label that is associated with both and.
A pivot operation consists of taking some pair and replacing with some
vertex adjacent to in, or alternatively replacing with some vertex adjacent to in. This has the effect of replacing some label of with some other label. The replaced label is said to be dropped. Given any label of, it is possible to drop that label by moving to a vertex adjacent to that does not contain the hyperplane associated with that label.
The algorithm starts at the completely labeled pair consisting of the pair of origins. An arbitrary label is dropped via a pivot operation, taking us to an almost completely labeled pair. Any almost completely labeled pair admits two pivot operations corresponding to dropping one or other copy of its duplicated label, and each of these operations may result in another almost completely labeled pair, or a completely labeled pair. Eventually, the algorithm finds a
completely labeled pair, which is not the origin. corresponds to a pair of unnormalised probability distributions in which every strategy of player 1 either pays that player 1, or pays less than 1 and is played with probability 0 by that player. Normalizing these values to probability distributions, one has a Nash equilibrium.
Properties
The algorithm can find at most different Nash equilibria. Any choice of initially dropped label determines the equilibrium that is eventually found by the algorithm.The Lemke–Howson algorithm is equivalent to the following homotopy-based approach. Modify by selecting an arbitrary pure strategy, and giving the player who owns that strategy, a large payment to play it. In the modified game, the strategy is played with probability 1, and the other player will play their best response to with probability 1. Consider the continuum of games in which is continuously reduced to 0. There exists a path of Nash equilibria connecting the unique equilibrium of the modified game, to an equilibrium of. The pure strategy chosen to receive the bonus corresponds to the initially dropped label. While the algorithm is efficient in practice, in the worst case the number of pivot operations may need to be exponential in the number of pure strategies in the game. Subsequently, it has been shown that it is PSPACE-complete to find any of the solutions
that can be obtained with the Lemke–Howson algorithm.