Equilibrium Tracing in Bimatrix Games

We analyze the relations of the van den Elzen-Talman al- gorithm, the Lemke-Howson algorithm and the global Newton method introduced by Govindan and Wilson. It is known that the global Newton method encompasses the Lemke-Howson algorithm; we prove that it also comprises the van den Elzen-Talman algorithm, and more generally, the linear tracing procedure, as a special case. This will lead us to a dis- cussion of traceability of equilibria of index +1. We answer negatively the open question of whether, generically, the van den Elzen-Talman algorithm is ∞exible enough to trace all equilibria of index +1. In this paper we investigate several algorithms for the computation of Nash equi- libria in bimatrix games. The Lemke-Howson and the van den Elzen-Talman algorithms are complementary pivoting methods; both have been studied exten- sively. The difierence between the two methods is that while the Lemke-Howson method only allows for a restricted (flnite) set of paths, the van den Elzen- Talman algorithm can start at any mixed strategy pair, called prior, and hence generates inflnitely many paths. This implies that the van den Elzen-Talman algorithm is more ∞exible than the Lemke-Howson method. An even more ver- satile algorithm is the global Newton method (1); it works for the more general case of flnite normal form games. We investigate the relations between those three algorithms: We show that the Lemke-Howson and van den Elzen-Talman algorithms difier substantially. However, both can be understood as special cases of the global Newton method. For the van den Elzen-Talman algorithm, this is a new result, which can be generalized to the statement that for N-player games, the global Newton method implements the linear tracing procedure introduced by Harsanyi (3). As a special case of the global Newton method, the van den Elzen-Talman algorithm can generically flnd only equilibria of index +1. This leads us to the ? Supported b y the EPSRC and the LSE Research Studentship Scheme. The author would like to thank Bernhard von Stengel for helpful comments and stimulating discussions.