Equilibrium tracing in strategic-form games

We analyze the relationships of the van den Elzen–Talman algorithm, the Lemke–Howson algorithm and the global Newton method for equilibrium computation by Govindan and Wilson. For two-player games, all three can be implemented as complementary pivoting algorithms. The algorithms by Lemke and Howson and by van den Elzen and Talman start at a pair of strategies: the first method at a pure strategy and its best reply, the latter anywhere in the strategy space. However, we show that even with the same starting point they may find different equilibria. Our second result is that the van den Elzen–Talman algorithm is a special case of the global Newton method, which was known only for the Lemke–Howson algorithm. More generally, the global Newton method implements the linear tracing procedure for any number of players. All three algorithms find generically only equilibria of positive index. Even though the van den Elzen–Talman algorithm is extremely flexible in the choice of starting point, we show that there are generic coordination games where the completely mixed equilibrium, which has positive index, is generically not found by the algorithm.