# Best reply structure and equilibrium convergence in generic games

## TL;DR

This paper analyzes how the complexity and competitiveness of two-player games influence the likelihood of convergence to equilibrium, showing that cycles dominate in more complicated and competitive scenarios, challenging equilibrium assumptions.

## Contribution

It provides a comprehensive analysis of the prevalence of best reply cycles in generic games, highlighting the impact of game complexity and competitiveness on convergence.

## Key findings

- Best reply cycles become more common in complex, competitive games.
- Convergence to equilibrium is unlikely in such games due to prevalent cycles.
- Six learning algorithms fail to converge in the presence of these cycles.

## Abstract

Game theory is widely used as a behavioral model for strategic interactions in biology and social science. It is common practice to assume that players quickly converge to an equilibrium, e.g. a Nash equilibrium. This can be studied in terms of best reply dynamics, in which each player myopically uses the best response to her opponent's last move. Existing research shows that convergence can be problematic when there are best reply cycles. Here we calculate how typical this is by studying the space of all possible two-player normal form games and counting the frequency of best reply cycles. The two key parameters are the number of moves, which defines how complicated the game is, and the anti-correlation of the payoffs, which determines how competitive it is. We find that as games get more complicated and more competitive, best reply cycles become dominant. The existence of best reply cycles predicts non-convergence of six different learning algorithms that have support from human experiments. Our results imply that for complicated and competitive games equilibrium is typically an unrealistic assumption. Alternatively, if for some reason "real" games are special and do not possess cycles, we raise the interesting question of why this should be so.