The Basic Setup

Whether one’s dealing with biology, economics, politics or a host of other fields, it’s common to encounter situations that can be modeled as involving two agents that repeatedly compete with each other. One imagines that at each step each agent can take one of a certain set of actions, and that then—in a classic game theory way—each agent (or “player”) gets a certain fixed “payoff” based on the action they and their opponent take. But how do the agents decide what action to take? We imagine that each agent has a certain fixed procedure—or “strategy”—for making its decisions. And we imagine that the input to each of those decisions is the sequence of past actions that the agent and its opponent have taken.

There’s been lots of work done over the course of nearly a century on particular choices of strategies. But something I’ve long been curious about is what happens if one systematically considers all possible strategies.…