Suppose you and I are playing a two-player zero-sum game repeatedly. There is a payoff matrix $(A\_{ij})$, and if I play action $i$ and you play action $j$ then I receive $A\_{ij}$ and you receive $-A\_{ij}$. I am not clever enough to compute my best strategy (my maximin strategy), but I can just manage to compute a best response to your mixed strategy, if you were to publicize it. Similarly you. We decide that at each turn (i.e., repetition of the game) we will just play a best response to our opponent's apparent strategy so far. To be more specific we each do the following:


* Turn $1$: Play something. Who cares. It's only one turn.
* Turn $n$: Play a best response assuming that the opponent will simply pick one of turns $1,\dots,n-1$ uniformly at random and replay that action.


This is an attractive rough and ready way to try to solve a two-player game when tackling the linear program head-on is difficult. But is it any good? That is, do our strategies converge to our respective maximin strategies? Williams's *Compleat Strategyst* says it works, but unfortunately it's not the sort of book that gives references.


For example, if we were playing rock-paper-scissors, the play might go as follows:


* rock, rock (we both start arbitrarily)
* paper, paper (paper is the best response to rock)
* paper, paper (paper is the best response to 1/2 rock, 1/2 paper)
* scissors, scissors (scissors is the best response to 1/3 rock, 2/3 paper)
* etc.


Clearly play is always symmetric here and we always draw, but that is not important (and wouldn't be true if we started with different actions). The only thing I care about is whether our distribution of actions converges to 1/3 rock, 1/3 paper, 1/3 scissors (the optimal strategy), and how fast. 



Iterative approximate solutions to games
The keywords I was looking for were "fictitious play" and "Brown--Robinson process". The short answer seems to be that it works, sometimes. For more information see <https://en.wikipedia.org/wiki/Fictitious_play>.


