Analysis of adaptive algorithms for repeated matrix games

In this paper we consider repeated matrix games. We assume that the first player is a computer program, and the second one can belong to one of two types: "hacker" is a player who knows the algorithm implemented in the program and optimally counteracts it; "simpleton" is a player who does not know the optimal strategy or is not able to implement it. In this case, the first player does not know the number of repetitions of the game, and his opponent as well. The problem is posed by limiting the possible value of loss to the hacker, to achieve significant gains from the simpleton. It is shown that determining the reaction of the program to opponent`s actions is a non-trivial problem. An example of an intuitively good algorithm is given, the optimal strategy for counteracting which nonetheless leads to a significant value of loss.