Can I afford to remember less than you? Best responses in repeated additive games

In this paper, we study best responses in repeated additive games among two players. A stage game is additive if each player’s payoff is the sum of two components, and each component only depends on the action of a single player. We suppose one player’s strategy depends on the co-player’s last $n$ actions. Then we prove that the other player has a best response that only depends on their own $n-1$ actions. That is, for an important sub-class of games and strategies, players can achieve maximum payoffs even with less memory than their opponent.