 Viewing a game in this way makes it possible apply the machinery of matrix algebra and Markov chains, which led Press and Dyson to the identification of the class of Zero-Determinant (ZD) strategies. (A simpler proof of Press and Dyson's central result, employing more modest mathematical machinery, is given in Appendix A of Hilbe et al.) A ZD strategy is a strategy by which a player can ensure a fixed linear relation between his own long-term average payoff and his opponent's. For example, TFT \((= \bS(1,0,1,0))\) turns out to be such a strategy for any PD. If I adopt TFT, then I guarantee that, whatever strategy you choose, we will get the same payoff. If you choose unconditional cooperation \((=\bS(1,1,1,1))\) or GTFT \((= \bS(1, .25,1,.25))\) we both get the reward payoff, if you choose unconditional defection we both approach an average of the punishment. For other choices, you may get a payoff between the punishment and reward. Whatever you choose, however, you will still get the same payoff as me.