2/3 II. What has been understood since Nowak & Sigmund 7/ Your point that no strategy simply "wins" is a theorem. Boyd & Lorberbaum (1987) showed that no pure strategy is evolutionarily stable in the repeated PD. van Veelen (2016) proved that no strategy is evolutionarily stable in repeated games with discounting. They also proved that, for a large enough continuation probability, no strategy is even robust against indirect invasions: a neutral mutant can increase in frequency by drift and then give a second mutant a selective advantage. (Errors change this, and the cleanest theorems are about finite automata with errors are due to economists like Fudenberg & Maskin in the 1990s.) 8/ Press & Dyson (PNAS 2012) discovered zero-determinant strategies. A memory-one player using one can unilaterally enforce a linear relation between her own long-run payoff and her opponent's, extortion included. 9/ Press & Dyson also derived a lemma about memory. Against a memory-one player X, any strategy Y with longer memory yields exactly the payoffs of some memory-one strategy, obtained by averaging Y's responses over the histories that occur in play against X. The memory-1 equivalent strategy of Y thus depends on X. The upshot is that within one pairing, longer memory gives no advantage against a memory-one opponent. 10/ The opponent-dependence in the Press–Dyson lemma explains Wolfram's finite-state machine with 10 states, which scores the maximum against every 2-state machine in matching pennies. Against each 2-state opponent its payoff equals that of some memory-one strategy, but a different memory-one strategy for each opponent. Wolfram reports the same thing directly: the machine uses a different subset of its states against each opponent. 11/ Stewart and Plotkin (PNAS 2013) asked which memory-one strategies are evolutionarily robust. Robust means that, in a population of size N, no single mutant fixes with probability above the neutral value 1/N. For the donation game under weak selection we classified all robust memory-strategies, against invaders of anuy memory length. 12/ We also proved, for strong selection and any 2×2 game, that a robust memory-one strategy must be one of four types: • it cooperates with itself; • it defects against itself; • it alternates against itself; • it is an equalizer, which fixes its opponent's payoff. We gave exact robustness conditions for each type. 13/ The ZD construction extends well beyond two actions. McAvoy & Hauert extended ZD strategies to arbitrary action spaces with discounting. Stewart and I extended it to arbitrary numbers of discrete actions, arbitrary memory length, and n-player games, etc. 14/ Your point about whether Axelrod's properties describe successful strategies now has a mathematical answer. Axelrod identified these properties by examining which strategies scored highest in his tournaments and simulations. We have derived them mathematically (2013, 2014) as necceasry and sufficient conditions for cooperative strategy to resist invasion: • nice: a robust ZD strategy cooperates with copies of itself; • retaliatory: a robust cooperative memory-one strategy must defect, with positive probability, after its co-player defects against its cooperation; • forgiving, within limits: a robust ZD strategy is generous, so a co-player who departs from cooperation earns more than it does, but the co-player's advantage is less than half of the generous strategy's own shortfall below R.