Bill "Numerical Recipes" Press and Freeman "Dyson sphere" Dyson have a new paper on iterated prisoner dilemas (IPD). Interestingly they found new surprising results:
It is generally assumed that there exists no simple ultimatum strategy whereby one player can enforce a unilateral claim to an unfair share of rewards. Here, we show that such strategies unexpectedly do exist. In particular, a player X who is witting of these strategies can (i) deterministically set her opponent Y’s score, independently of his strategy or response, or (ii) enforce an extortionate linear relation between her and his scores.
They discuss a special class of strategies - zero determinant (ZD) strategies of which tit-for-tat (TFT) is a special case:
The extortionate ZD strategies have the peculiar property of sharply distinguishing between “sentient” players, who have a theory of mind about their opponents, and “evolutionary” players, who may be arbitrarily good at exploring a ﬁtness landscape (either locally or globally), but who have no theory of mind.
The evolutionary player adjusts his strategy to maximize score, but doesn't take his opponent explicitly into account in another way (hence has "no theory of mind" of the opponent). Possible outcomes are:
If X alone is witting of ZD strategies, then IPD reduces to one of two cases, depending on whether Y has a theory of mind. If Y has a theory of mind, then IPD is simply an ultimatum game (15, 16), where X proposes an unfair division and Y can either accept or reject the proposal. If he does not (or if, equivalently, X has ﬁxed her strategy and then gone to lunch), then the game is dilemma-free for Y. He can maximize his own score only by giving X even more; there is no beneﬁt to him in defecting.
If X and Y are both witting of ZD, then they may choose to negotiate to each set the other’s score to the maximum cooperative value. Unlike naive PD, there is no advantage in defection, because neither can affect his or her own score and each can punish any irrational defection by the other. Nor is this equivalent to the classical TFT strategy (7), which produces indeterminate scores if played by both players.
This latter case sounds like a formalization of Hosfstadter's superrational agents. The cooperation enforcement via cross-setting the scores is very interesting.
Is this connection true or am I misinterpreting it? (This is not my field and I've only skimmed the paper up to now.) What are the implications for FAI? If we'd get into an IPD situation with an agent for which we simply can not put together a theory of mind, do we have to live with extortion? What would effectively mean to have a useful theory of mind in this case?
The paper ends in a grand style (spoiler alert):
It is worth contemplating that, though an evolutionary player Y is so easily beaten within the conﬁnes of the IPD game, it is exactly evolution, on the hugely larger canvas of DNA-based life, that ultimately has produced X, the player with the mind.