Why is the tat-for-tat strategy a Nash equilibrium in infinitely repeated games?

Question

Why is the tat-for-tat strategy a Nash equilibrium in infinitely repeated games, but not a Nash equilibrium in a finite scenario? Specifically for this matrix:

Assume higher payoffs reflect higher utility. It's a prisoner's dilemma situation.

Since tit-for-tat assumes we start at (Honor, Honor), and play the strategy that the other player last played in future rounds, I don't really see why it's a Nash equilibrium in an infinite scenario and not a finite scenario.

In a finite scenario (e.g. one round), wouldn't the players end up at the NE (Cheat, Cheat) because they follow their self-interest? And in an infinite scenario, wouldn't they end up at (Honor, Honor) (not a NE) assuming the discount factor is high enough?

Any clarity appreciated!

Answer 1

(i) In the 1 round case, tit-for-tat is not a NE. To see this notice that the tit-for-tat strategy, as you describe, dictates that the players play $(H,H)$ in the first (and only) round---as you point out, this is clearly not a NE, since either player can increase her payoff by changing her strategy from $H$ to $C$. Perhaps what you missed is that the strategy includes proscription in first round, which does not cohere from the 1-shot Nash behavior.

(ii) In the $N$-round case, again tit-for-tat is not a NE. This is a little more subtle that point (i) but not much. If both players play according to the strategy, then they will make it to the $N-1$ round playing $H$. At this point, the strategy dictates they play $H$ in the $N^{th}$, and final, round. For the same reason as in (i) this is not a best response, hence not an equilibrium (note that forward looking agents would anticipate this, so the strategy would break down immediately, but analyzing the last round is sufficient to see it is not an equilibrium).

(iii) In the infinitely-repeated case, tit-for-tat can persist, but it depends on how agents discount future utility in comparison with current utility. The general logic is that since there is no final period where things begin to unravel, players are always willing to forgo current utility to maintain good standing (the tit part of the tit-for-tat?) and therefore a higher future (continuation) payoff. Of course, if the players care about today's payoff far more than tomorrow's, they will defect and play $C$. See here for more about NE in infinitely-repeated games.

Answer 2

Edit: This is an incorrect answer. Please disregard and see comments below.

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I don't think tit-for-tat is NE for infinitely repeated games. Tit-for-tat simply is, on average, the best strategy against many other strategies.

Tit-for-tat will lose out, for example, if the only other strategy is to deviate all the time.

Read Dawkin's The Selfish Gene for more on this.

Answer 3

Let's first consider the case with the discount factor $\rho = 1$.

In any finitely repeated case, the NE is to play (cheat, cheat) simply because of backward induction. A sophisticated rational player would think that, in the last round, the other player has no chance to punish him by playing cheat in the next round because there's no next round, therefore, he will play cheat. Another sophisticated player of course can also anticipate this and play cheat also in the last round. Given this, in the second-last round, no player can punish the other one by playing cheat in the next round because they are both playing cheat in the next round, and thus, they will both play cheat in the second last round. The same process goes on and on until the first round, and both players will also play cheat. The key thing here is there is a last round to begin with in a finitely repeated case.

In the infinitely repeated case, there is no last round to begin with, so each player can punish the other player by playing cheat in the next round if the other player plays cheat in this round. The presence of punishment makes the "tit-for-tat" strategy become an equilibrium in this case.

However, when the discount factor $\rho$ is sufficiently small, the "tit-for-tat" strategy can not be an equilibrium any more. The discount factor $\rho$ is usually interpreted as how much one discounts future utility. In an infinitely repeated game, $\rho$ can also be interpreted as how likely one player is going to play the same game with the other player in the next round. And straightforwardly, the less likely they are going to interact again, the less credible the threat is of playing cheat in the next round.

PS: I previously thought the TfT strategy was the Grim strategy. Thanks to @VARulle for pointing this out.