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Consider the repeated Prisoner's Dilemma.

Every day, for many days, two players play this game: $\left(\begin{array}{ccc} \left(3,3\right) & \left(0,10\right) & (-2,-2)\\ (10,0) & (1,1) & (-1,-1)\\ (-2,-2) & (-1,-1) & (-2,-2) \end{array}\right) $

Let $X$ be the strategy where a player:

  1. chooses the first line for all but the last $k$ days, for some $k$;
  2. chooses the second line for the last $k$ days;
  3. disregards the above and always chooses the third line should the other player not follow this strategy

Then it can be shown that $(X,X)$ is a Nash equilibrium for high enough $k$. Indeed, if the game lasted $N$ days (assuming the "discount factor" $\delta = 0$) the total utility is $3(N-k) + 1k > 3(n-k-1)+10+k\cdot(-1)$ (the utility of a betrayal on the $k-1$-th last day) for e.g. $k\geq 4.$

However, this is where I stop copying from my lesson notes and start thinking.

Unless I'm mistaken the whole idea of a Nash Equilibrium is having a pair of strategies $(A,B)$ such that $A$ is the best response to $B$ and viceversa; nothing can be said about any other $(A,C)$ pair, for example.

Now, say that for some reason one of the two players changes his strategy mid-game. Say, he's under ransom and must collect that 10 utility payment or the cat gets it. (If I understand correctly this can be modeled by having a sufficiently high $\delta$ so that the weight of 10 tomorrow outweighs the combined weight of $-1$ every day in the future.

Except... why would the other player insist on its own strategy? One of the strategies changed, the Nash Equilibrium is no more and enacting the menace could satisfy some sense of justice or vengeance but not the axiom of rationality. The rational thing to do is to play the second line as well, falling down to the "traditional" (and disappointing) $(1,1)$ pure equilibrium.

The menace ultimately never does happen, and after all why would add two strictly dominated lines to a game theory problem change it? How is this anything more than a self-fulfilling mathematical artifice?

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What you seem to be hinting at is that the equilibrium is not sub-game perfect. See wikipedia's description.

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The menace would work if the strategy could be locked in at the start, with no opportunity to reconsider.

In military tactics, this sort of thing is called "burning's one boats". It changes the game.

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