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user16268
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Cross-posted from Mathematics Stack Exchange.

Let $\delta\in(0,1)$ be the discount factor. Consider the stage game in the infinitely repeated prisoner's dilemma game:

stage game

The goal is to derive conditions on $\delta$ such that the symmetric tit-for-tat strategy profile is a Nash equilibrium.

To recall, tit-for-tat is when a player cooperates (here plays Yield, $Y$) the first round and then every round after copies the action of their opponent the second round.

I am told that if A is playing tit-for-that then B's best possible replies would be to alternate between $N$ and $Y$ (playing $N$ first), to always play $N$, or to play tit-for-tat himself resulting in both players always playing $Y$.

We compute the expected payoffs of each depending on $\delta$ and then figure conditions on $\delta$ so that $B$ should play tit-for-tat. As the payoffs are symmetric, these conditions provide $A$ should also play tit-for-tat given $B$ is and we have a Nash.

My only question is why it must be that these are the only best possible replies. Why might not $B$ have some periods where he plays $N$ for a while then switches to only cooperating? Or vice versa. Or anything outside of these three.

Cross-posted from Mathematics Stack Exchange.

Let $\delta\in(0,1)$ be the discount factor. Consider the stage game in the infinitely repeated prisoner's dilemma game:

stage game

The goal is to derive conditions on $\delta$ such that the symmetric tit-for-tat strategy profile is a Nash equilibrium.

To recall, tit-for-tat is when a player cooperates (here plays Yield, $Y$) the first round and then every round after copies the action of their opponent the second round.

I am told that if A is playing tit-for-that then B's best possible replies would be to alternate between $N$ and $Y$ (playing $N$ first), to always play $N$, or to play tit-for-tat himself resulting in both players always playing $Y$.

We compute the expected payoffs of each depending on $\delta$ and then figure conditions on $\delta$ so that $B$ should play tit-for-tat. As the payoffs are symmetric, these conditions provide $A$ should also play tit-for-tat given $B$ is and we have a Nash.

My only question is why it must be that these are the only best possible replies. Why might not $B$ have some periods where he plays $N$ for a while then switches to only cooperating? Or vice versa. Or anything outside of these three.

Let $\delta\in(0,1)$ be the discount factor. Consider the stage game in the infinitely repeated prisoner's dilemma game:

stage game

The goal is to derive conditions on $\delta$ such that the symmetric tit-for-tat strategy profile is a Nash equilibrium.

To recall, tit-for-tat is when a player cooperates (here plays Yield, $Y$) the first round and then every round after copies the action of their opponent the second round.

I am told that if A is playing tit-for-that then B's best possible replies would be to alternate between $N$ and $Y$ (playing $N$ first), to always play $N$, or to play tit-for-tat himself resulting in both players always playing $Y$.

We compute the expected payoffs of each depending on $\delta$ and then figure conditions on $\delta$ so that $B$ should play tit-for-tat. As the payoffs are symmetric, these conditions provide $A$ should also play tit-for-tat given $B$ is and we have a Nash.

My only question is why it must be that these are the only best possible replies. Why might not $B$ have some periods where he plays $N$ for a while then switches to only cooperating? Or vice versa. Or anything outside of these three.

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user16268
user16268

Tit-For-Stat Strategy Best Replies

Cross-posted from Mathematics Stack Exchange.

Let $\delta\in(0,1)$ be the discount factor. Consider the stage game in the infinitely repeated prisoner's dilemma game:

stage game

The goal is to derive conditions on $\delta$ such that the symmetric tit-for-tat strategy profile is a Nash equilibrium.

To recall, tit-for-tat is when a player cooperates (here plays Yield, $Y$) the first round and then every round after copies the action of their opponent the second round.

I am told that if A is playing tit-for-that then B's best possible replies would be to alternate between $N$ and $Y$ (playing $N$ first), to always play $N$, or to play tit-for-tat himself resulting in both players always playing $Y$.

We compute the expected payoffs of each depending on $\delta$ and then figure conditions on $\delta$ so that $B$ should play tit-for-tat. As the payoffs are symmetric, these conditions provide $A$ should also play tit-for-tat given $B$ is and we have a Nash.

My only question is why it must be that these are the only best possible replies. Why might not $B$ have some periods where he plays $N$ for a while then switches to only cooperating? Or vice versa. Or anything outside of these three.