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Consider the down belowenter image description here

which I have trouble with solving.

For part 1) I have said that a possible outcome path is to play $(D,D)$ in the first round and for all rounds following until $i \leq 298$. Afterwards play $(C,C)$ forever. If anyone deviates then play $(D,D)$ forever. This is since we have that

$$0.01 ( \sum_{i=1}^n 0.99^{i-1} + 9 \sum_{i=n+1}^{\infty} 0.99^{i-1} ) = 1.4 \Leftrightarrow n \leq 298$$

Is this allright? However, then to show that it is a NE in 2) I have trouble with. I thought I could use a Proposition in my book saying that because $(D,D)$ is ne in stage game, then the strategy must be a SPNE. Is this alright?

For 3) I know what do just following the same as in 1) but I have no idea how to prove if it is a SPNE or not. Can you help me with that? The strategy is to play $(D,C)$ for $i \leq 92$ and $(C,C)$ aftwards. If anyone devaites, then play $(D,D)$ forever.

TIA.

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For the first part: correct, any NE in the stage game is a SPNE in an repeated game. In fact, it is the only SPNE if the game is repeated finitely many times.

For the second part: to check that a strategy is SPNE you can use the one-shot deviation principle. That is, check that for any strategy there is no profitable deviation only in one particular stage. Provided there is none, then you are fine.

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  • $\begingroup$ Hi Pedro. Thanks for the comment. Would I have to calculate the sums, i.e. what would happen if one player chose to deviate? I just have trouble with understanding how to. If you could show me how to it would be very much appreciated. $\endgroup$
    – Mathias
    Jun 24, 2021 at 10:41
  • $\begingroup$ Exactly: take one stage and suppose the player deviates and plays another action. Assume the other player sticks to the strategy. For the following periods they will both stick to the strategy. For the case of a grim trigger strategy it would be like this: 1) player 1 plays D instead of C 2) player 2 still plays D because that's what his strategy says 3) this triggers both of them to play D on the subsequent periods. Compute the payoff of such deviation. If it is not profitable then you're fine. $\endgroup$
    – Pedro Ignacio Martinez Bruera
    Jun 24, 2021 at 15:49
  • $\begingroup$ Hi Pedro. Would you check if my answer is alright? I posted it. $\endgroup$
    – Mathias
    Jun 26, 2021 at 12:57

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