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I'm reading the book "Strategic games" by Krzysztof R. Apt. I have a question about the strategies in Prisoner Dilemma repeated game. On page 63 there is expression: "In the first round each player has two strategies. However, in the second round each player’s strategy is a function f : {C,D} × {C,D} → {C,D}. So in the second round each player has 2^4 = 16 strategies and consequently in the repeated game each player has 2 × 16 = 32 strategies. Each such strategy has two components, one of each round." I understood like 32 strategies each player has after second round, but there is 2^16 (quantity of function strategies of each player after second round), I think. What is the meaning of "32 strategies"?

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    Can you explain a little bit why on the second round, the player has 16 different strategies? Aren't those affected only by the decisions made on the 1st round? If so shouldn't they be only 8?$(C_1,C_1)\rightarrow C_2,(C_1,D_1)\rightarrow C_2,(D_1,C_1)\rightarrow C_2,(D_1,D_1)\rightarrow C_2$ and $(C_1,C_1)\rightarrow D_2,(C_1,D_1)\rightarrow D_2,(D_1,C_1)\rightarrow D_2,(D_1,D_1)\rightarrow D_2$.2012-04-24
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    @chemeng: Strange coincidence; it looks like you got the correct count of $8$ strategies by an incorrect argument. The decisions are all independent; i.e. whether to play $C_2$ or $D_2$ after $(C_1,C_1)$ is a separate decision from whether to play $C_2$ or $D_2$ after $(C_1,D_1)$, so the number of strategies is $2^n$, not $2\cdot n$ as you counted. The reason the count is nevertheless only $8$ is that the player's own move in the first round isn't something to be responded to (see my answer).2012-04-24
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    @joriki: Though your answer is detailed, I'm a bit confused about the term $strategy$. The strategy of a player is the combination of only HIS choices no matter if choices by other players might affect thiose?For example in 2 rounds the choices a prisoner has are only 4 right?: {C,C};{C,D};{D,C};{D,D}2012-04-24
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    @chemeng: No. A strategy is an entire prescription for how to play the game, which includes a response to every possible move of the opponent. I suspect that you're confusing this with a game in which moves are played simultaneously; in that case a strategy is indeed only a choice for one player. However, a fully specified strategy for a game where decisions are made after other players' decisions have become known requires a response to each possible decision by the other players.2012-04-24
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    Yeah, i got it now!Thanks2012-04-24

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