Consider the following game, where player 1 decides whether to trust (T) or not

Question

Consider the following game, where player 1 decides whether to trust (T) or not trust (N) the opponent, and then player 2 chooses to either cooperate (C), or defect (L 0 0 2 If the game above is played repeatedly, it can be shown that for the following pair of strategies is a subgame perfect equilibrium. For player 1: "in period 1 I will trust player 2, and as as long as there were no deviations from the pair (T, C) in any period, then I will continue to trust him. Once such a deviation occurs then I will not trust him forever after." For player 2: "in period 1 I will cooperate, and as as long as there were no deviations from the pair (T, C) in any period, then I will continue to do so. Once such a deviation occurs then I will deviate forever after."3 Show that if instead player 2 uses the strategy"as long as player 1 trusts me I will cooperate" then the path (T, C) played forever is a Nash equi- librium for - but is not a subgame perfect equilibrium for any value of

Explanation / Answer

Consider the given problem, here in the given game there are 2 player each of them have 2 possible strategies, so “P1” have “N” and “T” and “P2” have “C” and “D”.

So, given the strategy mentioned in the game problem, if “P2” will cooperate then "P1" will trust "P2" and both will get “1” as payoff forever, if not then in the 1st stage “P2” will get “2” but after that “P2” will get “0” forever, because form to forever the 2nd stage "P1" will start to play "N".

So, now let “d” be the discounter factor here.

So, the payoff when “P2” decide to play “C”, “1 + 1*d + 1*d^2 + 1*d^3 + ……”.

Now, the payoff when “P2” decide to play “D, “2 + 0*d + 0*d^2 + 0*d^3 + ……”.

So, “P2” will decide to play “C” rather then “D” if the following condition hold good.

“1 + 1*d + 1*d^2 + 1*d^3 + ……” >= “2 + 0*d + 0*d^2 + 0*d^3 + ……”.

=> 1/(1-d) >= 2, => (1/2) >= 1-d, => d >= 1- ½ = ½, => d >= 1/2.

So, if “d>=1/2”, then (T,C) be the only SPNE in this repeated game, otherwise there don’t have any SPNE.

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