I am currently learning about SPNE in repeated game, and I came across this problem with the following payoff table (denote columns as A,B,C and rows as D E F.)
$$\begin{array}{|c|c|c|} \hline (-9, -9) & (0, -10) & (-15,-10) \\ \hline (-10,0) & (-1,-1) & (-15,-10) \\ \hline (-10,-15) & (-10,-15) & (-12,-12)\\ \hline \end{array}$$
Suppose this game is repeated twice. Player 1 stands for row. Player 2 stands for column.
Question Can (B,E) be supported in the first round? If it is, what is the discount factor?
My attempt
The answer is YES. My reasoning is following: $(B,E)$ is played in the first time, play $(A,D)$ in the second time, which is a Nash equilibrium of the stage game (the other equilibrium is $(C,F)$), the total payoff in period 1 for player 1 (and player 2 as well, due to symmetry in the payoff values) $= -1 -9\delta$. We can show that this is the highest payoff player 1 can obtain:
Playing $(A,D)$ twice or $(C,F)$ twice, or any mixed strategy between them, results in $-9-9\delta$ and $-12-12\delta$, and $[-9x - 12(1-x)](1+\delta)$ for $x\in (0,1)$. But all of these payoffs are less than $-1-9\delta$. So the strategy profile of playing (B,E) in the 1st period, and then (C,F) in the 2nd period, is a SPNE(?!).
At this point, I don't see why we need certain conditions on the discount factors (because the game is only repeated twice, not infinite), except maybe trying to bound the payoff of $-1 -9\delta$ between min-max payoff and max-min payoff?
Can someone please help review my solution above? I find the conclusion weird, but I am not sure what went wrong. Note that strategy F dominates strategy E, so we can get rid of F (but then $(B,E)$ would not exist anymore).
You’re comparing the payoff to a payoff that results if both players switch strategies. A (subgame-perfect) Nash equilibrium is defined by each player’s strategy being the best response to the fixed strategies of the other players. So you need to compare the putative equilibrium strategy profile to strategy profiles where player $1$ chooses a different strategy while player $2$ sticks to her strategy.
You haven’t specified a full strategy profile; you only wrote how the players act in the first round, and how they act in the second round given those actions in the first round – but a strategy profile also includes how they act in the second round if the other player acts differently in the first round. In the present case, it’s part of the subgame-perfect Nash equilibrium that both players threaten to play $C/F$ if the result of the first round is not $(B,E)$. They can do this because both $(B,E)$ and $(C,F)$ are Nash equilibria in the non-repeated game.
If player $1$ deviates from this putative equilibrium strategy profile and plays $A$ in the first round, both players will play $F$ in the second round. The resulting discounted payoff for player $1$ is $0-12\delta$. Comparing this to $-1-9\delta$ yields the desired bound on the discount factor.
By the way, the game is solved in these slides.