Tom and Jake are playing the game like the above. For example, if Tom select $B$, then Jake can select $x$ or $y$. If Jake selects $x$, Tom and Jake will get rewards $2$ and $6$, respectively. Also, if Jake choose $y$, Tom and Jake will get rewards $6$ and $2$, respectively. On the other hands, if Tom selects a strategy $A$, then they will play a game $\theta$.
In this game, I want to find (1) subgame perfect equilibria, and (2) one Nash equilibrium which is not a subgame perfect equilibrium.
(1)
- In the Game $\theta$, the Nash equilibrium strategy is only the $(b, \beta)$.
- In the game that Jake selects whether $x$ or $y$, if Jake selects $x$ he gets $6$, but if he selects $y$ he gets $2$, so Jake will select $x$.
- If Tom selects $A$, Tom gets $1$ because Tom and Jake will select $(b, \beta)$ in the Game $\theta$ as already found in process 1. In the other hand, if Tom selects $B$, he gets $2$ because Jake will selects $x$ as already seen in process 2.
- Therefore, Tom will selects $B$ because $2>1$ as seen in process 3.
To sum up with the above four processes, I conclude that $\{(B,b), (\beta,x)\}$ is a subgame perfect equilibrium.
Q) Is there anymore subgame perfect equilibrium?
(2)
The Nash equilibrium but not a subgame perfect game is $\{(B, b), (\beta, x)\}$