$4.$ Backward induction and subgame$-$perfect Nash equilibria

Backward induction is a useful tool while solving for the subgame$-$perfect Nash equilibrium $($SPNE$)$ of a sequential game. This problem walks you through how to find the SPNE in the following game using this method.

Suppose you are creating the invite list for an upcoming party, but you face a dilemma about two of your friends. Everyone knows that your friend Felix never misses a party if invited. However, if you invite Felix, you are concerned that your friend Janet may not attend because of their recent breakup. Since you are doing your invitation online, Janet can observe whether you choose to invite Felix. The following game tree represents the game you face. At each node, your payoff is shown first, Janet's second:

Left Good PathRight Good PathYouJanetJanetInvite FelixDo not invite FelixAttendDo not attendAttendDo not attend$28,-411,412,161,6$

To find the SPNE using backward induction, you must first start at the end of the game, or at Janet's decision. Since you act first and Janet can observe this action, she can make a different decision depending on which node she faces.

Suppose that you decide to invite Felix to the party. If Janet does not attend, she will receive a$$payoff than if she attends. Now that you have determined which action yields the higher payoff, mark the leg off the left node corresponding to the action that represents the good path on the diagram with an orange line $($square symbol$).($Note: You will not be graded on your manipulation of the diagram.$)$

Now suppose that you decide not to invite Felix. If Janet does not attend, she will receive a$$payoff than if she attends, given that you do not invite Felix. Again, mark the leg off the right node corresponding to the action that represents the good path on the diagram with a purple line $($diamond symbol$).($Note: You will not be graded on your manipulation of the diagram.$)$

It is now time for your decision. You have already determined what Janet will do$,$ given your decision to invite or not invite Felix. Since you can internalize this future decision of Janet's, you need only to examine your payoffs for the two good paths. Comparing these two payoffs, you will choose$$Felix to your party.

What is the subgame$-$perfect Nash equilibrium of this game?

You do invite Felix, and Janet attends if Felix is not invited and does not attend if Felix is invited.

You do not invite Felix, and Janet attends if Felix is not invited and does not attend if Felix is invited.

You do not invite Felix, and Janet attends.

You do not invite Felix, and Janet attends if Felix is invited and does not attend if Felix is not invited.

$4.$ Backward induction and subgame$-$perfect Nash equilibria

Backward induction is a useful tool while solving for the subgame$-$perfect Nash equilibrium $($SPNE$)$ of a sequential game. This problem walks you through how to find the SPNE in the following game using this method.

Suppose you are creating the invite list for an upcoming party, but you face a dilemma about two of your friends. Everyone knows that your friend Felix never misses a party if invited. However, if you invite Felix, you are concerned that your friend Janet may not attend because of their recent breakup. Since you are doing your invitation online, Janet can observe whether you choose to invite Felix. The following game tree represents the game you face. At each node, your payoff is shown first, Janet's second:

To find the SPNE using backward induction, you must first start at the end of the game, or at Janet's decision. Since you act first and Janet can observe this action, she can make a different decision depending on which node she faces.

Suppose that you decide to invite Felix to the party. If Janet does not attend, she will receive a payoff than if she attends. Now that you have determined which action yields the higher payoff, mark the leg off the left node corresponding to the action that represents the good path on the diagram with an orange line $($square symbol$).($Note: You will not be graded on your manipulation of the diagram.$)$

Now suppose that you decide not to invite Felix. If Janet does not attend, she will receive a payoff than if she attends, given that you do not invite Felix. Again, mark the leg off the right node corresponding to the action that represents the good path on the diagram with a purple line $($diamond symbol$).($Note: You will not be graded on your manipulation of the diagram.$)$

It is now time for your decision. You have already determined what Janet will do$,$ given your decision to invite or not invite Felix. Since you can internalize this future decision of Janet's, you need only to examine your payoffs for the two good paths. Comparing these two payoffs, you will choose Felix to your party.

What is the subgame$-$perfect Nash equilibrium of this game?

You do invite Felix, and Janet attends if Felix is not invited and does not attend if Felix is invited.

You do not invite Feli