# How to do backwards induction for this problem?

This is the tree: https://imgur.com/a/MJ9mmBN

I solved for the normal form equilibria: (SS, SS), (SS, SC), (SC, SS), (SC, SC), and (CC, CC)

However, I am not sure how to go about backwards induction.

My specific point of confusion is that if player 1 chooses C, then player 2 can choose S or C and get the same payoff (3).

How does player 1 factor this into his decision making? I'm not sure if the risk of 2 choosing S will make player 1 choose S instead of C, even though C can potentially get him a higher payoff if player 2 chooses C as well.

## 1 Answer

Since player 2 can choose S or C to receive the same payoff, he will choose the decision which results in a lower payoff for his opponent. As such, player 2 should choose S in order to give player 1 a payoff of 0 rather than 3. Knowing this, player 1 will choose S in his first turn in order to maximize his payoff of getting 1 instead of 3.

• Why is this? We were never taught that a player prefers the other player to be worse off. If player 1 doing worse gave player 2 extra utility, wouldn't that already be factored into his payoff? – StackO123 Oct 9 '18 at 7:43
• This is true, I think another way of looking at is that if player 2 chooses C over S then he opens himself up to lower payoffs if player 1 chooses S on his next turn. To avoid this, player 2 will still choose S as that maximizes his payoff and minimizes the risk of player 1 not letting him get a payoff of 3. – coronermclarson Oct 9 '18 at 7:45
• Thank you for the help, but I have to disagree that player 2 choosing C opens himself up to lower payoffs. Since it is stated that both the tree and players rationality are common knowledge, player 2 knows that if he picks C, they will end at the (3, 3) payoff. – StackO123 Oct 9 '18 at 7:47