There are two players in a game. At each round of the game, one player has to trust the other for a particular task. In the first round, Player 1 has to decide whether he will trust Player 2

If he does not trust Player 2, he will get one-third of the prize money, while Player 2 will get the rest of the prize money. If he trusts Player 2, Player 2 can either cooperate with him or defect. If Player 2 defects, Player 1 will earn $0, while Player 2 will get the entire prize money. If Player 2 cooperates, each of them will get half the prize money.
What will the equilibrium outcome of this game be if Player 1 can impose a guilt penalty of two-thirds of the prize money and is known to be a vengeful player?

If there is no guilt penalty, in equilibrium Player 1 will not trust Player 2. He will make this decision using the method of backward induction. If he trusts Player 2, Player 2 is likely to defect because Player 2 will earn the entire prize money in that case, while he will earn only half the money by cooperating. Given this information, Player 1 will not trust him as he will not earn money if he trusts him, while he will earn one-third of the money if he does not trust him.
However, if there is a guilt penalty of two-thirds of the prize money, Player 2 will choose to cooperate because he will earn a lower payoff by defecting. Given this information, Player 1 will trust him and each of them will get half the money.