Players 1 and 2 are promised $100 if they can agree how to divide it. Player 2 h
ID: 1105408 • Letter: P
Question
Players 1 and 2 are promised $100 if they can agree how to divide it. Player 2 has also been told that she will get $60 even if there is no agreement. Player 1 first makes an offer, which must be in units of a penny, specifying how much she keeps and how much goes to Player 2. Player 2 can accept, in which case the game is over and the payoffs correspond to the shares proposed by Player 1; or Player 2 can reject. If she rejects, the game is terminated with probability 0.5 and Player 1 gets 0 and Player 2 gets $ 60. With probability 0.5, the game continues, Player 2 makes an offer and Player 1 accepts or rejects. If Player 1 accepts, this means the proposals shares give the payoffs of the two players. If Player 1 rejects, she gets 0 and Player 2 gets $60.
(a) What does Player 2 offer if the game continues after a rejection, in a subgame perfect Nash equilibrium?
(b) What does Player 1 offer at the beginning of the game to Player 2, in the subgame perfect Nash equilibrium?
Explanation / Answer
Nash Equilibrium is a point in game theory when all players in the game are at their best possible position and have no incentive to change their current strategy to get a better payoff. A subgame Nash equilbirum is when there is a nash equilibrium in every subset of a game.
In the given question, Player 1 makes the first move and proposes a way to divide the $100. Player 2 can either accept or reject this. He/she gets $60 regardless of whether there is an agreement. If Player 2 rejects the offer and makes no further offers, then the game ends and player 2 gets $60 and Player 1 gets $0. The probability of this happening is 0.5. If player 2 rejects player 1's offer but makes a counter-offer of his/her own, then player 1 gets to accept or reject this offer. The probability of this happening is 0.5.
(a) To determine what is Player 2's offer after rejecting Player 1's offer, we can look at the following hypothetical situation that player 1 will not accept anything less than $25. Player 2 gets $60 even if they do not reach an agreement. So, he/she will expect atleast a payoff $60. But this is not an optimum situation for player 2. So, he/she will offer $5 dollars to player 1 and keep $95. However, player 1 will reject this as he/she is getting only $5 and would rather let go of this money and reject the offer. In this case, player gets only $60 and player 1 gets $0.
Player 2 may change his/her offer and give $25 to the player 1 and keep $75. This offer is likely to be accepted by player 1 as they get $25 which is much higher than $0 and player 2 also gets $15 more than if the offer is rejected. This will reflect a situation where both players are getting optimum from the current situation and have no incentive to change their strategy.
So, Player 2's offer if the game continues after a rejection, in a subgame perfect nash equilibrium is taking $75 and giving $25 to player 1.
(b) Since player 2 gets atleast $60 regardless of whether they reach an agreement or not, player 1 will offer to give an additonal $5 to player 2 and keep the rest. That is, he/she will to keep $35 and give Player 2 $65. This will be rejected by player 2 as he/she will not mind letting go of $5 and make a counter-offer.
So, player 1's offer at the beginning of the game to player 2, in a sub-game perfect Nash equilbirum is that he/she will keep $35 and give $65 to player 2.