## Three Player Ultimatum Game Infinite Horizon

In this game a proposer and a responder are chosen randomly. Then the proposer has the opportunity to play the ultimatum game with another player. In addition each person has discount rates, which basically rate how impatient he is. If a person has a high discount rate, he will want to close the deal quickly because there are important things he could be doing with that money. We are told that player one is more impatient than player two, who is more impatient than player three. This means that:

Where represents the discount rate of player i.

Each player has von Neumann Morgenstern preferences. That is, he wants more rather than less. That means the proposer and the asked will both agree that the dollar should *only *be split among the two of them. That is, the third player should get no money. For this reason we can neglect the third player. In other words, each round will have a proposer and a responder. We neglect the last player because he will always receive a payoff of zero.

We consider three separate cases:

(i) Player one is the proposer and player two is selected as the responder

(ii) Player two is the proposer and player one is selected as the responder

(iii) Player three is the proposer and player one is selected as the responder

These are the only possibilities because proposers will always select the player with the highest discount rate to bargain with. This is because that player will be the most eager to close a deal.

Below I represent offers in terms of (x, y) where x is the proposer amount and y is the responder amount.

Case (i):

If T=1, player two accepts any because if not the game will end and each will get nothing.

If T=2, player two accepts only. Knowing this player one will propose, which is accepted by player two.

If T=3, player two will accept any first proposal where. Note that if player two rejects this, he will be in the same position that player one was in in the T=2 game. Therefore player two will get as noted in T=2. Of course it will be discounted to. Player one knows this! So he actually proposes at t=1 and player two accepts.

We iterate this to any finite T and we get:

However the above is for a finite horizon game. Rubinstein first considered the infinite horizon game in 1982. In this game, if the players do not reach an agreement, each person receives a null payoff. We simply take the limit as.

The result is:

Case (ii):

By the same logic as in case (i), we get the result:

Case (iii):

By the same logic as in case (i), we get the result:

Section five of Fudenberg’s 1983 paper on infinite horizon game proved to be very useful to me. It is a bit technical, but at least this particular section is useful I think. See it here http://www.dklevine.com/papers/sgperfect.pdf.