Repeated games with terminating cells

Abstract

Buying a house can be a time consuming process. After the buyer places its first bid the seller may o
er a counter bid, which might be followed by a counter bid from the buyer. After a certain time period this can result in an agreement when the bid is accepted. These multiple rounds of bids can be described mathematically, and the scientific discipline dealing with them is called game theory. In this discipline the focus lies on the analysis of strategic choices of the players involved. When we go back to the example of the buyer and the seller we see that the action that both players can play is to place a bid. If the bid is not accepted by the other player the game continues and a counterbid has to be placed. However, if the bid is accepted the game ends and the payoff is paid from the buyer to the seller. When an agreement is made the payoff is equal to the final bid, otherwise, if the players do not come to a settlement the game may go on forever, in that case the payoff equals 0. A strategy for a player is a decision rule that prescribes a mixed action, that is, a probability distribution on the possible actions. The combination of these strategies yields some payoff for all players. We assume that all players seek a strategy that maximizes their payoff. In reality the bids of the buyer and seller tend to come closer to each other and to the value of the house as the game proceeds. In the end a price agreement is made someplace in the midst. In this paper we assume however that the strategies of the players are stationary and are played simultaneous. That is, they do not depend on the history of the game and in each round all players play their action at the same time.