Here, we see one of the bargaining strategies: the take-it-or-leave-it game, which you may know by another name: the ultimatum game.
Annie owns a car that she values at $8,000. Becky wants a car and values Annie’s car at $8500. In this game, Becky makes a single offer (ultimatum) on a price. Also, Becky knows how much Annie values the car. Annie will then accept (and sell) or reject the offer. In bargaining games, players always want to maximise their economic benefit. Here is the game tree of the situation.
Let’s see from the top. Becky is making offer X, between $0 and $8,500. The bottom part is what Annie will do—accept or reject. If Annie accepts, her payoff will be $X. Becky’s payoff will be the surplus—the difference between what she values and what she will pay to Annie. If Annie rejects, Becky will get nothing, and Annie will retain the car worth $8,000.
The problem is solved by backward induction. Annie has three choices on Becky’s offer.
X>$8000 – Annie accepts
X=$8000 – Annie is indifferent (case1: she rejects)
X<$8000 – Annie rejects.
Having built Annie’s choice, let’s work upward on Becky’s decision. So, if Becky decides to offer $8000 or below, her payoff will be zero.
On the other hand, Becky can offer X > $8000. Her payoff will vary but maximises at 499 if she offers $8000, one dollar more than the rejection level.
And Becky should just do that.
