Our first look at bargaining: a simple version. A wealthy man has decided to allocate his fortune between his two children when he dies. For fun, he decides to include them in on the process. To decide who gets how much, he tells his two children the following. • The two children must simultaneously submit claims to the amount of their father’s fortune that they feel they are entitled to. For example, one half of the fortune, one quarter, two thirds, one tenth, etc. • If the total amount the children ask for is less than or equal to the total amount (100%), then they each will receive that amount of the fortune. Any remainder will go to charity. • If the total amount the children ask for adds up to more than 100%, they each receive nothing and the father simply wills his entire fortune to charity. Since the children’s names are Veronica and Wallace, let v represent portion of the fortune Veronica asks for and w represent the portion of the fortune Wallace asks for. If you’d like, assume that the size of the father’s fortune is $20 billion (though that part is arbitrary).
(a) Is this game symmetric? How can you tell?
(b) What is the payoff to Veronica if v = 1 2 and w = 1 4 ?
(c) Can you identify any symmetric Nash equilibrium? If so, provide an example of one and prove that it is indeed a Nash equilibrium. If not, prove that there can be no symmetric Nash equilibrium.
(d) Can you identify any asymmetric Nash equilibrium? If so, provide an example of one and prove that it is indeed a Nash equilibrium. If not, prove that there can be no asymmetric Nash equilibria.
(e) Though the two games are not exactly the same, can you identify any similarities between this game and the incentives present in the classic Prioner’s Dilemma game?