1. ĂdĂĄm and BĂ©la like candies. They are good brothers, so whenever they (jointly) receive
a package of candy they never fight about how to share it, they prefer to bargain by using
their special version of the ultimatum game.
Their game typically goes as follows. When uncle JĂĄnos gives a package of candy to ĂdĂĄm
(the older brother) and asks him to share it with his brother. ĂdĂĄm becomes the proposer
and gets to make an offer to BĂ©la (the younger brother). Then BĂ©la, the responder, either
accepts or rejects the offer.
If BĂ©la accepts the offer, then the brothers share the package of candy exactly according
to ĂdĂĄm's proposal. If BĂ©la rejects the offer, uncle JĂĄnos gets really upset and takes the
package of candy back. In these situations, uncle JĂĄnos always gives 3 candies to BĂ©la and
eats the rest.
Suppose that there are exactly 6 candies in the package and that they are indivisible. Also
assume that in case BĂ©la faces a choice between two equally attractive actions (for himself),
he chooses the one that is better for his brother.
(a) Represent the game played by ĂdĂĄm and BĂ©la with a game tree. Write down sepa-
rately the brothers' strategy sets and compute how many strategies each strategy set
contains.
(b) Find the subgame perfect equilibrium of the game by backward induction.
(c) Aunt MĂĄrta does not believe in subgame perfection and wonders what are the Nash
equilibria of the game played by ĂdĂĄm and BĂ©la (in pure strategies). Show that the
following pair of strategies constitutes a Nash equilibrium of the game.
ĂdĂĄm: "offer exactly 4 candies to BĂ©la"
BĂ©la: "accept the offer if it has at least 4 candies and reject it otherwise"
Introduction to Game Theory
