SUMMER TERM 2022
CENTRALLY-MANAGED ONLINE EXAMINATION
ECON0027: GAME THEORY
1. When King Minos demanded fourteen Athenians to be sent to Crete to be devoured by the Minotaur, a hero called Theseus volunteered to meet with the monster.
Theseus sails to Crete and inds the entrance into the Labyrinth where the Minotaur lives. The Minotaur does not know if Theseus is strong enough to defeat him or not, but Theseus himself knows his own strengths and weaknesses. Theseus faces a choice: either to sail away or enter the Labyrinth. If Theseus sails away, he will get nothing and the Minotaur will keep terrorizing Athenians (which is worth 20 utility units to him).
If Theseus enters the Labyrinth, the Minotaur sees that. He can either stay and ight Theseus or lee and never return. If the Minotaur lees, he will stay alive, but will no longer strike fear into the hearts of Greeks (this outcome is worth 10 utility units to him). If the Minotaur ights Theseus and wins, he will enjoy the title of the most ierce monster of all Greece (which is worth 25 utility units to him), but if he loses, Theseus decapitates him (which is worth -10 utility units to the Minotaur).
If Theseus does not ight the Minotaur, he returns to Athens with nothing (which is worth zero utility units to him). If he wins the duel, he receives y utility units and if he loses one, he receives x utility units.
(a) Formalize this situation as a game of incomplete information by drawing the tree of the game.
(b) Show that for any x and y, there exists a fully separating equilibrium in this game. Charac- terize that equilibrium. Find the values of x andy for which the fully separating equilibrium is unique.
(c) Explain the role of the Minotaur’s beliefs in this fully separating equilibrium. How do these posterior beliefs depend on the prior beliefs?
2. Five amazon queens, Otrera, Hippolyte, Penthesilea, Myrina, and Thalestris, decide to marry ive Greek Kings: Alexander, Theseus, Menelaus, Agamemnon and Odysseus. All ive Amazon Queens have the same preferences over the ive Greek Kings and all ive Greek Kings have the same preferences over the ive Amazon Queens. Every King and every Queen prefers to get married over staying single.
(a) Show that the stable matching is unique.
(b) Show that the stable matching can be obtained by running a serial dictatorship algorithm.
(c) How many matchings are both Pareto efficient and unstable? Explain.
3. Heracles and Augeas are bargaining over the payment for cleaning the Augean stables. If the stables are cleaned right away, Heracles and Augeas will jointly get a surplus worth 999 gold coins. However, every day of delay is costly—it reduces the surplus by 333 gold coins. Starting with Augeas, the king and the hero make alternating ofers to each other. Only one ofer per day is allowed. If the ofer is accepted, Heracles cleans the stables and receives the agreed payment. If the ofer is rejected, the king and the hero have to wait one day before the new ofer is made. Each day, both Augeas and Heracles have an option of abandoning the enterprise of cleaning the stables for good and receiving a payof of zero.
(a) Formalize this problem as an extensive-form game.
(b) Find all subgame perfect Nash equilibria of this game.
(c) Find a Nash equilibrium of this game that is not subgame perfect. Using this equilibrium as an example, explain why subgame perfection is a reasonable requirement for equilibria in extensive form games.
4. King Leonidas and King Xerxes assemble their armies for a battle against each other. The king who brings more warriors wins the battle and gets control over Sparta. Both kings value the throne of Sparta the same amount.
King Leonidas can equip x warriors per 1000 gold coins spent and King Xerxes can equip y warriors per 1000 gold coins spent. Let x and y be drawn independently from a uniform distribution with the support [0, z], z > 0.
(a) Suppose that both x and y are observable. Show that there is no equilibrium in pure strategies in this game.
(b) Suppose that Leonidas privately learns x and Xerxes privately learns y. How many warriors ight in the battle between Leonidas and Xerxes?
(c) Keep the assumptions made in 4b. Suppose that the god of war Ares loves large battles.
Mighty Ares threatens to keep Sparta for himself if both kings bring less than a warriors each. Find a that maximizes the size of the winning army on average.