Posted: August 17th, 2015

1) What features are specified by the rules of a game? (15)

2) Consider this game between the USSR and the USA.

How small do x and y have to be in the game above for the US’s threat to enforce to be credible and sufficient to make the USSR withdraw? (15)

3) He and she prefer to go out together in the evening, but he wants to go to ballet and she wants to go to boxing. They chose strategies simultaneously. Here is the normal form of the game when they choose simultaneously.

She | |||

Ballet | Boxing | ||

He | Ballet | 4, 2 | 0 , 0 |

Boxing | 0 , 0 | 2, 4 |

Define sub-game perfect equilibrium. Compare the *pure strategy equilibria* in this game with the *sub-game perfect* equilibria in a similar game where he chooses first and she chooses second. Discuss any differences between the equilibria of the two games that you observe. (15)

4) A and B are playing divide the dollar. Denote A’s strategy by a (the amount he claims) and B’s strategy by b. If they can agree on a division of the dollar (a + b ≤ 1) they walk away with the share they have agreed; if they cannot agree (a + b > 1) they walk away with nothing. Show that any pair of strategies {a , b} such that a + b = 1, a ≥ 0 and b ≥ 0 is an equilibrium. What is the Nash bargaining outcome of this game assuming that each side’s cardinal utility function is linear in money? (20)

5) Under what conditions does Fearon’s bargaining model of war predict that two nations will negotiate rather than fight? (15)

6) What do Axelrod and Keohane (1985) mean by the ‘shadow of the future’? Explain why this affects whether nations will cooperate with each other. [Words will do, but use algebra if you like!] What factors do Axelrod and Keohane think affect the shadow of the future? (20)

7) Countries A and B have been at war. Country B has incomplete information about A’s type: it might be a “PD type” that has a dominant strategy of defecting (D) from a peace agreement; or it might be an “Assurance type” that wishes to cooperate (play C) if country B does but to defect otherwise. With probability p country A is a PD type; with probability (1-p) it is an Assurance type. Before negotiations begin on a possible peace agreement, country A can move troops away from the border or not. Then countries A and B simultaneously decide whether to cooperate or not. There is a peace agreement if both sides choose C. Moving troops away from the border costs c units if A goes on to choose C. [E.g. because this move generates short term risks in the conflict.] If A moves troops away from away from the border and goes on to choose D, the costs of this move are d, where d > c. This is because having made a cooperative gesture it then refuses to strike a peace agreement, which generates additional costs to its international reputation. The game tree is shown below. [The simultaneous move negotiation stage is shown as four different matrix games to make the figure more compact.

For what values of c and d is there a separating equilibrium in which:

- PD type doesn’t move troops and plays D;
- Assurance type moves troops and plays C;

- Country B plays C after observing the troops are moved and D if it observes they are not moved?

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