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EC220 Problem Set 1

October 12, 2020

1 Ye olde game of Chicken

Questions

Jane and Kate play a game of Chicken on the highway. They run their cars headlong towards each other, with just one chance to swerve out of the way. If neither swerves, a crash occurs, and they both die, getting −5 utility each. If just one swerves, they both live; the one who swerved is dubbed a ‘Chicken’ for her cowardice, getting −1 utility; the one who didn’t swerve gets praise, worth 2 utility. If both swerve, neither loses face, so they both get 0 utility. Both players make their choices simultaneously.

1.1 Draw the Normal Form representation of the game of Chicken.

1.2 Graph the utility for each of Kate’s pure strategies, as a function of Jane’s strategy.

1.3 Derive the Best-Response Correspondences for both players. Draw them.

1.4 Find all Nash Equilibria.

1.5 Suppose Jane recently took out life insurance, so that she instead gets x − 5 should she die. For what positive values of x does the game have a unique Nash Equilib-rium?

2 Auctions!

Questions

A first price, sealed-bid auction is an auction where

• All players simultaneously place their bids.

• The player with the highest bid wins the item at auction.

• In case of a tie, the winner is determined randomly (with equal probability among those tied).

• The winning player pays his bid.

Two players bid for a rare bottle of Laphroaig whiskey in a first-price, single-bid auction.

This is an auction in which they can bid any amount in R+. Suppose the two players both value the item at £500.

2.1 Write out each player’s utility function.

2.2 Write out the best-response correspondences (just for pure opponent strategies).

2.3 Find all pure-strategy Nash Equilibria.

Now, we’re going to change the rules a little bit. Suppose the auction instead is of the all-pay sort. In such an auction, all players, not just the winner, pay their bids.

2.4 Write out each player’s utility function.

2.5 Write out the best-response correspondences (just for pure opponent strategies).

2.6 Find all pure-strategy Nash Equilibria.

2.7 Find all symmetric mixed strategy Nash Equilibria where the strategies can be de-scribed by a pdf with support [0, 500]. (Symmetric equilibria are those where all players use the same strategy)

3 Cournot Price Competition

Questions

I hoped to cover this one in class, but felt like we didn’t have the time (it appears in the slides). Two firms, I = {1, 2}, produce a homogeneous good. They simultaneously choose quantities q1 ∈ [0, 3/2] and q2 ∈ [0, 3/2], respectively, to produce at no cost at all. The firms then sell their products at the market price which is set at P(q1, q2) = 1 − q1 − q2 for each unit of output. Each firm will therefore get profits equal to its own production, times the price.

3.1 Write the two firms’ utility functions.

3.2 Find the two firms’ best-response correspondences.

3.3 Find all Nash Equilibria.

3.4 Find any Iterated Elimination Equilibrium. (Note: in this problem, and any pos-sible exam problems, the order of elimination will not matter!)