Consider allocating an object to one of two players when each player's preferences are her private information. Player 1's value for the object, denoted by v1 is drawn from a continuous distribution with [1;2] as its support. Player 2's value for the object, denoted by v2, is likewise drawn from the interval [0;1]. 1. What is the equilibrium in dominant strategies if a second price auction is used to allocate the object. Is the outcome ex-post efficient? 2. For what pairs (v1;v2) should Player 1 and Player 2 respectively be allocated the object in a mechanism that maximizes expected revenue for the seller?

Jane is interested in buying a car from a used car dealer. Her maximum willingness to pay for thecar is 12 ($12,000). Bo, the dealer, is willing to sell the car as long as he receives at least 9($9,000). What is the Nash bargaining solution to this game?

Suppose that, in an ultimatum game, the proposer may not propose less than $1 nor fractional amounts, and therefore must propose $1, $2, …, or $10 (see image attached below). The responder must Accept (A) or Reject (R). Suppose, first, that this game is played by two egoists, for whom u(x,y)=x. Find all subgame-perfect equilibria in this game. Suppose, second, that this game is played by two altruists, for whom u(x,y) = ⅔(x)1/2 + ⅓(y)1/2. Find all subgame-perfect equilibria in this game.

Prove that in the variation on the centipede game given in figure 14.5(b) the unique sequential equilibrium described is, in fact, the unique Nash equilibrium. (Hint: Take some presumed Nash equilibrium and suppose information set 2n+ 1 [for player 2] is the first unreached information set. Derive an immediate contradiction. Then suppose that node (2n) t is the first unreached information set and derive a contradiction that is one degree removed from immediate.)

Olivia is thinking about opening a new bakery (the entrant). There is already a bakery open in her neighborhood (the incumbent), and the owner of the incumbent bakery makes it clear that if Olivia enters the market, they will cut their prices in an attempt to drive the new bakery out of business. Based on the payoff matrix below, is the incumbent’s threat credible? That is, if Olivia opens a new bakery, will the incumbent actually lower their prices? Note: the entrant chooses the row, the incumbent chooses the column      High price   Low price Enter   1, 2   -1, 1 Don’t enter   0, 10   0, 1   a. Yes, the threat is credible b. No, the threat is not credible

The US and Canada have overfished North Atlantic cod stocks nearly to the point of extinction. Both countries wish to preserve the cod industry (and hence the cod stocks), so the two countries sign an agreement to limit fish hauls. Each country has two possible strategies: comply with the agreement (limit fishing) or renege on the agreement (overfish). Each country’s payoffs for different strategy combinations are given in the matrix below. The numbers in the cells represent utilities, and the payoff ordering is (US, Canada). QUESTIONS: 1.If this is a one-shot game (i.e., it is played once), do the players have a dominant strategy? If so, what is it? Briefly explain your answer. 2.What is the equilibrium of this game? Is it Pareto-optimal? How do you know?

In the normal form game below, the payoff matrix depends on the parameter a. 1 1 2 ABC a 0,2 6,4 -2,4 b 0, -24 -2a, 2a 8, 2 C 4,0 0,4 0, 14 Find the values of a, for which at least one pure strategy Nash equilibrium exists. Compute the value of a for which the expected payoff is the same for both players when the mixed strategy profile (01,02) = ((1/3,2/3,0), (0,1/4,3/4)) is played. Find the best response of player 1 (as a function of the parameter a) to player's 2 mixed strategy o2 = (1/2,1/2,0). Assuming a = 0 eliminate iteratively all dominated strategies and find a mixed strategy Nash equilibrium in this game.

There is a city, which looks like chopped isosceles triangle, as shown below. Citizens live uniformly distributed all over the city.  Two ice-cream vendors, A and B, must independently set up stores in the city. Each citizen buys from the vendor closest to their location and when equidistant from both vendors they choose by coin toss. Each vendor’s aim is to maximize the expected number of customers. A choice of location by the two vendors is a Nash equilibrium if no vendor can do better by deviating unilaterally. Does this game have a Nash equilibrium? If so, describe it. If not, explain why not