ECON0027 Game Theory
Home assignment 4
1. Consider the following hide-and-seek game between two players. Player 1 puts a dollar bill into one of the three boxes without telling Player 2. Player 2 picks one box of his choice and opens it. Player 2 gets to keep the content of the box he opened and the content of the other two boxes remains with Player 1.
(a) Formulate this as a game in an extensive form. Argue that there is no equilibria in pure strategies.
(b) Solve for an equilibrium.
(c) Consider the following modiication of the game. At the beginning of the game Player 1 puts a dollar into a box of his choice without telling Player 2 the location of the dollar. Player 2 then takes one box, but does not open it yet. After that, Player 1 has to choose one of the two remaining boxes and show its content to Player 2. Once it is done, Player 2 can either keep the box that he previously selected or can reconsider and take one of the two other boxes. Once Player 2 makes his inal choice, the game ends. At the end of the game Player 2 gets the content of the box he obtained and the content of the other two boxes remains with Player 1. Solve for an equilibrium of the game. Does Player 2 always open the box he has originally chosen or does he reconsider? Give an intuition for your answer.
2. Consider the following model of presidential elections. There are three candidates. Each candidate can be either good or bad. The type of a candidate is drawn at random in the beginning of the game. The probability of drawing a good type is π . The type is a private information of a candidate.
There are two periods in this model. In the irst period, the irst two candidates participate in the election (the third candidate remains idle) and the candidate that wins becomes a president in the irst period. The president’s type is revealed to the public.
In the second period, the incumbent participates in the election against the third candidate, and the person that wins becomes a president in the second period. Each candidate is risk-neutral and values the office at v (this value is per term).
In the irst period, the candidates can inance their political campaigns using their own funds. The amount e spent on the campaign is publicly observable. You can assume that the candidates have quasilinear utility (and no budget constraints): each candidate maximizes
E[k]v - e,
where E[k] is the expected number of periods the candidate occupies the office.
Both candidates have to make their campaign inance decisions simultaneously and independently of each other. (Hint: the monies spent on the political campaign do not afect the quality of the candidates but they may afect the voters’ perception of the candidates’quality)
The voters are all identical. They prefer a good candidate over a candidate of unknown quality and they also prefer a candidate of unknown quality over a bad one. You can assume that if the voters are indiferent, they elect either candidate with probability 1/2.
(a) Find an equilibrium in which candidates do not spend any money on their campaign.
(b) Find an equilibrium in which some candidates spend money on their campaign.
(c) A parliament introduces a law that imposes an upper limit on the amount of money that a candidate can spend on his campaign. How does such a law afect the equilibrium payofs of the candidates and the welfare of the voters?
3. Consider the following Bayesian game. Nature chooses the type θ of player 1 from the set f1, 2, 3, 4g, where each type has equal probability. Player 1, the sender, observes his type and may send a costless message from the set fm1 , m2 , m3 , m4 g, that does not afect either player’s payofs.. Player 2, the receiver, does not observe player 1’s type, and must choose an action a from the set of real numbers. The sender’s payof is given by
U(θ, a) = 1.5a - (a - θ)2 .
The receiver’s payof is given by
V (θ, a) = -(a - θ)2 .
(a) Show that there is always an equilibrium where the sender plays the same action after every message. Interpret this equilibrium.
(b) Show that there cannot be an equilibrium with full separation of types.
(c) Solve for an equilibrium with partial separation of types. (Hint: Look for separation between unequally sized subsets of the set of types). Provide an argument why there cannot be any other equilibrium with partial separation apart from the one you ind.
4. Consider the following simpliied version of a game of poker. There are two players and three cards in a deck: Jack, Queen and King of spades. Player 1 always gets a Queen. Player 2 gets a Jack with probability 0.5 or a King with probability 0.5. In the irst round, player 2 can either Raise or See. If he plays See, the game ends and players see each other’s cards. Whoever has a lower card has to pay $1 to the opponent. If player 2 played Raise, the game continues to the second round in which player 1 can either play Pass or Meet. If he plays Pass, the game ends and player 1 has to pay $1 to player 2. If player 1 plays Meet, the game ends and the players see each other’s cards. Whoever has a lower card has to pay $2 to the opponent.
(a) Formulate the game in the extensive form and draw a tree of the game. What is an assessment in this game? Write it down in a parametric form (i.e., by denoting unknown strategies and beliefs by some letters).
(b) Under what conditions on the belief system does player 1 play Meet?
(c) Solve for all weak sequential equilibria of this game and classify each equilib- rium as pooling, partially separating or fully separating.