ECON 2015-1

Quiz 1

Carlos Hurtado Game Theory

1 O Ch 1.2 Example 5.2

A person is faced with the choice of three vacation packages, to Havana (H), Paris (P ), and Venice (V ). She prefers the package to Havana to the other two, which she regards as equivalent. a. [1.5 pt] Write down a payo function that represents the preferences of this person. Is this representation unique? b. [1.5 pt] The theory of rational agents is base on the transitivity of the preferences. Explain with an example why transitivity of the preferences can fail. Why it makes sense to have rational agents in economic models? Answer: a. Let H denote Havana, P denote Paris and V denote Venice. Dene u(H) = 1 and u(P ) = u(V ) = 0. This function represents the preferences of this person. This representation is not unique because any monotonically increasing transformation also represents the same preferences. b. Aggregation of considerations as a source of intransitivity; the use of similarities as an obstacle to transitivity; the dependence of the choice conditional on the bundle: frog legs, meat and chicken. Although the agents are not rational, in the economic models they act as if they are rational. Given that the economic models are like fables, we can conclude using this type of agents.

2 L5 Exercise 3.

Rock (R), Paper (P ) or Scissors (S) is a zero sum game. A player who decides to play rock will beat another player who has chosen scissors ("rock crushes scissors") but will lose to one who has played paper ("paper covers rock") (?); a play of paper will lose to a play of scissors ("scissors cut paper"). Assume that there are only two players. Assume further that the players prefer to win rather that tie, and tie rather than lose. a. [1 pt] Dene the structure of the game from the noncooperative point of view, that is, players, rules, outcome and preferences. b. [1 pt] Write down Rock-Paper-Scissors in extensive form. c. [1 pt] Write down Rock-Paper-Scissors in normal form. Answer: a. players: 1 and 2; rules: Each player simultaneously chooses between Rock, Paper or Scissors. Rock crushes scissors, scissors cut paper, and paper covers rock. If both players choose the same, the game is tied; Outcomes: Player i assigns a value of 1 to win, a value of 0 when tie and -1 to lose.; Preferences: Players prefer to win rather that tie, and tie rather than lose. b.

1 c. 1/2 R P S R 0,0 -1,1 1,-1 P -1,1 0,0 -1,1 S -1,1 1,-1 0,0

3 O Ch 2 Example 2.3

Two people wish to go out together. Two concerts are available: one of music by Bach, and one of music by Stravinsky. One person prefers Bach and the other prefers Stravinsky. If they go to dierent concerts, each of them is equally unhappy listening to the music of either composer. a. [1.5 pt] Write down the game in reduced form. b. [1.5 pt] Find the Pure Strategy Nash Equilibrium (or Equilibria) (PSNE). Answer: a. 1/2 B S B 2,1 0,0 S 0,0 1,2

b. The PSNE are: 1) P1 plays B and P2 plays B and, 2) P1 plays S and P2 plays S.

4 L3 Exercise 2 y 3 (g).

Consider the following reduced form game, where rows are the strategies for player one and columns are the strategies for player two. The payos, as usual, are listed such that the rst number represents the payo of the player one and the second represents the payo of player two.

a. [1 pt] Apply the iterated elimination of strictly dominated pure strategies to nd the equilibrium of the game. Say exactly in what order you eliminated rows and columns and the reasons for elimination. b. [1 pt] Apply the iterated elimination of weackly dominated pure strategies to nd the equilibrium of the game. Say exactly in what order you eliminated rows and columns and the reasons for elimination. c. [1 pt] Determine the set of rationalizable pure strategies for this game. Say exactly why the set of pure strategies is rationalizable.

5 MWG Proposition 8.B.1 (⇒)

[3 pt] Player i's pure strategy si ∈ Si is strictly dominated in the game ΓN = [I, {∆Si} , {ui(·)}] then, there exist another mix strategy σ˜i ∈ ∆Si such that ui(σi, s−i) > ui(si, s−i) for all s−i ∈ S−i.

2 Answer:

We have to prove that : si ∈ Si, is strictly dominated implies there exist another mix strategy σ˜i ∈ ∆Si such that ui(σi, s−i) > ui(si, s−i) for all s−i ∈ S−i. Given that si is strictly dominated we know that there exist another strategy s˜i ∈ Si, s˜i 6= si, such that, for all s−i ∈ S−i, it is true that ui(˜si, s−i) > ui(si, s−i). In particular we can view s˜i as a degenerate mix strategy (where the probability of playing s˜i is 1). Then, we know that there exist another mix strategy σ˜i ∈ ∆Si such that ui(σi, s−i) > ui(si, s−i) for all s−i ∈ S−i.

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