Question

Consider the game in Question 1(I wrote it at the end of this question. See below). Assume the only letter available are G, K and Q and that the game is played sequentially.

a. Write down the game in its extensive form (assume Bill moves first) and find the Rollback equilibrium(a) of the game

b. Specify the strategies of the two players

c. Draw the game in its strategic form and find the Nash Equilibria

d. Define a Subgame-Perfect Equilibrium and discuss the results at points b. and c.

THIS IS QUESTION 1 : Two players, Jack and Bill are put in separate rooms. Then each is told the rules of the game. Each is to pick on of six letters: G, K, L, Q, R, or W. If the two happen to choose the same letter, both get prizes as follows:

Letters : G K L Q R W

Jack get 3 2 6 3 4 5

Bill gets 6 5 4 3 2 1

If they choose different letters, each gets 0. This whole schedule is revealed to both players, and both are told that both know the schedules and so on

Answer 1

a)

The extensive form of a game corresponds to the game tree; where the action proceeds from left to right. Each node (shown as a dot on the tree) represents a decision point for the player indicated there. The first move in this game belongs to Bill; he must choose whether to play strategy G, K, or Q. Then Jack makes his decision. Payoffs are given at the end of the tree. The convention is for Bill’s payoff to be listed first, then Jack’s.

To find the roll back equilibrium each profitable payoff strategy of Jack is consider given Bills strategic node. tHE STRATEGY THAT IS ONLY PROFITABLE IS THE one where the terminal player chooses the exact same letter as the other player. Then the strategy that is profittable for Jack, is kept all other nodes are terminated. Then in the second stage the game reduced as follows:

Given this the profitable node for Bill is to choose the letter G. Then the equilibrium of the game is

==========================================================================

b)

The strategies of the two players is to choose one of the three letters G,K or Q. Then the strategy profile is

===================================================================

c)

The Nash equilibrium of the game occurs when both the player takes their best strategies in response to the strategies of the other players. The prisoner’s dilemma is a situation where the players chose a Nash equilibrium that is not associated with the highest payoff of the players.

The strategic form of the game is given below

==========================================================================

d)

To find the pure strategy Nash equilibrium we will use the underlining the “best response payoffs” method.

Step1: We underline the payoffs corresponding to Jack’s best responses. Jack’s best response when Bill plays strategy G is G; we underline the payoff corresponds to it. Jack’s best response when Bill plays strategy K is K; we underline the payoff corresponds to it. Jack’s best response when Jill plays strategy Q is Q; we underline the payoff corresponds to it.

Step 2: We follow the same procedure for Bill’s responses. We underline the payoffs corresponding to Bill’s best responses. Bill’s best response when Jack plays strategy G is G; we underline the payoff corresponds to it. Bill’s best response when Jack plays strategy K is K; we underline the payoff corresponds to it. Bill’s best response when jack plays strategy Q is Q; we underline the payoff corresponds to it.

Step 3: Now, we look for the box where the responses of both the Players are underlined. These are the cells (G, G), (K,K) and (Q,Q). This boxes corresponds to Nash equilibria. The given payoffs are (3,6), (2,5), and (3,3).