Mixed and behavioral strategies

Consider the sequential version of Bach or Stravinsky, called Version B in the lecture. First,

Player 1 chooses Bach (B) or Stravinsky (S). After observing Player 1's choice, Player 2 chooses

Bach (b) or Stravinsky (s). If both players choose Bach, the Player 1 receives 2 and Player 2

receives 1. If both players choose Stravinsky, Player 1 receives 1 and Player 2 receives 2. If they choose differently, both players receive 0.

In this game, Player 1 has two strategies, S1 = {B, S}, and Player 2 has four strategies, S2 =

{bb,bs, sb, ss} where the first letter denotes the action taken after Player 1 has chosen B, and the second letter denotes the action taken after Player 1 has chosen S.

First, consider the following mixed strategy profile.

Player 1 plays the strategy (, ) where the numbers are the probability of playing B and S, respectively.

Player 2 plays the strategy (1/6, 1/3, 0, ) where the numbers are the probability of playing bb, bs, sb, and ss, respectively.

a. Determine the probability of each outcome (end node) under this mixed strategy profile.

b. Find a behavioral strategy profile that leads to the same probability of each outcome.

Next, consider the following behavioral strategy profile.

Player 1 chooses B with probability 1/4

If Player 1 chooses B, Player 2 chooses b with probability 1/3

If Player 1 chooses S, Player 2 chooses b with probability 1/2

c. Determine the probability of each outcome under this behavioral strategy profile.

d. We want to find an equivalent mixed strategy profile, i.e. one that leads to the same probability

of each outcome. Denote player 1's strategy as (p,1p) and player 2's strategy as (x, y, z,1

x y z). What is value of p so that player 1 is playing an equivalent mixed strategy as her

behavioral strategy described above?

e. Write the probability of the outcome (B,b) and the probability of the outcome (S, s) in terms

of p, x, y, and z. Using the outcome probabilities found in part (c) and the value of p found in

part (d), write two equations in terms of three variables, x, y, and z. f. There is an equivalent mixed strategy profile where x = 0. Describe it. Is there another equivalent mixed strategy profile? If so, provide one. If not, explain why not.