1.  Consider again the Rock, Paper, Scissors game from problem 2 in Tutorial 1, and assume that payoffs are given by monetary transfers, as specified in part (ii) of the problem.

(i) If a player believes that his opponent chooses Rock, Paper and Scissors with equal probabilities  , compute the player’s expected payoffs for each of his 3 actions.

(ii) Assuming that the players can choose their actions randomly, by selecting each action with a given

probability, explain why when both players choose all 3 of their actions with equal probabilities  , no player will have an incentive to deviate from such a “randomised strategy.”

(iii) Randomised strategies as discussed in part (ii) are referred to as mixed strategies in game theory.

If a player believes that his opponent randomises uniformly between her actions, by choosing each action with equal probability, discuss how the player’s expected payoffs vary according to his own mixed strategy.

2.   Consider the Language Game presented in Figure  1.17 on page 36 of the textbook.   Assume that Aisha believes that Ben chooses Swahili with probability q, and English with probability 1 − q . Similarly, assume that Ben believes that Aisha chooses Swahili with probability p, and English with probability 1 − p.

(i) Find each player’s expected payoff resulting from choosing Swahili and from choosing English, given that the player’s opponent plays Swahili with probability q and p, respectively.

(ii) If the two players can themselves choose any mixed strategy p or q, respectively, carefully analyse

how each player’s optimal mixed strategy depends on the mixed strategy of her or his opponent.

(iii) Sketch Aisha’s optimal mixed best responses p as a function of her beliefs q, and Ben’s optimal

mixed best response q as a function of his beliefs p, then combine your graphs into one graph that has q on the horizontal axis and p on the vertical axis. Hint: You just need to diagrammatically represent your answers from part (ii).

(iv) Consider the graph you derived in part (iii), and explain why each point that lies on the inter-

section of the two best responses can be viewed as a Nash equilibrium in mixed strategies. How many such points are there?

3.  Two people select a policy that affects them both from three feasible policy choices X , Y and Z . They do so by alternately vetoing policies until only one remains, according to the following procedure: First, person 1 vetoes one of the three policies. If more than one policy remains, person 2 then vetoes a policy. If more than one policy still remains, person 1 then vetoes another policy. The process continues until a single policy remains unvetoed, which is then implemented.  Suppose that both persons have transitive preferences, and that person 1 prefers X to Y to Z, and 2 prefers Z to Y to X .

(i) Sketch a game tree for the sequential game defined by the given alternative veto procedure. Denote the action of vetoing policy X by vX, vetoing policy Y by vY , and vetoing policy Z by vZ . Don’t worry about assigning any game payoffs for now.

(ii) Carefully characterise all strategies that are available to both players in the sequential game.