91Ó°ÊÓ

(Incompetent experts) Consider a (realistic?) variant of the model, in which the experts are not entirely competent. Assume that each expert always correctly recognizes a major problem but correctly recognizes a minor problem with probability \(s<1:\) with probability \(1-s\) she mistakenly thinks that a minor problem is major, and, if the consumer accepts her advice, performs a major repair and obtains the profit \(\pi\). Maintain the assumption that each consumer believes (correctly) that the probability her problem is major is \(r .\) As before, a consumer who does not give the job of fixing her problem to an expert bears the cost \(E^{\prime}\) if it is major and \(I^{\prime}\) if it is minor. Suppose, for example, that an expert is honest and a consumer rejects advice to obtain a major repair. With probability \(r\) the consumer's problem is major, so that the expert recommends a major repair, which the consumer rejects; the consumer bears the cost \(E^{\prime} .\) With probability \(1-r\) the consumer's problem is minor. In this case with probability s the expert correctly diagnoses it as minor, and the consumer accepts her advice and pays \(I ;\) with probability \(1-s\) the expert diagnoses it as major, and the consumer rejects her advice and bears the cost \(I^{\prime} .\) Thus the consumer's expected payoff in this case is \(-r E^{\prime}-(1-r)\left[s I+(1-s) I^{\prime}\right]\) Construct the payoffs for every pair of actions and find the mixed strategy equilibrium in the case \(E>r E^{\prime}+(1-r) I^{\prime} .\) Does incompetence breed dishonesty? More wary consumers?

(Election campaigns) A new political party, \(A\), is challenging an established party, B. The race involves three localities of different sizes. Party \(A\) can wage a strong campaign in only one locality; \(B\) must commit resources to defend its position in one of the localities, without knowing which locality \(A\) has targeted. If \(A\) targets district \(i\) and \(B\) devotes its resources to some other district then \(A\) gains \(a_{i}\) votes at the expense of \(B\); let \(a_{1}>a_{2}>a_{3}>0 .\) If \(B\) devotes resources to the district that \(A\) targets then \(A\) gains no votes. Each party's preferences are represented by the expected number of votes it gains. (Perhaps seats in a legislature are allocated proportionally to vote shares.) Formulate this situation as a strategic game and find its mixed strategy equilibria. Although games with many players cannot in general be conveniently represented in tables like those we use for two-player games, three-player games can be accommodated. We construct one table for each of player 3 's actions; player 1 chooses a row, player 2 chooses a column, and player 3 chooses a table. The next exercise is an example of such a game.

Players 1 and 2 each choose a positive integer up to \(K\). If the players choose the same number then player 2 pays $$\$ 1$$ to player \(1 ;\) otherwise no payment is made. Each player's preferences are represented by her expected monetary payoff. a. Show that the game has a mixed strategy Nash equilibrium in which each player chooses each positive integer up to \(K\) with probability \(1 / K\). b. (More difficult.) Show that the game has no other mixed strategy Nash equilibria. (Deduce from the fact that player 1 assigns positive probability to some action \(k\) that player 2 must do so; then look at the implied restriction on player 1's equilibrium strategy.)