– Jane’s Utility decreasing by 2/3 implies that she only hires Roy if she thinks that thetha>2/3. Remember that not hiring gives Jane zero utility and hiring Roy with thetha2/3, then Roy will be hired in the second period, Zoe gets 5/6

if thetha1/2

So destroying the flowers is a dominated strategy… so much for a deterministic solution.

Zoe will randomize using f(y)=P(alpha=y) over (0,1] so that it will make Jane more uninformed, that is, x is still a signal of thetha, but is less precise, since now it is made of the product of 2 random variables, thetha and p.

The cheapest way to do so is to find a function f(y) such that the signal x is uninformative, and the posterior is such that Jane is indifferent between hiring Roy or not, that is E(thetha-2/3/x,f)=E(thetha-2/3/f)=0.

EJMR you are my favorite blog follower so I will give you a hint. A necessary condition for a prelim question to be considered good is that the solution must be elegant.

A further necessary condition for the question to be considered not just good but great is that the solution must have a touch of tragedy.

]]>After some clumsy crowdsourcing, 6493 provides a promising answer

http://www.econjobrumors.com/topic/great-prelim-question-cheap-talk

]]>1) alpha must be not be invertible (else Jane learns theta with certainty always, hence there is no gain to destroying crops in the first stage, hence since alpha is a scalar it must be random)

2) for any alpha such that alpha*theta>=2/3, Jane hires regardless of Zoe’s strategy.

3) the maximal deterministic strategy pays off 5/6 (all deterministic strategies except alpha=0 are invertible)

4) the optimal random strategy involves Jane hiring sometimes (alpha>2/3 cannot be in the support if theta=1 is to never be hired away, hence no strategy where Ray always stays can generate more than payoff 5/6)

5) Of course one can do better if alpha was known to Zoe: note that letting Ray go when theta=1 earns at most 1 (since Ray will be hired away), but he can be kept by mixing theta=1 and theta=1/3 in round 1 (letting alpha given theta=1 be equal to 1/3), earning 1.33>1. In general, if alpha is known to JZoe, she can earn (as a lower bound) an expectation of (1/6)*(1/3)+(1/2)*(1/3)+(1/2)*(1/3)+1/2=8/9>5/6.

6) With known alpha to Zoe, both the gain from strategic destruction, and the amount of crops destroyed, are increasing as theta increases above 2/3. The additional tradeoff with unknown alpha is that sometimes crops will be destroyed unnecessarily when theta is lower. ]]>