Friday, March 4, 2016

unravelling

I'm making less of an effort than usual with this post to be interesting to people who aren't me.  I'm just formalizing the idea of "unravelling" of certain employment markets, in particular markets where there's some sense in which it's best for all firms and workers to be making offers that are in some sense simultaneous, but where individual agents might have an incentive to cheat, making deals before the thick market; in particular, under what conditions might a firm make a take-it-or-leave-it offer to a prospective employee before the market rather than waiting for the market?  In particular, it should be clear that, if the offer were really enticing, the employee would have accepted it in the regular market, and if it's not enticing enough, the employee won't accept it anyway.

Suppose there are two prospective firms hiring an employee, and let's examine the consequences of their making offers sequentially or simultaneously (in the sense of game theory).  Ex ante, the employee attaches probabilities p1 and p2 of getting offers from them, and gets utility u1 or u2 from accepting an offer. (The value of unemployment is set to 0, and may be assumed lower than either ui.)  If firm 1 makes an offer that must be accepted or declined before receiving a response from firm 2, then declining the offer is worth p2u2; conditional on firm 1's offer, the expected payoff is
p2u2.
 max{u1,p2u2} or p2u2. If the offers are simultaneous, then those conditional values are max{u1, u1(1-p2)+p2u2} or p2u2.
If the first firm fails to make an offer, the structure of the game is immaterial. Similarly, if the first firm is the worker's first choice, the structure is immaterial. If the worker gets an offer from the first firm but u2>u1, then the simultaneous game payoff, which can be written as u1+max{0, p2(u2-u1)}, is u1+p2(u2-u1)=p2u2+u1(1-p2); the simultaneous game payoff in these circumstances is higher by min{p2(u2-u1),u1(1-p2)}.

The first firm gets the hire in the sequential game whenever it does in the simultaneous game, but also when
• It makes an offer
• firm 2 makes or would have made an offer
• u2 > u1 > p2u2
where the last condition is the condition in which the worker would choose firm 2 over firm 1 if given both choices but will choose firm 1 rather than take the chance that the other offer isn't coming.  This condition essentially captures the intuition of the last sentence of the first paragraph; by exploiting the worker's uncertainty as to whether a better offer would be forthcoming, the employer can induce the worker to take the "bird in the hand", but as the probability of another offer gets close to 1, there's an increasingly narrow range of offers the worker would accept now but not later.