1. QR 38 Signaling I, 4/17/07 I. Signaling and screening II. Pooling and separating equilibria III. Semi-separating equilibria

2. I. Signaling and screening Two ways to use Bayes’ rule to extract information from the actions of others: Signaling: undertaken by the more- informed player Screening: undertaken by the less- informed player

3. Signaling and screening example Consider a country trying to raise money on the international capital markets: Lenders unsure about the probability of repayment Borrowers can be reliable or unreliable If banks require countries to take costly steps before getting loans, that is screening

4. Screening and signaling example If countries are always careful to repay loans and develop a reputation as reliable, this is signaling. In practice, might be hard to distinguish signaling from screening.

5. Market screening example A classic. Potential employees are of two types: Able (A) Challenged (C) A is worth $150,000 to an employer C worth $100,000 How can the employer tell which is which?

6. Market screening Employer announces he will pay $150,000 to anyone who takes n hard courses. Otherwise the salary is $100,000. These courses are costly for potential employees to take. –For A, these courses cost $6000 each –For C, $9000

7. Market screening How many courses should employers require? Can’t set n too high or too low Need to make it worthwhile for A to take n courses but still separate A from C. For C, need 100,000 > 150,000-9000n 9n>50 n>5.6 (n>=6)

8. Market screening For A, need 150,000-6000n>100,000 50>6n n<=8 So we find that we need n>=6 to keep C from pretending to be A; and n<=8 to make it worthwhile for A to take the course. If n=6, this arrangement is worth 150,000- 6(6000) to A; $114,000. $100,000 to C.

9. Market screening Because it is possible to meet both incentive compatibility constraints, the types separate; self-selection. It is possible to meet both incentive compatibility constraints because A and C have substantially different costs attached to taking tough courses. Note that the presence of C means that A bears the cost of taking courses; this is just a cost, because the courses don’t add any value.

10. Market screening Pooling: this means that the types don’t separate –They both behave the same way, e.g., neither A nor C takes any classes. What would the employer be willing to pay if neither took any classes (pooled)? Assume that 20% of the population is A, 80% C.

11. Market screening The employer will offer all employees their expected value to him:.2(150,000)+.8(100,000) = $110,000 So A will not pool, because he does better under separation Pooling is not an equilibrium in this case

12. Market screening What if the population is split 50-50? Then the pooled salary is $125,000. A and C would both then prefer pooling to separating. But pooling still isn’t an equilibrium, because the employer could benefit by deviating –For example, offering $132,000 to anyone who took just one course.

13. Market screening A would accept this offer, but not C Employer then would only offer $100,000 to those who didn’t take the course Then C would agree to take the course Any arrangement would keep unraveling until we get back to the separating equilibrium identified above.

14. II. Pooling and separating equilibria Consider signaling dynamics in a deterrence game. Challenger (C) is of two possible types: strong or weak. The defender (D) has to decide whether to fight or retreat. Fighting a strong C is bad for D But if C is weak, D prefers fighting to retreating.

15. Deterrence game The probability that C is weak is w. What if C can do something to signal its strength, like spending money on its military? If C is strong, this step costs nothing If C is weak, this costs c

16. Deterrence game Nature Weak (w) Strong (1-w) C C No challenge Challenge and spend Challenge only Fight Retreat Fight Retreat -2, 1 2, 0 -2-c, 1 2-c, 0 D D Challenge and spend No challenge D 0, 3 2, -2 Fight Retreat 4, 0 0, 3

17. Deterrence equilibria In this signaling game, the equilibrium has to address both actions and beliefs. D updates w, using Bayes’ Rule, depending on whether or not C spends. This leads to a Bayesian perfect equilibrium.

18. Deterrence equilibria Three types of equilibria, depending in the values of w and c: 1.Separating 2.Pooling 3.Semi-separating Finding these equilibria is difficult, not required in this class. But once the equilibria are specified, we can check to make sure that they really are equilibria.

19. Separating equilibrium If c is high, the types will separate because the weak type won’t spend. Given these payoffs, the condition for separation is c>2. Equilibrium when c>2: C challenges iff strong; weak C does not spend. If D sees a challenge without spending, he infers that C is weak, and fights.

20. Separating equilibrium Note: in stating the equilibrium, have to specify what would happen off the equilibrium path. Check that this is an equilibrium: What if a weak C tries to spend and challenge? –Leads to payoffs that are dominated by not challenging. Payoff of -2-c if D fights, 2-c if D retreats. Because c>2, these are both <0, the payoff for no challenge.

21. Separating equilibrium What if a strong C chooses not to challenge? Leads to a payoff of 0, which is dominated by challenging (payoff 4). So it is an equilibrium for the types to separate fully when c>2.

22. Pooling equilibrium If c is small (<2), a weak C could bluff and pretend to be strong by spending. So both types would behave the same way; they would pool. But for this to be an equilibrium, also requires that w isn’t too high.

23. Pooling equilibrium If w is high, D would choose to fight, because C is likely to be weak. Knowing that D would fight, a weak C wouldn’t challenge. So need w<2/3 as well as c<2 for a pooling equilibrium to exist.

24. Pooling equilibrium Equilibrium when c<2 and w<2/3: All C challenge and spend; D always retreats. If C were to challenge but not spend, D would fight. Since all C challenge and spend, D can’t update beliefs about w. D’s expected payoff from fighting is 3w-2. With w<2/3, D is better off retreating.

25. Semi-separating equilibrium A semi-separating equilibrium arises when c 2/3. Can’t get full separation, because the temptation for a weak C to bluff is too high. But can’t get full pooling because w is high enough that D won’t then retreat. So a weak C can neither always challenge nor always not challenge in equilibrium. C must play a mixed strategy.

26. Semiseparating equilibrium Equilibrium when c 2/3: Weak C challenges and spends with probability p. D uses Bayes’ Rule to update beliefs about w, and responds to a challenge by fighting with probability q. Equilibrium is p=2(1-w)/w and q=(2-c)/4.

27. Updating beliefs about type How does D draw inferences about w? Knows that a strong C always spends, weak C sometimes. So when D observes spending has to update beliefs about w using Bayes’ Rule.

28. Updating beliefs about type D calculates p(weak|spend) and p(strong|spend). Then D calculates expected payoff from fighting using updated (posterior) probabilities This is equal to: (1)(p(weak)) + (-2)(p(strong))

29. Applying Bayes’ rule A strong type always spends So after observing spending, D calculates p(weak|spend)= p(sp|w)p(w)/(p(sp|w)p(w)+p(sp|strong)p(str)) =pw/(pw+1(1-w)) =pw/(pw+1-w)

30. Appling Bayes’ rule p(strong|spend)= p(sp|str)p(str)/(p(sp|str)p(str)+p(sp|w)p(w)) =1(1-w)/(1(1-w)+pw) =1-w/(1-w+pw) D now has updated probabilities for C’s type. Use these to calculate D’s expected payoff from fighting:

31. D’s payoffs D’s expected payoff from fighting= 1(p(weak) + (-2)(p(strong)) =(wp/(1-w-wp))-2((1-w)/(1-w-wp)) =(wp-2(1-w))/(1-w-wp) When D is using a mixed strategy, the payoff from fighting and not fighting must be equal.

32. D’s payoffs The payoff from not fighting is 0. So we can calculate the p that makes D indifferent between fighting and not fighting: wp-2(1-w)=0 (set the numerator equal to 0) wp=2(1-w) p=2(1-w)/w

33. How the probability of spending depends on the probability that C is weak Calculate what happens to p when w>2/3: p<(2(1-2/3))/(2/3)=(2-4/3)/(2/3) =(6-4)/2; p<1, as required. As w increases, p falls. So, as the probability that C is weak increases, the probability of a weak C spending falls.

34. C’s payoffs Given D’s strategies and inferences, now calculate a weak C’s payoff from challenging and spending: q(-2-c)+(1-q)(2-c)=2-c-4q This must be equal to 0, the payoff from not challenging: 2-c-4q=0; 4q=2-c; q=(2-c)/4 So, as the cost of spending decreases, the probability of D fighting increases.

35. Properties of semi-separating equilibrium Note that in the semi-separating equilibrium, a strong C ends up fighting with some positive probability. This decreases C’s payoff The presence of weak Cs imposes negative externalities on strong Cs

36. Summary of signaling game equilibria w<2/3w>2/3 c<2PoolingSemi- separating c>2Separating Probability that challenger is weak Cost of spending for weak challenger