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

QR 38 Signaling I, 4/17/07 I. Signaling and screening II. Pooling and separating equilibria III. Semi-separating equilibria. Two ways to use Bayes’ rule to extract information from the actions of others: Signaling: undertaken by the more-informed player

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

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

  2. 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 I. Signaling and screening

  3. 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 Signaling and screening example

  4. 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. Screening and signaling example

  5. 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? Market screening example

  6. 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 Market screening

  7. 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) Market screening

  8. 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. Market screening

  9. 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. Market screening

  10. 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. Market screening

  11. 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 Market screening

  12. 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. Market screening

  13. 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. Market screening

  14. 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. II. Pooling and separating equilibria

  15. 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 Deterrence game

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

  17. 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. Deterrence equilibria

  18. Three types of equilibria, depending in the values of w and c: • Separating • Pooling • 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. Deterrence equilibria

  19. 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. Separating equilibrium

  20. 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. Separating equilibrium

  21. 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. Separating equilibrium

  22. 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. Pooling equilibrium

  23. 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. Pooling equilibrium

  24. 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. Pooling equilibrium

  25. A semi-separating equilibrium arises when c<2 and w>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. Semi-separating equilibrium

  26. Equilibrium when c<2 and w>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. Semiseparating equilibrium

  27. 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. Updating beliefs about type

  28. 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)) Updating beliefs about type

  29. 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) Applying Bayes’ rule

  30. 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: Appling Bayes’ rule

  31. 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. D’s payoffs

  32. 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 D’s payoffs

  33. 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. How the probability of spending depends on the probability that C is weak

  34. 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. C’s payoffs

  35. 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 Properties of semi-separating equilibrium

  36. Summary of signaling game equilibria Probability that challenger is weak Cost of spending for weak challenger

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