QR 38

Signaling I, 4/17/07

I. Signaling and screening

II. Pooling and separating equilibria

III. Semi-separating equilibria

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

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

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.

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?

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

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)

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.

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.

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.

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

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.

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.

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.

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

Deterrence game

Nature

Weak(w)

Strong(1-w)

C

C

Nochallenge

Challenge and spend

Challenge onlyFight

Retreat

Fight

Retreat

-2, 1

2, 0

-2-c, 1

2-c, 0

D

D

Challenge and spend

No challenge

D

0, 3

2, -2Fight

Retreat4, 0

0, 3

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.

Deterrence equilibria

Three types of equilibria, depending in the values of w and c:

1. Separating2. Pooling3. Semi-separatingFinding 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.

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.

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.

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.

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.

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.

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.

Semi-separating equilibrium

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.

Semiseparating equilibrium

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.

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.

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))

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)

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:

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.

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

How the probability of spending depends on the probability that C is

weakCalculate 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.

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.

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

Summary of signaling game equilibria

w<2/3 w>2/3

c<2 Pooling Semi-separating

c>2 Separating Separating

Probability that challenger is weak

Cost of spending for weakchallenger