Incomplete information: signaling games We have seen an equilibrium concept for these games (weak PBE). • Two important subsets of games of incomplete information: ‐signaling games:

Incomplete information: signaling games

Game Theory and Political Theory (W4209) 1


Introduction – from last class

Informal definition:

Weak perfect Bayesian equilibrium (weak PBE):“It requires, roughly, that at any point in the game, a player’s strategy prescribe optimal actions from that point on given her opponents’ strategies and her beliefs about what has happened so far in the game and that her beliefs be consistent with the strategies being played.” (MWG, p. 283).

Even more informal:

Weak perfect Bayesian equilibrium (weak PBE):1.Actions are optimal at information sets given beliefs [Sequential rationality];2.Beliefs are formed by Bayes’ rule when possible, and when not possible the beliefs could be anything [Bayesian updating on the equilibrium path].


Introduction – from last classBayes’ rule:

10% of people carry gene X.

Prior belief I have gene X is Pr(X) = 10%

Test: + for 90% of people that have it (Pr(+|X)=90%), and + for 20% that do not have it (Pr(+|¬X)=20%).

I test positive. What is my belief now that I have gene X?

9 / 27

Pr(X|+) = Pr(+ and X) = 9/27 = 1/3Pr(+)

Bayes’ rule = Pr(X) * Pr(+|X) Pr(X)*Pr(+|X) + Pr(¬X)*Pr(+|¬X)

= 0.1 * 0.9 = 1/30.1 * 0.9 + 0.9 * 0.2

9 18

↑ 10 people have it ↑ 90 don’t have gene X•We update our beliefs from 10% to 33 1/3%.

‐+

‐

+


Introduction

• Games of incomplete information; one player knows something the other player does not.•We have seen an equilibrium concept for these games (weak PBE).

• Two important subsets of games of incomplete information:

‐ signaling games: the informed player moves first

‐ screening games: the uninformed player moves first

• George Akerlof’s second‐hand car market: signaling and screening.


Signaling games

• In signaling games an informed player (sender) moves first and sends a signal with which the uninformed (the receiver) may update his beliefs about the nature of the informed player’s information – his or her ‘type’.

• A signal can be separating, pooling or partially‐separating.

• Incomplete information raises a number of interesting issues:

‐ Can the sender send an informative signal? ‐ Does the sender send a truthful signal or not?‐ Is the signal credible?‐ Do any of the players have a strategic advantage based on the way information is allocated? ‐ Can informed players mislead uninformed players?


Signaling games

Frank and Jolene• Nature chooses the state of theworld θ. That is, Frank wears black or white trousers, θ = {BT,WT}• Frank sends a message, m = {b,w} • Jolene takes a black or a white sweaterto the party, x = {BS,WS}• Jolene wants the sweater to match thecolor of Frank’s trousers; Frank doesn’t.

What next?

NatureBT WT

p 1‐p

BS WS BS WS

m=b m=b

m=w m=w

Jolene Jolene

Frank Frank

Jolene Jolene

BS WS BS WS


Signaling games

• Introduce payoffs:uF (WS|θ=BT) > uF (BS|θ=BT)uF (BS|θ=WT) > uF (WS|θ=WT)uj (WS|θ=BT) < uj (BS|θ=BT)uj (BS|θ=WT) < uj (WS|θ=WT)

• For a PBE:We want F’s message to be optimal given J’s beliefs, which are based on F’s strategyby using Bayes’ rule. And we want J’sstrategy to be optimal given her beliefs.

Nature

p 1‐p

1,2 2,1 2,1 1,2

1,2 2,1 2,1 1,2

BT WT

BS WS BS WS

BS WS BS WS

m=b m=b

m=w m=w

Frank Frank

Jolene Jolene

Jolene Jolene


Signaling games – separating eq.

• Consider F’s strategy where F playsm=b if θ=BT and m=w if θ=WT.

•This strategy is truthful and fully revealing (it fully reveals the node at which J finds herself).

• For this to be a PBE, J’s beliefs mustbe based on F’s strategy. J’s actions must also be based on her beliefs.

•We use Bayes’ rule to develop these Beliefs of Jolene.

Nature

p 1‐p

1,2 2,1 2,1 1,2

1,2 2,1 2,1 1,2

BT WT

BS WS BS WS

BS WS BS WS

m=b m=b

m=w m=w

Frank Frank

Jolene Jolene

Jolene Jolene



• Using Bayes’ rule:

Pr(θ=BT |m=b) =(Pr(m=b | θ=BT)Pr(θ=BT)/Pr(m=b) =(1*p)/p = 1

Pr(θ=BT|m=w) =(Pr(m=w|θ=BT)Pr(θ=BT)/Pr(m=w) =(0*(1‐p)/(1‐p) = 0

Pr(θ=WT|m=w) =(Pr(m=w|θ=WT)Pr(θ=WT)/Pr(m=w) =1*(1‐p)/(1‐p) = 1

Pr(θ=WT |m=b) =(Pr(m=b|θ=WT)Pr(θ=WT)/Pr(m=w) =(0*p)/p = 0

•Given these beliefs:x=BS if m=b and x=WS if m=w

Nature

p 1‐p

Jolene 0 1 Jolene

Jolene 1 0 Jolene

1,2 2,1 2,1 1,2

1,2 2,1 2,1 1,2

BT WT

BS WS BS WS

BS WS BS WS

m=b m=b

m=w m=w

Frank Frank



•Given these beliefs:x=BS if m=b and x=WS if m=w

• If there exist a strategy where thereceiver is able to fully indentifythe sender’s type after receiving thesignal, then the equilibrium is knownas a separating equilibrium. The receiver is able to separate θ=BTfrom θ=WT.

• Do we really have a separating equilibrium here? Do we have a weak PBE here?

1. Bayesian updating on the equilibrium path? YES2. Sequential rationality? Is it optimal for Frank to send a truthful signal if x=BS if m=b and x=WS if m=w? NO

Nature

p 1‐p

1,2 2,1 2,1 1,2

1,2 2,1 2,1 1,2

BT WT

BS WS BS WS

BS WS BS WS

m=b m=b

m=w m=w

Frank Frank

Jolene 1 0 Jolene

Jolene 0 1 Jolene



• If Frank is truthful (and then x=BS if m=b and x=WS if m=w), he receives a payoff of 1.

• But if Frank lies and sends m=w in stateBT and sends m=b in state WT (indicated by the red lines), then given Jolene’s strategies, Frank will get a payoff of 2.

• Frank’s strategy is thus not optimalgiven Jolene’s strategy and beliefs.

Thus a truthful, fully‐revealing strategy is not an equilibrium for this game.

Nature

p 1‐p

1,2 2,1 2,1 1,2

1,2 2,1 2,1 1,2

BT WT

BS WS BS WS

BS WS BS WS

m=b m=b

m=w m=w

Frank Frank

Jolene 1 0 Jolene

Jolene 0 1 Jolene



• The other candidate for a separating equilibrium is a fully revealing, non‐truthful strategy by Frank (m=w when θ=BT and m=b when θ=WT).

•Does not exist.

• There are thus no separating equilibriums to this game.

•This is an example of cheap talk,where sending signals of differenttypes does not cost the sender anything.

Nature

p 1‐p

1,2 2,1 2,1 1,2

1,2 2,1 2,1 1,2

BT WT

BS WS BS WS

BS WS BS WS

m=b m=b

m=w m=w

Frank Frank

Jolene Jolene


Signaling games – pooling eq.

• The opposite of a separatingequilibrium is a pooling equilibrium,where the Jolene learns no information about Frank’s type from Frank’s signals.

• An example would be: m=b whenθ=BT and when θ=WT.

•Bayes’ rule to update on the equilibriumpath:Pr(θ=BT|m=b) =(Pr(m=b|θ=BT)Pr(θ=BT)/Pr(m=b) =(1* p)/1 = p

Pr(θ=WT|m=b) =(Pr(m=b|θ=WT)Pr(θ=WT)/Pr(m=b) =1*(1‐p)/1 = 1‐p

• Jolene does not update her beliefs. Prior = posterior beliefs

Nature

p 1‐p

1,2 2,1 2,1 1,2

1,2 2,1 2,1 1,2

p 1‐p

BT WT

BS WS BS WS

BS WS BS WS

m=b m=b

m=w m=w

Frank Frank

Jolene Jolene

Jolene Jolene



•We can make any assumption aboutq and 1‐q.

•Because Bayes’ rule gives us no way to calculate beliefs of the equilibrium path(i.e. one can’t divide by zero).

•Because the nodes m=w are never reached in equilibrium, any set of beliefs at these nodes willbe consistent with Frank’s strategy.

•Let’s assume: q = Pr(θ=BT|m=w) = p and 1‐q = Pr(θ=WT|m=w) = 1‐p.

Nature

p 1‐p

Jolene Jolene

1,2 2,1 2,1 1,2

1,2 2,1 2,1 1,2

p 1‐p

q 1‐q

BT WT

BS WS BS WS

BS WS BS WS

m=b m=b

m=w m=w

Frank Frank

Jolene Jolene



• Jolene has no new information from Frank’s signal. J’s beliefs are not updated.Thus Jolene chooses her optimalstrategy based upon an expected utilitycomparison:

EUj(WS) = p*(1) + (1‐p)*(2) = 2‐p

EUj(BS) = p*(2) + (1‐p)*(1) = 1+p

• So Jolene will play BS instead of WS if

EUj(BS) ≥ EU2(WS) ↔ p ≥ 1/2

Nature

p 1‐p

Jolene Jolene

1,2 2,1 2,1 1,2

1,2 2,1 2,1 1,2

p 1‐p

p 1‐p

BT WT

BS WS BS WS

BS WS BS WS

m=b m=b

m=w m=w

Frank Frank

Jolene Jolene



• For example, when p ≥ 1/2: x = BS.

• Sequential rationality for J: YES. • Bayesian updating on theequilibrium path: YES. •Is F’s strategy sequential rational given this?

• YES: Frank has no incentive to change the signal.

•We have found a weak PBE; apooling equilibrium.

•This equilibrium depends on the characterization of J’s beliefs at the off‐the‐equilibrium‐path nodes”.

Nature

p 1‐p

1,2 2,1 2,1 1,2

1,2 2,1 2,1 1,2

p 1‐p

p 1‐p

BT WT

BS WS BS WS

BS WS BS WS

m=b m=b

m=w m=w

Frank Frank

Jolene Jolene

Jolene Jolene



• Consider for example the casewhere p ≥ 1/2. Thus J plays BS.

• Assign beliefs to nodes on the off‐the‐equilibrium‐path:q = Pr(θ=BT|m=w) = 0 and 1‐q = Pr(θ=WT|m=w) = 1.

• Jolene’s optimal reaction to an off‐the‐equilibrium‐path signal (m=w)is x = WS.

Nature

p 1‐p

1,2 2,1 2,1 1,2

1,2 2,1 2,1 1,2

p 1‐p

q 1‐q

BT WT

BS WS BS WS

BS WS BS WS

m=b m=b

m=w m=w

Frank Frank

Jolene Jolene

Jolene Jolene



• BUT given this strategy, Frank could improve his payoff when θ=BT bysending m=w instead of m=b (indicatedby the red lines).

•Thus with these new beliefs a pooling equilibrium does not exist.

•Pooling equilibria exist as long asq ≥ 1/2.

• Insight: What happens off the equilibrium path and what people belief there matters for the equilibrium of the game!

Nature

p 1‐p

1,2 2,1 2,1 1,2

1,2 2,1 2,1 1,2

p 1‐p

q=0 1‐q= 1

BT WT

BS WS BS WS

BS WS BS WS

m=b m=b

m=w m=w

Frank Frank

Jolene Jolene

Jolene Jolene


Signaling games – Conclusion

• There are three main types of equilibria:

‐ Separating equilibrium: Receiver receives full information from the sender’s signal;‐ Pooling equilibrium: Receiver receives no information from the sender’s signal;‐ Semi‐pooling/ semi‐separating equilibrium: Receiver receives partial information from the sender’s signal.

• Important:The receiver’s beliefs must be based on the sender’s strategy, the receiver’s actions must be optimal based on his beliefs, and the sender’s strategy must be optimal given the receiver’s strategy.

Weak PBE:1.Sequential rationality2.Bayesian updating on the equilibrium path


Extra slides – illustration: entry deterrence in elections

See Epstein and Zemsky (1995):

• Incumbent candidates often raise a lot of campaign contributions – a campaign war chest – in order to deter challengers from entering the race.

• N = {I,C}, the incumbent I and the challenger C

• The incumbent is strong (weak) with probability p (1‐p)

• The challenger does not know the incumbent’s type

• Both types get 1 from holding office

• Challenger will win against a weak (strong) type with probability πw (πs), where πw > πs

• It cost k for the challenger to run, where πw > k > πs

• The incumbent can choose to build a war chest (WC), or not (¬WC). The challenger can enter (E), or not (¬E).

• There is a cost for the incumbent to raise money, which is greater for weak incumbents: cw > cs


Naturep 1‐p

S WIncumbent

WC ¬WC WC ¬WC

Challenger

E ¬E E ¬E E ¬E E ¬E

1‐πs‐cs 1‐cs 1‐πs 1 1‐πw‐cw 1‐cw 1‐πw 1πs‐k 0 πs‐k 0 πw‐k 0 πw ‐k 0

• Is there a separating equilibrium? That is, an equilibrium where the challenger can separate weak from strong types.

• If so, challengers would only run against weak types (πw‐k>0, and πs‐k< 0).

• Thus weak types have an incentive to pretend to be strong.

• The only way for a separating equilibrium to exist is if the costs of pretending are too high for a weak incumbent.



Naturep 1‐p

S WIncumbent

WC ¬WC WC ¬WC

Challenger



• Consider WC if θ=S and ¬WC if θ=W.

• Using Bayes’ rule the challenger updates his/her beliefs:

Pr (S|WC) =Pr(WC|S)Pr(S)/P(WC) =(1*p)/p = 1

and Pr (S|¬WC) = 0

Pr (W|¬WC) =Pr(¬WC|W)Pr(W)/P(WC) =(1*(1‐p))/(1‐p) = 1

And Pr (W|WC) = 0

• Given these beliefs the challenger will play E when ¬WC and ¬E when WC.

1 0 0 1




Naturep 1‐p

S WIncumbent

WC ¬WC WC ¬WC

Challenger



• Do we have a separating equilibrium? Do we have a weak PBE?

‐Bayesian updating on the equilibrium path: YES‐ Sequential rationality for the challenger: YES‐ Sequential rationality for the incumbent?

That depends.

1 0 0 1



Naturep 1‐p

S WIncumbent

WC ¬WC WC ¬WC

Challenger



• Strong incumbent can choose WC which gives 1‐cs or ¬WC which gives 1‐πs. For the strategy to hold: 1‐cs > 1‐πs, or πs > cs.

• Similarly, weak incumbent can choose between the payoffs 1‐cw (from WC) or 1‐πw (from ¬WC) . We thus need: cw >πw.

• Thus: if πs > cs and cw >πw a separating equilibrium exist.

• Intuitively:While a weak incumbent may want to bluff, it is too costly for him/her to do so (cw > πw).

1 0 0 1



Naturep 1‐p

S WIncumbent

WC ¬WC WC ¬WC

Challenger



• The other candidate for a separating equilibrium is ¬WC if θ=S and WC if θ=W.

• Using Bayes’ rule:

Pr (S|¬WC) =Pr(¬WC|S)Pr(S)/P(WC) =(1*p)/p = 1

and Pr(S|WC) = 0

Pr (W|WC) =Pr(WC|W)Pr(W)/P(WC) =(1*(1‐p))/(1‐p) = 1

and Pr(W|¬WC) = 0

• Given these beliefs the challenger will play E when WC and ¬E when ¬WC.

0 1 1 0



Naturep 1‐p

S WIncumbent

WC ¬WC WC ¬WC

Challenger



• Again, do we have a separating equilibrium? Do we have a weak PBE?

‐Bayesian updating on the equilibrium path: YES‐ Sequential rationality for the challenger: YES‐ Sequential rationality for the incumbent?

NO.

0 1 1 0



Naturep 1‐p

S WIncumbent

WC ¬WC WC ¬WC

Challenger



• Strong incumbent can choose WC which gives 1‐πs‐cs or ¬WC which gives 1. The strong type thus has no problem adhering to the strategy.

• The weak incumbent can choose between the payoffs 1‐πw‐cw (from WC) or 1 (from ¬WC) .

•Weak types will therefore always defect and play ¬WC.

• So: there does not exist a separating equilibrium of this kind, because the weak incumbents will always bluff (indicated in red) by not building a war chest to scare of the entrants.

0 1 1 0



Naturep 1‐p

S WIncumbent

WC ¬WC WC ¬WC

Challenger



•Do pooling equilibria exist?

• Consider a situation where the incumbent plays ¬WC when θ=S and when θ=W.

• In this case the challenger can’t update his/her beliefs about the type of incumbent: Pr(S|¬WC)=p, Pr(W|¬WC)=1‐p.

• The challenger therefore optimizes his/her expected utility:

EUc(E) = p*(πs‐k) + (1‐p)*(πw‐k) = pπs+(1‐p)*πw‐k

EUc(¬E) = p*(0) + (1‐p)*(0) = 0

EUc(E) > EUc(¬E) ↔p ≤ (k‐πw)/ (πs‐πw)

p 1‐p



Naturep 1‐p

S WIncumbent

WC ¬WC WC ¬WC

Challenger



• Also assume priors off‐the‐equilibrium path (Pr(S|WC)= q = p and Pr(W|WC)= 1‐q = 1‐p).

• A challenger enters only if p ≤ (k‐πw)/ (πs‐πw); this is sequentially rational.

•We also have Bayesian updating on the equilibrium path.

• Now when is the incumbent sequentially rational?

p p 1‐p 1‐p



Naturep 1‐p

S WIncumbent

WC ¬WC WC ¬WC

Challenger



•More complete: Are both type S and type W sequentially rational?

• Both type S and W will always play ¬WC, because 1‐πs > 1‐πs‐cs and 1‐πw > 1‐πw‐cw

Regardless of the incumbents action the challenger will always enter because he thinks it is unlikely that the incumbent is strong, θ=S. Because the challenger always enters, whatever happens, a war chest by either type of incumbent wastes resources. Thus neither type has an incentive to build one.

p p 1‐p 1‐p



For completeness, in this game there are:

• Separating equilibrium (the one we obtained in the slides): If cs ≤ πs and cw ≥ πw, the following strategies and beliefs constitute a separating equilibrium: type S plays WC, type W plays ¬WC, when observing ¬WC (WC) the challenger E (¬E), Pr(S|WC)=1 and Pr(S| ¬WC)=0.

• Pooling equilibrium 1: Suppose that p ≥ (k‐πw)/ (πs‐πw). The following strategies and beliefs constitute a pooling equilibrium: both type S and W play ¬WC, when observing WC (¬WC) the challenger E (¬E), Pr(S|¬WC)=p, and Pr(S|WC) ≤ (πw‐k)/ (πw ‐πs).

• Pooling equilibrium 2: Suppose that p ≥ (k‐πw)/ (πs‐πw). The following strategies and beliefs constitute a pooling equilibrium: both type S and W play ¬WC, the challenger never enters, Pr(S|¬WC)=p, and Pr(S|WC) ≥ (πw‐k)/ (πw ‐πs).

• Pooling equilibrium 3: Suppose that p ≥ (k‐πw)/ (πs‐πw), cs ≤ πs and cw ≤ πw. The following strategies and beliefs constitute a pooling equilibrium: both type S and W play WC, when observing WC (¬WC) the challenger ¬E (E), Pr(S|WC)=p, and Pr(S|¬WC) ≤ (k‐πw)/ (πs‐πw).

(more on the next slide)



• Pooling equilibrium 4 (the one we obtained in the slides): Suppose that p ≤ (k‐πw)/ (πs‐πw). The following strategies and beliefs constitute a pooling equilibrium: both type S and W play ¬WC, the challenger always plays E, Pr(S|WC)=p, and Pr(S|¬WC) ≤ (k‐πw)/ (πs‐πw).

• Semi‐pooling equilibrium: Let cw ≤ πw and p ≥ (k‐πw)/ (πs‐πw). Then the following strategies constitute a partial‐pooling equilibrium: S plays WC, W plays WC with probability q, after observing ¬WC the challenger plays E, and after observing WC the challenger plays E with probability r > 0. Where q ≤ 1, and r = (πw ‐ πw)/ πw

See pages 219‐226 in McCarty et al (2007) for a complete discussion.

References

Akerlof, George. 1970. The Market for 'Lemons': Quality Uncertainty and the Market Mechanism." Quarterly Journal of Economics, 84(3), 488‐500.

Epstein, David and Peter Zemsky. 1995. Money Talks: Deterring Quality Challengers in Congressional Elections. American Political Science Review, 89(2), 295‐308.

McCarty, Nolan and Adam Meirowitz. 2007. Political Game Theory. An Introduction. New York: Cambridge University Press.

Documents

Incomplete information: signaling games We have seen an equilibrium concept for these games (weak PBE). • Two important subsets of games of incomplete information: ‐signaling games: