Graduate Microeconomics IILecture 5: Cheap Talk

Patrick Legros

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Outline

Cheap talk

Crawford-SobelWelfare

Partially Verifiable Information

Multiple Senders

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Outline

Cheap talk

Crawford-SobelWelfare

Partially Verifiable Information

Multiple Senders

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Outline

Cheap talk

Crawford-SobelWelfare

Partially Verifiable Information

Multiple Senders

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Outline

Cheap talk

Crawford-SobelWelfare

Partially Verifiable Information

Multiple Senders

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Cheap talk and information transmission

Similar to signaling games except that the signal is not costly forthe sender. Hence if all types have the same order of preferencesover actions chosen by the receiver. (as the sender in the“beer-quiche” prefers not to have a fight), signals cannot beinformative.

A necessary condition for signals to be informative in cheap talkgames is therefore that different types have different orders ofpreferences over actions chosen by the receiver.

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Map

I One sender (Crawford-Sobel 1982)I Inefficiency, partition information equilibria prevent full

aggregation of information.

I Efficiency with partially verifiable informationI Many senders: on the benefits of competitionI Battaglini (2002): Generic efficiency with multiple dimensions

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Crawford-Sobel (1982)Simplified model

I One agent (sender) with private information about the stateof the world θ (the set of states is an interval Θ = [0, 1] .)

I The prior density on Θ is uniform.

I One agent (receiver) who has to choose a decision y (a realnumber)

I If y is chosen in state θ, the outcome is x = y − θ. Thesender and the receiver have preferences over outcomes:US(x) = − (x − u)2 and UR (y) = −x2; u and 0 are the“ideal points” for the sender and the receiver respectively.

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Since only the sender knows the state, it is more convenient towork with the state dependent utility functions

US (y , θ) = − (y − θ − u)2

UR (y , θ) = − (y − θ)2

Note that U i12 > 0 and U i

11 < 0.

I As u = 0,R and S have common interest : y∗ (θ) = θ

I As u 6= 0,divergent interest: S likes y = u + θ 6= y∗ (θ) .

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I Key assumption: the receiver cannot commit to a decisionrule.

I no commitment: makes it different from mechanism design.I message is not costly to send (“cheap talk”): makes it

different from the “signaling” literature, e.g., Spence labormarket example.

I Communication: the sender can use messages m ∈ M tocommunicate with the receiver. Assume that M is “richenough” in the sense that there exists a map Θ → M that isinjective (for any two different states there exist two differentmessages).

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Crawford-SobelGame

I The sender chooses a state dependent message strategyσ (m, θ) , that is a distribution (mixed strategy) over M. LetM (σ, θ) = {m : σ (m, θ) > 0} be the support of σ in state θand let M (σ) = ∪θ∈ΘM (σ, θ).

I The receiver chooses a message dependent decision ruley (m) . Since V is concave, the receiver will never use a mixedstrategy.

I Use Bayesian Nash equilibrium: there exist a belief structureµ (θ|m) that assigns a probability distribution on Θ as afunction of the received message m such that

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I If m ∈ M (σ) , µ must be consistent with Bayes’ law, that is

µ (θ, m) =σ (m, θ)∫

θ σ(m, θ

)d θ

.

I The message strategy is optimal given y , that is

σ (m, θ) > 0 ⇒ y (m) ∈ arg maxm− (y (m)− θ − u)2 .

I The decision rule is optimal given σ and given µ

y (m) ∈ arg maxy−

∫(y − θ)2 µ (θ, m) dθ.

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Crawford-SobelExamples of equilibria

I “Babbling”: the strategy of the sender is state independent,σ (m, θ) = σ (m) for all θ. Then Bayes law implies that for allm, µ (θ, m) = 1 (the initial distribution - assumed to beuniform on [0, 1]).

I “Fully revealing”: the sender uses a pure strategy that is 1-1:σ (θ) = m, and θ 6= θ implies σ (θ) 6= σ(θ). In this case, theposterior belief is for m ∈ M (σ) : µ (θ, m) = 1 whenσ (θ) = m.

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I For fully revealing equilibria, the receiver can attain the firstbest.

I Note that if there is a fully revealing equilibrium, there areplenty of others: the content of the message is not important,what matters is that the map from state to message is 1-1.

I “Partially revealing”: for each m ∈ M (σ) , the set{θ : σ (m, θ) > 0} is not a singleton but is not the full set Θeither. Hence the message transmits some information aboutthe state.

I Since the content of the message does not really matter, thereis no loss of generality in supposing that the set of messages isa copy of Θ.

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Crawford-SobelProperties of an Equilibrium

1. The equilibrium message strategy is (essentially) partitional:there exists k cutoff values θk , k = 0, . . . ,K such that foreach k ∈ [0,K ] ,

θ ∈ (θk , θk+1) ,

I σ (m, θ) is the uniform distribution on [θk , θk+1]

0 0θ = 1θ 2θ 3θ 4θ 5 1θ =θ

( )support( ,σ θimessage

state

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2. The maximum number of elements of a partition equilibriumis a function of the degree of conflict between R and S; herethe ideal point u is an index of conflict.

3. If N (u) is the maximum number of elements, for anyN ≤ N (u) , there exists an equilibrium with N elements.

4. Under a monotonicity condition can show that the N + 1equilibrium is more informative than the N equilibrium (finerpartition).

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Crawford-SobelSome Observations

I In equilibrium, to each θ corresponds at most two possibledecisions taken by R.

I Consider the set of messages that yield R to choose y

N (y) = {m : y (m) = y}

I Say that y is induced by θ if m ∈ N (y) is in the support ofthe message strategy of S∫

m∈N(y)σ (m, θ) dm > 0.

I Let Y be the set of all induced decisions. Immediate that if yis induced by θ that US (y , θ) ≥ maxy∈Y US (y , θ) [revealedpreferences]

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I By concavity US11 < 0 : US (y , θ) is maximum on Y for at

most two values.

I Thus θ can induce at most two decisions in any equilibrium.

I For any y and y in Y , |y − y | ≥ u

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I In the parametric example, the best state contingent decisionsare yS (θ) = θ + u and yR = θ

I Let two equilibrium decisions y and y where y > y . Let θinduce y and θ induce y . If θ = θ, thenUS (y , θ)− US (y , θ) = 0, and choose θ = θ. If θ 6= θ, byrevealed preferences

US (y , θ) ≥ US (y , θ)

US(y , θ

)≥ US

(y , θ

)⇔

US (y , θ)− US (y , θ) ≥ 0 ≥ US(y , θ

)− US

(y , θ

)with one strict inequality. By continuity there exists θ suchthat US

(y , θ

)− US

(y , θ

)= 0. Since U12 > 0, θ > θ and

θ ∈ [θ, θ].

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Since U11 < 0,y < yS

(θ)

< y

and moreover

all θ > θ prefer y to y

all θ < θ prefer y to y .

x y θ= −

( )Sy uθ θ= + yy

0uy uθ− − y uθ− − y uθ− −y uθ− −

ˆ preferred to

by all

y y

θ θ<

θ θ< θ θ<

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I It follows that the beliefs of R when he chooses y havesupport N (y) ⊂ [0, θ) and when he chooses y have supportN (y) ⊂ (θ, 1].

I Since y > y , and UR12 > 0,UR (y , θ)− UR (y , θ) is increasing

in θ. Using continuity, UR12 > 0, UR

11 < 0 and θ ∈(θ, θ

), we

conclude thatyR

(θ)

= θ ∈ [y , y ] .

Now,∣∣yR

(θ)− yS

(θ)∣∣ = u; therefore for any two induced

actions y 6= y , |y − y | ≥ u. Since u is greater than 0, the setof actions in Y must be finite.

I Since when y > y we have shown that N (y) > N (y) (setnotation), it must be the case that messages that induce yand y respectively generate supports for the beliefs µ that aredisjoint; since this is true for any two pairs in Y , thegenerated supports must form a partition of Θ.

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I Hence any equilibrium message strategy that does not havethe partitional property can be redefined as a partitionalstrategy.

I In practice, to conclude the proof, need to show that thereexists cutoffs θk , k = 0, . . . ,K withθ0 = 0, θK+1 = 1, θk < θk+1 such that when the messagestrategy is to use a uniform distribution on [θk , θk+1] whenthe type is θ ∈ [θk , θk+1] , that the best response propertieshold. Note that the belief structure is trivial in this case: ifreceives message m ∈ [θk , θk+1] , µ (θ, m) = 1

θk+1−θkwhen

θ ∈ [θk , θk+1] , and µ (θ, m) = 0 otherwise.

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I Let yk be the decision taken when the message is in(θk , θk+1) . If m ∈ (θk , θk+1) the receiver will choose

yk = arg maxy

∫ θk+1

θk

UR (y , θ) dθ. (1)

I Since by sending a message θk + ε the sender can generatedecision yk and by sending θk − ε he can generate decisionyk−1, the sender must be indifferent between yk and yk−1 atθk , or

US (yk , θk) = US (yk−1, θk) . (2)

Given the indifference relation (2), US12 < 0 implies that all

θ > θk prefer yk to yk−1 and all θ < θk−1 prefer yk−1 to yk .

I The question of an equilibrium boils down to finding cutoffvalues θk for with conditions (1) and (2) hold.

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In the example the conditions are:

I the receiver chooses the average state in the interval:

yk =θk + θk+1

2(3)

with yK = yK−1.I For the sender, since θk is increasing in k,

yk − θk > yk−1 − θk ; hence can have(yk − θk − u)2 = (yk−1 − θk − u)2 only ifyk − θk − u > 0 > yk−1 − θk − u, and (2) boils down to

yk + yk−1 = 2θk + 2u

⇒ θk+1 = 2θk − θk−1 + 4u

⇒ θk = θ1k + 2k (k − 1) u

I We need θK = 1, hence 2K (K − 1) u < 1, or

K ≤ 12 + 1

2

√1 + 2

u ; the right hand side is strictly less than 2

when u ≥ 14 . In this case K = 1 and the unique equilibrium is

babbling!

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I Given the boundary condition θK+1 = 1, we have

θ1 =1− 2K (K − 1) u

K

and therefore

θk =k

K+ 2uk (k − K ) , k = 0, . . . ,K .

I As long as K ≤ 12 + 1

2

√1 + 2

u , there exists a partitional

equilibrium with K intervals.

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Crawford-SobelWelfare

I From an ex-ante perspective, both R and S value informationtransmission.

I In equilibrium EUS = −∑K−1

k=0

∫ θk+1

θk(yk − θ − u)2 dθ and

EUR = −∑K−1

k=0

∫ θk+1

θk(yk − θ)2 dθ.

I Since yk = θk+θk+1

2 is the average state in the interval

[θk , θk+1] ,∫ θk+1

θk(yk − θ)2 dθ

θk+1−θkis the within variance and∑K−1

k=0

∫ θk+1

θk(yk − θ)2 dθ is the expected variance σ2 (K ).

Hence,

EUS = −(σ2 (K ) + u2

)EUR = −σ2 (K ) ,

I Both R and S prefer (from an ex-ante point of view) equilibriain which K is the largest since then the variance σ2 (K ) is thesmallest.

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Partially Verifiable Information

In Crawford-Sobel, there are no constraints on the type ofinformation that the sender can transmit. In particular, the set offeasible messages is independent of the state of the world.Information is soft.

Often, there are natural possibilities to verify a claim.

I Someone announcing that he has 10 dollars in his pocketshould be able to show 10 dollars. Note that if he has 15dollars, he may still decide to announce that he has only 10dollars, but he cannot pretend that he has 20 dollars.

I Accounting rules allow firms to release verifiable informationabout profits in many ways; for a given financial stateannounced profits can vary depending on the use that is madeof accounting principles. However there are limits to this andpartial verifiability.

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I Consider a simple restriction on the set of messages: if thestate is θ, then M (θ) = [0, θ] , that is the sender canunderstate the state but not overstate it. In this case, for anyvalue of u (the index of conflict between S and R), it ispossible to achieve full revelation and the first best for R.Suppose that u > 0, that is that the sender prefers decisionyS (θ) = θ + u > yR (θ) in any state.

I “Trick”: pessimistic belief about the state in the sense thatthe receiver takes the message that is sent at face value: if Ssends m, R believes that the state is m. In this case, in stateθ, if S sends m, the receiver chooses m.

I Since S is constrained to send m ≤ θ, his payoff (given thebeliefs of the receiver) is

US (m, θ) = − (m − θ − u)2

By concavity of US and the fact that m ≤ θ < yS (θ) , it isimmediate that US (m, θ) is maximized at m = θ.

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This is a special form of what is called the unraveling theorem : ifthere is a bound on messages that is state dependent andmonotone in the state, skeptical beliefs from the receiver (alwaysbelieving that the state is the worth possible state consistent withthe announced message) induces revelation of information by thesender.

More general formulations (announcements can be intervals butmust be truthful in the sense that the true state belongs to theinterval): Milgrom-Roberts 1986, Shin 2003,

Green-Laffont (1986,87) consider general restrictions in amechanism design framework.

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Multiple Senders

I Literature on multiple referralsMilgrom-Roberts 1986, Battaglini 2002, Ottaviani-Sorensen2000, Garinaco-Santos 2003, ...

I Remember if only i sends information, the maximumtransmission of information is N (ui ) , decreasing in ui .

I Note here a significant cost of no-commitment. If R couldcommit to a decision rule as a function of the messagesreceived, he could induce full revelation (mechanism design).

I Competition helps for information revelation but still residualinefficiencies.

I Most work done with one dimensional states and decisions.

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MultidimensionsLimits to competition with one dimension

Two senders with ideal points ui ; wlog assume quadratic utilityfunctions US

i (y , θ) = − (y − θ − ui )2 for the senders

UR = − (y − θ)2 for the receiver. Assume that u1 ≤ 0 ≤ u2 andthat Θ =

[−θ, θ

], θ > 0.

Battaglini 2002 A necessary and sufficient condition for theexistence of a fully revealing equilibrium is thatu2 − u1 ≤ θ.

I Therefore, as long as the ideal points of the two senders arenot “too far apart”, there exists a fully revealing equilibrium.

I An example of equilibrium construction when the condition ofthe proposition is true.

y (m1,m2) =

(m1 + m2)/2 if m1 ≤ m2

θ if m1 > m2 and m1 < 2u2 − θ−θ if m1 > m2 and m1 ≥ 2u2 − θ.

I The beliefs are such that µ (θ;m1,m2) = y (m1,m2).

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I Suppose that 2 is truthful in state θ. If 1 is truthful, thedecision is θ and 1’s payoff is US

1 = −u21 .

I Suppose that m1 < θ; then the decision is m1+θ2 and the

outcome is m1−θ2 < 0; therefore US

1 < −u21 .

I If m1 > θ, it must be that m1 ≥ 2θ − θ (the casem1 < 2m2 − θ is inconsistent with m1 > θ and θ ≤ θ),therefore the decision is −θ and the outcome is −θ − θ < 0,hence US

1 < −u21 . This shows that 1 does not want to deviate

from truth-telling

I Note that belief structure is “extreme”

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Multiple dimensionsGeneric revelation of information

I Battaglini (2002) considers multiple dimensions for states anddecisions. Turns out that there exist generically fully revealingequilibria.

I State space is Θ ⊂ <2. Two agents (senders) with ideal pointsu1 and u2 observe θ and transmit a message m ∈ Θ to areceiver with ideal point of 0. Decisions are y ∈ <2 and theoutcome is x = y − θ where θ is the true state of the world. Ifideal point is u =

(u1, u2

), preference over decision y in state

θ is − (y (1)− θ (1)− u (1))2 − (y (2)− θ (2)− u (2))2 .

proposition Suppose that u1 and u2 are not on the same ray fromthe origin, then there exists a fully revealingequilibrium.

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I Idea: For fully revealing messages, the policy is y = θ. Sinceu1 and u2 are not on the same ray from the origin, thetangent I1 to the indifference curve of sender 1 at x = 0 andthe tangent I2 to the indifference curve of sender 2 at x = 0form a cone. For any line parallel to Ii , there exists a numberx that identifies uniquely this line (since there exists a uniquevector (0, x) belonging to this line); let Ii (x) be the lineparallel to Ii passing through (0, x) . Then for any x1 and x2

and y , I1 (x2) ∩ I2 (x1) exists and is unique.

I Messages are for i to announce a number xi . The decision isto choose y ∈ I1 (x2) ∩ I2 (x1) . That is agent 2 announces theline parallel to I1 going through θ and agent 1 announces theline parallel to I2 going through θ.

I If 2 is truthful in state θ, that is announces the number x2

such that θ ∈ I1 (x2) , then by announcing x1 6= x1, sender 1generates outcomes on I1 (0) which are by construction worsethan the outcome under truth telling.

I The argument generalizes to higher dimensions.

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I Sequential transmission of information

I Endogenous communication system

I Loss due to no-commitment in one vs. many senders, uni vs.multi-dimensions?

I Noisy information: value of duplication of transmission ofinformation?

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