Persuasion and Learning by Countersignaling Kim-Sau Chung

Peter Eso

Department of Economics University of Minnesota 4-101 Hanson Hall, 1925 4th Street South Minneapolis, MN 55455-0462 U.S.A. [email protected] Tel: +1-612-624-4060 (corresponding author)

Department of Economics University of Oxford Manor Road Building Manor Road Oxford, OX1 3UQ United Kingdom [email protected]

Abstract We model countersignaling (i.e., very high types refraining from signaling) arising from the tradeoff between persuasion and learning in a signaling game. We assume the agent has imperfect private information regarding his productivity, which the signaling action provides additional, verifiable information about. A higher-type agent benefits more from providing such objective, albeit imprecise “proof” for the market, but may also gain less from learning about his productivity. When the latter effect dominates the former for the very high types, the equilibrium exhibits countersignaling: very high and low types pool on refraining from signaling, and only the medium types signal. Under certain conditions the countersignaling equilibrium is the unique pure-strategy Perfect Sequential Equilibrium.

JEL Classification Codes: D82, D86. Keywords: signaling, countersignaling, persuasion, learning

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1

Introduction

Monotonic (or Spencian) signaling models, originating in Spence (1973), have become standard in social and biological sciences. In such models an agent who is privately informed about his productivity can credibly indicate that he has a high type by taking an observable action that is less costly or more beneficial for higher types. Besides Spence’s original example of educational signaling, other applications include Myers and Majluf’s (1984) theory of equity underpricing, Gambetta’s (2009) analysis of excessive violence among criminals in sociology, Zahavi’s (1975) handicap principle explaining the evolution of costly sexual ornaments and behaviors in biology, and so on. Veblen’s (1899) much earlier, informal analysis concerned conspicuous consumption. Careful anthropological and sociological studies have also noted the phenomenon of countersignaling: the truly rich need no symbols of status (Veblen), top crime bosses do not act tough (Gambetta), and so on. Casual observation agrees: some very talented people, from Bill Gates and Steve Jobs to Tom Hanks and Woody Allen, did not bother to finish college. The literature appears to be using two, nested definitions of countersignaling. According to the weaker notion, neither low nor high productivity types engage in the signaling action (only medium types do); however, high and low types may still separate using distinguishing cues, indices, contextual signs. The stronger definition requires that outside observers cannot distinguish non-signaling high types from non-signaling low types. Feltovich, Harbaugh and To (2002) use the former, weaker definition. They set up a Spencian signaling model with three types, and add an action that reveals the agent’s type with high probability. For appropriate parameter values there is an equilibrium in which only the medium type engages in traditional signaling, whereas the high type stochastically separates from the low type using the type-revealing action.1 In contrast, Araujo, Gottlieb and Moreira (2007) build an educational signaling model with a multidimensional agent type (e.g., cognitive and non-cognitive abilities). When the two aspects of ability affect the agent’s productivity and cost differently, a higher productivity type with a lower cost parameter may choose a lower level of the signaling action. In their model different types of the agent that pool on the same action cannot be distinguished by the market (strong countersignaling). In our model an agent has imperfect private information about his productivity, which his signaling action provides additional, verifiable information about. A higher-type agent would benefit more from providing a more precise, objective “proof” for the market, but may 1

In Hvide’s (2003) two-sector labor market model there is an equilibrium where high-type workers enter the talent-intensive sector, low-type workers enter the other sector, and medium types invest in education before deciding. This may also be considered a model of weak countersignaling.

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also gain less from learning about his productivity. When the latter effect outweighs the former for the highest types the result is strong countersignaling: medium types signal, but very high and very low types pool on refraining from signaling. All separating equilibria must exhibit countersignaling, and under certain conditions no other outcome is consistent with Perfect Sequential Equilibrium. A distinguishing and potentially testable feature of our model is that countersignaling can only arise if the average productivity of non-signaling (extreme) types exceeds that of the signaling (medium) types. A stylized example is an artist’s choice of his debut project that can be either traditional (e.g., portraiture), or non-traditional (e.g., an installation).2 Assume that the success or failure of the traditional project generates a more precise verifiable public signal about the artist’s talent than the fate of the experimental project. The artist is unsure about his talent, so the reception of his debut also informs him about the wisdom of pursuing art as a career. Under the right conditions countersignaling occurs: very talented and untalented artists go for non-traditional projects, whereas medium types choose traditional ones. Interestingly, such equilibria exist only if the average talent of artists with non-traditional projects exceeds that of those with traditional projects. The choice of a traditional project is viewed by the market in the short run as a signal that the artist has self-doubt. Our model is related, but in the key aspect of countersignaling differs from, reputational models of Brandenburger and Polak (1996), Morris (2001), Ottaviani and Sørensen (2006), in which separating equilibria are impossible due to the agent’s reputational concerns.

2

Model

We describe a two-period, two-player game between an agent and the market. Before the � � game starts, the agent privately observes his type, θ ∈ 0, 12 , 1 , where Pr[θ = 0] = p0 and Pr[θ = 1] = p1 are commonly known parameters. Then Nature draws the state, ω ∈ {0, 1} with Pr[ω = 1] = θ; the realization of ω is not observable to anyone. The interpretation is that ω is the agent’s binary productivity (1 or 0 by normalization), and θ his private information on how likely it is that his productivity is high.3 The agent may be sure he is unproductive (type θ = 0), or sure he is productive (type θ = 1), or have a 50-50% belief (type θ = 12 ). For an uninformed observer (e.g., the market), the ex-ante probability that the agent’s productivity is 1 is E[θ] = 12 (1 − p0 + p1 ). 2

Alternatively, think of the agent as a candidate from a top PhD program in economics, the signaling action being the topic and methodology of his job market paper (mainstream or non-traditional). 3 We may think of the model as one where ω is drawn first, unobserved by all. Then the agent observes a private signal about ω represented by the posterior generated by the signal, θ.

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In period 1 the agent chooses an observable action, a ∈ {a0 , a1 }. Action a1 generates a verifiable signal y ∈ {0, 1}, whereas a0 does not. Assume that y is known to be correct � � about the state of nature with probability π = Pr [ y = ω | ω, a1 ] ∈ 12 , 1 . At the end of the first period the market pays the agent W , the market’s expectation of his productivity given all public information. Denote the agent’s (potentially mixed) strategy of choosing the first-period action as a function of his type by α(θ). Then W = E [ ω | α(θ) = a, y ] , where a is the action chosen by the agent and y the realization of the signal provided a = a1 . If the conditioning event has zero probability (e.g., if α(θ) ≡ a1 but a = a0 ) then W can be set anywhere between 0 and 1. The agent’s equilibrium choice of α(·) will be the central question of the analysis of the next section. As we will see, it is the first-period wage that motivates the agent to try to persuade the market of his high productivity by the choice of his signaling action and the generated verifiable signal. In period 2 the agent makes a decision, In or Out, and receives an additional payoff, T . � � If the agent’s decision is In then T = ω. If his decision is Out then T = k, where k ∈ 12 , π is a commonly known “outside option”. One may interpret the second-period decision as a choice made by the agent over the long run, after any short-run signaling activity has already taken place. As we will see, it is the second-period payoff, T , that motivates the agent to learn about his own productivity. The agent is sequentially rational, obeys Bayes’ rule, and maximizes the sum of his payoffs over both periods, W + T . The market is a dummy player whose only role is to observe the agent’s first-period action and the signal generated by it, and pay him a rational-expectations wage (W , also in the first period), as outlined above. In what follows equilibrium refers to Perfect Bayesian Equilibrium.

3

Analysis

First, we characterize the agent’s optimal decision in the second period. Let θˆ denote the agent’s posterior belief about ω = 1 given his private information and the realization of signal y. This is the agent’s expected payment in period 2 from choosing In. If θ = 1 then θˆ = 1, whereas if θ = 0 then θˆ = 0, for any action chosen in period 1 and any realization of y. Therefore in period 2 the agent with type θ = 1 chooses In for payoff T (1) = 1 no matter which action he chose in period 1, whereas with type θ = 0 he chooses Out for payoff T (0) = k, again, no matter what. If type θ = 12 picked action a0 in period 1 then θˆ = θ = 12 ; he weakly prefers Out over In because k ≥ 12 . If type θ = 12 picked action a1 in period 1 then by Bayes’ rule he updates 4

to θˆ = π conditional on y = 1 and θˆ = 1 − π conditional on y = 0. Therefore, in period 2 he picks In iff y = 1. Thus the expected period-2 payoff of the agent with type θ = 12 is � � � � T 12 = π iff a = a1 and y = 1, and T 12 = k otherwise. Therefore, before y is realized, the agent’s expected period-2 payoff from choosing a1 is k conditional on θ = 0; it is 1 conditional on θ = 1; and it is 12 (π + k) conditional on θ = 12 and a = a1 . (The latter is computed by noting that when type θ = 12 chooses a = a1 , his period-2 payoff is π if y = 1 and k if y = 0, each occurring with 50% chance.) In contrast, the agent’s expected period-2 payoff from choosing a0 is 0 conditional on θ = 0, it is 1 conditional on θ = 1, and it is k conditional on θ = 12 . Note that the informative action generates an option value for the agent when his type is θ = 12 . Next, we compute the agent’s wage in period 1. Given his first-period strategy α(·), denote xi the expected type of the agent choosing ai , that is, xi = E [ θ | α(θ) = ai ] for i = 0, 1. By the law of iterated expectations E [ ω | α(θ) = ai ] = xi for i = 0, 1; that is, xi is the market’s expectation of the agent’s productivity, knowing the agent’s equilibrium strategy α(·), and observing the action chosen by him in period 1. Denote Wi (θ, xi ) the expected first-period wage of type θ conditional on having taken action ai with the market’s beliefs being xi , for i = 0, 1. If action a0 is chosen in period 1 then the agent’s short-run wage is W0 (θ, x0 ) ≡ x0 irrespective of his type. If the agent picks action a1 then signal y is generated, which further updates the market’s expectation of ω. The market wage conditional on a1 is therefore  w(x ) 1 E [ ω | α(θ) = a1 , y ] = E [ ω | x1 , y ] = w(x ) 1

if y = 1, if y = 0,

where, by Bayes’ rule, w(x1 ) =

πx1 (1 − π)x1 and w(x1 ) = . πx1 + (1 − π)(1 − x1 ) (1 − π)x1 + π(1 − x1 )

(1)

It is easy to see that w(x1 ) ≥ w(x1 ) by π > 12 , and the inequality is strict for x1 ∈ (0, 1). Type θ = 1, who is sure the state is ω = 1, expects y = 1 to occur with probability π and y = 0 with probability (1 − π), hence his expected period-1 wage from action a1 is W1 (1, x1 ) = πw(x1 ) + (1 − π)w(x1 ). Type θ = 0, who is sure the state is ω = 0, expects y = 1 to occur with probability (1 − π) and y = 0 with probability π, hence his expected period-1 wage from action a1 is W1 (0, x1 ) = (1 − π)w(x1 ) + πw(x1 ). 5

Type θ = 12 expects each realization of y with 50-50% chance, hence his expected period-1 � � wage from action a1 is W1 12 , x1 = 12 [w(x1 ) + w(x1 )] . The agent’s total payoff, W + T , given his type θ and the action chosen in period 1, is summarized in the following table: Agent’s type

W + T from a0

θ=0

x0 + k

1 2

x0 + k

θ=1

x0 + 1

θ=

W + T from a1 (1 − π)w(x1 ) + πw(x1 ) + k

1 2

[w(x1 ) + w(x1 )] + 12 (π + k)

πw(x1 ) + (1 − π)w(x1 ) + 1

By 12 ≤ k < π < 1 and w(x1 ) ≤ w(x1 ), if either type θ = 1 or θ = 12 weakly prefers a0 to a1 then type θ = 0 also prefers a0 to a1 , and his preference is strict provided x1 ∈ / {0, 1}. If type 1 θ = 0 weakly prefers a1 to a0 then type θ = 2 strictly and θ = 1 weakly (if x1 ∈ / {0, 1} then strictly) prefers a1 to a0 . We are ready to state, with its proof relegated to the Appendix, Proposition 1. All equilibria are of the following forms: (i) Pooling on a0 . In period 1 all types of the agent choose a0 . Hence x0 = E[θ], and x1 is set sufficiently low (e.g., x1 = 0). In period 2 only type θ = 1 picks In. (ii) Pooling on a1 . In period 1 all types choose a1 ; hence x1 = E[θ], x0 is sufficiently low (e.g., x0 = 0). In period 2 type θ = 1 and type θ = 12 with y = 1 pick In. (iii) Countersignaling. In period 1 type θ = 0 plays a0 for sure, type θ = 1 plays a0 with positive probability, and type θ = 12 plays a1 with positive probability. In period 2 type θ = 1, as well as type θ = 12 observing y = 1, pick In. In any equilibrium of this form x1 < E[θ] < x0 . Proposition 1 implies that every separating equilibrium must involve countersignaling. In a pure-strategy countersignaling equilibrium types θ = 0 and θ = 1 pool on a0 and type θ = 12 plays a1 . The equilibrium beliefs are x∗0 = E [ θ | θ ∈ {0, 1} ] = p1 /(p0 + p1 ) and x∗1 = 12 , hence w(x1 ) = π and w(x1 ) = 1 − π by (1). Then the condition that type θ = 1 (and hence θ = 0) weakly prefer a0 to a1 but type θ = 12 weakly prefers a1 to a0 can be written as π 2 + (1 − π)2 ≤

p1 1 ≤ (1 + π − k) . p1 + p0 2

(2)

Condition (2) holds for some p0 and p1 whenever π 2 + (1 − π)2 ≤ 12 (1 + π − k). After rearranging, this is equivalent to k ≤ (4π − 1)(1 − π). Hence we have proved: Proposition 2. If k ≤ (4π − 1)(1 − π) then there exists a pure-strategy countersignaling equilibrium for some p0 , p1 ∈ (0, 1) with p0 + p1 < 1.

A necessary condition for the hypothesis of Proposition 2 to hold is that π ≤ 34 because k ≥ 12 . Condition (2) requires p0 < p1 because π 2 + (1 − π)2 > 12 by π > 12 . But p0 < p1 6

implies x∗1 = 12 < p1 /(p0 +p1 ) = x∗0 , that is, in any pure-strategy countersignaling equilibrium the average expected productivity (or short-run average wage) of agents choosing to signal must be less than that of agents choosing the uninformative action. Using the informative action to signal is a “stigma” that the medium type is willing to bear in the short run, hoping that learning more about his productivity will benefit him in the long run. This is another specific, testable prediction of our model. The pure-strategy countersignaling equilibrium is socially efficient: the medium type experiments (as he should), whereas agent types that are sure of their productivity do not. If the informative action had a small cost then this would be the only equilibrium outcome that is socially optimal – with no direct cost of a1 , pooling on a1 is also efficient. Next, we argue that under certain conditions countersignaling is the focal equilibrium outcome. Clearly, since both actions are played in a countersignaling equilibrium, it is robust to all refinements of out-of-equilibrium beliefs.4 We show that pooling on a1 is robust to stability-type refinements (such as D1, D2, NWBR), whereas pooling on a0 fails them whenever a countersignaling equilibrium exists. However, under certain conditions neither pooling outcome can be sustained in Perfect Sequential Equilibrium (PSE), hence in those circumstances countersignaling emerges as the unique robust outcome. Proposition 3. (i) Pooling on a1 satisfies D1, D2 and NWBR. Pooling on a0 fails these refinements whenever a countersignaling equilibrium exists. (ii) Pooling on a0 is not PSE. Pooling on a1 is not PSE provided a pure-strategy countersignaling equilibrium exists and πw(E[θ]) + (1 − π)w(E[θ]) < p1 /(p0 + p1 ). The proofs are relegated to the Appendix. A numerical example in which the countersignaling equilibrium is the unique pure-strategy PSE is obtained using parameter values k = 12 , π = 23 , p0 = 0.1 and p1 = 0.135.

4

Conclusions

We have analyzed the tradeoff between persuasion and learning in a simple signaling game. In the model all separating equilibria exhibit countersignaling. Such equilibria exist under certain conditions; in particular, the average productivity of extreme, non-signaling agent types must exceed that of the medium, signaling type. Moreover, countersignaling emerges as the unique pure-strategy Perfect Sequential Equilibrium outcome for a positive measure of parameter values. 4

See van Damme (1991) for the definitions and properties of equilibrium refinements.

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5

Appendix: Omitted proofs

Proof of Proposition 1. The existence of the pooling equilibria is straightforward; in what follows assume the equilibrium is different from those in parts (i) and (ii) of the proposition. First suppose θ = 0 plays a1 in equilibrium with positive probability; then W1 (0, x1 ) ≥ x0 . By W1 (1, x1 ) ≥ W1 (0, x1 ) and π > k, type θ = 12 strictly prefers a1 over a0 , hence he chooses a1 with probability 1. As a result x1 ∈ / {0, 1}; therefore W1 (1, x1 ) > W1 (0, x1 ), and type θ = 1 also strictly prefers and chooses a1 over a0 . If type θ = 0 plays both actions with positive probability then x0 = 0, and his indifference implies W1 (0, x1 ) = x0 = 0. However, W1 (0, x1 ) > 0 by x1 > 0, a contradiction. This proves that all equilibria in which type θ = 0 plays a1 with positive probability are of type (ii), i.e., pooling on a1 . In all other equilibria type θ = 0 plays a0 with probability 1, hence x0 ≥ W1 (0, x1 ). Next we characterize all such equilibria that are different from pooling on a0 . If type θ = 0 plays a0 for sure and in equilibrium a1 is played with positive probability, then type θ = 12 must play a1 with positive probability. This is so because otherwise only type θ = 1 would play a1 with positive probability, implying x1 = 1. Therefore W1 (0, x1 ) = W1 (1, x1 ) = 1 > x0 , which contradicts that type θ = 0 prefers a0 to a1 . In any equilibrium where type θ = 0 plays a0 for sure and type θ = 12 plays a1 with positive probability it must be the case that type θ = 1 plays a0 with positive probability. To see this, suppose towards contradiction that type θ = 1 plays a1 with probability 1. Then type θ = 12 must indeed strictly mix a0 and a1 (i.e., play a0 with positive probability as well) because otherwise x0 = 0 and x1 > 0, hence type θ = 0 could not prefer a0 to a1 . So, we must have x1 > 12 (because only types θ = 1 and θ = 12 play a1 ), and x0 < 12 (because only types θ = 0 and θ = 12 play a0 ). The indifference condition of type θ = 12 can be written as x0 = 12 [w(x1 ) + w(x1 )] + 12 (π − k). The first term on the right-hand side exceeds 12 iff x21 > (1 − x1 )2 by straightforward direct calculation, which holds iff x1 > 12 . From this and π > k we get x0 > 12 , a contradiction. So type θ = 1 plays a0 with positive probability. There are no more cases to check. To conclude the proof we establish x0 > x1 ; the expected value of θ falls in between being a weighted average of the two. Note that W1 (θ, x1 ) = θW1 (1, x1 ) + (1 − θ)W1 (0, x1 ) is increasing in θ and x1 , moreover W1 (x1 , x1 ) = [πx1 + (1 − π)(1 − x1 )] w(x1 ) + [(1 − π)x1 + π(1 − x1 )] w(x1 ) = x1 . By type θ = 1’s equilibrium condition we have x0 ≥ W (1, x1 ) > x1 , where the last strict inequality follows from the fact that x1 < 1. � 8

Proof of Proposition 3: (i) Consider a “pooling on a1 ” equilibrium. On the equilibrium path x1 = E[θ] ∈ (0, 1). Recall that the set of beliefs x0 such that type θ = 0 strictly prefers a0 to a1 strictly includes the set of x0 such that type θ = 12 (similarly, θ = 1) weakly prefers a0 to a1 . Therefore under D1, D2 or NWBR the market’s beliefs must put zero weight on type θ = 12 or θ = 1 deviating to a0 , implying x0 = 0, which supports the equilibrium. Consider a “pooling on a0 ” equilibrium. Here x0 = E[θ]. Recall that the set of beliefs x1 such that type θ = 12 strictly prefers a1 to a0 strictly includes the set of x1 such that type θ = 0 weakly prefers the same. Therefore, by either D1, D2 or NWBR the market’s beliefs � � must put zero weight on type θ = 0 deviating to a1 . Therefore x1 ≥ 12 , and so W1 12 , x1 ≥ 12 . � � The equilibrium condition for type θ = 12 is W1 12 , x1 + 12 (π + k) ≤ x0 + k, which implies 1 (1 + π − k) ≤ E[θ]. In any countersignaling equilibrium the beliefs satisfy x∗0 > E[θ] > x∗1 ; 2 type θ = 12 weakly prefers a1 to a0 , hence x∗0 ≤ 12 [w1 (x∗1 ) + w1 (x∗1 )] + 12 (π − k). Therefore 1 1 1 [w1 (x∗1 ) + w1 (x∗1 )] + (π − k) ≤ (1 + π − k), 2 2 2 contradicting the pooling equilibrium condition derived above. E[θ] < x∗0 ≤

(ii) Consider a “pooling on a0 ” equilibrium, so x0 = E[θ] = 12 (1 − p0 + p1 ). For each type θ let bθ be the infimum of the set of x1 such that type θ prefers action a1 to action a0 . We know that b1 < E[θ] < b0 < 1 and b 1 < b0 < 1, which in particular implies b1 < � � � � 2 θ ≡ E θ|θ ∈ { 12 , 1} = p1 + 12 (1 − p0 − p1 ) /(1 − p0 ). We now consider three exhaustive cases, and in each case construct a subset of types deviating to a1 , with average type x �1 , 5 that breaks the equilibrium according to the PSE criterion. Case 1: b0 ≤ θ. The deviating subset of agent types consists of type θ = 1, type θ = 12 , and fraction λ of type θ = 0, such that p1 + 12 (1 − p0 − p1 ) b0 = x �1 ≡ . 1 − (1 − λ)p0 Such λ ∈ [0, 1) exists because E[θ] < b0 ≤ θ. Case 2: b 1 , b1 < θ < b0 . The deviating group consists of type θ = 1 and type θ = 12 ; their 2 average productivity is x �1 = θ. Case 3: b1 < θ ≤ b 1 . The deviation coalition consists of type θ = 1 and fraction µ of 2 type θ = 12 such that p1 + 12 µ(1 − p0 − p1 ) b1 = x �1 ≡ . 2 p1 + µ(1 − p0 − p1 ) Such µ ∈ [0, 1) exists because θ ≤ b 1 < 1. We conclude that pooling on a0 is not PSE. 2

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PSE eliminates an equilibrium if a deviation by a subset of agent types benefits exactly these types provided the market believes the deviation has come from them and updates the prior accordingly. If a type in the deviating subset is indifferent then it may mix in a way anticipated by the market.

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Assume there exists a countersignaling equilibrium with beliefs x∗0 and x∗1 . Consider a “pooling on a1 ” equilibrium, so x1 = E[θ]. Suppose that types θ = 0 and θ = 1 with average productivity x �0 = x∗0 deviate to a0 . If the market’s beliefs are updated to x∗0 upon seeing a0 then type θ = 0 prefers a0 to a1 because x∗0 > x1 > W1 (0, x1 ), and type θ = 1 also prefers a0 to a1 whenever the condition in part (ii) of the proposition is satisfied. However, type θ = 12 prefers to play a1 because this type would also prefer to play a1 in the countersignaling equilibrium where the market’s beliefs are x∗1 = 12 < x1 and x∗0 . Therefore the proposed deviation indeed breaks the equilibrium according to the PSE criterion. �

Acknowledgements We thank an anonymous referee, participants at various seminars, as well as Yeon-Koo Che, Hanming Fang, Sven Feldman, Drew Fudenberg, Renato Gomes, Stephen Morris, Alessandro Pavan, Nicolas Sahuguet, Kane Sweeney, and Jeroen Swinkels for comments.

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