Strategic stability of equilibria in multi-sender signaling games

Abstract

The concept of unprejudiced beliefs equilibrium is simple: out-of-equilibrium beliefs should be consistent with the principle that multiple deviations are infinitely less likely than single deviations. Our questions are: does there always exist such an equilibrium and what can be done if there are multiple such equilibria?

To select a unique equilibrium, this notion is usually coupled with the intuitive criterion. The simultaneous usage of these concepts is ad hoc, unjustified, and again might eliminate all the equilibria.

We show that coupling these notions is legitimate, as both are implied by strategic stability (Kohlberg and Mertens (1986)), hence a desired equilibrium always exists. The intuitive criterion is trivially implied by stability. We show that in generic multi-sender signaling games stable outcomes can be supported with unprejudiced beliefs. It follows by forward induction that stable sets contain an equilibrium which is unprejudiced and intuitive at the same time.

Introduction

It is well known that in extensive form games restricting the out-of-equilibrium beliefs can eliminate equilibria which are not sensible. In this paper we investigate the usage of the extremely simple and powerful restriction of Bagwell and Ramey (1991), dubbed unprejudiced beliefs, in signaling games with multiple senders. In several applications, see e.g., Bagwell and Ramey (1991), Bester and Demuth (2015), Schultz (1996), (1999), and Hartman-Glaser and Hebert (2019), this restriction is used together with versions of the intuitive criterion (see Cho and Kreps (1987)), so as to be able to eliminate (or to justify) undesirable pooling, yet unprejudiced equilibria.¹ Some of these papers report the non-existence of pure equilibrium outcomes which can be supported both by unprejudiced and by intuitive beliefs. Unprejudiced beliefs and the intuitive criterion are seemingly unrelated concepts. Hence, their simultaneous usage, even though successful and frequent, seems to be ad hoc, unjustified and can yield to eliminate all the (pure) equilibria. Moreover, there can be undesirable equilibrium outcomes which can be supported by both types of beliefs but cannot be supported with a belief which is unprejudiced and intuitive at the same time (see our example in section 1.4).

Our question is: is it legitimate to couple these concepts? Does there always exist an equilibrium satisfying both of these concepts? Our answer and main contribution is: yes, in the sense that both are implied by strategic stability (see Kohlberg and Mertens (1986)). More precisely, a strategically stable set contains an equilibrium which is unprejudiced and intuitive (or D1 etc.).

A strategically stable set of equilibria (henceforth: stable) always exists, exhibits desirable properties and narrows down the set of equilibria. Although the consequences of stability are well understood when there is a single-sender (see Banks and Sobel (1987), Cho and Kreps (1987), and Cho and Sobel (1990)), the properties of stable outcomes of multi-sender games have not yet been analyzed.² It is obvious, by the application of forward induction, that the multi-sender versions of the intuitive criterion are also implied by stability. We show that stable equilibrium outcomes have the following additional desirable property. They can always be maintained with unprejudiced beliefs. It follows that the simultaneous usage of unprejudiced beliefs and the intuitive criterion is justified. The existence of an equilibrium (possibly in mixed strategies) satisfying both of these concepts is also ensured.

First, in sections 1.1, 1.2, 1.3 we explain by an example with economic content: the main difference between the structures of the out-of-equilibrium events in the single and the multiple sender case, what unprejudiced beliefs are, and the gist which is common in all the applications mentioned above. In this example we concentrate on pure equilibria, just as in the papers mentioned above. Pooling equilibria are eliminated solely by a version of the intuitive criterion, and then unprejudiced beliefs select a unique equilibrium among the separating equilibria. This method is successful, just because the senders are assumed to have only two types, exactly as in all the papers mentioned above.

Second, to provide further motivation and to show that something new can be done using our result, in section 1.4 we extend our previous example and allow for the possibility of a third type of the senders. We show that there is an undesirable equilibrium outcome which can be supported by unprejudiced beliefs and also by beliefs satisfying various versions of the intuitive criterion. However, none of these intuitive beliefs are unprejudiced. Hence, the outcome could only be eliminated by requiring that the supporting beliefs are unprejudiced and intuitive at the same time. Doing this elimination safely, i.e. without risking to lose the existence of a desirable equilibrium, is extremely subtle. In short, one must use our result and the full power of Proposition 6 in Kohlberg and Mertens (1986) (we discuss this in detail in section 1.4). It is not surprising that in the applications listed above the senders cannot have more than two types. Eliminating an equilibrium outcome solely on the ground of some versions of the intuitive criterion xor solely on the ground of unprejudiced beliefs is something that many authors have done already. Contrary to other scholars, we still believe that doing so is unjustified and risky without knowing our result. But eliminating an equilibrium outcome because it cannot be supported with beliefs which are unprejudiced and intuitive at the same time has never been done. Yet, using our result, this can be safely done without risking the non-existence of a desired equilibrium.

Summing up, our result ensures that one can always find an equilibrium that can be supported by intuitive and unprejudiced beliefs and provides a safe and effective tool for analyzing and solving models having more than two types.

We conclude the introduction with section 1.5 in which we restate our main contribution and describe the structure of the paper.

To understand the notion of unprejudiced beliefs and the main differences between the single- and the multi-sender case, consider a variant of the job market signaling model of Spence (1973) similar to that of in Cho and Kreps (1987). Let us start with a single worker whose marginal product t can be 0 (low) or 1 (high), with equal probabilities, and t is known to the worker but unknown to the firm. The worker chooses a level of education m, to signal his type, which changes his marginal product to $4tm$. The firm, after observing m, forms a belief about the worker's type t and pays a wage equal to his expected marginal product. On the equilibrium path beliefs must be formed using Bayes rule, out-of-equilibrium the beliefs can be arbitrary. Education is costly for the worker and its cost is given by $(3-t)m^2$. Hence, given m and that the firm's belief, conditional on m, about t is $\tau (m)(t)$, the type t worker's payoff is $4m{\sum }_{t}t\tau (m)(t)-(3-t)m^2$. It is well known that if the set of available signals is sufficiently rich, there is a unique equilibrium outcome that survives a version of the intuitive criterion (called D1) and that this is the unique stable outcome (see for example Cho and Sobel (1990)). In this equilibrium outcome the types fully separate: the low type chooses $m=0$, gets 0 wage, and 0 utility. The high type's equilibrium signal is the lowest possible signal that the low type does not want to mimic. This is the signal 4/3, and the high type gets 16/3 wage and 16/9 utility. The firm's belief about t is 0 after observing the equilibrium signal 0, and it is 1 after the equilibrium signal 4/3. For any unsent, out-of-equilibrium signal $m\in (0,4/3)$, the firm believes that the worker is of low type with high enough probability.³ This is necessary for maintaining the equilibrium. To see this, for example, should the firm assign 0 probability to the low type after observing the out-of-equilibrium signal $m=1$, the high type can deviate and achieve his first best utility level (given that the firm knows his type) $2>16/9$ by sending the signal 1.

In this section, we argue that in some situations the firm should indeed attach 0 probability to the low type at the unexpected event when observing the signal $m=1$. Assume there is another worker with the same (perfectly, positively correlated) type, with the same equilibrium behavior as described in the single sender case? Suppose the firm observes the out-of-equilibrium signal 1 from one of the workers and the equilibrium signal 4/3 from the other at the same time? Should not the firm infer that the workers are high type from the fact that one of the workers is sending the equilibrium signal 4/3 of the high type? This brings us to the concept of unprejudiced beliefs.

First, we describe the game with two senders which we derive from the game with a single sender described above. Second, we describe an equilibrium of this game in which both senders and the receiver behave exactly as in the equilibrium we discussed in the single sender case. Third, we argue that this equilibrium is not sensible. We question the plausibility of the out-of-equilibrium beliefs with which this equilibrium can be maintained. We argue that the justification of these beliefs is too complicated (or prejudiced). We show that there are other simpler (or unprejudiced) beliefs which are much easier to justify given the putative equilibrium. However, with these simpler beliefs this equilibrium cannot be maintained, it falls apart.

Hence, assume that there are two workers, husband and wife, who are assortatively matched, i.e., their types are perfectly, positively correlated which we denote also by $t\in \{0,1\}$. That is, either both have low types or both have high types. Assume that the husband's preference is the same as described above and it does not depend on the signal sent by the wife neither it depends on the wage the wife receives. Assume the same for the wife's preferences. Further assume that signals are sent simultaneously and that the firm forms a belief about t and pays each worker his and her own expected marginal product. Clearly, the firm cannot ever believe that the workers have different types. The signaling strategy profile, in which both workers signal according to the equilibrium described above in the single sender case, can be maintained as a sequential equilibrium outcome. Namely, there is an equilibrium in which the firm either observes the signal pair (0, 0) and concludes that both of them have low type or observes the signal pair (4/3, 4/3) and concludes that both of them have high type. Individual deviations from this equilibrium behavior are deterred just as in the single sender case. Namely, after any out-of-equilibrium signal pair, say for example after (1, 4/3), the firm believes that the workers are of low type with high enough probability and pays a sufficiently low wage which makes these deviations unattractive.

We do not believe in the plausibility of this equilibrium because we find the out-of-equilibrium beliefs of the firm rather strange (prejudiced). To demonstrate this, consider an unilateral deviation and suppose that the high type husband sends the signal 1 instead of his equilibrium signal 4/3, while, of course, the high type wife sends her equilibrium signal 4/3. As discussed above in the single sender case, to deter such a deviation of the husband, the firm must believe that the couple has low type with positive probability after observing the out-of-equilibrium signal pair (1, 4/3), where the first coordinate is the signal of the husband.⁴ We as many others (see the references above) question the plausibility of such a belief, which is called prejudiced, as the wife is clearly signaling that their type is high. To recapitulate, the firm cannot believe that the wife has high type and the husband has low. Types are perfectly and positively correlated. Clearly, this situation cannot emerge in the single sender case.

Unprejudiced beliefs, as defined in Bagwell and Ramey (1991), require that the firm should not attach positive probability to double deviations when an out-of-equilibrium event can be explained by a single deviation.⁵ Given the equilibrium above and given the out-of-equilibrium signal pair (1, 4/3), such a simpler, unprejudiced explanation is that the couple is of high type and only the husband was deviating. Believing the couple is of low type is prejudiced, more complicated, and less likely. For such a prejudiced belief, the firm must assume that both workers were deviating at the same time, i.e. they are low type, the husband has sent the signal 1 instead of his equilibrium signal 0 and the wife has sent the signal 4/3 instead of her equilibrium signal 0. This equilibrium can only be maintained with prejudiced beliefs and cannot be maintained by simple, unprejudiced beliefs. Hence, the restriction to unprejudiced beliefs eliminates this equilibrium.

In our example, if the signal space is sufficiently rich, the unique pure strategy stable equilibrium outcome is the same as the unique pure equilibrium outcome which survives a version of the intuitive criterion and where the beliefs are unprejudiced. This outcome is the efficient one: low types are sending the signal 0 and high types are sending the signal 1 and achieve their first best utility level (conditional on that their type is known to the firm).

Notice that just as in the single sender case, solely a version of the intuitive criterion (a multi-sender version of D1) eliminates all the pooling equilibria. We give a formal definition of an even stronger criterion in the next subsection. For now, it is enough if one thinks of a pooling equilibrium in which both players pool in pure strategies and considers a deviation of a single player. Then the intuitive criterion or D1, as defined and applied in the single sender case, directly applies in the very same way here as well. This is because the signal of the other player does not change with his or her type, hence the incentives of the different types of a sender are clearly comparable.

Now we show how unprejudiced beliefs select a unique equilibrium among the separating equilibria. Assume w.l.o.g. that the husband separates. Then the low type wife must send the signal 0, otherwise deviating to 0, by unprejudiced beliefs, she must get her first best outcome (since the firm must believe that their type is low given the low type husband's separating signal). Also, for the same reason, the high type wife must send the signal 1. Given that the wife also separates the husband's separating equilibrium signals must be also 0 and 1. To recapitulate, clearly, if one sender separates the other must choose his or her first best given that the firm knows his or her type (given that the spouse separates, unprejudiced belief dictates that the firm concentrates its belief on the true type). Hence, in the unique equilibrium low types must send the signal 0 and high type must send the signal 1.⁶ Any deviation to a non-equilibrium signal is deterred by the unique unprejudiced belief formed by using the equilibrium signal of the other player who fully separates and sends his or her equilibrium signal. Deviations to the equilibrium signal of another type are deterred by for example the lowest possible unprejudiced belief which is the belief that the senders' type is 0 with probability 1.

Consider the same game as in section 1.2 with the difference that now the senders may have 3 different types, $t=0,1$ and also 1.5, with equal probabilities. Let us also restrict the signal space of both senders to $[0,2]$. We claim that the following partially pooling equilibrium outcome can be supported with a version of intuitive beliefs and also with unprejudiced beliefs, however, it cannot be supported with beliefs which are intuitive and unprejudiced at the same time.

The equilibrium outcome is as follows: Type 0 senders send the signal 0, type 1 and type 1.5 senders pool by sending the signal 1, hence separate from type 0 senders. The firm uses Bayes rule to calculate its belief τ on the equilibrium path and payments are made accordingly. Type 0 senders obtain 0 utility, type 1 senders obtain 3 utility, and type 1.5 senders obtain a utility of 3.5. For example, for any out-of-equilibrium signal pair, choosing the lowest unprejudiced belief supports the given outcome as an equilibrium.

Notice that in any unprejudiced beliefs supporting this outcome, the firm must believe with positive probability that the type of the senders is 1 after the out-of-equilibrium signal pair (2, 1), i.e. $\tau (2,1)(1)>0$. Clearly, by unprejudiced beliefs it must be that $\tau (2,1)(0)=0$. Furthermore, if we had that $\tau (2,1)(1)=0$ then $\tau (2,1)(1.5)=1$ in which case type 1.5 husband finds it profitable to deviate to signal 2 and obtains $6=4\cdot 2\cdot 1.5-(3-1.5)\cdot 2^2>4\cdot 1\cdot 1.25-(3-1.5)\cdot 1^2=3.5$. However, such a belief does not survive the multi-sender version of the D1 criterion. We can simply compare the incentives of type 1 and type 1.5 husbands, as those types of the wife pool. Clearly, whenever type 1 husband is weakly better off by sending signal 2 relative to his equilibrium payoff, type 1.5 husband always strictly prefers to send the signal 2 rather than his equilibrium signal. Hence, there are no unprejudiced beliefs which are intuitive and support the outcome at the same time.

Yet, while believing that the type of senders is 0 after the signal pair (2, 1) is clearly prejudiced, it does satisfy the multi-sender version of the D1 criterion and deters type 1.5 husband from deviating and sending the signal 2. To see this, we invoke and tailor to our games the definition of Never a Weak Best Response (NWBR) criterion used in the single sender case in Cho and Kreps (1986).⁷

An equilibrium outcome together with supporting beliefs τ (i.e. an equilibrium) is called intuitive if for all out-of equilibrium signal pair $(m_H,m_W)$, where at least one of the signals is sent in equilibrium, $\tau (m_H,m_W)(t)=0$ if either $m_H$ or $m_W$ is never a weak best response for the husband or for the wife of type t.⁸

It is easy to see, using Proposition 6 in Kohlberg and Mertens (1986), that any stable outcome is an intuitive equilibrium outcome. Of course, not all intuitive equilibrium outcomes are stable as we will see it in the next subsection.

Our equilibrium is intuitive. For example, the firm can believe after the out-of-equilibrium signal pair (2, 1) that the type of the senders is 0, i.e. we can choose $\tau (2,1)(0)=1$. Indeed, we can set ${\tau }_H(2,0)(1.5)=1,{\tau }_W(0,1)(1)=3/4,{\tau }_W(0,1)(0)=1/4$, and one can set the rest of the beliefs arbitrarily for both ${\tau }_H$ and ${\tau }_W$ while supporting the outcome. Indeed, type 0 husband and type 0 wife both expect 0 by sending the signal 2 and the signal 1 under ${\tau }_H$ and ${\tau }_W$, respectively, and there are no profitable deviations. One can similarly justify the belief concentrated on type 0 after any out-of-equilibrium signal pair.

Can we safely eliminate this equilibrium by the fact that the equilibrium cannot be supported with beliefs which are both unprejudiced and intuitive at the same time? It is still not obvious that the above, partially pooling equilibrium outcome is not a stable outcome. The subtle point is the following. It is not trivial to rule out the possibility that stable sets contain some unprejudiced equilibria and some others which are intuitive but do not contain an equilibrium which is unprejudiced and intuitive at the same time. We argue now that this indeed cannot be the case. The trick is to apply Proposition 6 in Kohlberg and Mertens (1986). It states that a stable set contains a stable set of the game obtained by the deletion of all the strategies which are never a weak best responses. In the resulting set all the equilibria are intuitive. Applying our theorem to this smaller stable set, we can find an unprejudiced equilibrium which is also intuitive. Notice, that we used the full force of forward induction of Proposition 6 in that after deleting NWBR strategies we still find a stable set. It follows that eliminating an equilibrium outcome with the combination of these concepts is safe, and that in any stable set one can find such a desired equilibrium. The partially pooling equilibrium outcome can be then eliminated by simply observing that no unprejudiced equilibrium is intuitive. On the one hand, there is no need to test and compare the incentives of types which are outside the pool because the belief set at (2, 1) is irrelevant for the incentives of these types. On the other hand, testing and comparing types from the pool is just as simple as in the single sender case.

After elimination of the partially pooling equilibria, one can select again the efficient, non distorted equilibrium which is the unique pure strategy equilibrium with unprejudiced and intuitive beliefs. The strategies of types 0 and 1 are just like in the two type case, and types 1.5 send the signal 2 which is their first best signal given that the receiver knows their type as required by unprejudiced beliefs. Any deviation to a non-equilibrium signal is deterred by the unique unprejudiced belief formed by using the equilibrium signal of the other player who fully separates and sends his or her equilibrium signal. Deviations to the equilibrium signal of another type are deterred by the lowest possible unprejudiced belief ($\tau (2,1)(1)=\tau (1,2)(1)=1,\tau (0,1)(0)=\tau (1,0)(0)=1,\tau (2,0)(0)=\tau (0,2)(0)=1$).

Neither consistency nor the intuitive criterion nor forward induction or properness necessarily imply unprejudiced beliefs, but we show that stability does.

The converse is not true. First, in pooling equilibria any belief is unprejudiced. Second, these pooling outcomes are not stable in our example or in the applications mentioned above, once the signal space is sufficiently rich. This is exactly the reason why in the applications cited above unprejudiced beliefs have to be coupled with a version of the intuitive criterion, i.e., to be able to eliminate pooling equilibria.

By showing that stable outcomes can be supported with unprejudiced beliefs and observing that stable outcomes satisfy the various versions of the intuitive criterion, we can justify the coupling of these concepts. It follows that instead of working with the complicated machinery of strategic stability one can reach sharp predictions in games by the safe and simultaneous application of these simple concepts, even if there are more than two types of the senders. We repeat, in our example with three types above one can select again the efficient, non distorted equilibrium which is the unique pure strategy equilibrium with unprejudiced and intuitive beliefs. The strategies of types 0 and 1 are just like in the two type case, and types 1.5 send the signal 2 which is their first best signal given that the receiver knows their type as required by unprejudiced beliefs.

In our example, types are perfectly correlated. If the prior probability that the type of the husband is low or the type of the wife is high and vice versa is positive, then consistency already implies unprejudiced beliefs. We prove (see our Proposition 1) that any sequential equilibrium has unprejudiced beliefs if the prior has full support. For general supports, when some type profiles have 0 probability under the prior this is not the case. Sequential or even proper equilibria well can have prejudiced beliefs.

In our Theorem 1, we show that in generic multi-sender signaling games every stable outcome can be supported by unprejudiced beliefs.⁹

The paper is structured as follows. In Section 2, we define finite multi-sender signaling games, sequential equilibria, and its refinements: unprejudiced sequential equilibrium and stable sets of equilibria, and we also state Proposition 1 and Theorem 1. In Section 3, we provide the proof of Theorem 1. In section 4 we conclude and give a hint how and to what extent our result can be directly generalized for arbitrary extensive form games with perfect recall and generic payoffs.