Abstract
We model aggregate information release, in a dynamic setting with random matching, as a conscious, preference-driven choice. We introduce a “planner”, who possesses and selectively reveals aggregate information. Aggregate information is gathered slowly, by taking small samples from the population, and can only be revealed after the dynamic process has stabilized. By selectively revealing information, the planner may upset a given self-confirming equilibrium, in order to achieve a preferred outcome for him. Hence, some self-confirming equilibria are “unstable” relative to public information release. We show that only equilibria supported by heterogeneous beliefs can be information-unstable. We provide several real-life examples of manipulation by means of public information, showing the relevance of the theoretical analysis.
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Notes
In such games, the steady states of the dynamic recurring interaction, with no strategic links across repetitions of the interaction, have a close relationship with the equilibria of the static game (Fudenberg and Levine 1993b).
The word “personally” is very important here, since members of the same population may have different experience, hence different beliefs. Individuals do not necessarily share the knowledge that other members of their group have acquired by interacting with other social groups. Battigalli (1987), and Kalai and Lehrer (1993) also introduced concepts similar to self-confirming equilibrium, but they ruled out this type of heterogeneous beliefs within populations.
This is especially relevant in modern societies, where the media can easily convey public information. The public availability of aggregate data might depend on the incentives for disclosing them.
When officials do not cooperate, they illegally try to expropriate rents from the investors.
Strictly speaking, this is not the state of the system, because a specification of beliefs is also necessary, in order to determine the future evolution of the system. We will use the same convention as Fudenberg and Levine (1998), calling \(\sigma \) “the state”.
This is captured by officials’ negative payoffs from expropriating rents. Their pure monetary payoffs might be positive, but their overall utility is negative.
We take the costless knowledge of aggregate statistics by the planner as given. Our setting can easily take into account costs of aggregate information acquisition, but this would not change the main results.
They thus provide a specific example of a “planner” and how he selectively reveals information about the aggregate data to maximize his objective value.
The experimental literature has also addressed the issue of whether revealing aggregate information matters, and whether expectations can be manipulated. Roth and Schoumaker (1983) and Harrison and McCabe (1996) directly manipulated subjects’ expectations about others’ play in an ultimatum game, with significant and lasting effects. Berg et al. (1995) and Ortmann et al. (2000) performed experiments of one-round trust games, and found some support for the notion that information revelation of aggregate data can push the economy to desirable equilibria.
By the term “individual” or “agent”, we shall refer to a particular person who belongs to some population. On the contrary, the word “player” will denote a whole player-role (corresponding to a population of individuals).
This captures the fact that our social interactions are anonymous. A individual need not know, and need not have strong beliefs about, the payoff functions of other individuals that belong in any population (even her own population). For example, an individual official in our introductory example need not know whether other officials share his preferences. He might believe that other officials are corrupt, so they get a higher payoff by not cooperating. In general, learning models emphasize the fact that agents form beliefs by observing others’ behavior, rather than by introspection.
Note the specific interpretation of mixed strategies here. Each individual is assumed to play a pure strategy, but each population as a whole randomizes across strategies, since individuals in the same population may be choosing different pure strategies.
We need to emphasize from the beginning that we assume that individuals do not infer the behavior of others based on the behavior of the planner. The reason is that individuals need not know that this type of selective information revelation is taking place. Even if they do, they do not know or have strong beliefs about the objective function of the planner. This is fully consistent with the overall learning foundation underlying our model and the literature on self-confirming equilibrium.
This can be thought as a benchmark case for analysis. Our key insights would not change if we assume that a given fraction \(\alpha \) of each subgroup believes the planner’s announcements, and another fraction \( 1-\alpha \) ignores the announcements. Clearly, the quantitative results depend on the parameter \(\alpha \), but the qualitative ones carry over if we assume that only some people believe the planner, so that \(\alpha \) is not zero. The assumption that the planner is credible is more convincing in some real economies, such as advanced democracies, than others, such as totalitarian regimes. Note that by always selectively revealing true information, the planner can also develop a reputation for truth-telling.
In other words, only a sample gathered over many periods can be released, but if behavior changes in every period, such a sample cannot be representative of true behavior. Therefore, only information that describes behavior in a steady-state may be released.
If the best responses do not constitute a self-confirming equilibrium, it is not clear where the system will stabilize.
In our analysis, we will use the following assumption. If, after public information is released, some individual is indifferent between his pre-revelation strategy and a different one, she adheres to what she was doing before the information was revealed. In a similar spirit, in the dynamic learning processes at the background of our analysis, we visualize each agent as unwilling to shift actions, unless there is a strict incentive to do so.
In the absence of this condition, one could find examples of mixed Nash equilibria that would qualify as unstable, where it is not clear why aggregate information would change anybody’s behavior.
More formally, an argument such as the one used in Proposition 6.3 of Fudenberg and Kreps (1993) (p. 345) can be made. In particular, players maintain their transition beliefs until “sufficient evidence” against it accumulates. Then it can be shown that an arbitrarily small probability exists such that after the “sufficient evidence” accumulates, the resulting beliefs will differ more that some given \(\epsilon \) from a convex combination of the transition beliefs and beliefs that assign probability one to the mixed actions induced by \(\sigma ^{*}\) in \(\overline{H}(s_{i}^{*},\sigma _{-i}^{*})\) and by \( \sigma \) outside \(\overline{H}(s_{i}^{*},\sigma _{-i}^{*})\).
Note that this is just one of infinitely many self-confirming equilibria in this game. Any purely mixed strategy of player \(1\), coupled with pure strategy \(C\), is a self-confirming equilibrium profile (and so is the pure profile \(\{E,C\}\)). We use this specific numerical example only for concreteness.
Note that this example represents a special case, where beliefs in the periods after the announcement are exactly a convex combination of transition beliefs and beliefs that correspond to the “final” profile. In more general examples, where this need not be true, a similar argument can still be valid, as explained in footnote 19.
This is true because each information set on the equilibrium path is reached by all individuals of all populations.
We also assume that for any population \(i\), each individual’s beliefs \(\mu _{i}\) are such that, for all \(h \in \overline{H}(\sigma ) \bigcap H_{-i}\), \(p(h/\mu _{i},s_{i})>0\) for some \(s_{i} \in S_{i}\). This rules out the possibility that a player could infer something about behavior in an opponents’ node about which no information is revealed, by the mere fact that another node is reached with positive probability.
This arguably restrictive assumption is made because of its great benefit in terms of simplicity. If information regarding only a subset of the family of a player’s information sets were revealed, then others’ beliefs for the remaining information sets of that player would have to be updated. This leads to several complications, especially problems involving updating when probability zero events occur. Since the main message of this paper does not depend on our simplifying assumption, we prefer to avoid these complications.
Note that we assume that all individuals in a given subgroup choose the same best response, even if there exist multiple best responses.
Note, however, that the ensuing profile of best responses does not constitute a self-confirming equilibrium.
This example illustrates our assumption that individuals do not predict others’ behavior based on a priori knowledge of their payoffs. If individual \(1\)’s and \(2\)’s knew that the distribution of actions is common knowledge, and also knew \(3\)’s payoffs, they could infer the change in \(3\)’s behavior, but they do not do so in our setting.
Notice that half of individual \(1\)’s who “pass”, according to this profile, believe that \(2\)’s “take” with probability \(\frac{3}{4} \) and the other half believe that \(2 \)’s “take” with probability \(\frac{1}{2}\). As these agents interact with player \(2\)’s (and assuming that the \(2\)’s will all keep playing “pass”), they will gradually adjust their expected fraction of “\(P_{2}\)” upwards, towards \(1\). Notice that for the range of beliefs between their old and new ones, their best response does not change. Moreover, individual \(3\)’s believe that \(1\)’s and \(2\)’s pass with probability \( \frac{1}{2}\), and they will gradually adjust their expectations regarding \(P_{1}\) and \(P_{2}\) upwards. Again, \(3\)’s immediate best response to the announcement is clearly optimal throughout the adjustment process.
Assume that groups \(1\) and \(2\) correspond to producers who could invest (\(P_{1},P_{2}\)) or not invest (\(T_{1},T_{2}\)), group \(3\) to criminals who could steal (\(P_{3}\)) or not (\(T_{3}\)), and group \(4\) to the police who could punish crime (\(T_{4}\)) or shirk (\(P_{4}\)). The State would like to reveal the fact that crime is not rampant, but not to reveal the inadequacy of the police, which would induce more crime.
For an anecdote, according to a Dutch journalist, there is an implicit agreement in the Dutch press to refrain from overemphasizing the occurrences of sports violence and hooliganism, since hooligans are often “proud” of their violent acts.
An astute reviewer correctly pointed out that although the endogeneization of information revelation by examining the decisions of a planner who selectively reveals aggregate information is key a idea behind this paper, in fact information is not really endogenous here. The objective of this paper is to use this underlying idea in order to derive a refinement of self-confirming equilibrium in an environment with a high degree of generality (unlike the approaches of Esponda 2008 and Jehiel 2011). The important problem of modeling strategic information revelation—and thus of truly endogeneizing it—requires a more specialized environment in order to be addressed.
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Acknowledgments
This paper is based on Chapter 2 of my doctoral dissertation at UCLA. I am particularly grateful to David K. Levine for his incredible support and guidance throughout this project. I greatly benefited from lengthy discussions with Bill Zame, Jean-Laurent Rosenthal and Paolo Battigalli. I am also indebted to seminar participants at UCLA, U Bocconi, U of Maastricht, U of Granada, the University of Athens and participants at the La Pietra-Modragone workshop at the EUI. Two anonymous referees and an associate editor greatly contributed with insightful comments. During this project, I received financial support from the Greek State Scholarships Foundation (IKY) and the Cournot Centre for Economic Research.
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Maniadis, Z. Selective revelation of public information and self-confirming equilibrium. Int J Game Theory 43, 991–1008 (2014). https://doi.org/10.1007/s00182-014-0415-0
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DOI: https://doi.org/10.1007/s00182-014-0415-0