EX-Ante Information Heterogeneity in Global Games Models with Application to Team Production

1 Introduction

Since the seminal work by Carlsson and van Damme (1993) and Morris and Shin (1998), global games models have been a widely used tool to explain various economic circumstances with strong complementarity such as bank-run, financial crisis, and collapse of exchange rates. This class of models is particularly characterized by incomplete information on the fundamental value, denoted by θ, which is introduced in the form of dispersed information; instead of directly observing the fundamental value, each player is assumed to receive private information (and/or public information) with a noise, which yields ex-post information heterogeneity across the players. In reality, however, there also exists ex ante information asymmetry (or inequality) across players; some players possess more precise information and/or they have comparative advantages in processing additional information than others. In addition, information inequality might have become widened in the last few decades as it is associated with income inequality (Kennedy and Prat 2019), raising the importance of considering information inequality in the analysis. The extent to which this feature influences the equilibrium properties, however, has not yet been extensively studied in the context of global games. For instance, whether a more-informed agent is more aggressive/active in action than a less-informed one is an unexplored but important question from the perspective of a policymaker who would like to keep the status quo. Team production is another example: A worker might have different information on the success probability of the project from other workers, and this might result in coordination failure. Manager of the firm would then need to implement a strategy to avoid the coordination failure; suppose that the manager can control the precision of private information à-la Moriya and Yamashita (2020). In this case, what would be the solution to optimal information allocation problem in the context of global games? This paper aims to add to the literature by addressing the above issues.

Toward that end, we assume that there are two types of players in the game, especially in the form of a regime attack game; the first type is an “expert type,” who receives an informative private signal with precision α x e > 0 . The other type is an “inexpert type,” who receives a private information with precision α x i > 0 . In order to introduce ex ante heterogeneity in information, we further assume that α x e = ( 1 + μ ) α x i > α x i with μ > 0 so that experts receive more informative private information. This captures the information asymmetry across the market participants. In this regard, three papers are closely related to our paper. First, James and Lawler (2012) consider a coordination game with two types of players who have different private information. The major difference between our work and theirs is that we consider a model with strong complementarity that yields multiple equilibria while they consider a coordination game with weak complementarity and hence there is no such an issue. Second, Corsetti et al. (2004) and Bannier (2005) consider global games with two types of players. In contrast to our framework, their main consideration was the size, not information heterogeneity; they study the role of “large investor” in determining the equilibrium properties while we assume that each type consists of infinitesimal agents. This enables us to isolate the informational effect from the size effect. While the symmetric approach introduced in Bannier (2005) yields a similar equilibrium outcome to ours, our work is differentiated from theirs mainly by providing conditions under which equilibrium outcome alters when informational gap changes. In this sense, this paper complements and extends the insights of Bannier (2005) to an environment in which none of the players have advantages in size but in information. Furthermore, we apply the equilibrium properties to a different context, optimal information allocation problem.

We first prove the important property of this class of models^[1] that there exists a unique symmetric perfect Bayesian threshold equilibrium (Proposition 1). In doing so, we first show that there are two threshold levels of private signals under which each type attacks the regime when he/she receives a signal below the thresholds. In addition, we argue that there exists a unique threshold level for the fundamental value under which the regime collapses when the fundamental is lower than the threshold value.^[2] With these two properties, it is proved that there exists a unique symmetric perfect Bayesian threshold equilibrium when private information of inexpert type (and hence expert type) is sufficiently precise, which nests Morris and Shin (1998) as a special case.^[3]

The main result, Proposition 2, shows that less-informed players’ behaviors are more extreme than more-informed players’ in two plausible and tractable cases. We call the environment with (1) low cost of effort (or attacking) and (2) low realized public signal, which the players observe, “low-cost low-p” environment and focus on this case. Predictions from the opposite case (“high-cost high-p”) are exactly the opposite.^[4] We first prove that the threshold level for more-informed worker is lower than that of less-informed worker in the low-cost low-p environment. As greater threshold value enlarges the range of private signal under which the worker puts effort, whether more-informed worker puts more effort or not crucially depends on the status of the underlying parameter values. If public signal is sufficiently low, all workers perceive the chance of success to be high because the fundamental is likely to be weak. Less-informed workers would rely more on this public information than the more-informed ones, and hence their propensity to attack the regime increases as the cost of attacking is low.

We then exploit the virtue of the equilibrium uniqueness by establishing comparative statics. In particular, we consider team production problem following Moriya and Yamashita (2020) as an application of the general model discussed above: The economy consists of two types of workers: Each worker is either more-informed (expert type) or less-informed (inexpert type) on the success probability of the team production. A worker either works hard (a _i = 1) or not (a _i = 0) with i denoting each worker and the project in which both workers are involved succeeds when aggregate effort (A = ∫a _i di) is greater than a threshold level (θ). In addition, there exists a manager who would like to exert the best efforts from workers to maximize A. The manager can control the information on the success probability of the project to both types.

In this environment, greater differences in private information between the more- and less-informed workers, measured by greater μ, lowers the threshold value regardless of the type in the low-cost low-p environment (Proposition 3). Hence, both types become less active, resulting in low probability of project success. This extends the finding by Metz (2002) to a situation in which only a fraction of players in the economy receives unevenly better private information, which improves the average precision of private information in the economy. This result is preserved when μ is fixed while the measure of more-informed workers becomes greater. This finding provides an interesting policy implication: A policymaker has a policy tool to keep the status-quo in addition to previous policies deciding only the extent of agents receiving better information: In the low-cost low-p environment, for example, it might exaggerate the informational gap across the agents by providing sophisticated information without changing the portion of more-informed workers. It only enhances precision of more-informed workers’ private information and lowers the success probability.

We finally characterize the conditions under which the manager chooses an extreme information structure: In our model setup, it is optimal to provide all the workers with no-information (resp. full-information) when public information takes significantly low (resp. high) values. This implies that symmetric information allocation can be optimal, which is a finding different from Moriya and Yamashita (2020) who prove that full-information is never optimal while no-information and asymmetric information allocation can be optimal. In this regard, our finding confirms the idea that more work might be necessary to derive the optimal information allocation strategy along different dimensions of games (Moriya and Yamashita 2020). Moreover, our finding implies that the extreme information structure can hold even if the information designer does not assume the worst scenario (Inostroza and Pavan 2022; Morris, Oyama, and Takahashi 2022).

2 The Model

In this section, we introduce the main model, the regime attack game, which is an extended version of Metz (2002).^[5]

There are players who are uniformly distributed on [0, 1]. Each player maximizes his/her expected utility. In particular, each player i chooses either to attack the status-quo (denote as a _i = 1) or not to attack (denote as a _i = 0) and utility function is defined as

where 1 is an indicator function that takes the value of one (resp. zero) when the size of aggregate attack (A = ∫a _i di) is greater (resp. lower) than the fundamental value (θ). We assume that θ is drawn from an improper distribution and is not directly observed by each player. c ∈ (0, 1) is the cost of attacking.

Instead of directly observing θ, each player receives a (noisy) public signal and a (noisy) private signal. Public signal is described as follows.

where α _p > 0 denotes precision of the public signal and ɛ ∼ N(0, 1).

Differently from the previous literature, we assume that there is an ex ante heterogeneity in private information across the players. In doing so, we assume that players are divided into two types; an expert type with measure λ ∈ [0, 1] and an inexpert type with measure 1 − λ.

The inexpert type, distributed on (λ, 1], is a player who is not specialized in accessing information on the fundamental.^[6] One can think of small investors participating in coordination games, who do not have superior private information on the market. This type receives the private signal with precision α x i > 0 :

where η _i ∼ N(0, 1) and η _i  ⊥ ɛ.

Expert type, distributed on [0, λ], denotes players who care more about economic situations and hence process more information on the fundamental. This type might include full-time investors and traders in the financial firms such as investment banks: Since income of these players rely more on the outcomes from the coordination game, this type would receive more private information. Or alternatively, one can consider a situation in which some players have better skills to exploit valuable information from the common sources of information; they might have greater capacity to process information (Sims 2003) than others, resulting in better private information. Denoting precision of private signal received by this type as α x e > 0 , private signal can be described as follows:

where η _e ∼ N(0, 1) and η _e  ⊥ η _i  ⊥ ɛ.

The following equation captures the idea that this type receives more precise private information than the inexpert type:

where μ > 0 measures the information discrepancy between the two types.

Notice that this model nests Morris and Shin (1998) and Metz (2002) as special cases when μ = 0. Importantly, while the size of the expert type is fixed as λ, the measure of this type who chooses to attack the status-quo is not exogenously determined, which makes our model be distinguished from Corsetti et al. (2004) and Bannier (2005). In their model, a size of the particular group (large investor) is predetermined and hence there is no strategic uncertainty around the size of the attack from the group. On the contrary, this feature, which is an important characteristic of this class of models, is preserved in our model.

3 Equilibrium Characterization and its Uniqueness

To be consistent with the previous literature (Bannier 2005; Metz 2002 as examples), we consider a symmetric perfect Bayesian threshold equilibria in this paper.^[7] In particular, inexpert (resp. expert) type would attack the regime if and only if the realization of her private signal is less than some threshold x i * (resp. x e * ). In this section, we first describe how the threshold values are determined and then provide a sufficient condition for equilibrium uniqueness of the symmetric perfect Bayesian threshold equilibria. Since proof techniques we exploited are well established in the literature, proofs are relegated to the Appendix. Under the assumption that all players take threshold strategies, we can first prove the following lemma.

Lemma 1 hence determines the threshold θ*: The regime will survive only when its fundamental value, θ, is sufficiently high. We first note that this is a generalization of the previous findings with ex ante homogenous players (Morris and Shin 1998). Importantly, the above lemma is distinctive from that of Corsetti et al. (2004): There are two threshold levels for θ in their model because the large investor can succeed in the attack even when there is no small investors who choose to attack. As a result, one additional threshold level would be obtained in the equilibrium. On the contrary, there also exists a strategic uncertainty among the expert type in our model framework. Hence, from the perspective of each individual player, it is required to consider one threshold level for θ when making a decision.

The equilibrium should ensure that two indifferent conditions under which both inexpert- and expert type would be indifferent between attack the regime and not to attack are met, which is summarized by Lemma 2.

Hence, Lemma 2 is again a generalization of the previous results with only one type of players. With above lemmas, we can prove the following proposition.

In the sense that the player attacks the regime only when her private signal is lower than the threshold level, we can say the player becomes more active (resp. passive) in attacking the regime when the threshold level increases (resp. decreases). With better private information, would the expert be more active or passive than the inexpert type? The following proposition, which is the main result of our paper, summarizes the answer to this question.

In order to obtain not only interpretable results but also an economic intuition behind the above proposition, we consider a case in which (1) cost of attacking is low ( c ∈ [ 0 , 1 2 ) ) and (2) the realized public signal is low enough (p ≤ θ*), which is abbreviated as a “low-cost low-p” environment.

All players perceive the chance of regime collapse high when public signal is sufficiently low. With low cost of attacking, an inexpert type would rely more on this public information than the expert type and hence it increases their propensity to attack the regime compared to the expert type, resulting in the inexpert type to be more active.^[8] In contrast, the expert type knows more accurately about the fundamental. This makes them to rely less on public information than the inexpert type does even when p is sufficiently low. As a result, smaller size of expert type is required to make the regime collapse (in the “high-cost high-p” environment, on the contrary, there is an incentive for the expert type to act like a first penguin to enable regime collapse before the inexpert type based on more accurate private information, which is impossible without their action). This is an interesting result that provides potential implications on the market behavior of investors in the sense that informational advantage does not make the expert type always more or less aggressive/active than the inexpert type: More precise private information makes this type to be less dependent on the underlying fundamental (both of cost structure and the public signal) than the inexpert type. Hence, its position at the market (game) would be less volatile than that of the inexpert type, which can be possibly tested with the empirical data.

4 Comparative Statics: Application to Team Production

In this section, we study how the changes in key parameter values affect the equilibrium of this economy under the assumption that unique equilibrium is achieved. In other words, Proposition 1 is assumed to be satisfied. In order to highlight new findings due to ex ante information heterogeneity, we present findings that extend previous results in the Appendix A.

We particularly consider a team production model à-la Moriya and Yamashita (2020) as an application of the general model discussed in the previous section. The economy consists of two types of workers who work together to make a project succeed: Each worker is either more-informed (expert type) or less-informed (inexpert type) on the success probability of the team production. A worker either works hard by providing enough efforts (a _i = 1) or not (a _i = 0) when i denotes each worker and the project in which both workers are involved succeeds only when the aggregate effort (A = ∫a _i di) is greater than a random variable, θ, which is unknown to the workers. Providing effort is costly (c > 0). In addition, there exists a manager who would like to exert the best efforts from workers to maximize A. The manager’s problem is to control the information on the success probability of the project to both types or one type or not in order to maximize the success probability of the team production.

We first address the extent to which ex ante information heterogeneity affect the success probability in Proposition 3, which is the second main result of our paper:

In order to understand the above proposition, we first focus on the effect of information gap (μ) on the threshold level of the more-informed worker using equation (14) in the low-cost low-p environment ( c ∈ [ 0 , 1 2 ) and p ≤ θ*). Given fixed θ*, lower μ increases x e * . This is because lower private information makes the more-informed worker to rely more on public information, which signals that θ is likely to be low. Then the size of the aggregate effort increases, which will increase θ* at the equilibrium. Hence, as more-informed one loses its superiority in private information, the likelihood of success would surge since both types rely more on the public signal that serves as a coordination device.

The effect of λ on θ* can be understood by considering Proposition 2. In the low-cost low-p environment, x e * < x i * , implying that less-informed worker tends to put more effort. With lower λ, numbers of workers putting efforts increase while that of non-working workers decreases, resulting in high θ*.

We now characterize the extent to which λ and μ affect the threshold levels of private signals as presented in Corollary 1.

The above finding provides a policy implication to keep the status-quo in addition to a usual policy to provide unclear public information in the low-cost low-p environment (Proposition A3): As the success probability becomes higher when (i) informational gap between the agents and/or (ii) the numbers of the expert type is small in the low-cost low-p environment, the manager (policymaker) might increase the success probability by narrowing the information gap or lowering the number of more-informed workers. For instance, this could be achieved by providing more education on less-informed workers, which can result in smaller information gap. In the high-cost high-p environment, on the contrary, the manager might impose the opposite policy. This discussion leads to the following Corollary on the optimal information allocation problem, which is faced by the manager.

From the perspective of the manager, Corollary 2 suggests that her optimal information allocation decision should be to make all the workers identical with respect to information. Since the opposite holds in the high-cost high-p environment, what is suggested to the manager is to choose an extreme information structure in some stylized parameters’ region: The manager offers any workers no further private information in the low-cost low-p environment, while she offers every player perfectly precise private information in the high-cost high-p environment.^[9] Such extreme information structure is also observed in Inostroza and Pavan (2022) and Morris, Oyama, and Takahashi (2022). They prove that the optimality and existence of binary signal structure which makes all players attack or not when the information designer assumes the worst scenario that agents choose a suboptimal action for the designer. While the manager in this paper does not assume the worst scenario, we show that extreme information structure can still be optimal. In this regard, our finding contributes to the literature on information design by proving that the extreme information structure robustly holds in global games. This finding is also important in the sense that it complements Moriya and Yamashita (2020): They show that providing full information to workers is always suboptimal. However, Corollary 2 implies that there is an equilibrium in which full information can also be optimal when both effort cost and public signal are substantially high.

5 Conclusions

In this paper, we extend the typical regime attack game to exhibit ex ante information heterogeneity across players and characterize optimal information structure encouraging more players to attack the regime. Importantly, we prove that there exists a unique threshold equilibrium in this economy. This uniqueness result allows us to do comparative statics in the context of team production: In particular, we analyze the extent to which ex ante information heterogeneity (numbers of more-informed workers and information gap between the two types) influences the equilibrium result.

Our setting provides a tool to understand the behavior of small investors in the market, who possess less precise private information than professional traders. According to our model, informational superiority does not fix the position of the investor; depending on the market environment, small investors can be more or less active than professional ones. For instance, the rally of the stock price indices, including GameStop, in early 2021 were lead by the inexpert type, which can be interpreted as an attack to the regime in our setup. We also provide a tool to understand the optimal information allocation problem in the context of team production model. We leave more applications of our model to different and interesting circumstances as future works.