Communication and behavior in organizations: An experiment

Abstract

We design a laboratory experiment to study behavior in a multidivisional organization. The organization faces a trade‐off between coordinating its decisions across the divisions and meeting division‐specific needs that are known only to the division managers, who can communicate their private information through cheap talk. While the results show close to optimal communication, we also find systematic deviations from optimal behavior in how the communicated information is used. Specifically, subjects' decisions show worse than predicted adaptation to the needs of the divisions in decentralized organizations and worse than predicted coordination in centralized organizations. We show that the observed deviations disappear when uncertainty about the divisions' local needs is removed and discuss the possible underlying mechanisms.

1

Introduction

Coordination problems play a central role in organizations. Firms coordinate production decisions across divisions, districts in federal systems coordinate policies, and NGOs coordinate their decisions across countries. Often, such problems are complicated by privately known motives of the decision makers. Two division managers attempting to coordinate their business strategies, for instance, might have incomplete knowledge of each other's goals. When this is the case, coordination can be facilitated by a communication channel between the managers, such as that established in General Motors by Alfred Sloan in the 1920s (Alonso, Dessein, and Matouschek ()).

While the manner in which private information is communicated and used to coordinate decisions has been explored in recent theoretical work, key predictions of these models remain to be tested. The present paper uses a laboratory experiment to provide a first attempt, focusing on the question below:

Main Question

What effect does the structure of an organization have on (i) how precisely private information is communicated and (ii) how the communicated information is used?

Following Alonso, Dessein, and Matouschek (), the experiment makes use of two types of organizational structures, centralized and decentralized, operationalizing them as simple coordination games. A decentralized game is played between two agents, with a single decision to be made by each. An agent has private information about her local conditions, which affect the payoff the agent receives from her own decision. She incurs an adaptation loss if her decision fails to adapt to her local conditions, and a coordination loss if her decision is not perfectly aligned with the decision of the other agent, therefore, facing a trade‐off between adaptation and coordination. The agents can communicate with each other before making their decisions.

In a centralized game, decision rights are delegated to an unbiased coordinator, referred to as the principal, who maximizes joint profits and is uninformed about both local conditions. The agents can communicate their private information to the principal before the decisions are made. Because the answer to our main question above in theory depends on the size of incentives to coordinate, the experimental treatments independently manipulate the structure of the game (centralized vs. decentralized) and the importance of coordination (high vs. low) for individual payoffs.

Our results show that models in the organizational economics literature capture key features of how subjects communicated in the experiment but provide an incomplete explanation of how the communicated information was used. Furthermore, the direction in which subjects' behavior deviated from the theory was closely tied to how authority was allocated within the game. Thus, the principal behaved as if the importance of coordination was smaller than it actually was, that is, focused too much of her efforts on adapting to the agents' privately known states, while the agents focused too much of their efforts on coordination. We summarize these deviations from the theory as follows:

1

Main Result

The importance of coordination was overweighted by the agents under decentralization and underweighted by the principal under centralization.

Estimating the effect of the observed over and underweighting on payoffs, we find that $94\mathrm{\%}$ of the difference between the optimal and the observed losses can be explained by the distortions in decision rules, with the remaining $6\mathrm{\%}$ due to communication. This is our second main result:

2

Main Result

Most payoff losses were due to distortions of decision rules rather than miscommunication.

Our starting point in explaining the observed distortions is the observation that uncertainty enters the players' payoff functions differently under centralization and decentralization. Under decentralization, adaptation involves no uncertainty (since own states and decisions are known), while coordination involves the other agent's potentially uncertain decision. Under centralization, coordination involves no uncertainty (since both decisions are known), while adaptation involves the two agents' unknown states. Thus, one way to interpret Main Result above is that the subjects overweighted the uncertain part of their payoff functions in all treatments.

To test the hypothesis that the observed deviations were driven by uncertainty, we use data from additional treatments with complete information and unique equilibrium predictions that were otherwise identical to their counterparts in the first stage of the experiment. Consistent with the hypothesis, we find no significant distortions in decision weights on average in these additional treatments: 3

Main Result

With complete information, there was no over or underweighting of the importance of coordination on average.

Section of the paper provides several possible channels for how uncertainty might have led to the observed deviations from equilibrium behavior. We show there that the results are consistent with ambiguous communication in the presence of ambiguity‐averse message receivers as well as a simpler explanation based on gift‐exchange. We also argue in Section that the observed deviations from equilibrium behavior were unlikely to be caused by social preferences or risk aversion.

Our paper makes several methodological contributions to the experimental cheap talk literature. First, we elicit subjects' beliefs about their matched subjects' states and use the elicited beliefs to construct an empirical counterpart to the residual variance of communication, a measure commonly used in theoretical work. This allows us to formulate predictions about how well subjects communicate without relying on assumptions about how they do it. Second, we use the elicited beliefs together with the equilibrium decision rules to study how subjects decide conditional on the communicated information. This allows us to test theoretical predictions about subjects' decision rules directly. Third, we use the elicited beliefs to perform a detailed payoff analysis that decomposes subjects' losses into a component due to miscommunication and a component due to deviations from equilibrium behavior.

The closest experimental study to ours is Brandts and Cooper (). While they also compare centralized and decentralized coordination games, they do not investigate the role of communication in coordinating multiple decisions. Moreover, the games they use are different from ours. Specifically, the agents are symmetrically informed about each others' local conditions as well as a global state of the world, which affects the payoffs of each player, while the principal is uninformed about the global state but informed about the agents' local conditions. Unlike Brandts and Cooper, we focus on communication of private information and its effect on coordination.

Our paper also contributes to the experimental literature investigating strategic information transmission in the spirit of Crawford and Sobel (). Most of this literature has focused on one sender‐one receiver games, with more recent work investigating the case of multiple senders (Vespa and Wilson ()). While we also consider the case of multiple senders, our focus is on using communication to coordinate multiple decisions as opposed to information aggregation.

The implications of uncertainty in coordination problems have only recently begun to be studied in communication games with incomplete information. We show experimentally that uncertainty biases subjects' decision rules in a manner that depends on how authority is allocated within an organization, and our results suggest a promising direction for future work.

2

Experimental design

Our experimental design is based on the models of Alonso, Dessein, and Matouschek () and Rantakari (). Every treatment of the experiment has two players, 1 and 2, and two decisions, $d_1\in D$ and $d_2\in D$, to be made. The set D is a discretization of the interval $[-1,1]$ in increments of 0.01; that is, $D=\{-1,-0.99,-0.98,\dots ,0.98,0.99,1\}$. The payoff of Player $i\in \{1,2\}$ is given by [Image Omitted. See PDF] where ${\theta }_i$ is Player i's state, or local conditions. The first component of the payoff function captures the adaptation loss arising from the mismatch between $d_i$ and ${\theta }_i$. The second component captures the coordination loss arising from the mismatch between the two decisions. The parameter $\gamma \in [0,1]$ measures the importance of coordination for the players. It is common knowledge that ${\theta }_1$ and ${\theta }_2$ are drawn independently from the set $\Theta =D$, with each state being equally likely.

The experiment has four initial treatments, Decentralized‐High, Centralized‐High, Decentralized‐Low, Centralized‐Low. In the two Decentralized treatments, Player 1 makes decision $d_1$, Player 2 makes decision $d_2$, and each match consists of two players. In the two Centralized treatments, the decisions $d_1$ and $d_2$ are made by an additional Player 3 (the principal), whose payoff is given by the average of the payoffs of Player 1 and Player 2: [Image Omitted. See PDF]

Each subject starts the session with an initial endowment and loses points in each period based on the decisions made in the period. In the High treatments, the points lost by Player 1 and Player 2 in each period of the game are determined by the following formula: [Image Omitted. See PDF] which corresponds to a choice of $\gamma =3/4$. Thus, the High treatments place a higher weight on coordinating $d_1$ and $d_2$ than on adapting to each state ${\theta }_i$. The Low treatments place a high weight on adaptation to ${\theta }_i$ ($\gamma =1/4$): [Image Omitted. See PDF]

The timing in the decentralized treatments is as follows. First, Player 1 and 2 privately observe their local conditions; that is, Player i observes ${\theta }_i$, but not ${\theta }_j$, $j\ne i$. Then each player is asked to send a message $m\in M=\Theta$ to the other player. The framing of the screen is intentionally left neutral to avoid any suggestion on how to use the messages. This is followed by an empty box and an OK button in the bottom right corner of the screen. After both messages are sent, they are simultaneously revealed to both players. Then the players are asked to make their decisions. After the decisions are made, but before they are made public, the players make incentivized conjectures of each other's states: Player 1 guesses ${\theta }_2$, and Player 2 guesses ${\theta }_1$. At the end of each match, the players receive feedback.

In the centralized treatments, Player 1 and Player 2 also start each match by privately observing their local conditions. Player 3 observes neither ${\theta }_1$ nor ${\theta }_2$. Then Player 1 and 2 are each asked to send a message to Player 3. The screens that Player 1 and 2 see at this stage are identical to those displayed in the decentralized treatments. While the senders decide what messages to send, Player 3 waits. After both messages are sent, they are simultaneously revealed to Player 3, and this player is asked to make the two decisions. After the decisions are made, the messages are made public, and all players make conjectures about the states not known to them: Player 1 guesses ${\theta }_2$, Player 2 guesses ${\theta }_1$, and Player 3 guesses both ${\theta }_1$ and ${\theta }_2$. As in the decentralized treatments, these conjectures are incentivized. At the end of the match, all players receive feedback.

In all of our treatments, the state, message, and decision spaces are restricted to be equal to each other. Thus, the ${\theta }_i$, $m_i$, and $d_i$ variables are all selected in increments of 0.01 from the set $\{-1,-0.99,-0.98,\dots ,0.98,0.99,1\}$. Restricting the message space to be equal to the state space, as in Cai and Wang (), can be motivated by the observation that subjects tend to interpret messages in cheap talk games using a natural language (see, e.g., Blume et al. ()).

We use subjects' elicited beliefs to measure the quality of communication in the experiment (Section ) and analyze how the communicated information is used in the decision‐making stage (Section ). Subjects' conjectures of each other's states are obtained with quadratic scoring rules (Nyarko and Schotter ()). For Player 1 and Player 2, the points lost for the guesses are equal to the square of the distance between the conjecture and the true value of the state being guessed. In the centralized treatments, Player 3 also guesses the states of both Player 1 and Player 2. We interpret subjects' elicited conjectures as proxies of their posterior beliefs. While Section of the paper discusses some of the issues associated with belief elicitation, we also note that our main results survive robustness checks that do not use subjects' elicited beliefs.

After analyzing the data from the initial treatments, we ran two additional treatments to test our explanations of the observed deviations from the theory. The treatments are identical to Centralized‐High and Decentralized‐High in all respects except that local conditions are common knowledge to all players. We provide more details in Section .

2.1

Implementation

The experiment was conducted at Instituto Tecnológico Autónomo de México in Mexico City between October 2014 and September 2015 using the software z‐Tree (Fischbacher ()). After entering the laboratory, sitting down at their computer terminals, and signing the consent forms, the subjects are distributed their treatment's instructions. At the same time that the subjects are reading the instructions, a quiz is displayed on their computer screens. The subjects are informed that they have 20 minutes to read the instructions and complete the quiz.

The treatments are implemented between subjects. Each session of the experiment consists of 2 practice periods followed by 15 periods that count toward each subject's earnings. In each period, subjects are randomly and anonymously matched with randomly‐assigned roles at the beginning of each period.

The subjects' earnings are determined as follows. Every subject is guaranteed a 30 Mexican pesos (≈ US$2) show up fee in addition to the earnings from the quiz. These earnings are called the subject's “guaranteed earnings.” In addition, each subject is given 210 Mexican pesos (≈ US$15). In each (nonpractice) period of the game, each subject loses a number of points due to her decisions and the decisions made by the subjects with whom she is matched during the period. In addition to losing points from the game, the subjects lose points from their conjectures of other players' states in accordance to the quadratic scoring rule described above. The subject's “additional earnings” are determined as follows: [Image Omitted. See PDF] Each subject's total earnings are given by the sum of the guaranteed and additional earnings. It is explained to the subjects that any subject losing more than 50 cumulative points (150 Mexican pesos) would be excluded from further matches, and that in the event this happens, the remaining subjects will be rematched with each other, with some randomly chosen subjects sitting out in each subsequent match. In practice, this never happened, but the program we used allowed for the contingency.

2.2

Predictions

The communication rule of sender i is a mapping ${\mu }_i:\Theta \to \mathrm{\Delta }{M}_i$ from local conditions to probability distributions over messages. Under decentralization, the decision rule of receiver i is a mapping $d_i^D:\Theta \times M_1\times M_2\to \mathrm{ℝ}$, $i\in \{1,2\}$, from local conditions and messages to decisions. Under centralization, the decision rule of the sole receiver (Player 3) is a pair of mappings $d_i^C:M_1\times M_2\to \mathrm{ℝ}$, $i\in \{1,2\}$, where $d_i^C$ maps a pair of messages $(m_1,m_2)$ to a decision for Player i. The belief functions of receiver i are the mappings ${\eta }_j:M_j\to \mathrm{\Delta }\Theta$, $j\in \{1,2\}$, each denoting the probability assigned by the receiver to each state ${\theta }_j\in \Theta$ after receiving message $m_j$ from sender j.

A communication equilibrium is defined by communication rules for Player 1 and Player 2 (${\mu }_1(m_1|{\theta }_1)$ and ${\mu }_2(m_2|{\theta }_2)$), decision rules for the decision makers ($d_i^D(m_1,m_2,{\theta }_i)$ under decentralization, and $d_i^C(m_1,m_2)$ under centralization), and belief functions for the receivers (${\eta }_1({\theta }_1|m_1)$ and ${\eta }_2({\theta }_2|m_2)$) such that the communication rules are optimal given the decision rules, the decision rules are optimal given beliefs, and the beliefs are derived from the communication rules using Bayes' rule whenever possible.

We define $E[{\theta }_i|m_i]$, $i=1,2$, as the posterior belief held by the receiver of message $m_i$ about local conditions ${\theta }_i$. Following the theoretical literature, we measure the quality of communication through the residual variance of the posterior belief, defined as $E[{({\theta }_i-E[{\theta }_i|m_i])}^2]$. Higher residual variance means more dispersion in the posterior beliefs, and thus lower quality of communication. The advantage of such a measure in experimental settings is that the residual variance of communication is defined independently of the partitional structure of equilibrium. Therefore, it can be used to measure communication quality whether or not players are conforming to any particular equilibrium or even exhibiting nonequilibrium behavior.

It is well known that communication games admit a multiplicity of equilibria. We therefore formulate most of our predictions about equilibrium communication around the Most Informative Equilibrium (MIE). We base our predictions on MIE for several reasons: (i) it is the equilibrium selection rule used in the theoretical literature; (ii) MIE maximizes ex ante expected payoffs and it is therefore the right benchmark to compare payoff losses in the experiment; (iii) it leads to clear theoretical predictions. A different equilibrium selection rule might generate potentially different predictions given the large number of possible equilibrium choices across our treatments. Ultimately, we address the question of what, if any, equilibrium is played using the experimental data.

Theory predicts the following about communication in MIE:

1

Prediction

Communication

• 1.The residual variance of communication is lower under centralization than decentralization, for any $\gamma \in (0,1)$.
• 2.As the importance of coordination increases, the residual variance of communication increases under centralization while it decreases under decentralization.

The logic behind Prediction is the following. Since the agents maximize their own individual payoffs, while the principal cares about the payoffs of both agents, the incentives of the agents are more aligned with the incentives of the principal than they are with each other. This leads communication to be more informative under centralization than decentralization, as long as $\gamma <1$. As γ increases under centralization, the principal cares less about adapting to local conditions and information becomes less relevant, distorting incentives of senders toward exaggeration. As γ increases under decentralization, the increased consequences of coordination failure provide incentives for better communication.

Recall that $E[{\theta }_i|m_i]$ denotes the posterior expectation about i's state held by the receiver of the message following message $m_i$. Under decentralization, Player i makes the following decision in equilibrium after receiving the message $m_j$: [Image Omitted. See PDF] The agent's decision rule is a linear function of her own state ${\theta }_i$, her own posterior $E[{\theta }_j|m_j]$, and the other agent's posterior $E[{\theta }_i|m_i]$. When the importance of coordination is low, both agents put large weights on their own local conditions and small weights on the other pieces of information. As coordination needs increase, both players increase the weight on the information provided by the other player, and decrease the weight on their own private information, which leads to more coordination in equilibrium. We estimate the decision rule in equation () as part of our statistical analysis of the data to quantify the magnitude and direction of the deviations from optimal behavior.

Under centralization, the principal makes the following decisions after receiving the message $m=(m_1,m_2)$: [Image Omitted. See PDF] The decision rules are functions of the principal's posterior beliefs about the players' states. When the importance of coordination is low, the posterior about Player i's state has a much larger weight in determining $d_i$ than the posterior about the state of Player j. As the importance of coordination increases, the weights on the two posteriors become closer to each other. These comparative statics can be summarized as follows:

2

Prediction

Optimal Decisions

As the importance of coordination increases:

• 1.Under centralization, Player 3 puts more weight on the information communicated by Player j when making decision $d_i$.
• 2.Under decentralization, Player i, $i=1,2$, puts less weight on her own local conditions and a larger weight on the information communicated by Player j.

We can also use the decision rules in equation () and equation () to formulate predictions about average degrees of adaptation and coordination in the experiment. The advantage of centralization lies in the principal's ability to perfectly control the degree to which the decisions are coordinated with each other. The principal, however, lacks complete knowledge of the other players' local conditions, which makes adaptation difficult. By contrast, under decentralization, the players can perfectly control the degree of adaptation of their decisions to their own local conditions, but coordination is difficult because each player only controls her own decision.

Let ${\mathit{CL}}_k=E[{(d_1-d_2)}^2]$, $k\in \{C(\mathit{entralized}),D(\mathit{ecentralized})\}$, denote the expected (normalized) coordination loss. In MIE, the principal's comparative advantage at coordination generates a smaller coordination loss under centralization than under decentralization. Moreover, as γ increases, the coordination loss falls regardless of how authority is allocated. This implies the following prediction, which is proved in Proposition 1 of Appendix B in the Online Supplementary Material:

3

Prediction

Coordination Losses

• 1.The average coordination loss is larger under decentralization than centralization.
• 2.As the importance of coordination increases, the average coordination loss decreases under both centralization and decentralization.

Similarly, we can compute the expected (normalized) adaptation loss in MIE for an arbitrary Player i, $i=1,2$, denoted by ${\mathit{AL}}_k^i=E[{(d_i^k-{\theta }_i^k)}^2]$, $k\in \{C,D\}$. The agents' comparative advantage at adaptation leads to larger adaptation losses under centralization than under decentralization. As coordination becomes more important, the adaptation loss rises regardless of how authority is allocated. The following prediction is proved in Proposition 2 of Appendix B in the Online Supplementary Material:

4

Prediction

Adaptation Losses

• 1.The average adaptation loss is larger under centralization than decentralization.
• 2.As the importance of coordination increases, the average adaptation loss increases under both centralization and decentralization.

Thus, the games lend themselves to a rich array of predictions about communication quality, decision rules, and adaptation/coordination losses. We now describe the experimental data.

3

Results

For the initial treatments with incomplete information, we collected data from 238 undergraduate students recruited from introductory level classes. A total of 14 experimental sessions were conducted with a minimum of 11 students and a maximum of 21 students per session. The distribution of subjects among treatments is shown in Table . More subjects participated in the centralized treatments to ensure that the amount of observations (e.g., for $d_1$ and $d_2$) is not too unbalanced. A session lasted 75 minutes on average.

1

Subjects per treatment (initial treatments with incomplete information).

	Low γ	High γ
Decentralized	3 sessions	3 sessions
	N = 48	N = 56
Centralized	4 sessions	4 sessions
	N = 66	N = 68

In what follows, we first discuss how subjects communicated in the experiment (Section ), and second how the communicated information was used (Section ). Section quantifies the payoff consequences of deviations from equilibrium behavior, as well as the payoff consequences of miscommunication. Section describes the design and results of additional treatments with complete information. All of our results focus on nonpractice periods of the experiment.

3.1

How private information was communicated

The quality of communication is in theory measured in terms of the residual variance of the receiver's posterior, $E[\theta |m]$, around the sender's privately known state, that is, $E[{(\theta -E(\theta |m))}^2]$. To obtain a unit‐free measure, we divide this variable by 1/3 (the residual variance in the babbling equilibrium). We then assess treatment effects on the empirical analogue of this object, by defining $3\times {(\mathit{Other}\mathrm{\_}{\mathit{State}}_{it}-{\mathit{Guess}}_{it})}^2$ as three times the squared distance between receiver i's guess in period t and the true value of the state. If the mean of this variable is equal to 0.5, for instance, the interpretation is that one‐half of the residual variance associated with the uninformative equilibrium is observed in the data.

We regress $3\times {(\mathit{Other}\mathrm{\_}{\mathit{State}}_{it}-{\mathit{Guess}}_{it})}^2$ against treatment dummies, allowing the error term in the regression to be correlated for observations coming from the same session. While the number of sessions is small, we assume within‐session correlations because this assumption can be applied in the vast majority of our econometric analysis, and when possible, provide robustness checks with subject‐clustered standard errors in the Appendix in the Online Supplementary Material.

The results are shown in Table . Consistent with Prediction , we find that the residual variance of communication was lower under centralization than decentralization when the importance of coordination was low ($P<0.05$). The residual variance decreased with the importance of coordination under decentralization ($P<0.05$), also as predicted. There was no significant difference between the residual variance under centralization and decentralization when γ was high ($P=0.435$), which is also in line with Prediction , according to which the quality of communication converges to the same level in both organizational structures as γ increases. Inconsistent with Prediction , we find no significant effect of γ on the residual variance under centralization ($P=0.351$).

2

Treatment effects on residual variance of communication. The session‐clustered standard errors for the pairwise differences are reported in parentheses.

	Decentralized		Centralized
γ = 0.25	0.4272	>	0.1796
	(0.0574)	(0.0987)	(0.0803)
	∨		∧
	(0.0976)		(0.2250)
γ = 0.75	0.2164	<	0.3971
	(0.0789)	(0.2245)	(0.2102)

Note: Session‐clustered standard errors in parentheses.

Appendix C provides some robustness checks of the result that the residual variance of communication was significantly higher under decentralization if and only if the importance of coordination was low. As shown there, the result is observed in later periods of the experiment taken separately, suggesting that the underlying effects were not learned away. It is also observed if residual variance is measured using subjects' messages as proxies for guesses. That is, even if the analysis does not use subjects' elicited beliefs, which in principle may be biased, to form a measure of communication quality, we see some of the theoretically predicted comparative statics in the data. Moreover, the result is reflected in distributions at the level of individual subjects and not just within‐treatment averages.

These results suggest that MIE was at least partially predictive of the treatment effects observed in the communication data. Directly comparing the predicted and observed residual variances of communication using the standard errors in Table , we find that the residual variances were not significantly lower than predicted in any of the treatments, and significantly higher than predicted in Decentralized‐Low ($P<0.01$). While this suggests at least some undercommunication relative to MIE, it is consistent with behavior in other, less informative, equilibria. On the other hand, the way in which the communicated information was used deviated from the predictions of any communication equilibrium, not just MIE. These deviations, which are demonstrated in Section below, are the mainstay of our paper.

3.2

How the communicated information was used

While our predictions about the optimal use of information are centered on subjects' decision rules (equation () and equation ()), it is instructive to first study the observed adaptation and coordination losses. This allows us to assess treatment effects without resorting to involved econometric analysis and to see if systematic deviations from optimal coordination and adaptation are present in the data. After noting and describing these deviations, we turn to a more rigorous analysis of subjects' decision rules.

Subjects' adaptation and coordination losses are plotted in Figure . It can be seen from Figure that, as predicted, subjects coordinated more and adapted less as coordination became more important, and that this was true both under centralization and decentralization. Specifically, as γ increased, the coordination loss decreased under centralization ($P<0.001$), while the adaptation loss increased under both centralization ($P<0.01$) and decentralization ($P<0.05$). While the decrease in the coordination loss under decentralization was not statistically significant ($P=0.1024$), the effect was in the right direction. That adaptation losses increased but coordination losses were little affected suggests a possible coordination failure in the Decentralized‐High treatment.

Fig. 1. Observed misadaptation and miscoordination (dark grey) and MIE predictions (light grey).

Because MIE maximizes agents' ex ante expected payoffs, it provides a useful benchmark for how well subjects should adapt and coordinate. The treatment comparisons show that, with low γ, the coordination loss was higher under centralization than decentralization ($P<0.05$), while the adaptation loss was higher under decentralization than centralization ($P<0.01$). Both of these observations directly contradict Prediction and Prediction , as well as the intuition that centralized organizations are better at coordinating while decentralized ones are better at adapting. When γ is high, the coordination loss was statistically indistinguishable under centralization and decentralization, as was the adaptation loss (both $P>0.1$). These observations are also at odds with Prediction and Prediction . Comparing the data to the MIE predictions directly, the coordination loss ${(d_1-d_2)}^2$ was significantly greater than predicted for both values of γ under centralization ($P<0.001$ in both cases), while the adaptation loss ${(d_i-{\theta }_i)}^2$ was significantly greater than predicted for both values of γ under decentralization ($P<0.001$ in both cases). Taken together, the results in this paragraph suggest that decision makers underadapted in decentralized treatments and undercoordinated in centralized ones.

To directly estimate the equilibrium decision rule under decentralization (equation ()), we regress the decision made by subject i in period t (${\mathit{Decision}}_{it}$) against subject i's state (${\theta }_{it}$), the guess of subject i's partner about subject i's state ($\mathit{Guess}\mathrm{\_}\mathit{of}\mathrm{\_}\mathit{the}\mathrm{\_}{\mathit{State}}_{it}$), and the guess of subject i about her partner's state ($\mathit{Guess}\mathrm{\_}\mathit{of}\mathrm{\_}\mathit{the}\mathrm{\_}\mathit{Other}\mathrm{\_}{\mathit{State}}_{it}$). To accommodate the effect of γ on subjects' decisions, we interact the explanatory variables with a dummy that takes on the value of one for treatments with $\gamma =\frac{3}{4}$. We also place the restriction that the weights add up to one both when $\gamma =\frac{1}{4}$ and when $\gamma =\frac{3}{4}$. The results are shown in the first column of Table . As can be seen from the interaction terms, subjects qualitatively responded to incentives to coordinate and adapt roughly as theory predicts. As the importance of coordination increased, the weight on ${\theta }_{it}$ decreased ($P<0.05$), and the weight on $\mathit{Guess}\mathrm{\_}\mathit{of}\mathrm{\_}\mathit{the}\mathrm{\_}\mathit{Other}\mathrm{\_}{\mathit{State}}_{it}$ increased ($P<0.001$). Subjects' decisions put a smaller weight on the state and a larger weight on posterior beliefs when the incentive to coordinate was greater.

3

Estimated decision weights.

	Decentralized	Centralized
High (dummy = 1 if $\gamma =\frac{3}{4}$)	−0.00735	0.0149
	(0.0131)	(0.0149)
State (θ)	0.493
	(0.0775)
Guess of the state	0.162	0.946
	(0.0224)	(0.0196)
Guess of the other state	0.345	0.0544
	(0.0572)	(0.0196)
θ × High	−0.270
	(0.115)
Guess of the state × High	0.0576	−0.290
	(0.0939)	(0.0323)
Guess of the other state × High	0.213	0.290
	(0.0598)	(0.0323)
Constant	0.0197	0.00615
	(0.0123)	(0.0118)
Observations	1560	1320

Note: Session‐clustered standard errors in parentheses.

We use equation () as a guide to estimate an analogous model for the centralized treatments. That is, we regress the principals' decisions against their elicited posterior beliefs with the restriction that the weights sum up to one. The results, reported in the second column of Table , suggest that the principals' decision rules responded in the predicted direction to changes in incentives to coordinate. Thus, the weight on $\mathit{Guess}\mathrm{\_}\mathit{of}\mathrm{\_}\mathit{the}\mathrm{\_}\mathit{State}$ decreased and the weight on $\mathit{Guess}\mathrm{\_}\mathit{of}\mathrm{\_}\mathit{the}\mathrm{\_}\mathit{Other}\mathrm{\_}\mathit{State}$ increased when the importance of coordination was high ($P<0.001$ in both cases). That is, as coordination became more important, the principal weighted the belief about i's state less and the belief about the state of i's partner more when making a decision on behalf of subject i.

Importantly, while the responses to changes in incentives to coordinate were in the right direction both under centralization and decentralization, subjects' decisions showed significant and systematic quantitative deviations from equilibrium (Table ). In Decentralized‐Low, subjects underweighted their own states ($P<0.001$), overweighted their partners' posteriors ($P<0.001$), and overweighted their own posteriors about their partners' states ($P<0.05$). In Decentralized‐High, subjects overweighted their own posteriors ($P<0.001$). Thus, subjects in the decentralized treatments underweighted their own states and overweighted their own and their partners' beliefs. In both Centralized‐Low and Centralized‐High, the principal put too much weight on the belief about ${\theta }_i$ and too little weight on the belief about ${\theta }_j$ when making decision $d_i$ ($P<0.001$ in both cases). These deviations are consistent with the hypothesis that decision makers overweighted the importance of coordination under decentralization and underweighted it under centralization.

4

Predicted and estimated weights in subjects' decision rules. The session‐clustered standard errors for the differences between actual and predicted weights are reported in parentheses.

Decentralized Treatments	${\theta }_i$	$E[{\theta }_i|m_i]$	$E[{\theta }_j|m_j]$
Low (Predicted)	0.75	0.05	0.2
Low (Actual)	0.49	0.16	0.34
	(0.0775)	(0.0224)	(0.0572)
High (Predicted)	0.25	0.32	0.43
High (Actual)	0.22	0.22	0.56
	(0.0857)	(0.0912)	(0.0175)
Centralized Treatments	$E[{\theta }_i|m_i]$	$E[{\theta }_j|m_j]$
Low (Predicted)	0.71	0.29
Low (Actual)	0.95	0.05
	(0.0196)	(0.0196)
High (Predicted)	0.54	0.46
High (Actual)	0.66	0.34
	(0.0256)	(0.0256)

To quantify the degree to which coordination was over or underweighted, we structurally estimate the γ implied by the subjects' decisions under the null hypothesis of equilibrium. We do this separately for each of the experimental treatments using nonlinear least squares and session‐clustered errors. The estimation results are shown in Table . The estimated γ's are significantly higher than what they should be (i.e., those specified in the instructions) in both Decentralized‐Low and Decentralized‐High ($P<0.05$ and $P<0.01$, resp.). For example, when $\gamma =\frac{3}{4}$, the agents acted as if adaptation is almost irrelevant. In Centralized‐Low and Centralized‐High, the weights are significantly lower than what they should be ($P<0.001$ in both treatments). For example, when $\gamma =\frac{1}{4}$, the principal acted as if the weight on coordination were almost zero. We summarize these findings as follows:

5

The estimated distortions of γ.

	Decentralized	Centralized
$\hat{\gamma }$ when γ = 0.25	0.517 > 0.25	0.0296 < 0.25
	(0.103)	(0.0116)
$\hat{\gamma }$ when γ = 0.75	0.9396 > 0.75	0.356 < 0.75
	(0.0375)	(0.0548)
Observations	1560	1320

Note: Session‐clustered standard errors in parentheses.

1

Main Result

The importance of coordination was overweighted by the agents under decentralization and underweighted by the principal under centralization.

To see how robust the result above is to learning, we modify the nonlinear least squares model used in Table by allowing the estimated $\hat{\gamma }$'s to differ across periods in every treatment of the experiment. This reveals a significant time trend in the Decentralized‐Low treatment, where the estimated $\hat{\gamma }$ loses significance by period 15 ($P=0.345$). On the other hand, the overweighting in this treatment is still positive in period 15, and the lack of significance might simply be a question of power. We find no significant time effects on $\hat{\gamma }$ in Decentralized‐High ($P=0.496$), Centralized‐Low ($P=0.273$), or Centralized‐High ($P=0.266$). Taken together, these results suggest that the distortions identified in Table are not easily learned away.

To study between‐subject heterogeneity in deviations from equilibrium, we estimate subject‐level regressions analogous to those in Table . The subject‐level estimates of γ in the four treatments are reported in Figure . The red vertical lines represent the predicted γ's of 0.25 and 0.75. As seen in the figure, a large fraction of subjects in the Decentralized‐High treatment acted as if γ was close to one, while a large fraction of subjects in the Centralized‐Low treatment acted as if γ was close to zero. It can also be seen that many subjects used a γ above 0.25 in the Decentralized‐Low treatment and a γ below 0.75 in Centralized‐High. While we omit a description of the relevant statistical comparisons here, it is argued in Appendix D.1, available in the Online Supplementary Material, which compares the observed medians and means to their predicted values, that Main Result is reflected not only in overall averages but also in distributions at the level of individual subjects.

Fig. 2. Heterogeneity in decision weights.

While the analysis so far has made use of subjects' elicited beliefs, there is another way to identify deviations from equilibrium decision rules in our experiment. Specifically, the experimental design allows us to derive subjects' implicit beliefs under the assumption of equilibrium behavior. For the centralized treatments, because we observe the principal's decisions, the equilibrium decision rules in equation () form a system of two equations in two unknowns which can be used to solve for $E({\theta }_1|m_1)$ and $E({\theta }_2|m_2)$. For the decentralized treatments, a similar procedure can be employed given knowledge of the decisions made by each player, the true states, and the equilibrium decision rules given by equation (). When we do this, we find that $43.19\mathrm{\%}$ of implicit beliefs lie outside the interval $[-1,1]$ overall, with $50.96\mathrm{\%}$ under decentralization and $34.02\mathrm{\%}$ under centralization. This provides corroborative evidence of suboptimal behavior in both treatments.

Appendix D of the Online Supplementary Material provides additional robustness checks of Main Result . Specifically, Table S.14 reestimates the weights in Table using subjects' messages in place of elicited beliefs; Table S.15 reestimates the weights in the centralized treatments replacing the beliefs of Player 3 with those of Players 1 and 2; Table S.16 reestimates the weights with standard errors clustered at the level of the decision maker; Appendix D.1 of the Online Supplementary Material carries out the above‐described analysis of heterogeneity in decision rules. We find qualitatively similar distortions of decision weights in every case.

3.3

Payoff consequences of deviations from equilibrium

We now turn to an analysis of subjects' losses in the experiment. On average, subjects earned 163.5 Mexican pesos (≈ US$11), excluding the show‐up fee and the payment from the quiz. Of the 46.5 pesos subjects lost on average from playing the game and making guesses about their partners' states, $91\mathrm{\%}$ were lost from the game and $9\mathrm{\%}$ from the guesses. While the average losses from playing the game were moderate, several subjects came close to bankruptcy (losing more than 150 pesos by period 15) in the course of the experiment, and there was significant variation across subjects, as shown in Figure S.2 of Appendix E in the Online Supplementary Material. Because only good decisions ensured subjects from incurring substantial losses, we believe that our experiment provided subjects with effective incentives.

For each team in our data set, we define the total relative payoff loss as the loss observed in the data minus the loss predicted by MIE: [Image Omitted. See PDF] To compute $L^{\mathrm{MIE}}$, for every treatment, we calculate an explicit solution for the most informative partitional equilibrium provided by Alonso, Dessein, and Matouschek (, pp. 171–172). We then derive the posterior beliefs that receivers would form in MIE. Using these posterior beliefs, we compute the optimal decisions (using () and ()). Then, using (), we compute the decision makers' utilities given the decisions. The payoffs of Player 1 and Player 2 are then added to calculate $L^{\mathrm{MIE}}$.

We can further decompose the relative payoff loss as follows: [Image Omitted. See PDF] $L^{\text{reported beliefs}}$ is the team's payoff loss given the decision makers' reported beliefs. To compute it, we calculate the equilibrium predictions for $d_i^C$ and $d_i^D$ conditional on subjects' elicited beliefs. We then use equation () to calculate individual utilities and add the utilities to calculate $L^{\text{reported beliefs}}$. This gives the total amount of points that each team would have lost if the decision makers followed the equilibrium decision rules using their reported beliefs. $L^{\text{observed}}-L^{\text{reported beliefs}}$ is labeled as the loss due to distortions in equation (). We interpret this variable as the loss in payoffs due to subjects' decisions deviating from the equilibrium decision rules in equation () and equation (). $L^{\text{reported beliefs}}-L^{\mathrm{MIE}}$ is labeled as the loss due to miscommunication. This variable captures the payoff loss due to subjects' posterior beliefs deviating from those suggested by MIE.

The total relative payoff loss ($L^{\text{observed}}-L^{\mathrm{MIE}}$) was on average 1.39 points. To get a reference for the size of this number, note that the observed loss ($L^{\text{observed}}$) of 1.90 (or $1.39+0.51$) points is approximately 3.75 times as large as the MIE benchmark ($L^{\mathrm{MIE}}$) of 0.51 points. The relative loss due to distortions was 1.31 points, while the relative loss due to miscommunication was 0.08 points. Thus, the distortions accounted for $94\mathrm{\%}$ of the overall relative loss in payoffs. We highlight this as one of our main results:

2

Main Result

Most payoff losses are due to distortions of decision rules rather than miscommunication.

3.4

Complete information treatments

Uncertainty enters the players' payoff functions differently under centralization and decentralization. Under decentralization, adaptation involves no uncertainty (since own states and decisions are known), while coordination involves the other agent's potentially uncertain decision. The opposite is true under centralization, where coordination involves no uncertainty (since the principal controls both decisions), while adaptation is uncertain because it involves the two agents' unknown states. It follows that one way of restating Main Result is that the uncertain component of the payoff function was overweighted by the decision maker under both centralization and decentralization.

To shed light on the role of uncertainty in generating the distortions in the initial treatments of the experiment, we ran two additional treatments in September 2015: Decentralized‐Complete Info ($N=30$, one session with 10 and one with 20 subjects) and Centralized‐Complete Info ($N=30$, one session with 12 and one with 18 subjects). The treatments were identical to Decentralized‐High and Centralized‐High in all respects but the following. First, every player observed every state ${\theta }_i$ ($i=1,2$) before making any decision. Second, the players did not make any guesses about the states of other players. In particular, the agents still sent messages to each other under decentralization and to the principal under centralization. Thus, the only difference between the complete and incomplete information treatments is uncertainty about ${\theta }_i$.

Because the messages are irrelevant in theory, we make no predictions about residual variance of communication in Centralized‐Complete Info and Decentralized‐Complete Info. The complete information decision rules are the same as those in equation () and equation (), with the modification that each posterior belief $E[{\theta }_i|m_i]$ is replaced by the true value of the corresponding state ${\theta }_i$. While the decision rule under decentralization is the unique equilibrium solution, the decision rule under centralization is the unique solution to the principal's decision problem: [Image Omitted. See PDF] [Image Omitted. See PDF]

Estimating the γ's in equation () and equation () using nonlinear least squares, we find a weight of 0.774 in Decentralized‐Complete Info and a weight of 0.687 in Centralized‐Complete Info. Neither of these estimates is significantly different from the predicted value of 0.75 ($P=0.814$ in Decentralized‐Complete Info and $P=0.5257$ in Centralized‐Complete Info). We also find little significant differences in predicted and observed decision weights when the weights are estimated at the level of individual subjects. Indeed, as shown in Figure , the distribution of individually estimated decision weights shifts to the left under decentralization and to the right under centralization.

Fig. 3. Distributions of decision weights, incomplete versus complete information.

In Decentralized‐Complete Info, the mean estimate of γ is 0.76, the median is 0.853, and neither is significantly different from 0.75. In Centralized‐Complete Info, the mean estimate of γ is 0.802, and the median is 1. While the mean is not significantly different from 0.75 ($P=0.365$), the median is significantly higher ($P<0.001$).

Thus, regardless of how the data is analyzed, the deviations from equilibrium behavior are significantly reduced in the complete information treatments. Specifically, no underweighting of γ is observed under centralization, and no overweighting is observed under decentralization. While behavior of the median subject deviates from equilibrium under centralization, it deviates in the direction of overweighting the importance of coordination, the reverse of Main Result . Moreover, we find no significant deviations from the predicted decision weights on average. We summarize these findings as follows:

3

Main Result

With complete information, there was no over or underweighting of the importance of coordination on average.

The results in the complete information treatments are consistent with the hypothesis that the deviations from equilibrium observed in the initial treatments were caused by uncertainty. We discuss the possible underlying mechanisms in Section below.

4

Discussion and conclusion

Risk aversion cannot rationalize our findings. Thus, while the distortions associated with risk aversion are qualitatively the same as those observed in the experiment, they are not nearly of the right magnitude. To develop some intuition for the effect of risk aversion on decision rules, consider the case of centralization. Noise in messages introduces uncertainty about the principal's adaptation losses. While a risk‐neutral decision maker only cares about the posterior expectations of the states, a risk‐averse principal also cares about the variance of her conditional expectation and might undercoordinate to decrease this variance. Because the models of Alonso, Dessein, and Matouschek () and Rantakari () have no closed‐form solutions with risk‐averse preferences, predictions have to be obtained through simulations. We do this in Appendix H in the Online Supplementary Material, where we assume that a decision maker has a utility function of the form $U(x)=-{(-x)}^{\alpha }$ over her point losses $x<0$, where the parameter $\alpha >0$ determines the decision maker's risk attitudes (Tversky and Kahneman ()). We argue that only $4\mathrm{\%}$–$17\mathrm{\%}$ of the observed under‐coordination in the centralized treatments can be accounted for by reasonable degrees of risk aversion.

Ambiguity in communication provides a better explanation of our findings. Consider again the game under centralization. If the principal is uncertain about the communication rules used by the agents, she needs to form subjective beliefs about how the states are communicated through subjects' messages. Assuming ambiguity‐neutral preferences, the principal can easily form posterior beliefs $E({\theta }_i|m_i)$, and ambiguity about communication rules has no predictive power. In the presence of ambiguity aversion, however, the predictions of the model change. In Appendix J of the Online Supplementary Material, we perform simulations to solve the problem of an ambiguity‐averse principal under centralization. The simulation assumes that the principal has preferences of the maxmin sort, where the $\min$ is taken over different beliefs about the agents' states. We find that ambiguity aversion substantially amplifies the distortions due to risk aversion. Thus, with the Tversky and Kahneman ()'s functional form, a risk aversion parameter of $\alpha =2$, and ambiguity‐averse preferences, the model accounts for $16\mathrm{\%}$ to $50\mathrm{\%}$ of the undercoordination observed in the data. Moreover, if the principal is ambiguity‐averse, the model can generate distortions in the right direction even if her utility function is moderately risk‐seeking.

Although informative, the simulations described above do not assume equilibrium behavior. We now sketch a simple variant of the models of Alonso, Dessein, and Matouschek () and Rantakari () that allows for ambiguous communication in the presence of ambiguity‐averse message receivers. Suppose that each division manager has access to an Ellsberg urn. An Ellsberg urn is an urn which contains red and black balls but whose color composition is unknown to every player. Let $\rho \in [0,1]$ denote the fraction of red balls. Each division manager privately observes her local conditions and a draw from her Ellsberg urn before communication takes place. Ambiguous communication occurs when the sender conditions her message on the urn realization. While seemingly abstract, this construct can be used to capture both intentional and unintentional vagueness in communication.

While solving the model is beyond the scope of this paper, recent theoretical work by Kellner and Le Quement () shows that ambiguous communication can be sustained in equilibria of sender‐receiver games of the Crawford and Sobel () type. More precisely, for any informative communication equilibrium without ambiguity aversion, there exists an ambiguous communication equilibrium which strictly Pareto‐dominates it. In the latter equilibrium, communication is more informative and the receiver takes actions which are more accommodating toward the preferences of the sender. Conjecturing that similar results can be extended to the framework of Alonso, Dessein, and Matouschek () and Rantakari (), the latter observation would provide a theoretical rationale for the result in our experiment that the decision rules are more accommodating toward the message senders' private needs than predicted by the baseline model. Focusing on MIE, coordination losses would be larger than predicted in the ambiguous game under centralization while adaptation losses would be larger than predicted in the ambiguous game under decentralization. While it is not clear that the ambiguity‐averse extension of the model would capture all elements of the data, it provides a promising avenue for an explanation.

A simpler explanation for the observed distortions is gift exchange. For example, it might be the case that the message receivers reward the message senders for communicating private information, where the reward comes in the form of putting a higher weight on the message. Under decentralization, this would lead to a larger adaptation loss than predicted; moreover, it could lead to coordination failure (as we observe in the Decentralized‐High treatment) if the weight on the other player's message is sufficiently high. Under centralization, gift exchange would lead the central manager's decisions to be less coordinated than predicted, as we observe in the data.

Another possibility is that the decision maker cares not only about her own monetary payoffs but also the monetary payoffs of the other players, for instance using the functional form in Levine (). Our results are not consistent with social preferences for two reasons. First, because the principal maximizes the agents' joint payoffs, social preferences cannot explain the distortions observed in the centralized treatments. Second, if social preferences caused the distortions in Decentralized‐Low and Decentralized‐High, we should observe similar distortions in Decentralized‐Complete Info.

Our experimental framework can be used to study a wide class of related coordination problems. For example, Alonso, Dessein, and Matouschek () show that the quality of communication under decentralization can be worse if decisions are made sequentially as opposed to simultaneously. This is because the player in the role of the follower has higher incentives to misreport in order to influence the decision of the leader, which makes coordination more difficult in theory. In light of our results, for example, the players overweighting the importance of coordination under decentralization, sequential decision making might make coordination easier in practice. It would also be interesting to study how our results on communication and behavior are affected by asymmetries in coordination needs or partial centralization (where only one of the decisions is controlled by the principal). A different project could investigate coordination in teams with incomplete information. Thus, each player in our experiment can be identified with a team of several subjects who need to agree on which message to send and which decision to make. Feri, Irlenbusch, and Sutter () find that team decision making can lead to higher coordination on efficient outcomes, a testable prediction. An open question is whether the distortions observed in our treatments will be robust to or alleviated by decisions being made in teams. Another experiment could nest the basic framework in a repeated game. This would bring the setup closer to real world organizations in which the same agents interact for longer periods of time and allow the dynamics of communication and coordination to be investigated.

²Several literatures build on this insight. Carlsson and Van Damme () applied this idea in the context of global games; Baliga and Sjöström () in the context of games of conflict; Dessein and Santos () in the context of organizational economics.

³See, for example, Alonso, Dessein, and Matouschek (, ), Rantakari (), Dessein, Garicano, and Gertner ().

⁴While intermediate cases in which the principal retains some, but not all, of the decision making authority can also be considered (Rantakari ()), the two extreme cases provide the sharpest contrast in theoretical predictions and are therefore particularly well suited to implementation in the lab.

⁵In applications of the model, an agent could be a manager in charge of a division or a function within a firm, a local district, a state government, etc.

⁶We also perform robustness checks of our results that do not rely on belief elicitation.

⁷Experimental economists have long been interested in coordination problems (see, e.g., Van Huyck, Battalio, and Beil (), Brandts and Cooper ()). Some existing studies also explore the role of communication as a coordination device (see, e.g., Cooper, DeJong, Forsythe, and Ross (), Blume and Ortmann ()).

⁸See, for example, Dickhaut, McCabe, and Mukherji (), Blume, DeJong, Kim, and Sprinkle (), Cai and Wang (), Sánchez‐Pagés and Vorsatz (), and Wang, Spezio, and Camerer ().

⁹See Wilson and Vespa (), who found that strategic uncertainty in a repeated cheap talk game leads to a failure to coordinate on efficient equilibria. Behavior is consistent with a repeated babbling equilibrium even when Pareto‐superior equilibria exist in which the sender uses a truthful strategy.

¹⁰The decision space is restricted because allowing the decisions to be elements of $\mathrm{ℝ}$ would make it possible for a player who behaves randomly, or simply makes a mistake while typing, to sustain enormous losses, making the experiment infeasible. While $D=\mathrm{ℝ}$ in Alonso, Dessein, and Matouschek () and Rantakari (), the restriction to $[-1,1]$ does not affect the theoretical predictions. A discretization of $[-1,1]$ is used because decisions in experiments can only approximate continuous variables.

¹¹We also ran some additional treatments, which we describe later.

¹²For Player 3, the payoff is equal to the average to ensure that the losses do not substantially differ in magnitude from those of Player 1 and Player 2.

¹³Specifically, the sender sees the following information displayed on her screen: “You are Player X. Your number [local conditions] is X. Send your message.”

¹⁴The feedback information consists of the other player's state, the other player's decision, own points lost due to the decisions made, own points lost from the conjecture about the other player's state, points lost in the period, points lost so far, and pesos lost so far.

¹⁵The feedback information consists of the unknown state(s), Player 3's decisions, own points lost from the decisions made, own points lost from the conjecture(s) about the other state(s), points lost in the period, points lost so far, and pesos lost so far.

¹⁶While we thought about allowing for free communication in the experiment, we decided in favor of a more restricted communication protocol (i) because this allowed for a more direct test of the theory, (ii) because this approach was followed by other experimental studies in the strategic communication literature (e.g., Cai and Wang ()), and (iii) because we do not think that free communication would eliminate strategic uncertainty, multiplicity of equilibria, or the possibility of ambiguous communication. Even if communication were unrestricted, subjects could in principle use ambiguous messages such as “my state is close to zero” or “my state is 0.2 units away from zero,” etc. In the experiment with free communication, the sender's interpretation of a message would still be conditional on beliefs about what communication rule is being used. Strategic uncertainty is not the result of our experimental design but rather an intrinsic feature of cheap talk games.

¹⁷Quadratic scoring rules incentivize risk‐neutral subjects to report their mean beliefs truthfully.

¹⁸Formally, denote Player i's conjecture about ${\theta }_j$, conditional on having received message $m_j$, by $p({\theta }_j|m_j)$. The points lost for the conjecture are given by ${(p({\theta }_j|m_j)-{\theta }_j)}^2$.

¹⁹For this player, the points lost are equal to the average of the two squared distances to ensure that the losses for the guesses do not strongly differ from those of Player 1 and Player 2.

²⁰The instructions can be found in the Online Supplementary Material (Evdokimov and Garfagnini ()). While the sample instructions are in English, the actual instructions were administered in Spanish.

²¹The quiz has 8 questions that are identical for all treatments, and 4 that differ across the decentralized and centralized treatments. The quiz tests the subjects' understanding of statistical independence, how they are to be matched in the experiment, the conversion of points to pesos, and the game's basic structure. The answers to all of the quiz questions are incentivized: each subject gains one Mexican peso for each quiz question correctly answered. The questions are included in the Online Supplementary Material.

²²The instructions provide subjects with several examples of final earnings as a function of points lost, and the quiz tests their understanding of the payment rules with yet another example.

²³Alonso, Dessein, and Matouschek () and Rantakari () showed that any equilibrium in which a finite number of messages is possible is economically equivalent to one in which the communication rules take a partitional form: a sender partitions the state space and only communicates which element of the partition the realized state belongs to.

²⁴It is also well known that an uninformative equilibrium always exists in such games.

²⁵Chen, Kartik, and Sobel () provided conditions which uniquely select MIE in cheap talk games.

²⁶The closed‐form solutions for the residual variance of communication under MIE can be found in Appendix A of the Online Supplementary Material.

²⁷Prediction is robust to social preferences. Alonso, Dessein, and Matouschek () considered a variation of the model described above in which Player 1 maximizes $\lambda {\pi }_1+(1-\lambda ){\pi }_2$ and Player 2 maximizes $(1-\lambda ){\pi }_1+\lambda {\pi }_2$, where $\lambda \in [\frac{1}{2},1]$. Although the payoff functions used in the experiment set $\lambda =1$, it is in principle plausible that Player 1 and Player 2 assign strictly positive weights to each other's payoffs. However, for a fixed γ, it can be shown that the difference in the quality of communication under centralization and decentralization, while shrinking as λ approaches $\frac{1}{2}$, remains strictly positive. Similarly, for any fixed λ, the arguments regarding the effect of γ on the quality of communication remain valid.

²⁸It can be shown that the decisions converge to each other as $\gamma \to 1$.

²⁹See Alonso, Dessein, and Matouschek () and Rantakari () for derivations of the decision rules described in this section.

³⁰This quantity is normalized by γ, and hence does not represent actual utility or point losses.

³¹The model is sufficiently tractable to also allow the computation of correlations between decisions, and between decisions and states, in closed form. However, we base our predictions on coordination and adaptation losses (see below) because they are easier to test compared to predictions based on correlation coefficients.

³²This has no immediate implication for welfare because an increase in γ also implies that adaptation losses have a lower impact on welfare.

³³The number of participants in the Centralized‐High treatment is not divisible by three. This is because one of the subjects experienced a health issue while the instructions were being administered and had to leave the room. We recalibrated the program in this session to accommodate 11 subjects rather than 12. This was accomplished by matching 9 people in every period of the game, with two remaining participants sitting out randomly. We also informed the participants in this session about the new rematching procedure. Our results do not significantly change if this session is excluded from the analysis.

³⁴We use all elicited guesses (in nonpractice periods) of all subjects in our analysis of the quality of communication. In centralized treatments, the correlation between the guesses of Player 3 and those of Player 1 and Player 2 has a coefficient of $\rho =0.855$.

³⁵Notice that subjects in the role of Player 3 made two guesses in every period.

³⁶For example, even under the assumption that residuals are independent across subjects, clustering by subject or using subject‐level random effects is not appropriate when the unit of observation is a game, as in Section below.

³⁸One possible source of bias is risk aversion. The results in this section are difficult to interpret if risk aversion on the part of the message receivers is assumed, as the direction of the bias would depend on assumptions about the message receivers' beliefs.

³⁹We compute the standard errors by regressing ${(d_i-{\theta }_i)}^2$ and ${(d_1-d_2)}^2$ against treatment dummies, with standard errors clustered by session. The regression results can be found in Table S.18 in the Appendix in the Online Supplementary Material.

⁴⁰Relative to MIE, we also find that the coordination loss was significantly higher than predicted in this treatment.

⁴¹Formally, the decision $d_i$ made by Player i depends on her state ${\theta }_i$, her belief about j' state, $E[{\theta }_j|m_j]$, and her second‐order belief about the belief held by Player j about state ${\theta }_i$ after having received message $m_i$, $E[{\theta }_i|m_i]$. We proxy this second‐order belief with the belief reported by Player j. While this second‐order belief equals $E[{\theta }_i|m_i]$ in any equilibrium, the two quantities might differ in the presence of deviations from equilibrium. To alleviate this concern, we perform a robustness check in the Appendix, available in the Online Supplementary Material, by redoing our analysis with received messages in place of reported beliefs and find similar results. Unlike elicited beliefs, messages are common knowledge even in the presence of equilibrium deviations.

⁴²The results are qualitatively similar if this restriction is removed. As in the rest of our analysis, we allow the error terms ${\mathrm{ϵ}}_{it}$ to be correlated within a session. Robustness checks using subject‐clustered standard errors can be found in the Appendix in the Online Supplementary Material.

⁴⁴To conserve the same variable names, we define ${\mathit{Decision}}_{jit}$ to be the decision made by principal j in period t on subject i's behalf, $\mathit{Guess}\mathrm{\_}\mathit{of}\mathrm{\_}\mathit{the}\mathrm{\_}{\mathit{State}}_{jit}$ the principal's guess about subject i's state in period t, and $\mathit{Guess}\mathrm{\_}\mathit{of}\mathrm{\_}\mathit{the}\mathrm{\_}\mathit{Other}\mathrm{\_}{\mathit{State}}_{jit}$ the principal's guess about the other subject's state in period t.

⁴⁵We provide a number of robustness checks below, including ones in which coefficients are estimated at the level of individual subjects.

⁴⁷The econometric details and estimation results are described in Appendix D in the Online Supplementary Material.

⁴⁸The model produces no estimate for one of the subjects in Centralized‐High because this subject set each of the decisions to zero. As adaptation was ignored and full coordination achieved, we manually set $\hat{\gamma }=1$ for this subject.

⁴⁹At the time of the experiment, the minimum wage in Mexico was about 70 pesos per day, which is arguably a poor reference point for students at a private research university such as ITAM. For a better one, consider that the cost of a 15 km Uber ride was around 80 pesos.

⁵⁰Our simulation approximates the most informative equilibria by partitional equilibria with 231 elements.

⁵¹This represents the per‐period average.

⁵²Appendices F and G in the Online Supplementary Material provide an additional analysis of subjects' payoffs.

⁵³We parametrized γ to be equal to 0.75 in the complete information treatments because the observed distortions in this case were most robust to learning, as shown in Appendix D in the Online Supplementary Material.

⁵⁴While we could have removed communication from the complete information treatments, we avoided doing this so that uncertainty and communication are not manipulated at the same time.

⁵⁵As pointed out by a referee, strategic uncertainty about whether or not an opponent is playing the right game could still be present in the complete information treatments.

⁵⁶Because we only have two sessions per treatment, the standard errors for the coefficient estimates are clustered at the level of the decision maker. The comparisons remain insignificant with session‐clustered errors.

⁵⁷$P=0.831$ for the mean and $P=0.414$ for the median.

⁵⁸As we argue in Appendices H and I in the Online Supplementary Material, risk seeking preferences generate distortions in the opposite direction than observed in the data and, therefore, also cannot explain our findings. There are several possible explanations for why our results differ from those of Tversky and Kahneman (). First, our experiment differs substantially from the choice‐theoretic experiments in the prospect theory literature or related experiments such as Myagkov and Plott (). The noise in our game is not objective but derived from other subjects' communication rules. Second, it is possible that subjects' reference points are not zero (Kahneman and Tversky ()). Because incurring no losses is difficult in our environment, subjects might have formed expectations accordingly (Kőszegi and Rabin ()). Third, as we show below, ambiguity‐aversion can generate distortions in the right direction even with moderate risk‐seeking preferences.

⁵⁹In principle, risk aversion might also have influenced subjects' elicited beliefs, which we used to compute a measure of communication quality and estimate subjects' decision rules. In a regression of elicited guesses against true states, $R^2$ is approximately 0.7, which suggests that subjects' guesses of unknown states were quite good on average. That is, if risk aversion biased the reported beliefs, the resulting bias was small, which is consistent with our observation that MIE rationalizes the communication data well. Second, as we show in Appendix C in the Online Supplementary Material, several of our results pertaining to communication quality are robust to defining a measure of residual variance based on subjects' messages as opposed to elicited beliefs. Likewise, our main results regarding the biases in subjects' decision rules are reflected in the analysis of subjects' adaptation and coordination losses, which does not rely on belief elicitation.

⁶⁰Ambiguity aversion, for instance, would bias the belief‐elicitation procedure, complicating our results regarding communication quality. Moreover, it is not clear if equilibria exist of the ambiguous game where the decision makers try but fail to accommodate the message senders' needs, as we observe in the data.

⁶¹We thank an anonymous referee for pointing out this explanation.

⁶²For example, in the extreme case, a decision maker might completely ignore coordination, setting her decision equal to the other player's message.

⁶³For example, Player 1 may be maximizing $\lambda {\pi }_1+(1-\lambda ){\pi }_2$ and Player 2 may be maximizing $(1-\lambda ){\pi }_1+\lambda {\pi }_2$, where $\lambda \in [\frac{1}{2},1]$.

⁶⁴Both of these extensions are theoretically explored in Rantakari ().

⁶⁵Will subjects communicate private information more truthfully and trust more in longer relationships? Does the quality of communication decrease over time in a finite game? Does the way in which communication feeds into subjects' decision rules change over time?