Ill-defined versus precise pre-play communication in the traveler's dilemma.

Chakravarty, Sujoy ; Dechenaux, Emmanuel ; Roy, Jaideep 等

1. Introduction

Day-to-day language abounds with imprecise terms like "high," "low," "good," and "bad." A natural question to ask is whether there are situations in which economic agents find it beneficial to use language that is imprecise. The question becomes even more interesting if we abstract away from the likely costs involved in using extremely precise languages, such as the effort spent defining each and every term that can be used. Surprisingly, little attention has been paid to the impact that the forms of language used in conversations between economic agents may have on their decisions.

A form of imprecise communication where messages are signals that reveal finer probability distributions to opponents over the sender's private information has received a fair amount of attention in theoretical work on sender-receiver games. This type of cheap talk can be credible in these environments with asymmetric information, as was first proved by Crawford and Sobel (1982), applied in Stein (1989) and discussed in Farrell and Rabin's (2000) survey. In such games, non-binding imprecise signaling of type or intended play generates equilibria that do not exist without communication. However, in a recent contribution, Lipman (2009) shows that if precision can be obtained costlessly, then in any game with no conflicts of interest between players, the use of precise language Pareto dominates the use of imprecise language. Note that in Lipman (2009) imprecise language generates probability beliefs over some support of truth as in Crawford and Sobel (1982).

In this article, we focus attention on the effect of communication in a two-player game with conflicting interests. Specifically, we compare the effect of precise and imprecise modes of pre-play communication on behavior in the Traveler's Dilemma (Basu 1994). However, the form of imprecise language we consider is not the one that generates probability beliefs, like in Crawford and Sobel (1982) or Lipman (2009), but one that is difficult to represent by formally well-defined information operators. (1) We believe that words like "high" and "good" used in natural language fall in this category, which we call ill-defined. Furthermore, the Traveler's Dilemma is an example of a normal form game in which rationality and reasonable behavior are in conflict with each other. Indeed, the game's unique Nash equilibrium is Pareto dominated by all other symmetric strategy profiles. In such environments, where the deductive reasoning underlying equilibrium behavior leads players to slightly undercut their rival's choice, numerical pre-play communication is often ineffective because such messages are not self-committing. That is, borrowing language from Blume and Ortmann (2007), a sender has no incentive to conform with his declaration if he expects to be believed. In this context, vagueness regarding the players' intentions may be beneficial and lead to outcomes that Pareto dominate those attained without pre-play communication. Therefore, it would be interesting to investigate the ways in which externally induced ill-defined categories, in the form of pre-play communication that is "vague" or "qualitative" by design, perform in achieving more efficient outcomes for the players.

We now formally define the simultaneous moves game referred to as Traveler's Dilemma. In this game, there are two players, 1 and 2, indexed by i, with a common pure strategy space [S.sub.i] = {40, 41, ..., 200}. A pure strategy for player i is denoted by [s.sub.i]. For any R > 0, the payoff to player i at the strategy profile ([s.sub.1], [s.sub.2]) is

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for i, j [member of] {1, 2} and j [not equal to] i. The game has a unique Nash equilibrium at (40, 40), which is strict and is also the unique rationalizable outcome. Yet, as we argue here, this outcome seems to be the most unreasonable. The experiment conducted by Capra et al. (1999) provides evidence that the behavior of human players is highly dependent on the size of the reward R and that the equilibrium outcome is attained only when R is sufficiently high. They show that an equilibrium model of boundedly rational behavior helps to explain this interesting effect. Cabrera, Capra, and Gomez (2007) designed an experiment to study introspective reasoning in the Traveler's Dilemma. Their data show that even after subjects are given advice on how to best respond to the other player's claim, observed claims in subsequent periods move away from best response behavior. This seems to suggest that factors other than bounded rationality are responsible for non-equilibrium behavior.

Our experiment differs from existing studies on the Traveler's Dilemma because we let the players engage in pre-play communication, including what we refer to as communication in ill-defined categories. (2)

More specifically, when pre-play communication is in ill-defined categories, before choosing his strategy in the Traveler's Dilemma, a player may decide to send a message consisting of the word "high" to the other player. Importantly, note that such a message may be self-committing. That is, there are many interpretations of the message under which player i is better off playing a high number if he expects his rival to believe him. In the experiment, we compare submitted claims when communication in ill-defined categories is available both to claims in the standard Traveler's Dilemma, in which no pre-play communication is allowed, and to the case where the allowed form of communication is precise (as subjects announce an integer). Of course, we know from Capra et al. (1999) that for a wide range of R values, subjects in experiments consistently submit claims in excess of Nash equilibrium claims. This article contributes to the literature on the Traveler's Dilemma by providing a ranking of claims among games in which pre-play communication in ill-defined categories is allowed, in which no communication is possible, and in which communication can only be precise. Based on this discussion, we make the following conjectures: Claims under communication in ill-defined categories are higher than in the absence of communication and cannot be lower than those under precise communication.

In this study, we find no significant difference between claims without communication and claims with pre-play communication in ill-defined categories. That is, communication in ill-defined categories does not foster cooperation. However, we find that precise communication helps increase claims above the levels attained when communication is not feasible at all. These results do not support our conjectures but are in line with Lipman (2009) in the sense that costless precise communication leads to higher payoffs for both players than imprecise communication.

By studying precise, numerical communication, we contribute to the literature on cheap talk in experiments. Sally (1995) and Crawford (1998) provide surveys of early contributions. It is well known from this literature that precise numerical cheap talk does not help players reach more efficient outcomes in games where interests are not aligned (see for instance, Sally 1995; Wilson and Sell 1997; Chapman Moore, Morgan, and Moore 2001). (3) Indeed, in such environments, messages are neither self-signaling, nor self-committing (Farrell and Rabin 2000). Duffy and Feltovich (2002, 2006) present evidence that corroborates these early findings. They compared the effectiveness of numerical cheap talk in three different games including the Prisoners' Dilemma and Stag Hunt. They show that precise pre-play communication improves coordination in Stag Hunt, where interests are aligned, but it does not increase the frequency of cooperative outcomes in the Prisoners' Dilemma.

Based on previous studies, the type of precise communication that helps to foster cooperation in social dilemmas seems to involve a degree of sophistication that is impossible to achieve with simple numerical messages. Sally's (1995) survey of early experiments on the Prisoners' Dilemma shows that cooperation rates increase when subjects are allowed to discuss choices prior to committing to an action. Similarly, in Isaac and Walker's (1988) experiments on the Voluntary Contribution Mechanism (VCM), providing the subjects the opportunity to discuss choices before making binding decisions improved contribution rates. More recently, Bochet, Page, and Putterman (2006) also find that free-form communication improves efficiency in the VCM, while simple numerical cheap talk has no effect (see also Frohlich and Oppenheimer 1998). Finally, Charness and Dufwenberg (2006, 2010) show that free-form non-binding communication helps subjects to sustain cooperation in repeated trust games, while communication whereby subjects are constrained to either promise or not promise to cooperate fails to achieve the same result.

Our study complements this body of work by focusing on costless communication with simple messages rather than open conversations. Based on Crawford and Sobel (1982), Lipman (2009), and our own conjectures, instead of making communication more precise than numerical cheap talk, we study the effect of letting communication be less precise while holding the cost of sending a message constant, since, in both cases, subjects merely have to click a button in order to send a pre-fabricated message. In our data, we do find that precise communication helps to raise claims above those submitted when communication is not feasible. Claims are also higher under precise communication than with communication in ill-defined categories. Whereas communication in ill-defined categories appears to have no effect in the aggregate, subject that receive a message from the other player indicating his intention to play "high" do submit higher claims.

The remainder of the article is organized as follows. Section 2 describes the experimental design and procedures. In section 3, we test our hypotheses and analyze data on messages in the experiment. Section 4 presents concluding remarks, and the Appendix contains the subject instructions used in the experiment.

2. Experimental Design and Procedures

The primary goal of this article is to shed light on the ways in which pre-play communication affect cooperation in the Traveler's Dilemma. Our hypotheses are listed below. Hypotheses H1 and H2 test whether some mode of communication helps subjects to achieve more efficient outcomes. Hypothesis H3 is a comparative test of the effectiveness of different modes of communication at generating more efficient outcomes for the players (see, for instance, Lipman 2009). Clearly, if our tests reject neither H1 nor H2, then they will not reject H3. However, tests may reject H1 and H2 when there is no evidence against H3.

(H1) Pre-play communication in ill-defined categories in the Traveler's Dilemma leads to higher claims than those submitted in the absence of communication.

(H2) Precise pre-play communication in the Traveler's Dilemma has no effect on claims.

(H3) Pre-play communication in ill-defined categories in the Traveler's Dilemma leads to claims that are no lower than when precise pre-play communication is allowed.

We report results for eight experimental sessions in which the lower bound of the set of claims is equal to 40 and the reward/penalty parameter R is equal to 30. Note that in the Traveler's Dilemma, in order to guarantee that payoffs are always positive, the lower bound of the set of claims must be greater than or equal to R. In Capra et al.'s (1999) study, the lower bound on claims is 80, which allows for a maximum reward/penalty parameter of 80. In this article, because we implement communication in ill-defined categories, the lower bound must be a sufficiently low number so as to avoid diluting the meaning contained in a message of "high." However, this implies that the reward/penalty parameter R can be no higher than 40 in order to ensure that subjects earn positive payoffs. Finally, we use R = 30 rather than R = 40 for two reasons. First, the earnings of a subject submitting a losing claim close to 40 remain above 10 experimental francs (or 1 rupee), and second, the number 40 coincides both with the lower bound of the set of claims and the Nash equilibrium claim. This may induce an undesirable focal point at the number 40 and bias claims or lead to undue confusion.

Each session had 16 subjects who made decisions for 45 periods. The three treatments we considered were No Communication (baseline), Precise Communication, and Ill-Defined Communication. In every session, the subjects made decisions under two different treatment conditions. Sessions were of the type A-X-A or X-A-X, where A is the baseline No Communication treatment and X is either the precise communication treatment or the treatment with communication in ill-defined categories. Each of three sequences in a session lasted 15 periods. Note that we observed decisions in the baseline treatment in all eight sessions, while data for each of the communication treatments come from four sessions. See Table 1. Our design allows for both within- and between-session comparisons of the Communication/ No Communication treatments and also between-session comparisons of the Precise and Ill-Defined Communication treatments. An important aspect of our design is that no subject took part in a session both with precise communication and communication in ill-defined categories. This was in order to ensure that experience in one treatment did not impact behavior in the other. In the baseline treatment, a subject's only decision was to submit an integer claim between 40 and 200. In the Precise Communication treatment, a subject could first send a numerical message to his paired participant using the computer (there was no verbal communication). The message consisted of an integer between 40 and 200. After receiving his paired participant's message, if one was sent, a subject then proceeded to submit a claim. In the Ill-Defined Communication treatment, a subject could send a message consisting of the word "high" to his paired participant using the computer. Again, after receiving his paired participant's message, if one was sent, a subject proceeded to submit a claim. Like in Duffy and Feltovich (2002), we implemented the perfect stranger matching protocol for each sequence of 15 periods. However, in all treatment conditions, subjects learned the actual claim submitted by their participant at the end of each period.

All sessions were run at the Indian Institute of Technology Delhi in August 2007. Subjects were recruited from undergraduate classes in engineering, science, and the social sciences. Upon arrival, they were seated randomly at visually isolated computer terminals. Each subject received a set of written instructions and record sheets, which are included in the Appendix. By reading the instructions aloud, we hoped to make them common knowledge among the subjects. The experiment was programmed and conducted with the software z-Tree (Fischbacher 2007). At the end of the session, subjects were paid privately and in cash by converting their earnings from experimental francs into rupees at a known and fixed rate of I0 experimental francs for 1 rupee. Each session lasted about one hour. Average earnings were 730 rupees (roughly equivalent to $15) per subject.

3. Observed Claims

Table 2 provides mean claims in each of the eight sessions. (4) We exclude the first five periods of a sequence because of potential treatment switchover effects. Figure 1 shows average claims as they evolve over time. Each series is constructed using the data from the two sessions of each type. It is clear both from Figure 1 and Table 2 that, overall, subjects did not submit claims equal to the Nash equilibrium claim of 40. In the No Communication treatment, a signed rank test using session averages rejects the null hypothesis that average claims are equal to 40 (one-tailed, p < 0.01, n = 8). This is consistent with Capra et al. (1999), who also find that average claims are in excess of the equilibrium prediction when R is not "too high." Turning to treatment comparisons, at first glance, precise communication seems to lead to higher average claims. For instance, the latter is true in three out of four sessions when compared to No Communication and in all four sessions when compared to Communication in Ill-Defined categories.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Figure 2 provides a clear illustration of our results. The figure compares the empirical cumulative distributions of claims across treatments, again using data from the last ten periods of each sequence. Note that when comparing communication in ill-defined categories to precise communication (bottom graph), we use data from different sessions, while the other two graphs use data from the same sessions. The distributions of claims in the baseline and the ill-defined categories treatments are barely distinguishable. On the other hand, the distribution of claims under precise communication is clearly concentrated on higher values than the distribution of claims in the baseline or that under communication in ill-defined categories.

Communication versus No Communication

Because we implement a 2 x 2 x 2 design with no communication as the baseline, we use within-session signed rank tests for comparisons of the communication treatments to the baseline. Using session averages, we find that neither communication in ill-defined categories nor precise communication leads to higher claims than in the baseline (one-tailed, p = 0.13, n = 4 for both tests). Our design allows us to account for potential sequencing effects. Accordingly, we compare average claims in treatment X to the average claims in the treatment A that directly preceded X. That is, for A-X-A sessions, we compare the mean claim under treatment X in periods 21 to 30 to the mean claim under treatment A in periods 6 to 15. For X-A-X sessions, we compare the mean claim under treatment X in periods 36 to 45 to the mean claim under treatment A in periods 21 to 30. In a separate test, we also compare average claims in treatment X to the average claims in the treatment A that directly followed X. The results are shown in Table 3.

Because the non-parametric tests rely on a small number of independent observations, to further test hypotheses H1 and H2, we resort to regression analysis. Our econometric model is given by the following,

E[[c.sub.it], | [[Z.sub.t]] = [[beta].sub.0] + [[beta].sub.1] comm + [[beta].sub.2] Seq2 + [[beta].sub.3] Seq3 + [[beta].sub.4] (1/period), (1)

where [c.sub.it] is subject i's claim in period t of a session, [Z.sub.t] = (comm, Seq2, Seq3, 1/period), Seq2 is a dummy variable equal to 1 if period t is between periods 16 and 30 of the session (sequence 2), Seq3 is a dummy variable equal to 1 if period t is between periods 31 and 45 of the session (sequence 3), period is a variable numbered from 1 to 10, where period = 1 if the observation is from the sixth period of a 15 period sequence, period = 2 if it is from the seventh period, and so on, and comm is a dummy variable equal to 1 if pre-play communication was allowed in period t.

We estimate Equation 1 separately for ill-defined and precise communication using the Tobit estimator. We assume subject-specific random effects for the error term to account for likely unobserved heterogeneity (see for instance, Duffy and Feltovich 2002). Our hypotheses are as follows. Under HI, the estimate of [31, the coefficient on the communication dummy, will be positive and statistically significant when using data from sessions in which communication was in ill-defined categories. On the other hand, under H2, the estimate for this coefficient is predicted not to be significantly different from zero when using data from precise communication sessions. Estimation results are given in Table 4. The table shows two specifications for each treatment comparison. The only difference between the two specifications is the inclusion of lagged claims.

In the ill-defined categories sessions, the communication dummy is statistically significant and positive, but the effect is quite small, and it is not robust to the inclusion of lags. The precise communication regressions also show a small significant and positive effect of communication, but this effect is robust to the inclusion of lags. We close this section by summarizing our results in Finding 1.

FINDING. (a) Communication in ill-defined categories has a weak positive effect on claims. Regression analysis shows that this effect is not robust to the inclusion of lags. Hence, there is weak support for H1. (b) By contrast, precise communication has a robust positive effect on claims. Therefore, H2 is rejected in favor of the alternative that precise communication helps to raise claims.

Ill-Defined versus Precise Communication

In this section, we compare claims under the two alternative modes of communication. Our design does not generate within-session variation between ill-defined and precise communication. Therefore, we compare claims under the two different modes of communication across sessions. A Wilcoxon Mann Whitney test rejects the null hypothesis that average claims are the same under precise communication as under communication in ill-defined categories (one-tailed, p = 0.03, n = m = 4). (5)

[FIGURE 3 OMITTED]

However, it is clear from the summary statistics in Table 2 as well as from Figure 2 that claims are generally higher in sessions with precise communication than in sessions with ill-defined communication. This observation holds true even when the baseline No Communication treatment is the first treatment of the session, which suggests that session effects unrelated to the mode of communication are in part responsible for the ranking of claims across communication treatments. To account for these confounding effects, we apply a weight to average claims in the following way. Define [bar.a] as the mean claim in the first five periods of Session 8, where [bar.a] = 197. Then, for each session i = 1, ..., 7, we define an index [W.sub.i] = [bar.a]/[a.sub.i], where [bar.g] is the mean claim in the first five periods of Session i. Finally, for each session, we define the variable [[??].sub.i] = [W.sub.i] [c.sub.i], where [c.sub.i] is the mean claim in the communication treatment of Session i. The values of [[??].sub.i] are shown in Table 2. Note that the adjustment raises the mean claim substantially in some of the ill-defined communication sessions. We then run a Wilcoxon-Mann Whitney test using the (i variable. The non-parametric test rejects the hypothesis that the adjusted mean claims under ill-defined communication are equal to the adjusted mean claims under precise communication, in favor of the one-sided alternative that such claims are higher under precise communication (p = 0.03, n = 4, m = 3).

FINDING 2. Claims are lower when communication in ill-defined categories is allowed than when subjects can use precise communication. Hence, H3 is rejected.

Messages

In view of our results, it seems natural to ask whether the subjects used their option to send a message and, if so, whether the actual claims were affected by the exchange of messages. In the precise communication treatment, subjects sent a message 93% of the time, while they did so 81% of the time in the treatment with communication in ill-defined categories. As a result, messages are sent by both players in a pair in 85% of the cases under precise communication but only in 64% of the cases under communication in ill-defined categories (the differences in frequencies are statistically significant; Fisher exact test, n = m = 4, p = 0.03).

Precise Communication

The average claim in a message is equal to 196, with a median of 200. Moreover, over 75% of the messages sent are equal to 200. However, the mean and median claims submitted by a subject who sent a message are equal to 169 and 179, respectively (with a standard deviation of 30). In fact, Table 5 reveals that subjects only rarely stick to their intended claim, except perhaps in Session 8. Furthermore, most deviations are downward, but often, such downward deviations follow from receiving a lower intended claim than that submitted. The average and median deviations (i.e., claim minus message) are equal to -25.99 and -20, respectively. Figure 3 shows the distribution of deviations. Figure 3 (left) includes data from all four precise communication sessions. The mode is at 0 with 244 observations, and most deviation amounts are in the range [-100,0]. In Figure 3 (right), we exclude Session 8, in which communication was often truthful. The distribution becomes nearly bimodal, with a spike in the [-60, -50] range and most of the observations again lying between -100 and 0.

Finding lb is thus not surprising. Most subjects send a message, and three-quarters of the time, this message is equal to 200. Furthermore, more than 50% of senders deviate from their message by less than 30. It follows that for a subject, a claim of roughly 170 is relatively safe in terms of securing a win. The messages, although basically uninformative, seem to allow the subjects to anchor their claims, and as a result, these claims are higher.

Communication in Ill-Defined Categories

For a subject sending a message of "high," the average submitted claim is 111, while it is equal to 96 when no message is submitted. For a subject receiving a message (but not necessarily sending one), the average claim is 110, while a subject who did not receive a message claimed 98 on average. Therefore, in the aggregate, claims appear to be higher when either a message was sent or one was received. However, a one-sided signed rank test does not reject the null hypothesis that average claims submitted after a message was sent were equal to average claims in the No Communication treatment (p = 0.13, n = 4). By contrast, when both subjects in a pair send an ill-defined message, claims are higher than when no message is sent (n = 4, p = 0.03) or than in the baseline (n = 4, p = 0.06). However, we note that claims submitted after both subjects sent an ill-defined message are much lower in the third sequence (106.66) than in the first (143.86) or the second sequences (121.15). Importantly, we do not observe this pattern under precise communication. This last mode of communication appears to have a more robust positive effect on claims than communication in ill-defined categories.

To further investigate the effect of ill-defined messages on claims, we ran Tobit regressions based on an equation similar to Equation 1 with a subject's lagged own claim and his rival's lagged claim but using ill-defined communication data only. The relevant coefficient estimates are reported in Table 6. (6) These results reveal that ill-defined messages matter to some extent because the coefficient on the Received dummy variable is positive and significant at the 5% level. On the other hand, the coefficient on the Sent dummy is not significantly different from zero. (7)

Thus, the subjects do take ill-defined messages into account. While a message sent is not necessarily associated with a higher claim, subjects do submit higher numbers when they receive a message from their rival. This indicates that they interpret a message of "high" as a signal that their rival's claim will be higher than if no message was sent. Results in Table 4 and Finding l a show that, in the aggregate, this is insufficient to raise claims above their levels in the baseline treatment without communication.

4. Conclusion

To the best of our knowledge, our experiment is the first attempt to design an environment that allows a test of whether communication in ill-defined categories fosters efficiency in the Traveler's Dilemma. We implemented a communication structure in order to help subjects coordinate claims by using messages in ill-defined categories. Our findings indicate that compared to the game without communication, communication in ill-defined categories has a weak positive effect on claims, while precise cheap talk leads to substantially higher claims.

Regarding our two communication treatments, our analysis of messages shows that subjects submit higher claims both when they send as well as when they receive a message under precise communication. On the other hand, when communication is ill-defined, claims are solely positively affected by messages that are received. Hence, the effect of pre-play communication when this communication is ill-defined is tenuous at best, whereas precise pre-play communication leads to stable outcomes, in the sense that they persist throughout the experiment, with claims in excess of their levels compared to conditions in which communication is not allowed.

Since the precision with which one wishes to communicate is often a choice variable in the field, a natural extension of our design is a treatment that allows the subjects to choose among precise, ill-defined, and no communication prior to submitting claims in the Traveler's Dilemma. Clearly, in all subgame perfect Nash equilibria of this extensive form game, players choose claims equal to the lower bound of the strategy space. However, our results suggest that stable behavioral patterns whereby the subjects choose to send precise messages and submit claims close to the upper bound of the strategy space are likely.

Some important questions remain open. First, a formal representation of ill-defined categories is needed so that a theory of equilibrium in such categories can be built. Second, it would be of interest to know how precise versus ill-defined communication compares in games of common interests as opposed to games where interests are in conflict. Third, Capra et al. (1999) show that in the absence of pre-play communication, behavior in the Traveler's Dilemma is highly dependent on the size of the reward R. In this article, we do not study the effectiveness of each mode of communication as R is varied. However, such an analysis would likely yield important insight into the determining factors in how successful each of these modes is at improving efficiency in a game with a conflict of interest between players.

Appendix: Sample Instructions Excluding Screen Shots

Precise Communication

Instructions

General

Thank you for agreeing to participate in this experimental study on decision-making. The instructions are simple, and by following them carefully, you may earn a considerable amount of money. In this experiment, you will be given a Rs 50 "show up fee" for coming on time. You will also have the opportunity of making decisions and earning money based on your decisions and those of the other participants in this experiment. All transactions in today's experiment will be in experimental francs. These experimental francs will be converted to Rupees at the end of the experiment at the rate of 10 experimental francs = Rs 1. Notice that the more experimental francs you earn, the more Rupees you earn. All this money will be yours to keep, and your earnings will be paid to you in cash today at the end of this experiment.

Your earnings are your own business, and you do not have to discuss them with anyone. It is important that you do not talk and discuss your information with other participants in the room until the session is over. If you have a question while the experiment is going on, please raise your hand, and one of us will come to your workstation to answer it.

Today's experiment consists of three phases of 15 periods each. New instructions will be read after period 15 and again after period 30. We will start by reading the instructions for Phase 1, and then you will have the opportunity to ask questions about the procedures described for this phase.

Instructions for Periods 1-15

Decisions and earnings

These instructions deal with the first (15) periods in this session. In each period, you will be randomly matched with another participant in this room (sometimes referred to as "your paired participant" in these instructions). Over the course of these 15 periods, you will not be paired with the same person more than once. The identity of the participant you are matched with in any specific period will not be revealed to you during or after this experiment. Every period, the decisions that you and your paired participant make will determine the amount earned by each of you.

In each of these 15 periods, you will use the computer to choose an amount between 40 and 200 francs that you wish to claim. That is, your claim can be any number like 40, 41, 42 ..., 199, 200.

Before you submit an actual claim, you will first have the opportunity to send a message to the participant you are paired with. See Figure 1 for an example screen. This message must be a whole number greater than or equal to 40 francs and less than or equal to 200 francs. You may use this message to indicate your intention regarding the actual claim that you will make in the stage described below. You may also decide not to send any message, in which case simply click the OK button without entering any number in the message box.

Figure 1: Message Screen (Phase 1)

After you and the participant you are paired with have made your decisions regarding messages, you will learn the message sent to you by the other participant. If you observe no message, it means that your paired participant decided not to send a message.

You will then have to submit an actual claim. See Figure 2 for an example screen. Remember that your claim must be a whole number between 40 and 200 francs.

Note that you bear no obligation to claim the amount that you may have communicated in your message to the other participant. You may still claim any amount you want between 40 and 200, and, even if your actual claim differs from what you communicated, your earnings in this experiment will NOT be affected.

Figure 2: Decision Screen with Message (Phase 1)

Your earnings for that period will be determined in the following way:

(i) If the claims are equal, then you and your paired participant each receive the amount claimed.

(ii) If the claims are not equal, then each of you receives the lower of the two claims, AND

(iii) in addition, the person who makes the lower claim earns a reward of 30, and the person with the higher claim pays a penalty of 30.

Thus, unless the claims are equal, you will earn an amount equal to the lower of the two claims, plus a reward of 30 if you are the participant making the lower claim in your pair, or a penalty of 30 if you are the participant making the higher claim in your pair.

There is no penalty or reward if the two claims are equal, in which case you and your paired participant receive what yon claimed.

Example: Suppose that your claim is a, and the claim of the participant you are paired with is b.

If a = b, you get a, and your paired participant gets b.

If a > b, you get b minus Rs 30, and your paired participant gets b plus 30.

If a < b, you get a plus Rs 30, and your paired participant gets a minus 30.

That is, suppose a = 120 and b = 100. Then you earn 100 - 30 = 40 francs, and your paired participant earns 100 + 30 = 130 francs.

Box 1: How earnings are determined.

At the end of each period, your earnings are computed and displayed on the outcome screen as shown in Figure 3. Once the outcome screen is displayed, you should record all of the information, your message (if any), the other participant's message (if any), your actual claim, the other participant's actual claim, the reward you obtained (if any), the penalty you had to pay (if any), and your earnings for the period. Then click on the button on the lower right corner of your screen to begin the next period. Recall that you will be randomly re-matched with a new participant every period.

Figure 3: Outcome Screen with Messages (Phase 1)

Summary of Instructions for Periods 1-15

1. At the beginning of each of 15 periods, you are randomly paired with another participant in this room.

2. You may send a message consisting of an amount between 40 and 200 to the other participant or decide not to send a message.

3. After observing the other participant's message to you (if one was sent), you must submit a claim between 40 and 200.

4. If your claim is lower than your paired participant's, your earnings for the round are equal to Your claim + 30.

5. If your claim is higher than your paired participant's, your earnings for the round are equal to The Other's claim 30.

6. If the claims are equal, your earnings are equal to Your claim.

7. The messages mentioned above will have no consequence on your earnings.

At this time, do you have any questions about the instructions or procedures? If you have a question, please raise your hands and one of us will come to your seat to answer it.

Quiz on Instructions for Phase I

Before we begin this phase, we would like to confirm that everyone understands the instructions. Please, answer the following questions and if you need further clarification, please raise your hand.

1. Which of the following is correct? Circle a. or h.

a. If I send a message of 125 to my paired participant, then, when I submit my actual claim, I have to submit a claim equal to 125.

b. If I send a message of 125 to my paired participant, then, when I submit my actual claim, 1 can still submit any claim between 40 and 200.

2. Which of the following is correct? Circle a. or b.

a. My earnings depend both on my message and my actual claim.

b. My earnings do not depend on my message, but do depend on my actual claim.

For all the questions below, suppose that in a given period, you have made a claim of 83.

3. If the participant who is paired with you this period made a claim of 130, then your earnings for the period are equal to

4. If the participant who is paired with you this period made a claim of 65, then your earnings for the period are equal to

5. If the participant who is paired with you this period made a claim of 83, then your earnings for the period are equal to

Instructions for Periods 16-30 (Phase 2)

We will now begin the second phase of this experiment. These instructions deal with periods 16-30 of this experiment. You will again have the opportunity to make decisions and earn money based on your decisions and those of the other participants in this experiment.

We will start by reading the instructions for Phase 2, and then you will have the opportunity to ask questions about the procedures described.

Decisions and earnings

The phase also consists of 15 periods. In each period, you will be randomly matched with another participant in this room (sometimes referred to as "'your paired participant" in these instructions). Over the course of these 15 periods, you will not be paired with the same person more than once. The identity of the participant you are matched with in a specific period will not be revealed to you during or after the course of this experiment. Every period, the decisions that you and your paired participant make will determine the amount earned by each of you.

The only difference between Phase 2 and Phase I is that in the following 15 periods, you will NOT have the opportunity to send a message to your paired participant before submitting your actual claim. Hence, your only decision consists of submitting your claim. Again this claim must be a whole number between 40 and 200 francs. See Figure 4 for an example screen.

Figure 4: Decision Screen (Phase 2)

Your claim and your paired participant's claim will determine the amount earned by each of you. Your earnings will be determined like in the previous phase of this experiment. See Box 1 in the instructions for Phase 1.

At the end of each period, your earnings are computed and displayed on the outcome screen as shown in Figure 5. Once the outcome screen is displayed, you should record all of the information, your claim, the other participant's claim, the reward you obtained (if any), the penalty you had to pay (if any), and your earnings for the period. Then click on the button on the lower right corner of your screen to begin the next period. Recall that you will be randomly re-matched with a new participant every period.

Figure 5: Outcome Screen (Phase 2)

Summary of Instructions for Periods 16-30

1. At the beginning of each of 15 periods, you are randomly paired with another participant in this room.

2. At the beginning of each period, you submit a claim between 40 and 200.

3. If your claim is lower than your paired participant's, your earnings for the round are equal to Your claim + 30.

4. If your claim is higher than your paired participant's, your earnings for the round are equal to The Other's claim 30.

5. If the claims are equal, your earnings are equal to Your claim.

Instructions for Periods 31-45 (Phase 3)

This phase is identical to Phase 1 (Periods 1 15). For the instructions, please refer to the instructions provided for Phase 1.

Ill-Defined Communication: Partial Instructions

Instructions

General

Thank you for agreeing to participate in this experimental study on decision-making. The instructions are simple, and by following them carefully, you may earn a considerable amount of money. In this experiment, you will be given a Rs 50 "'show up fee" for coming on time. You will also have the opportunity of making decisions and earning money based on your decisions and those of the other participants in this experiment. All transactions in today's experiment will be in experimental francs. These experimental francs will be converted to Rupees at the end of the experiment at the rate of 10 experimental francs = Rs 1. Notice that the more experimental francs you earn, the more Rupees you earn. All this money will be yours to keep, and your earnings will be paid to you in cash today at the end of this experiment.

Your earnings are your own business, and you do not have to discuss them with anyone. It is important that you do not talk and discuss your information with other participants in the room until the session is over. If you have a question while the experiment is going on, please raise your hand, and one of us will come to your workstation to answer it.

Today's experiment consists of three phases of 15 periods each. New instructions will be read after period 15 and again, after period 30. We will start by reading the instructions for Phase 1, and then you will have the opportunity to ask questions about the procedures described for this phase.

Instructions for Periods 1-15

Decisions and earnings

These instructions deal with the first (15) periods in this session. In each period, you will be randomly matched with another participant in this room (sometimes referred to as "your paired participant" in these instructions). Over the course of these 15 periods, you will not be paired with the same person more than once. The identity of the participant you are matched with in any specific period will not be revealed to you during or after this experiment. In every period, the decisions that you and your paired participant make will determine the amount earned by each of you.

In each of these 15 periods, you will use the computer to choose an amount between 40 and 200 francs that you wish to claim. That is, your claim can be any number like 40, 41, 42, ..., 199, 200.

Before you submit an actual claim, you will first have the opportunity to send a message to the participant you are paired with. You may either send no message or send a message to indicate that you wish to claim a high amount of francs in the stage described below. See Figure 1 for an example screen. If you send such a message, your paired participant will receive the word High as a message. You may also decide not to send any message, in which case, you choose the option "'No message."

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Sujoy Chakravarty, Centre for Economic Studies and Planning, School of Social Sciences, Jawaharlal Nehru University, New Delhi 11006% India: E-mail sujoy@mail.jnu.ac.in.

Emmanuel Dechenaux, Department of Economics, Kent State University, Kent, OH 44242, USA; E-mail edechena@kent.edu; corresponding author.

Jaideep Roy, Department of Economics, JG Smith Building, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK; E-mail j.roy.l@bham.ac.uk.

The data and instructions for the experiment are available at http://www.personal.kent.edu/-edechena/.

We thank Kaushik Basu for several insightful conversations, as well as the editor, Laura Razzolini, an anonymous referee, Jason Aimone, Joel Elvery, and Justin Sydnor for helpful comments. Preliminary versions of this article were presented at the Northeast Ohio Economics Workshop in Cleveland (November 2007) and the Southern Economic Association Annual Meeting in Washington, D.C. (November 2008). We are also grateful to Avinash Bhardwaj, who provided valuable assistance with administering the experimental sessions, and Dr. Syamala Kallury for being so generous with laboratory facilities at the Indian Institute of Technology, Delhi. Dechenaux and Roy wish to acknowledge the hospitality of the Planning Unit at the Indian Statistical Institute (Delhi), where part of this work was completed. This research has been entirely supported by a 2007 Small Grant Scheme (SGS) research grant from Lancaster University, UK.

Received September 2008; accepted October 2009.

(1) See also Farrell (1988, 1993), Myerson (1989), Crawford (2003), Kartik, Ottaviani, and Squintani (2007), and Demichelis and Weibull (2008) for models of how language affects actions in strategic environments. However, note that in all of these articles, messages have a well-defined preexisting meaning. Our work is therefore a departure from this aspect of language.

(2) Basu (1994) coined the term ill-defined category. In an argument referred to as "Possibility 3," he suggests that there may exist a rational explanation for non-equilibrium behavior of the type that Capra et al.'s experiment would later produce. The argument introduces the notion of ill-defined categories and reads as follows: "... suppose that player 1 believes that player 2 will play a large number. Then, if player 1 were simply deciding whether he himself should play a large number or not, it would be in his interest to play a large number. Thus (large, large) seems to be a kind of Nash equilibrium in ill-defined categories." (Basu 1994, p. 395). We do not directly test this conjecture since doing so would likely require eliciting the subjects' beliefs.

(3) An exception is Silk, Kaldor, and Boyd (2000), who observe honest, low-cost signals among female rhesus macaques engaged in interactions reminiscent of the Prisoners' Dilemma. See also Parks, Henager, and Scamahorn (1996).

(4) The variable [[??].sub.i] is defined and used in the "Ill-Defined versus Precise Communication" subsection of section 3.

(5) The result is identical if we use the robust rank order test instead.

(6) The other regressors for which we do not report a coefficient are identical to those in Equation 1, comm dummy excluded. Estimates are available from the authors upon request.

(7) We also ran a model replacing the Received and Sent dummies with Both Message, a variable that equals 1 if both subjects in the pair sent a message and 0 otherwise. The results are shown in Table 6. The estimated coefficient on Both Message is similar in size and standard error to that on Received. Furthermore, a likelihood-ratio test of the difference between a model that includes Received and Both Message and one that includes Received only shows that the models are statistically indistinguishable (p = 0.18). Table 1. Experimental Design A-X-A X-A-X X = Precise comm. 2 sessions 2 sessions (32 subjects) (32 subjects) X = Ill-defined comm. 2 sessions 2 sessions (32 subjects) (32 subjects) A is the baseline, No Communication treatment. X is either Precise Communication or Ill-Defined Communication. Table 2. Summary Statistics of Claims in the Last 10 Periods of Each Sequence for Each Session Precise Communication No Communication Adjusted Mean Claim Session Type Mean Claim Mean Claim [[??].sub.i] 1 CAC 57.4 (160) 2 ACA 76.3 (320) 3 ACA 139.6 (320) 4 CAC 129.1 (160) 5 BAB 132.8 (160) 160.9 (320) 163.2 6 ABA 135.6 (320) 145.1 (160) 158.0 7 ABA 156.1 (320) 159.2 (160) 160.4 8 BAB 196.4 (160) 194.1 (320) 194.1 Ill-Defined Communication Adjusted Mean Claim Session Type Mean Claim [[??].sub.i] 1 CAC 67.7 (320) 108.9 2 ACA 84.7 (160) 134.0 3 ACA 132.2 (160) 137.9 4 CAC 148.8 (320) 156.7 5 BAB 6 ABA 7 ABA 8 BAB The number of observations is in parentheses next to the average claim. Table 3. Signed Rank Tests for Within-Session Comparisons Baseline First Baseline Second All (n = 4) (n = 4) (n = 4) Test p p p No comm = Ill-defined 0.13 0.13 0.06 No comm = Precise 0.13 0.47 0.13 For All, all observations in the last 10 periods of a sequence are used. Baseline First (Second) uses data from pairs of sequences in which the communication treatment directly followed (preceded) the baseline treatment. Table 4. Random Effects Tobit Regressions for Within-Session Comparisons of the Communication and No Communication Treatments Individual Claim Ill-Defined Constant 90.67 *** 9.37 ** (3.91) (4.75) Comm 7.64 *** 0.97 (1.35) (1.22) Seg2 -3.51 -2.35 (2.67) (3.54) Seg3 -16.03 *** -7.75 ** (3.13) (3.12) 1/period 363.00 *** 16.47 (34.37) (33.7) lagged_claim 0.52 *** (0.03) lagged_rival 0.37 *** (0.02) No. of observations 1920 1920 Left censored 244 244 Right censored 66 66 Log likelihood -8044.30 -7734.40 Individual Claim Precise Constant 145.14 *** 48.04 *** (2.53) (4.87) Comm 12.10 *** 3.87 *** (1.24) (1.06) Seg2 -15.36 *** -3.16 ** (1.36) (1.27) Seg3 -22.62 *** -5.72 *** (1.35) (1.36) 1/period 286.16 *** 76.48 *** (17.95) (17.55) lagged_claim 0.43 *** (0.03) lagged_rival 0.27 *** (0.02) No. of observations 1920 1920 Left censored 2 2 Right censored 416 416 Log likelihood -7174.91 -6927.39 Levels of significance: *** = 1%, ** = 5%, and * = 10%. Table 5. Messages under Precise Communication % Downward % Who Stuck Deviations in % Who Sent to Their % Downward Response to a Message Message Deviations Lower Message Session 5 94.7 7.3 96.6 88.8 Session 6 91.2 16.4 93.4 71.9 Session 7 89.4 11.9 96.0 71.1 Session 8 92.8 30.9 95.8 80.4 Table 6. Random Effects Tobit Regression Using Ill-Defined Communication Data Only. The Other Regressions for Which We Do Not Report a Coefficient Are Identical to Those in Equation (1), comm Dummy Excluded Individual Claim Ill-Defined Only Constant -9.23 * -6.21 (5.49) (6.1) Sent 2.42 (2.14) Received 4.44 ** (2.05) Both Message 4.23 ** (1.74) No. of observations 960 960 Left censored 131 131 Right censored 30 30 Log likelihood -3789.94 -3790.50 Levels of significance: ** = 5% and * = l0%.