Central Bank Transparency and Speculative Attacks: An Overview and Insights from a Laboratory Experiment in Tunisia

Abstract

We propose the use of experimental economics as an innovative tool to introduce economic issues. The basic game of the experiment is a simple beauty contest model by Morris and Shin. Precisely, the paper contributes to the continuous debate on the effect of transparency in a context of a speculative attack using an experimental approach. In the spirit of subsequent protocols of Heinemann et al. and Cornand such as a laboratory experiment is designed to test theoretical predictions of static coordination games, players have access to heterogeneous information sets and according to which they have to decide between a risky action whose payoff depends on the decision made by the other players, and a safe action that generates a fixed gain. Results indicate perfect coordination in 61.75% of the total situations. However, non-parametric tests reveal no evidence that players really differentiate between public and private information, and of a destabilizing effect of public information due to self-fulfilling beliefs. The findings have policy implications regarding optimal tools for information disclosure. We performed the experiment on students who do not have any previous knowledge about game theory or the context.

1. Introduction

Economic experiments are becoming increasingly popular as an innovative tool to introduce economic issues. Basically, they are used to test theoretical predictions derived from the game theory. In this paper, we give a special focus to the coordination games under heterogeneous information (public versus private) as sparked in the influential paper by Morris and Shin [1]. Such a category of games is considered as a “structure of economic decision problems” in the words of Heinemann et al. [2]. More precisely, speculative attacks on a currency peg can be modeled as a coordination game. With respect to the theoretical framework of Morris and Shin [1], central bank transparency stands for the precision of public information, but transparency, in general, includes also the precision of private information. Receiving different information in nature has consequences on agents’ beliefs, which, in turn, impact the likelihood of a speculative attack occurrence: (see Trabelsi [3] for an econometric analysis). According to Morris and Shin [1], increasing the precision of public information is not always good, and generally leads to ambiguous effects on social welfare.¹

Two main experiments that explore the use of private (specific to each agent) versus public information (observed by all) are by Heinemann et al. [2] and Cornand [5] who extends the analysis of the former to allow for signals of different nature. Szkup and Tervino [6] resume the experimental design of Heinemann et al. [2], but more precise private information is made available to agents at a certain cost. Costain et al. [7] conduct an experiment with a sequential move while previous actions are observed. They show that there is a specific region where the event of “all players attack” or “no play attacks” happens with a probability of more than 1%. Experiments on speculative attacks generalize the static setting to a dynamic one. For example, Cheung and Friedman [8] propose an entry game in continuous time. At each point in time, each player chooses to attack or not. The attack is made at a certain cost, which is cumulative as long as the player has attack status. Fehr et al. [9] make an important observation of sunspot equilibria when analyzing a coordination game of agents deciding between different assets and receiving noisy signals about them. Battiston and Harrison [10] suggest a novel approach that makes information about other players’ behavior observable through a connected network. They find that information about sunspots improves coordination, while information about other players’ behavior hampers it. Other works (See for instance Trabelsi and Hichri [11]) provide an experiment in a closer context by testing treatment of fragmented information (a common information per sub-group of agents) against a treatment of partial publicity (public information observed by a fraction of agents) that was introduced by Cornand and Heinemann [12]. Other experiments that apply coordination games can be found in Trautmann and Vlahu [13] to study strategic loan default. More recent studies complement this setting by analyzing information transmission in networks [14, 15].

Although the protocol is close to Cornand [5] – which has also been already inspired by the experimental design of Heinemann et al. [2] − three main differences (while the other parameters are kept the same) between our protocol and theirs are detected:

• The number of subjects (players) in the game;

• The use of a discrete uniform distribution of the true state and the signals for the sake of simplicity (See details in the main text of the following sections);

• The absence of real-monetary incentives.

Usually, economic experiments are computerized. The advantage of using fewer numbers of subjects allows us to perform the experiment by paper and pencil. Each player writes his or her own contribution on a sheet of paper. Interpreting the results with caution is, thus, appealing since the coordination games are sensitive to variation in the subjects’ pool. Except the monetary incentive, all the conditions of implementing a standard laboratory experiment are respected. The experiment is run on students who do not know the context or the purpose previously.

The fact that earnings (expressed in the Experimental Currency Unit) are not really paid is explained to the participants in advance. But to maintain interest, we passed out forms on which students can keep track of their hypothetical earnings [16]. Our experiment turns up an interesting attempt by applying the argument of Read [17] on the non-necessity of monetary payments in experimental economics. On p. 266, he asserts the following: “My view is that monetary incentives are not an experimental magic bullet. They are one part of the experimentalist’s arsenal [..]”. Read [17] concludes further that “[…] there is no basis for requiring the use of real incentives to do experimental economics”. He departs from the fact that people have powerful intrinsic stimulations to do their best and money decreases their cognitive exertion. According to the same author, the use or not of monetary payments is akin to whether the objective of the experimenter is to arouse extrinsic or intrinsic motives of the players. In our case, we choose to make subjects rely on their intrinsic motives to play the game.

The rest of the paper is structured as follows; Section 2 describes the theoretical framework on which the experiment is based and Section 3 presents the experimental protocol. We discuss the main results in Section 4. We provide post-experiment discussion in Section 5 and Section 6 concludes.

2. Theoretical deliberations of the economic game

This section provides a theoretical background that motivates the economic game. First, based on the theoretical predictions of the game-theoretic model of Morris and Shin [1], we develop the framework that will be used in the experimental design. Then, within this framework, we derive the main results.

2.1 Speculative attack as a coordination game under heterogeneous information

Speculative attacks can be modeled as a coordination game with strategic complementarities. In the experiment, we employ a simple beauty contest model of Morris and Shin [1] with a finite number of economic agents N who decide simultaneously whether to attack or not.² Let us Y denote the fundamental state of the economy. The higher Y is, the better the state of the economy.

The gains of agents are given as follows:

• If an agent chooses to attack, he/she gets:

  • Y, if the proportion of agents who attach exceeds a(Y).

  • 0, if the attack fails

• If an agent chooses to not attack, he/she gets a fixed reward equal to T.

Agents decide to attack or not based on two types of information (signals) given the state of the economy Y. Hence, the central bank disseminates posterior private and public information, where the public signal $Z/Y\to N\left(0,{\sigma }_{\eta }^2\right)$. Individually, they receive private signals $x_i\to N\left(0,{\sigma }_{\epsilon }^2\right)$ . Morris and Shin [19] and Hellwig [20] show that there is a unique equilibrium with a critical state below which currency crisis occurs with certainty, that provided private information is sufficiently precise relative to the public information. Metz [21] showed that the signals’ interaction may have adverse effects. Her theoretical predictions can be summarized as follows:

• If the state of the economy is bad (Y↓) and the public information is more precise than the private signal, then an attack is likely to occur.

• If the state of the economy is good (Y↑), more precise public information reduces the probability of an attack.

3. Experimental design matching theoretical model

3.1 Payoff function and signals

An unknown number Y is selected randomly from a range of values {10, 11…, 90}. Players do not know this number but have to guess it based on two types of information (signals). The first signal is public (Pu), common to all players in the same game session. The second is private (Pr). It is specific to each subject (player). Both signals are randomly selected from the same interval [Y-10, Y + 10]. The decision consists of choosing between a safe action A that brings a fixed payoff T or a risky action B that generates a payoff Y under certain conditions:

• If a player chooses a secure action A, he/she earns T =

  • 20 in stage 1 of sessions 1 and 2 and in stage 2 of sessions 3−5

  • 50 in stage 2 of sessions 1 and 2 and in stage 1 of sessions 3−5

• If a player chooses a risky action B, he/she earns:

  • Y, in these cases:

If the unknown number Y is in the interval	Then at least…of the participants have to select B in order to have a positive gain
[10,39]	3
[40,59]	2
[60,90]	1

• 0, otherwise.

3.2 Treatment and session description

A sole treatment of Morris and Shin [1] (hereafter MS) is implemented for this exercise. The treatment consists of 2 stages. Each stage comprises 8 periods (The stages differ only by the amount allocated to the choice of a safe action A). In each period and for each of the 10 situations, 3 participants have to decide between a safe action A that generates a fixed gain (T = 20 in stage 1 and T = 50 in stage 2) and a risky action B whose payoff depends on two factors: The value of the true state Y and the choice of the participants playing in a game session. This game is repeated twice (Sessions 1–2). For sessions 3−5, the participants undergo the same game but we reverse the stages’ order: in stage 1, T = 50, and in stage 2, T = 20 (cf. Table 1).

3.3 Procedural considerations

The experiment was carried out at the Higher Institute of Management of Tunis. The simulations were run using R.2.0.6 software. The 15 subjects who participated in this experiment were split into 5 groups (N = 5). Five sessions were devoted to one treatment of MS, producing a total of five independent observations per treatment.

Subjects are inexperienced students from the unit of applied and quantitative analysis (UAQUAP) switched to the social and economic policy analysis laboratory (SEPAL) and from the laboratory of operational research, decision and process control (LARODEC), both located at the Higher Institute of Management in Tunisia. Sessions lasted about one hour and a half. In each period and for each of the 10 situations, subjects have to make a decision on the true state Y given the signals.

The procedure was kept the same throughout the experiment. Three subjects were seated apart so communication was not possible. The same group of three individuals played for 16 periods. At each individual’s place, there was an instruction sheet, one response table and a piece of scrap paper on which subjects can take notes about their choices after each decision and information phase. Instructions (given in detail in Appendix A) were distributed in written form (in French) to the subjects and were read out loud before the beginning of each session. It was made sure that these instructions were well understood. Subjects were asked to raise their hands if they had any questions, and answers were given privately by the experimenter (See Appendix B for details and Table 2 for an example). To make sure that players understand the game rules, a quiz was distributed to the participants at the beginning of each session.³

4. Results

The raw data are available upon request from the author. In what follows, we use non-parametric tests and logistic regressions in order to analyze our results.⁴

4.1 Some general considerations about individuals’ behavior

4.1.1 Existence of threshold strategies

According to Heinemann et al. [2], the behavior is consistent with undominated strategy if this behavior, during a period, is consistent with the existence of thresholds.

In other terms,

• B is dominated by A if max(x_i, Z) < T-ε

• A is dominated by B if min(x_i, Z) > T+ε

Figure 1 displays the number of cases where subjects played undominated strategies. This is very predominant especially in the sixth and twelfth periods, with 65 and 62 cases in total, respectively. More generally, a threshold strategy means that subjects were not able to predict sometimes others’ choices.

Although signals differ in nature (public versus private), different interpretations of public information lead somewhat to this common information to be a private one. Consequently, common information does not lead necessarily to common knowledge. In other terms, different interpretations of public information are considered also as private ones.

Given the fact that the number of players is low, the level of reasoning of players increases sometimes but with strategies change.

4.1.2 The determinant signal of individual behavior

We estimate the probability with which a subject chooses B by fitting a logistic distribution function to observed choices. The cumulative logistic distribution is given by:

$$P\left(B\right)=\frac{1}{1+exp\left(-a-\mathit{bPr}-\mathit{cPu}\right).}$$

We analyse whether the public signal is used as a focal point by the subjects. We define the null hypothesis as follows:

H0: the weight assigned to the public signal (c) is not different from the weight assigned to the private signal (b).

We test H0 owing to a Wilcoxon matched pairs signed rank test and both of them allowed us to derive these observations:

• Players seem to be indifferent between the two types of signals. P-value is equal to 0.9594. So, there is no evidence of using public information as a focal point by the agents. In other terms, subjects give equal weights to both signals (cf. Table 3).

4.2 The threshold to successful attacks

Players have a tendency to choose the safe action (A) for low values of Y and to decide on the risky action (B) for higher values of Y (cf. Figure 2).

Table 4 gives an indication on states in which B was a successful strategy. We define the interval of indeterminacy; whose lower bound is the lowest value of Y and above which the attack is successful, and whose upper bound is the highest value of Y until which B fails. The middle point of that interval gives information about the probability of a successful attack and the amplitude of this interval indicates the predictability of a successful attack.

4.3 Probability and the predictability of a speculative attack

4.3.1 The aggregate behavior analysis

4.3.1.1 Probability of a successful attack

Table 5 displays some statistics relative to the probability and the predictability of an attack. The mean threshold to success, as well as the average width of an interval of indeterminacy, is higher when the gain associated with the safe action increases (T = 50). That observation coincides with OLS estimations.⁵ We tested the impact of control variables (cf. Table 6 for description and definition of data) on mean thresholds Y* through linear regression (Eq. 1).

$$Y_i^{\ast }={\gamma }_0+{\gamma }_1{T}_i+{\gamma }_2{\mathit{Ord}}_i+{\gamma }_3{\mathit{\text{TO}}}_i+u_i$$E1

Y* is the threshold state that characterizes the behavior of a player (an agent) who attacks if he/she “gets a signal above this threshold, and does not attack otherwise” ([7], p.2). The payoff associated with the safe action T has a significant and positive impact on the probability of an attack. Order (Ord) adds a high and significant value to the level of threshold. An interaction term is used to control for nonlinearity in the payoff function. The variable TO is included to capture the different sizes of the order effect in the two stages. It is negative but insignificant. Overall, the controls explain about 89% of the total regression.

4.3.1.2 Predictability of an attack

We regress the interval that separates between the highest state up to which action B always failed and the lowest state from which action B was always successful ($∆Y^{\ast }$) on some controls.

$$∆Y_i^{\ast }={\gamma }_0+{\gamma }_1{T}_i+{\gamma }_2{\mathit{Ord}}_i+{\gamma }_3{\mathit{\text{TO}}}_i+u_i$$E2

Eq. (3) in Table 7 shows that only order variable (Ord) has a significant impact on $∆Y^{\ast }$.

4.3.2 The individual behavior analysis

The results of individual behavior analysis are in accordance with those of aggregate behavior. We estimate the proportion of agents who choose a risky action B by using a logistic estimation, for each session as follows:

To estimate the probability of choosing an action B, we use a logit model:

$$P\left(\mathit{De}\right)=\frac{1}{1+exp\left(-\mathit{De}\right)}$$

The relationship between De and the independent variables is supposed to be linear:

$$\mathit{De}=a+b\mathit{Pr}+c\mathit{Pu}+u,\mathrm{u}\text{is the error term}$$

$$\text{Mean}=-a/\left(b+c\right)\text{and standard deviation}=\pi /\left(b+c\right)\sqrt{3}.$$

The mean of the function ($-a/\left(b+c\right)$) is an indication of the probability of an attack and its standard deviation ($\pi /\left(b+c\right)\sqrt{3}$) measures the predictability of a speculative attack (cf. Table 8).

The results can be interpreted in two ways:

1. The estimated probability that players choose B is conditional on Y and x_i, respectively;

2. The estimation of individual thresholds;

As in our previous findings, the mean of thresholds is higher when T = 50 (cf. Table 9), but the standard deviation of thresholds is lower when the payoff is lower.

4.3.2.1 Probability of a successful attack

Eq. (3) estimates the effect of control variables on the estimated mean threshold.

$$-a/\left(b+c\right)={\gamma }_0+{\gamma }_1{T}_i+{\gamma }_2{\mathit{Ord}}_i+{\gamma }_3{\mathit{\text{TO}}}_i+u_i$$E3

The payoff of the secure action contributes significantly and positively to the estimated mean threshold. The result is in accordance with the preliminary analysis following Table 9. Order effect is negative and significant. The interaction term (TO) is, however, insignificant.

4.3.2.2 Predictability of an attack

It is clear that the average standard deviation is always higher when the payoff of the secure action is high.

$$\pi /\left(b+c\right)\sqrt{3}={\gamma }_0+{\gamma }_1{T}_i+{\gamma }_2{\mathit{Ord}}_i+{\gamma }_3{\mathit{\text{TO}}}_i+u_i$$E4

Eq. (4) shows that this is indeed significant only at 10%. The interaction term (TO) regains its significance but the order variable (Ord) turns out to be insignificant although it has its expected sign.

4.4 Coordination failure

In this section, we study the impact of the informational structure on coordination. We distinguish two extreme cases: Perfect coordination versus total coordination failure.

We define perfect coordination as the situation in which all agents choose the same action. Total coordination failure is the situation in which two players choose the same action. There are a total of 494 cases where subjects played the same action (cf. Table 10).⁶ Following Cornand [5], a deep interpretation of the coordination concept can be observed through a measure of the former (i.e., number of regrettable decisions). Indeed, when individual behavior fails, this means that the subject had difficulty in interpreting the signals he/she received. Therefore, he/she was unable to predict whether an attack was successful or not. This can happen in two situations:

1. He/she chose B and the gain =0 (failed attack)

2. He/she chose A while B would have brought a higher payoff (a missed opportunity to attack)

We proceed to calculate the number of regrettable decisions (the player regrets his choice according to the situations described above). Figure 3 shows that this number is particularly high in the first periods (periods 2 and 3) of stage 1. The number of regrettable decisions decreases over the following periods and regains a higher peak − though a bit lower than in periods 2 and 3 − at the end of stage 1 (period 8). Overall, the number of regrettable decisions clearly decreases in stage 2.

5. Post experiment discussion

As mentioned earlier, students had no knowledge of game theory or of experimental economics. Most of them have majors in Finance, Economics, and Management Applied Computer Sciences. At each session’s end, we begin the discussion by introducing (briefly) the theoretical background, which involves three topics and explain them in a simple manner:

• First of all, we presented to the students the structure of the game as an economic decision problem in the real world. Hence, they know that the instructor plays the role of the central bank as a provider of public information, while subjects represent the speculators (economic agents). The unknown number Y fits the state of the economy. The private signal x_i (Pr) represents the insider information about Y. It could correspond to any information that an individual has observed, such as news received through private discussions. The second signal Z (Pu) is commonly shared by all agents. The public signal can represent information gleaned from newspaper articles or other sources that report on central bank procedures.

• Given the game structure is clear for all students, we moved to the performance analysis. They observe that their own gains depend not only on their self-beliefs but also on their beliefs about other players’ actions. Furthermore, they realize that the other players’ behavior also relies on their own actions and so on. This is the most important feature of the beauty contest game.

• Students capture the idea that when the state of the economy (Y) was bad (let us say less than 50 and that they are in the stage where the payoff of a safe action brings 50), then rational behavior would follow a secure action A rather than a risky action B even if the attack was successful.

• Students confirmed that they relied on their intrinsic motives to learn a new and novel tool (i.e., experimental economics) although they did not engage for external rewards such as real monetary payments, but because they found the experimental game interesting and gratifying. This makes our learning challenge more successful than when the topic is explored through classical courses.

6. Concluding remarks

We presented an experiment with application on the role of transparency in currency crisis models. The originality of our contribution stems from the fact that such an exercise has never been implemented (to our knowledge) as a usual laboratory economic game with a pedagogical objective afterwards: (see Holt and Tanga [16], Alba-Frenandez et al. [22]). We find that public information plays somewhat a role in subjects’ choices, but could not be considered obviously as a focal point in determining their decisions. Overall, transparency as measured by the precision of public information gives the central bank more control over traders’ beliefs. However, the ability of the central bank to predict an attack is reduced as the traders get private information (uncontrolled by the central bank) from other sources. We further note that this experiment is a “première” in Tunisia. All published or unpublished (few) papers in the literature are about experiments conducted after ours. This adds value to our contribution and encourages us to carry out further economic experiments in our country about other scientific research objectives.

Acknowledgments

The author is thankful to the students of the Higher Institute of Management of Tunis for their participation to the experiment, to Camille Cornand for the feedback after the work’s realization and to Prof. Razali Haron for useful comments.

Appendices

A.1 General instructions (Translation of the French instructions)

Today, you will participate in an experiment. Your gain will depend upon your decision and the decisions of other participants in your group.

Since your gains depend on the decisions that you will make during the experiment, it is important to understand the instructions. Read them carefully. If you have any questions, raise your hand and the experimenter will come to your desk and provide answers.

Your main task

You and 2 other individuals are asked to choose between a safe action (A) and a risky action (B) for each of the 10 situations in the Table. All you have to do is to submit A or B. Gains will be determined by the following two factors:

• The true value Y;

• The actions (A or B) chosen by the others.

At the time when you make your decisions, you will not know either of these two factors. You will not know the decision made by your two other playmates because your playmates are deciding at the same time as you. You will not know the value of the true state Y because you have to make your decision before this one is revealed. Therefore, you will need to decide based on the information that will be made available to you.

Guessing the true state Y

At the beginning of each period, you and the 2 participants in your session will receive two signals that will provide you with information about the true state Y. Both signals are randomly drawn given the true state from the interval [Y-10, Y + 10]. Because signals are randomly drawn, it is impossible to precisely predict the true state Y given the signals. However, they will give you an idea of a range where the true state Y might be. The examples below show you how signals should be interpreted:

Example 1

• The public (common) signal drawn for all the 3 participants is 54, and then every participant will know that the true state Y will be between 44 and 64.

• Suppose that you received a private signal60. On top of that, you know that the true state Y is between 50 and 70.

Based on both signals, you can deduce that the true state Y is between 50 and 64.

Example 2

• The public (common) signal drawn for all the 3 participants is 5, then every participant will know that the true state Y will be between 10 and 15.

• Suppose that you received it as a private signal12. On top of that, you know that the true state Y is between 10 and 22.

Based on both signals, you can deduce that the true state Y is between 10 and 15. Reminder: The value Y cannot be less than 10!

Guessing the action of your playmates

In the previous section, we explained how to guess the true state given the information that you receive (public and private). However, your gain will depend also on how well you can guess the actions chosen by the 2 other participants. The decisions of others are made by humans; therefore, your best option would be to try to predict the action the other participants are going to decide given their information. Here is what you know and what you do not know about the information available to other players in your group:

• They receive 2 signals, just like you do;

• You know the first signal that everyone receives. It is public (common). All players in your session will have the same signal.

• You do not know the second signal that they receive. The second signal is a private signal. It means that you cannot see private signals received by the two other players. It also means that they cannot see the private signal that you receive.

• You do know that the private signals of other players are generated in the same way as your private signal. Most importantly that they are also centred around the true state Y. Use your knowledge about the information that other players have to predict the action they will choose (A or B). Based on that, you can make your final decision for each of the 10 situations.

A.2 Instructions relative to the first stage of the MS treatment (the two stages differ only by the gain associated with the choice of action A)

For each of the 10 situations, a number Y is drawn randomly from a discrete uniform distribution {10, 11, 12…., 90}. This number is the same for all 3 participants. When you make your choice, you will not know the true value of Y until you decide.

Your gain

As explained above, your earnings depend upon the information you receive and the action you choose:

• If you choose A, you earn T = 20.

• If you choose B, you earn:

• Y, in these cases:

If the unknown number Y is in the interval	Then at least…of the participants have to select B in order to have a positive gain
[10,39]	3
[40,59]	2
[60,90]	1

• 0, otherwise.

A.3 Training questions (before the game starts)

A.3.1 Training questions (Sessions 1–2)

Tick the right answer

1- During a situation of a certain period, you receive 2 signals: Public (common) signal = 45, private signal = 38, then Y is:	between 35 and 48 □ between 28 and 48 □ between 35 and 55 □
2- At stage 1 and during the first 8 periods, you choose a safe action A, then your gain is:	20 □ 50 □
3- How many periods will you play during the game?	8 □ 16 □

A.3.2 Training questions (Sessions 3–5)

Tick the right answer

1- During a situation of a certain period, you receive 2 signals: Public (common) signal = 40, private signal = 32, then Y is:	between 22 and 42 □ between 30 and 42 □ between 30 and 50 □
2- At stage 1 and during the first 8 periods, you choose a safe action A, then your gain is:	20 □ 50 □
3- How many periods will you play during the game?	8 □ 16 □

A.4 List of Figures and Tables

Figure 1.

Number of cases where behavior is consistent with undominated strategies.

Figure 2.

Combined data from all 8 periods of one stage of a session. There are 80 values of Y selected in one stage. Dots indicate the number of subjects who played B. The hurdle function a(Y) is the minimal number of players needed for getting a reward while playing B. Dots below the hurdle function indicate states at which there was no successful attack. Dots on or above the hurdle function indicate successful attacks. Two points indicate the highest state up to which action B always failed and the lowest state from which B was always successful.

Figure 3.

Evolution of coordination.

Sessions	Treatment	Players per session	Stages	Periods
1–2	MS	3	1- T = 20	8
			2- T = 50	8
3–5	MS	3	1- T = 50	8
			2- T = 20	8
Total players			15

Table 1.

Experiment: Summary.

Situation	Pr (xi)	Pu (Z)	Choice	Y	Gain
1	36	41
2	52	57
3	22	27
4	20	21
5	41	29
6	53	59
7	19	19
8	3	3
9	82	81

Table 2.

Example of a paper sheet received by a player during a certain round of the game (T = 20).

Note: Each period is divided into a decision phase and an information phase. In the decision phase, each participant receives a table containing 10 situations of the true state Y. When all participants made their choices, the decision phase is finished and an information phase begins, where the value of Y is indicated for each situation your own decision and your gain. After the players return their papers, a new period begins and participants cannot see the information from the previous period. Nevertheless, they are allowed to take notes.

b	c	c-b	sign
0.11	0.08	−0.03	(−)
0.12	0.2	0.08	(+)
0.05	0.01	−0.04	(−)
0.03	0.08	0.05	(+)
0.06	0.07	0.01	(+)
0.14	0.12	−0.02	(−)
0.09	0.01	−0.08	(−)
0.07	0.08	0.01	(+)
0.08	−0.02	−0.1	(−)
−0.02	0.34	0.36	(+)

Table 3.

Wilcoxon matched pairs signed rank test.

Note: number of positive differences = 5.

Session	T = 20	T = 50
1 20/50	40–60	52–67
2 20/50	41–49	49–61
3 50/20	33–40	52–59
4 50/20	39–43^*	53–60
5 50/20	39–49	53–65

Table 4.

Indeterminacy interval.

Note:^*indicates sessions where states with successful and failed attacks can be clearly divided.

MS Treatment	T = 20	T = 50
Mean threshold to success (Y^*)	43.3	57.1
Average width of the interval of indeterminacy (ΔY^*)	9.8	10.6
Standard deviation	4.45	1.83
Number of sessions	5	5

Table 5.

Observed mean threshold to success and an average width of the interval between the highest state, up to which action B always failed and the lowest state from which, action B is always successful.

Notation	Nature	Definition
T	dummy	0: payoff of the safe action T = 20	1: T = 50
Ord(er)	dummy	0: session beginning with T = 50	1: session beginning with T = 20
TO	dummy	0: if Ord = 0 or T = 20	1: if Ord = 1 and T = 50
De(cision)	dummy	0: if a player chooses A	1: if a player chooses B
Ri(sk)	integer	Number of agents who are not risk-averse
xi	integer	Private signal in sessions (MS)
Z	integer	Public signal in sessions (MS)
Y^*	real	Middle point between the highest value of Y above which action B failed and the lowest value of Y above which B is always a successful strategy.
ΔY^*	integer	Indeterminacy interval width
- a/(b + c)	real	Estimated mean threshold
π/(b + c)√3	real	Estimated standard deviation of thresholds

Table 6.

Definition of variables.

N°	# observations	Independent variables: Coefficients				R^2
		Constant	T	Ord	TO	R^2 adjusted
1	10	40.5^***	14.5^***	7^**	−4.75	0.89
		(29.64)	(5.73)	(2.47)	(−1.19)	0.83
2	10	7^**	1.67	7^*	−2.17	0.44
		(2.82)	(0.47)	(1.78)	(0.39)	0.16
3	10	37.65^***	16.24^***	11.34^**	−6.17	0.85
		(14.74)	(4.5)	(2.81)	(−1.08)	0.77
4	10	8.25^*	12.53^*	11.64	−21.34^*	0.41
		(1.67)	(1.8)	(1.49)	(−1.93)	0.13

Table 7.

Results of the estimation by MCO.

Note: t-student statistics are between (). ^*, ^**, ^*** denote statistical significance at 10%, 5%, 1%, respectively.

N°session	Stage	Information	Order	payoff T	Estimated parameters^*			Estimated mean	Estimated standard deviation
					a	b	c	−a/(b + c)	π/(b + c) √3
1	1	MS	20/50	20	−9.94	0.11	0.08	52.32	9.55
1	2	MS	20/50	50	−19.62	0.12	0.20	61.31	5.67
2	1	MS	20/50	20	−2.74	0.05	0.01	45.67	30.23
2	2	MS	20/50	50	−6.25	0.03	0.08	56.82	16.49
3	1	MS	50/20	50	−6.55	0.06	0.07	50.38	13.95
3	2	MS	50/20	20	−8.27	0.14	0.12	31.81	6.98
4	1	MS	50/20	50	−5.63	0.09	0.01	56.30	18.14
4	2	MS	50/20	20	−5.68	0.07	0.08	37.87	12.09
5	1	MS	50/20	50	−3.30	0.08	−0.02	55.00	30.23
5	2	MS	50/20	20	−13.85	−0.02	0.34	43.28	5.67

Table 8.

Results of logistic regressions.

Note: Estimated parameters are based on the last four periods of each stage.

MS Treatment	T = 20	T = 50
Average estimated mean thresholds	42.19	55.96
Average estimated standard deviation of thresholds	12.9	16.9
Number of sessions	5	5

Table 9.

Average estimated means and standard deviations of individual threshold to action B.

Sessions	Perfect coordination	Total coordination failure
MS	61.75%	38.25%

Table 10.

Percentage of situations in which all subjects played the same action (perfect coordination) and in which 1 or 2 played the same action (total coordination failure).

Jel Classification

C93, D82, F3