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An Experimental Study of Cheap Talk Game:
One Sender or Two Senders?
Name: Yifan Tian Student number: 11009217
Track: Managerial Economics and Strategy ECTS: 15 credits
2 Statement of Originality
This document is written by Student Yifan Tian who declares to take full responsibility for contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
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Abstract
I present an experimental study of cheap talk game. I investigate what is the most informational communication way with this study. Therefore, I design a one-sender game and a two-one-sender game to examine if the receiver can get more information with two senders. In two-sender game, I distinguish two variations: simultaneous and sequential to study which is a better mechanism in information provision. Finally, I include a Choice Game to investigate the amount of senders that the receiver wants. Results show that in one-sender game, senders tell truthful messages the most often. If there are two senders, simultaneous senders have a higher truth-telling level, compared with sequential senders. When the receiver has the right to choose the amount of senders, most receivers do not show
preferences to one sender or two senders. For receivers who show this preference, most of them choose two senders.
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1. Introduction
Correct decisions made in organizations must be based on relevant specific knowledge. In many organizations, specific knowledge is spread out among agents in organizations, while the right to decide about a project initially resides with the principal. Typically, agents are better informed and the principal is not necessarily informed. Therefore, the principal has to acquire information from agents. However, agents sometimes do not transfer all information to the principal, and they even transfer false information to maximize their own payoffs. When the principal makes decisions, she consider the interest of the organization and herself. The principal, to some extent, neglects interest of agents. Hence, there exist interest contradictions between agents and the principal. It is important for the principal to get appropriate information from agents to make correct decisions and it is meaningful to investigate what is the most informational way.
Strategic communication, introduced by Crawford & Sobel (1982), is a way to model the game playing by the more informed agent and uninformed principal. The sender is informed about the true state. He sends a noisy message to the receiver whose action determines payoffs for both participants. Crawford & Sobel find that communication is more important when interests are more aligned. Some other theoretical articles include another sender in the game and investigate whether a more sender will make messages more informational. When two senders have like biases, including one more sender is not helpful in improving informational level (Krishna & Morgan 2001b, Li 2008).
5 Another topic is about the ways to transmit messages. Since there are two senders, there are different communication mechanisms: simultaneous and sequential. Li (2010) favors that simultaneous communication is more informational. Gurdal et al (2013) shows that receivers are more likely to choose simultaneous communication if they have the right to choose to play simultaneous communication or sequential communication. Ishida & Shimizu (2012) prove that sequential communication cannot be more informational than one message.
I run an experiment to investigate what is the most informational communication method: one-sender games or two-sender games, and how many senders the receiver prefers If the receiver has the right to choose from one-sender games and two-sender games. Besides, I also study whether simultaneous communication is more informational or sequential communication is more informational when there are two senders. Therefore, I design four different cheap talk games: cheap talk game with one sender, cheap talk game with two senders (including two games, the Simultaneous Game and the Sequential Game) and a Choice Game. Senders have totally same interest alignment, while they have opposing interest alignments with the receiver. In the first game, one message is sent to the receiver and the receiver makes the final decision. In the following game, subjects are divided into two games evenly. Half subjects play simultaneous game and the other half do sequential game. After first two games, the receiver moves first in the third game. She chooses whether to receive messages from one sender or two senders at the beginning of each period, then the game will go back
6 to the process of game 1 or game 2, selecting on choices of the receiver.
My experiment yields several results. First of all, the truth-telling levels are 57%, 55% and 46% in games with one sender, the Simultaneous Game and the Sequential Game respectively. It means that 57% of senders report truthful messages in the cheap talk game with one sender. 55% of senders send truthful messages in the Simultaneous Game and 46% senders transmit truthful messages in the Sequential Game. Secondly, the frequency that receivers believe messages is higher than 50%. It is obvious in the game with one sender and in the Simultaneous Game. The trust level is exact 50% in the Sequential Game. The final result is about receivers’ choices in the Choice Game. Most receivers do not have clear preferences to one-sender game and two-sender game. Receivers who show evident preferences usually choose two-sender games.
This paper has two main contributions. The first is that this is the first experimental study combining a cheap talk game with one sender and two senders together. Most former literature either studies one-sender game or studies two-sender game. The other contribution is that this is the first experimental study investigating receivers’ choices. Former literature assumes that receivers choose one sender or two senders and then runs experiments, they never discuss this topic from receivers’ point of view.
The rest of my paper is structured as follows: Section 2 includes most related literature. Section 3 introduces the model of this experiment and derives some hypotheses. There
7 are cheap talk games with one sender and two senders. Section 4 exhibits the experimental design and experimental procedures. So the following section 5 reports the experimental results. Section 6 incorporates some discussions and conclusions of this paper.
2. Literature Review
Since the seminal contributions of Crawford & Sobel (1982), there are many theoretical literatures on strategic communication. Crawford & Sobel develop a model of strategic information transmission. This communication involves two agents: A Sender who has private information germane to both and a Receiver who determines their welfare. The Sender sends a possibly noisy signal and the determination of the Receiver is based on the information included in the signal. This is the reason that it is defined as strategic communication and it plays an important role in the game. To find the equilibrium in the decision rules about reactions to the information for the Sender and reactions to the noisy signal for the Receiver, Crawford & Sobel employ the Bayesian Nash equilibrium. The model attempts to portray agents’ rational behavior in the case where two-person communication is possible. It hints that direct communication is likely to be more important when goals of agents are closer. Additionally, the rational behavior can be achieved only when agents’ interests are completely equal. When their interested are misaligned, there will be no perfect communication.
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2.1 Theoretical literature
Some previous articles extend this model into the situation where there are more than one Sender and they consider it with various directions. Gilligan & Krehbiel (1989) identify equilibria for a two-sender game in a political setting. They create a model consisting of three crucial factors. One is information cannot be collected perfectly, which means that there still exists uncertainty in this model. Another one is information is not distributed evenly. In another word, information is asymmetric. The other one is they suppose that legislative committees (senders) have heterogeneous preferences. Then they discuss the model under open, modifies and closed rules1 desperately to achieve equilibria. They find that when preferences are heterogeneous, uncertainty cannot be expunged by any rule, for the strategic communication. However, to get informational benefits, preferences cannot be too extreme. Another finding is that restrictive rules are superior to less-restrictive rules in informational benefits. Finally, they conclude that heterogeneous preferences and restrictive rules (i.e. closed rule) are informational substitutes. Krishna & Morgan (2001a) base their model on Gilligan & Krehbiel (1989) model, but get contrast conclusions. They also study informationally efficient equilibrium when committees have asymmetric preferences under these three rules. Differences are that senders are perfectly informed. Their analysis starts from heterogeneous committees and the efficiency criterion is informational efficiency. They find that it is possible to achieve full informational efficiency under open rule and
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Under the open rule, two committee members make proposals and the legislature (Receiver) chooses policies; The modified rule is similar with the open rule, committee members make proposals, but amendment rights are taken from the legislature; Under the closed rule, the policy set is more restricted.
9 modified rule. However, this outcome is precluded with additional restrictions, which means that it is impossible to attain efficiency under closed rule. Then they analyze homogeneous committees and get the conclusion that equilibria are informational inefficient under both open and closed rule in this case.
Battaglini (2002) analyses conditions to realize full information revelation. He sets up a model with multiple players in multidimensional cheap talk, which is an extension of previous unidimensional cheap talk. Battaglini uses an example of environmental bill to explain multidimensions. An important aspect for an environmental bill is the direct impact, but Congress should also think about other dimensions as well: profits or employment. Hence, this bill makes a multidimensional affect. He concludes that full information revelation is possible in this setting. Lai et al. (2015) empirically test the model in Battaglini (2002) model and they confirm that more information is transferred in a multidimensional setting. Ambrus & Takahashi’s article (2008) consider the same structure with Battaglini (2002) model, but restricts the state space. The restricted state space arises in two situations. The one is available policies are restricted, and the other one is that some policies would not be selected by the receiver under any circumstances.
In my paper, I include a second sender in Crawford and Sobel (1982) model and investigate if the receiver will receive more informational messages from two senders. Besides, I want to investigate which communication mechanism results in most information when there are two senders.
10 David Austen-Smith (1993) studies multiple referrals under open rules, which means that there is one uninformed House (Receiver) and two specialist committees (Senders) in the model and the information transmission is a “cheap talk”. Although two committees are not fully informed, they still have superior information than the House. They know the state of nature while the House does not. Furthermore, their interest preferences are not aligned. There are three ways for the House to refer a bill: joint, sequential and split. David Austen-Smith focuses on the first two ways in this article. A joint referral denotes that referral to two committees are simultaneous, while a sequential referral means that the second committee gets the information from the first one. In equilibrium, multiple referrals are more informational than single referral and committees will reveal more information in sequential referral than that in the single referral. However, committees reveal the information more often under single referral, compared with joint referral. Austen-Smith concludes that the most informative transfer happens in the case that the relatively extreme committee transfers information first when preferences are asymmetric in the sequential referral.
Krishna & Morgan (2001b) precisely include another expert in Crawford & Sobel (1982) model and examine the informational efficiency in a political setting. They investigate informational efficiencies under different rules when interest preferences are homogeneous or heterogeneous. The analysis implies that preference divergence between committees (Sender) is not always harmful to information efficiency. The condition to realize this is that legislative rules are not too restrictive. For instance, full
11 informational efficiency can be achieved when rules are open and modified in heterogeneous committees. Krishna & Morgan also study if the information revelation will be improved when the decision maker consults two experts sequentially. They find that the decision maker cannot benefit if these two experts have like biases, while more information is revealed if experts have opposing biases.
Hori (2006) sets up a model with multiple agents to compare the communication and decision making procedures in Hierarchical communication and Horizontal communication. Senders are not perfectly informed in the model. In Horizontal communication, senders send messages simultaneously and privately. In Hierarchical communication, the sender in a lower tier sends his message to the sender in the upper tier of the hierarchy, then the upper sender sends the message to the principal (Receiver). The principal cannot communicate directly with the sender in a lower tier in hierarchical communication. He analyses optimal communication procedures in situations where bias is large or small, and the bias are heterogeneous or not. The finding is that Horizontal communication is the most informational procedure if the principle is able to commit to her action. Otherwise, Hierarchical communication is more informative than other procedures.
Li (2008) considers a model with two perfectly informed experts (Sender) and one uninformed principal (Receiver) and compares the information the principal can get when she consults one expert and two. In this model, experts are biased and their biases
12 are private. Li shows that the principal may get less information when she solicits from two experts. Li (2010) further analyses this model to study the efficiency of information transmission in various communication mechanisms and the efficiency is measured by the decision maker’s expected utility. When the decision maker (Receiver) chooses to ask information from two experts, she has following options: Direct communication mechanisms and Hierarchical communication. Direct communication includes two ways, the one is the decision maker consults experts simultaneously, the other one is she consults sequentially. Compared with previous literature, this paper assumes that biases of experts are uncertain. The findings are as follows: simultaneous consultation is the most efficient mechanism, and hierarchical consultation sometimes does better than sequential consultation.
Gick (2008) extends Sobel (2008) model by adding a second sender and investigates if fully revealing equilibrium exists. The finding is that partially revealing equilibrium exists in this situation. However, this paper proves that more information is revealed with a second sender even if there are biases. Ishida & Shimizu (2012) construct a model of strategic information transmission where the decision maker is endogenous to choose how many experts (one or two) she solicits information from. When the decision maker chooses two experts, they will transfer information sequentially to her. They show transferred information is less when the decision maker cannot resist the temptation to ask for a second expert and she is indecisive.
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2.2 Experimental literature
Experimental articles are much fewer than theoretical ones, but there are still some valuable articles. Dickhaut et al. (1995) use a laboratory experiment to examine strategic information transmission with single sender and single receiver. They vary the preference divergence of the sender and the receiver to examine the truth revelation and action choices. They address the influence of preference alignments. The probability that the receiver’s actions are responding to the state reduces and information disclosures from the sender are less with a larger preference misalignment.
Gneezy (2005) runs an experiment to study the relationship between payoff and strategic information. He concludes that changes in wealth is a motive for people to lie. When people make the decision to lie or not, they are sensitive to what they can gain from lying. Another conclusion is that when people make decisions, they not only care about wealth of themselves, but also care about wealth of others. However, this unselfish motivation diminishes with the size of wealth.
Sánchez-Pagés & Vorsatz (2007) run an experiment to study overcommunication in a sender-receiver game. There is one sender and one receiver in the model and the sender is more informed than the receiver. In this experiment, the interest misalignments between the sender and the receiver are increased. To explore if a tension between the incentives to lie and social norms can explain the overcommunication phenomenon, Sánchez-Pagés & Vorsatz include a cheap talk game and a punishment game in the
14 experiment. In punishment game, the receiver can punish the sender if sender lies. It shows that there exists excessive truth-telling in the cheap talk game, but this phenomenon vanishes with the increase of interest misalignments. The excessive truth-telling also decreases if subjects who prefer to tell the truth are excluded, which is found in the punishment game.
Gurdal et al. (2013) run a model based on Sánchez-Pagés & Vorsatz (2007) model. They consider a similar setting, but include an additional sender, who has the same information as the other sender. Subjects participate in three games. Senders transfer information simultaneously in the first game, and sequentially in the second game. In the third game, the decision maker has the right to determine which communication mechanism is applied. They conclude that transferred information is very different in the simultaneous game, when compared to the sequential game. Sender 2 conditions his behavior on sender 1 in sequential game, especially sender 1 tells lies. This increases the frequencies for sender 2 to tell the truth. As a result, there exist more conflictive messages in the sequential game than those in the simultaneous game.
3. The model and theoretical predictions
This paper is closely related to the chap talk game presented in Sánchez-Pagés & Vorsatz model (2007) and Gurdal et al. model (2013). My theoretical predictions are largely derived from models presented in Sánchez-Pagés & Vorsatz and Gurdal et al.
15 Below I will present a summary of the predictions that are relevant for my research. I consider a setting with one receivers and two senders (in line with Gurdal et al. (2013)) and compare this to a setting with one sender and one receiver (in line with Sánchez-Pagés & Vorsatz, 2007). The aim of my study is to investigate if the receiver (she) can get higher qualified information with an additional sender (he) and what kind of communication mechanism is more efficient when there are two senders in cheap talk games. In this model, the interests of senders are the same, but senders and the receiver are not aligned in interests. Besides, senders are more informed about information than the receiver. At the beginning of this game, Nature randomly
chooses a payoff table θ from Table A and Table B (see Table 1), which means p(A) = p(B) = 0.5. Senders can observe which payoff table is chosen by Nature and then decide to send a message from the message set M = {A, B} to inform the receiver about θ. I denote p and q as the probabilities that senders send message A to the receiver. After observing messages sent by senders, the receiver has to choose an action from Action Up and Action Down ({U, D}). γ is labeled as the probability that U is played by the receiver. Earnings are realized based on the actual payoff table and the receiver’s action.
Table 1a. Payoff Table A
Table A Sender 1 Sender 2 Receiver Action U
Action D
4.5 4.5 1 0.5 0.5 9
16 Table 1b. Payoff Table B
I consider four different games: one-sender game, two-sender game and the Choice Game. In two-sender game, there are two variations. The one is the Simultaneous Game and the other one is the Sequential Game. Next, I introduce each game as follows. In the one-sender cheap talk game, there is one sender and one receiver. The sender knows the actual payoff table and sends a message to the receiver. The
receiver gets the message and then chooses action Up or action Down to realize earnings. In two-sender cheap talk game, two variations are set as follows. Half of the participants participate in the simultaneous variation. These participants first play the one-sender cheap talk game, then play the simultaneous game and finally play the Choice Game. The other half of the participants participate in sequential variation. These participants first play the one-sender game, then play the sequential game and finally play the Choice Game. Whatever variation participates play, there are always two senders and one receiver in a group. In the Simultaneous Game, sender 1 and sender 2 transmit messages simultaneously and privately. The receiver observes their messages and make her decision. In the Sequential Game, sender 1 transmits his message to sender 2 and the receiver first. Then, after observing the message from sender 1, sender 2 sends his message to the receiver. The receiver chooses an action
Table B Sender 1 Sender 2 Receiver Action U
Action D
0.5 0.5 9 4.5 4.5 1
17 based on the two messages. The last game is a Choice Game. Subjects will have a general idea about their earnings in each game. Therefore, the receiver moves first and decides the number of senders that they want to get information from. Then they will proceed various games according to the receiver’s choices.
3.1 The one-sender cheap talk game
In the one-sender game, there is one sender and one receiver. When Nature selects payoff table A, the sender communicates with the probability 𝑝$= p (M=A | θ=A) that payoff table A is selected. Thus, he lies with the probability 1- 𝑝$= p (M=B | θ =A). Similarly, when the actual payoff table is B, the sender transmits the truth with probability of 1- 𝑝&= p (M=B | θ=B) and he lies in this case with the probability 𝑝&= p (M=A | θ=B). Next, I discuss the receiver’s belief system. When the receiver observes message A from the sender, the probability that she believes that the actual payoff table is A is 𝜇$ = 𝜇 (θ=A | M=A) whereas the receiver believes with the probability 1- 𝜇$ = 𝜇 (θ=B | M=A) that payoff table B determines earnings. When m = B, the receiver thinks with the probability 𝜇& = 𝜇 (θ=B | M=A) that the sender tells a lie and with probability 1- 𝜇& = 𝜇 (θ=B | M=B) that payoff table B is the actual payoff table. Then, the receiver should make a choice from {U, D}. I describe the probability that the receiver chooses action U as 𝛾. It means when the receiver observes message A, she will play action U with the probability 𝛾$= p (U | A), while play action D with the probability 1- 𝛾$= p (D | A). Similarly, if message B is sent to the receiver, she will choose U with the probability 𝛾&= p (U | B) and D with
18 probability 1- 𝛾&= p (D | B). Finally, earnings realize and subjects receive their
payoffs.
(Based on Sánchez-Pagés & Vorsatz (2007))
Any sequential equilibrium of the Benchmark Game satisfies 𝑝$= 𝑝& = p ∈ [0, 1]; with the supporting belief system (𝜇$, 𝜇&) = (*+, *+).
In the appendix A, I include a proof following Gurdal et al. (2013)
(Based on Sánchez-Pagés & Vorsatz (2007))
The probability of observing an untruthful message by the sender in an sequential equilibrium is 1 2.
Sánchez-Pagés & Vorsatz show that 𝑝$ = 𝑝& = p. The probabilities of telling lies are (1 − 𝑝$) and 𝑝& when actual payoff tables are table A and B respectively and Therefore, the expected probability that the receiver will see an untruthful message is given by the following equation:
*+(1 − 𝑝$)+ *+𝑝&= *+−*+𝑝$+*+𝑝&= *+
19 Following Gurdal et al. (2013), I will next consider the setting with two senders and one receiver. As my setup is the same as Gurdal et al. (2013), I will now summarize their theoretical predictions. The proofs are available in Gurdal et al. (2013).
3.2.1 The Simultaneous Game
In Simultaneous game, after observing the chosen payoff table, both senders submit their messages privately and simultaneously. Each sender has two message sets, {A} and {B}. The receiver then makes a choice between U and D. The remaining process is the same with the Benchmark Game.
Taken from Gurdal et al. (2013) Proposition:
The set of the sequential equilibrium of the Simultaneous Game is given by the set of strategies
𝑝$= 𝑝& = p ∈ [0, 1]; 𝑞$= 𝑞& = q ∈ [0, 1];
with the supporting belief system 𝜇34= *+, for every 𝑖, 𝑗 ∈ {𝐴, 𝐵}.
Proof. See Gurdal et al.’s article (Cheap talk with simultaneous versus sequential
messages) in 2013.
20 The probability of observing an untruthful message by any sender in an sequential equilibrium is 1 2.
For sender 1, the probabilities of telling lies are (1 − 𝑝$) and 𝑝& when actual payoff tables are table A and B respectively and 𝑝$ = 𝑝& = p. Therefore, the expected probability that the receiver will see an untruthful message is given by the following equation:
*+(1 − 𝑝$)+ *+𝑝&= *+−*+𝑝$+*+𝑝&= *+
With the same logic, the expected probability of observing an untruthful message from sender 2 is also 1 2.
3.2.2 The Sequential Game
The difference between the Sequential Game and the Simultaneous Game is that senders in the Sequential Game transmit their messages sequentially and publicly. Therefore, sender 1 still has two message sets, but sender 2 has four, because sender 2 can observe the messages sent by sender 1. I denote the strategy of sender 2 as 𝑞<(𝑖). 𝑛 ∈ {𝐴, 𝐵} represents the chosen payoff table and 𝑖 ∈ 𝐴, 𝐵 is the message from sender 1. Others remain the same with simultaneous game.
Taken from Gurdal et al. (2013) Proposition:
The set of the sequential equilibrium of the Sequential Game is given by the set of strategies
21 𝑝$= 𝑝& = p ∈ [0, 1];
𝑞$ 𝐴 = 𝑞& 𝐴 = 𝑞* ∈ 0, 1 ; 𝑞$ 𝐵 = 𝑞& 𝐵 = 𝑞+ ∈ 0, 1 ; with the supporting belief system 𝜇34= *+, for every 𝑖, 𝑗 ∈ {𝐴, 𝐵}.
Proof. See Gurdal et al.’s article (Cheap talk with simultaneous versus sequential
messages) in 2013.
Taken from Gurdal et al. (2013) Corollary:
The probability of observing an untruthful message by any sender in an sequential equilibrium is 1 2.
Sender 1 faces the same message sets in Sequential Game and Simultaneous Game, so it is straightforward that he tells the truth with the probability of 50%. Sender 2 faces four messages sets in this game and the expected probability of observing an
untruthful message for the receiver is expressed as:
*+ 1 − 𝑝$ 1 − 𝑞$ 𝐵 + 𝑝$ 1 − 𝑞$ 𝐴 + *+𝑝&𝑞& 𝐴 + 1 − 𝑝& 𝑞& 𝐵 = *+
3.3 The Choice Game
Since the qualities of messages in equilibria are the same and expected earnings for the receiver are also the same, the receiver should be impartial to choose between
22 games with one sender and two senders. After the receiver’s choice, one of the
equilibria of the chosen game is played.
3.4 Hypotheses
Based on theoretical predictions presented above, I can derive the following hypotheses about the truth-telling level of senders and trust levels of the receiver.
Hypothesis 1a. The truth-telling levels of the senders are the same, 50%, in all
games.
Hypothesis 1b. In the Sequential Game, messages from sender 1 will not influence
the messages from sender 2.
Senders can observe selected payoff tables after games start, but this cannot guarantee they can get most earnings. The receiver is the one who has the right to make final decisions. Therefore, the realization of senders’ earnings is dependent on the receiver’s choice. However, senders are uncertain about the receiver’s belief. To maximize senders’ earning, they should always send untruthful messages if they know that the receiver always trusts messages. If senders think that the receiver never believes messages, they should send truthful messages. Hence, in the situation that senders do not know about the receiver’s belief, the best way is to send truthful messages with 50%. This is also the reason that messages from sender 2 will not
23 affected by messages from sender 1. Sender 2 reports truthful messages in 50% cases to maximize his own earnings.
Hypothesis 2. The receiver will trust messages with 50% probability when
observing a non-conflictive message, independent Simultaneous Game or Sequential Game.
Non-conflictive messages occur in two-sender games. There are two messages from two senders in this game. If the receiver observes that two messages are both {A, A} or {B, B}, these messages are non-conflictive. The intuition is similar with hypothesis 1. When senders are uncertain about the receiver’s belief, the receiver is also
uncertain about senders’ beliefs. It means that the receiver does know that observed messages are truthful or not, especially two messages have the same content. Therefore, to maximize her own earnings, the receiver will trust messages from senders in 50% level.
Hypothesis 3. Receivers are indifferent between playing games with one sender or
two senders.
Since the truth-telling level of senders in equilibria is the same and expected earnings for the receiver are also the same, the receiver should be impartial to choose between games with one sender and two senders.
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4. Experimental Design and Procedures
This experiment was conducted with pens and paper in a junior high school in China. The subjects are students from 1st grade in this school. I ran this experiment in their Information and Technology classes, which guaranties that all students attended this experiment. As I explained above, there are four games in this experiment: The one-sender cheap talk game, the Simultaneous/ Sequential Game and the Choice Game. Each game contained 6 periods, 18 periods in total. Students from class 1 played the role “sender 1”, students from class 2 were “sender 2” and students from class 3 were labeled as the “receiver”. Notice that subjects were randomly distributed to proceed the Simultaneous Game and the Sequential Game in the two-sender game. Hence, the process could be either the cheap talk game with one sender-Simultaneous-Choice Game or the cheap talk game with one sender-Sequential-Choice Game. Before the experiment started, subjects were randomly assigned to groups of 3 and they kept their roles throughout the experiment.
In the one-sender game, subjects played the following game: First subjects read the instruction for the experiment and learned their roles. Then senders observed the payoff table (A or B) and sender sent a message about M to the receiver. This
message could be either “The payoff table in this period is A” or “The payoff table in this period is B”. Next, the receiver was informed about the message and chose the
25 action “U” or “D” to realize payoffs. She would answer the question that “Which action do you want to take?”
In the two-sender game, subjects were equally divided into two variations. Half participated in the Simultaneous Game: Senders learned the chosen payoff table in this period and transmitted their messages to the receiver privately and
simultaneously. Messages could be either “The payoff table in this period is A” or “The payoff table in this period is B”. Next, the receiver learned about the messages and took an action “U” or “D” to realize payoffs. She had to answer the question that “Which action do you want to take?” The other half participated in the Sequential Game. The distinction between these two variations is that sender 1 sent his message to sender 2 and the receiver after he learned about the chosen payoff table. Sender 2 observed sender 1’s message first, and then sent his message to the receiver. The rest of the timing is the same as in Simultaneous Game. At the end of this game, receivers would receive a summary of their earnings in the first two games. The purpose of this step is to give them a reference for their choices in the last game.
In the Choice Game, the receiver moved first and decided the game that she would like to play. Specifically, the receiver chose to play the one-sender game, or Simultaneous/Sequential Game at the beginning of each period. Following this choice, the game was proceeded as above. After collecting choice from receivers, the
26 experimenter distributed answer sheet to senders according to receivers’ choices. Senders chose messages to send and the receiver then made decisions.
After these games, subjects were distributed a questionnaire, which included some basic information and subjects’ strategies. Finally, three students (one in each class) were randomly chosen to be paid and the others got candies.
5. Experimental results
Generally, there are 29 groups in this experiment. Hence, there are 87 students participating in the experiment. 35 students are girls and the other 52 students are boys. Their ages are either 14 or 15 years old. Among 29 groups, 14 groups play simultaneous game and 15 groups play sequential game when there are two messages. In addition, there are 18 boys and 24 girls in the simultaneous game, while 34 boys and 11 girls in the sequential game. According to my hypotheses, I calculate the truth-telling level and the trust level for all groups. Results are shown below.
5.1 Truth-telling level
I first describe the behavior of senders in the cheap talk game with one sender and two senders. In the hypotheses, the behavior of senders remains the same in all games. Table 2 shows the behavior of senders in the first game.
Table 2. Behavior of One Sender
Game with one sender
% Sender is truthful 57*
p-value 0.0810
Number of groups 29
27 The truth-telling level is 56.90% in the cheap talk game with one sender. It means that 57% of the senders report the payoff table truthfully, which is significantly different from 50% (p-value is 0.0810 in a Wilcoxon signed-rank test).
Then I portray the behavior of senders in the cheap talk game with two senders. There are two distinct communication mechanisms in the second game. Table 3a describes senders’ behavior in the Simultaneous Game and part of senders’ behavior in the Sequential Game. Column 2 reports senders’ behavior in the Simultaneous Game and Column 3 exhibits behavior of senders in the Sequential Game. I compare the behavior of senders in the Simultaneous Game and Sequential Game in Column 4. I describe and compare senders’ behavior in three dimensions. The first dimension is a generalization of senders’ behavior. From the second row, I learn that senders present excess truth-telling level in the Simultaneous Game. 55% of senders tell the truth in the Simultaneous Game, which is significantly higher than 50% level (p-value is 0.0804 in the Wilcoxon signed-rank test). In the Sequential Game, only 46% of senders transmit the payoff table truthfully, which is not significant from 50% (p-value is 0.7789 in a Wilcoxon signed-rank test). Although messages from Simultaneous Game are more likely to be truthful than those from Sequential Game, Column 4 shows that this difference is not significant in statistics (p-value is 0.1794 in the Wilcoxon rank-sum test). The second dimension is the percentage that messages sent from senders are non-conflictive. Non-conflictive means that messages from two senders are the same. They are either {A, A} or {B, B}. Percentages from the two communication mechanisms are pretty similar. In Simultaneous Game, 42% of sent messages are either non-conflictive, where the non-conflictive level is 39% in the Sequential Game. Levels are not significantly different, because the p-value is 0.7829 in the Wilcoxon rank-sum test. The last dimension to describe senders’ behavior is the truth-telling level of non-conflictive messages. As I explain above, non-non-conflictive messages could be {A, A} and {B, B}, my purpose is to show the percentage of {A, A} messages. In the Simultaneous Game, 63% non-conflictive messages report payoff tables truthfully, while there are 40 truthfully non-conflictive messages in the Sequential Game. The table shows that both are not significantly different from 50% (in the Simultaneous Game, p-value is 0.1355 in a Wilcoxon signed-rank test and 0.4690 in the Sequential Game). Additionally, the probabilities that non-conflictive messages are correct in two
28 mechanisms are not significantly different (p-value is 0.1211 in a Wilcoxon rank-sum test).
Table 3a. Sender Behavior in the Two-sender Game Simultaneous
Game Sequential Game
P-value Sim = Seq % Sender is truthful 55* 46 0.1794 % Messages are non-conflictive 42 39 0.7829 % Non-conflicting messages are correct 63 40 0.1211 P-value % Sender = 50% 0.0804 0.7789 —— P-value % Correct messages = 50% 0.1355 0.4690 —— Number of Groups 15 14 ——
*significant at 10% confidence level
Next, I describe the rest part of senders’ behavior in the Sequential Game. Table 3b offers a summary of the rest characteristics of senders’ behavior in the Sequential Game. As senders privately make decision in the Sequential Game and sender 2 observes sender 1’s message before making decision, I test truth-telling levels of sender 1 and sender 2 separately. 44% of sender 1s send messages the payoff table truthfully, which is not significantly different from 50% (p-value is 0.7048 in a Wilcoxon signed-rank test). For sender 2, 48% sender 2s report truthful messages, which is not significantly different from 50% as well (p-value is 0.9317 in a Wilcoxon signed-rank test). Truth-telling levels of sender 1 and sender 2 are similar, and the levels are also close to the overall level in the Sequential Game.
29 Table 3b. Sender Behavior in the Sequential Game
Sequential Game p-value
% Sender 1 is truthful 44 0.7048
% Sender 2 is truthful 48 0.9317
% Sender 2 is truthful
when sender 1 is truthful 35* 0.0781
% Sender 2 is truthful
when sender 1 lies 58 0.2817
% Sender 2 lies when
sender 1 is truthful 65** 0.0485
% Sender 2 lies when
sender 1 lies 42 0.3086
Number of groups 14
*significant at 10% confidence level **significant at 5% confidence level
I next examine if messages from sender 1 affect messages of sender 2. Since theory predicts that messages from sender 2 are independent from messages from sender 1, the probability that sender 2 reports truthful messages is 50% in the case that sender 1 tells the truth and the probability of telling the truth is also 50% when sender 1 lies. Similarly, the probability that sender 2 tells lies is 50% no matter sender 1 tells the truth or not. Table 3b shows that when sender 1 transfers messages truthfully, the frequency that sender 2 is 35%, which is significantly lower than the theoretically predicted level of 50% (p-value is 0.0781 in a Wilcoxon signed-rank test). After observing truthful messages from sender 1, the frequency that sender 2 lies to the receiver is 65%, which is significantly larger than 50% (p-value is 0.0485 in a Wilcoxon signed-rank test). It indicates a tendency that sender 2 tries to revert messages from sender 1. Situation changes when sender 1 lies to the receiver. Although sender 2 still behaves the tendency to revert sender 1’s messages, the tendency cannot be supported in statistics. When sender 2 learns that sender 1 lies, 58% of sender 2s choose to tell actual payoffs. This percentage is not significantly from 50% (p-value is 0.2817 in the Wilcoxon signed-rank test). 42% sender 2s also lies to the receiver when sender 1 lies, which is not
30 significantly different from 50% (p-value is 0.3.86 in a Wilcoxon signed-rank test). Besides, the percentages that sender 2 reports payoff table truthfully are significantly different in situations that sender 1 tells the truth or lies. The difference is significant in the Wilcoxon rank-sum test (p-value is 0.0468), which also proves the reverting tendency.
Table 3 shows that, in general, receivers are more likely to get truthful messages in the Simultaneous Game than in the Sequential Game. Moreover, I investigate whether the cheap talk game with one sender is better or the Simultaneous Game is better for receivers in terms of truth-telling level. Table 4 exhibits the comparison results. When I compare truth-telling levels of the cheap talk game with one sender and two senders, I use the average truth-telling level of two senders in the two-sender game. The difference between truth-telling levels of one sender and two senders is not significant (p-value is 0.1015 in the Wilcoxon signed-rank test). However, this result is possibly affected by truth-telling level in the Sequential Game. As I discussed above, the behavior of sender 2 relates to the behavior of sender 1, so messages from the two senders are not independent. Furthermore, I compare truth-telling levels of one-sender game and the Simultaneous Game. In the game with one sender, I use data from 14 simultaneous groups in the second game. For Simultaneous Game, I calculate the average truth-telling frequency. Since senders make decisions privately, messages from different senders will not affect each other. The p-value is 0.0323 in the Wilcoxon signed-rank test, and it means that the truth-telling levels of these two games are significantly different.
Table 4. Comparisons of One-sender and Two-sender Game
One-sender game Two-sender game p-value one = two % Sender is truthful % Sender is truthful in simultaneous groups 57 68 51 55 0.1015 0.0323
31
5.2 Trust level
Given messages from senders, the receiver will choose an action from {U, D}. Since payoffs are the same for subjects in all three gamess, I assume that subjects are rational and their preferences remain constant. Hence, I construct a measure for trust level of receivers. The measure will equal 1 in two situations. The one is that the receiver takes the action D after observing message A. The other one is the receiver takes the action U after observing message B. Similarly, the measure will equal 0 in situations that the receiver takes the action D when message is B and action U when message is A. The trust levels in each game are shown in Table 5.
In the first game, the receiver gets one message. On average, the percentage she trusts the message is 70% of the cases. It means that most receivers believe that transferred messages are true, which is significantly different from the theoretical prediction of 50% trust level (p-value is 0.0048 in the Wilcoxon signed rank test).
In the second game, there are two messages from two senders. Besides, there are two communication mechanisms. To calculate the trust level of receivers, I first focus on cases that senders transmit non-conflictive messages. Non-conflictive messages refer to messages that are both truthful or both untruthful. Table 5 is a summary of overall trust levels. The overall trust level in the Simultaneous game is 68% and 50% in the Sequential Game, which are not significantly different from 50% (p-value is 0.1658 in a Wilcoxon signed-rank test for Simultaneous Game).
Table 5. Trust Level with Non-Conflictive Messages
Games Trust level (%)
The one-sender game 70***
The overall level in Simultaneous Game 65
The overall level in Sequential Game 50
***significant in 1% confidence level
Then I study trust levels when sent messages are conflictive. Theoretically, the probabilities the receiver believes sender 1 and sender 2 are the same, 50%. The
32 statistical results in Table 6 are consistent with the theoretical predictions. In the Simultaneous Game, frequencies the receiver trust sender 1 and sender 2 are really similar and difference between two frequencies is not significant (p-value is 0.8737 in a Wilcoxon signed-rank test ). Trust levels are 52.5% and 47.5% respectively. None of them is significantly different from 50%, which is shown in row 4. In the Sequential Game, the receiver trusts sender 1 in 64% cases, while she trusts sender 2 in 36% cases. Nevertheless, both are not significantly different from 50% (p-values are shown in the last row) and the difference between the two frequencies is still not significant (p-value is 0.1140 in a Wilcoxon signed-rank test). Trust-levels seem to be pretty distinct and the receiver trusts sender 1 more often than sender 2. The potential reason that the distinction is not significant in statistics is that samples are too small. There are only 14 simultaneous groups and 15 sequential groups.
Table 6. Trust Level with Conflictive Messages
Games Sender 1 Sender 2 p-value, S1=S2
Simultaneous 52.5 47.5 0.8737
Sequential 64 36 0.1140
P-value Sim* 0.8488 0.8989 ——
P-value Seq** 0.1076 0.1208 ——
* Sim is the abbreviation of Simultaneous ** Seq is the abbreviation of Sequential
5.3 The choice game 5.3.1 Receivers’ choices
Generally speaking, receivers can obtain more truthful messages from one sender not two. Table 7a displays receivers’ preferences in the simultaneous groups, considering with earnings. There are 5 groups that receivers got more earnings in the game with one sender (𝐸* > 𝐸+), but only one group chose one sender more often than two senders. Another 1 group chose one sender in 3 periods and two senders in the other 3 periods as well. The other groups were more often to choose to receive messages from two senders. When the receivers got the same amount earnings from one sender and two senders (𝐸* = 𝐸+), they all chose to play with one send and two senders in 3 periods
33 respectively. In the case that receivers got less earnings with one sender (𝐸* < 𝐸+), there were 7 groups facing this situation. Two of them were more often to choose games with two senders, and two groups were more often to choose games with one sender. The other 3 groups chose one sender and two senders equally. For simultaneous groups, 6 of 14 groups chose one sender and two senders 3 periods respectively. There was also a large amount of groups (5) that were often to get messages from two senders. Only 3 groups chose one sender more often than two senders.
Table 7a. Choices of Groups in the Simultaneous Game
𝐸* > 𝐸+* 𝐸* = 𝐸+** 𝐸* < 𝐸+*** Number of Groups in Total 10 8 11 Number of Groups 5 2 7 Number of Groups: More One-sender 1 0 2 Number of Groups: 3 One-sender 1 2 3 Number of Groups: Less One-sender 3 0 2
*The receiver gets more earnings in games with one sender than in the games with two senders; **The receiver gets the same earnings in games with one sender and two senders
***The receiver gets less earnings in games with one sender than in the games with two senders.
Table 7b shows the choices related to receivers’ earnings in the Sequential Game. There were 5 groups that receivers get more earnings in the game with one sender (𝐸* > 𝐸+), but none of these groups chose one sender more often than two senders. 2 groups chose one sender in 3 periods and two senders in 3 periods as well. The other 3 groups were more often to choose to receive messages from two senders. When the receivers got the same amount earnings from one sender and two senders (𝐸* = 𝐸+), half of the 6 groups chose one send and two senders 3 periods respectively. Two groups were more often to choose one sender to get messages, while one group was more often to choose two
34 senders. In the case that receivers got less earnings with one sender (𝐸* < 𝐸+), there were 4 groups facing this situation. However, none of these groups was more often to choose games with two senders. One receiver was more often to choose games with one sender. The other 3 groups chose one sender and two senders equally. In sequential groups, most groups chose one sender and two senders 3 periods respectively. 4 of the 15 groups were often to get messages from two senders, while 3 groups chose one sender more often than two senders.
Table 7b. Choices of Groups in the Sequential Game
𝐸* > 𝐸+* 𝐸* = 𝐸+** 𝐸* < 𝐸+*** Number of Groups in Total 10 8 11 Number of Groups 5 6 4 Number of Groups: More One-sender 0 2 1 Number of Groups: 3 One-sender 2 3 3 Number of Groups: Less One-sender 3 1 0
*The receiver gets more earnings in games with one sender than in the games with two senders; **The receiver gets the same earnings in games with one sender and two senders
***The receiver gets less earnings in games with one sender than in the games with two senders.
Next, I further analyze receivers’ choices in each period. Table 8a and Table 8b describe the relation between the receiver’s choices and average earning in each period. In Table 8a, the first column shows the number of period that one-sender game is preferred by receivers. The second and third columns show comparisons between average earnings of receivers in the one-sender game and two-sender games in simultaneous groups. Similarly, the fourth and fifth columns show comparisons in sequential groups. In groups that less one-sender games were preferred, for example, the number of periods equals to 0, 1 and 2, receivers got more earnings in one-sender games. In other words, receivers have realized that one message brings them more earnings, they still chose
35 two senders in the Choice Game. This phenomenon happens in both simultaneous groups and sequential groups. When receivers chose to play 3 periods of one-sender game, they could get more earnings in two-sender games. In groups that receivers always chose to get two messages, the average earnings in two-sender games were much higher than those in one-sender games. There are 2 sequential groups do so.
Table 8a. Average Earning of the Receiver in Each Period Number of Period One Sender is Preferred One-sender Game* Simultaneous Groups One-sender Game** Sequential Groups 0 6 4 0 0 1 0 0 4 2 2 6 5 6 4 3 6 7 4 4.5 4 5 5 6 6 5 0 0 0 0 6 0 0 4 7
*average earning of the receiver in one-sender games in simultaneous groups **average earning of the receiver in one-sender games in sequential groups
From Table 8b, I learn that about half groups have the same preferences to choose one sender and two senders. In other words, 15 receivers think there are no differences for their expected earnings to choose one sender or two senders. 9 groups prefer two senders to send messages, and the other 5 groups need one sender. These 14 receivers hold a clear belief about the choice that will bring them higher expected earnings.
In general, I cannot conclude that receivers are totally rational and they always make decisions to maximize their expected earnings. Most receivers do not differentiate one-sender games and two-sender games, because they choose to get messages from one sender and two senders equally, no matter their earnings. For receivers who show evident preferences between one-sender game and two-sender games, they prefer to receive messages from two senders.
36 Table 8b. Receivers’ Choices in the Choice Game
Number of Period One
Sender is Preferred Number of Groups
Number of Sim-Groups Number of Seq-Groups 0 2 2 0 1 2 0 2 2 5 3 2 3 15 7 8 4 3 2 1 5 0 0 0 6 2 0 2
5.3.2 Truth-telling level and trust level
Then I discuss the truth-telling level of senders and trust level of receivers in the Choice Game. For one-sender game, the average truth-telling level is 44%. It means that 44% of senders report truthful messages in the one-sender game. The following Table 9 offers a brief summary of senders’ behavior in two-sender games. The truth-telling levels decrease in the Simultaneous Game. 51% senders in simultaneous groups send truthful messages and the percentage that non-conflicting messages are correct is 41%. However, the average truth-telling level remains the same in the Sequential Game and 46% of senders send truthful messages to the sender. The frequency that non-conflictive increases to 52% in the Choice Game. Table 10 specifically shows senders’ behavior in sequential groups. There are no obvious changes for sender 1, but the percentage that sender 2 reports the truthful messages increases to 59. Besides, sender 2 tells the truth more often, independent messages from sender 1. However, there are time intervals between the first two games and the last Choice Game and I do not know if subjects would discuss the experiment with each other. Hence, results in this part could offer a reference, but it is hard to guarantee to what extent they are valid.
37 Table 9. Sender Behavior in the Choice Game
Simultaneous Games Sequential Game
% Sender is truthful 51 46
% Messages are
non-conflictive 45 52
% Non-conflicting
messages are correct 41 52
*Observations under the Choice Game only includes groups and periods that the receiver preferred two senders to play with her.
Table 10. Sender Behavior in the Sequential Game Sequential Game
% Sender 1 is truthful 43
% Sender 2 is truthful 59
% Sender 2 is truthful when sender 1 is
truthful 63
% Sender 2 is truthful when sender 1
lies 56
% Sender 2 lies when sender 1 is
truthful 37
% Sender 2 lies when sender 1 lies 44
*Observations under the Sequential Game only includes groups and periods that the receiver preferred two senders to play with her.
Table 11 shows trust level of receivers in the Choice Game. Trust levels in the Simultaneous Game and the Sequential Game are calculated with non-conflictive. When the receiver observes non-conflictive messages, to what extent she believes that observed messages are true. Table 11 exhibits that trust levels are pretty similar in all games in the Choice Game and they are all around 50%, which is consistent with theoretical predictions. Trust levels decline heavily in the one-sender game and the
38 Simultaneous Game. It decreases to 53% in the one-sender game and 50% in the Simultaneous Game. However, there are no significant changes in the Sequential Game.
Table 11. Trust Level in the Choice Game
Games Trust Levels (%)
The one-sender game 53
The Simultaneous Game 50
The Sequential Game 54
6. Discussion and Conclusion
In the cheap talk game with one sender, there is one sender and the truth-telling level stays above 50%, which is significantly higher than 50%. Hence, there exists excess truth telling and overcommunication by senders in this game. After including another informed sender, the truth-telling level lowers decreases, but situations are distinct between the two variations (simultaneous and sequential variation). The truth-telling level in the Simultaneous Game is 55%, while it is 46% in the Sequential Game. Although there is no significant difference between the two variations, the frequency that senders transfer truthful messages in the Simultaneous Game is much higher than that in the Sequential Game. Besides, the truth-telling level in the Simultaneous Game is significantly higher than what theory predicts. Except truth-telling level, there exists unique phenomenon in the Sequential Game which is sender 2 makes the choice under the influence of sender 1. Furthermore, sender 2 is more likely to revert the choice of sender 1, not follow it. In addition, the truth-telling levels of non-conflicting messages are also pretty different. 63% non-conflicting messages from two senders are correct in the Simultaneous Game, while only 40% non-conflicting messages from two senders reveal true payoff tables in the Sequential Game. The tendency for sender 2 to revert messages from sender 1 could explain the percentage (40%) in the Sequential Game. In general, the Simultaneous Game is more informational than the Sequential Game. Comparing the truth-telling level of the Simultaneous Game with the truth-telling level of one-sender game, the truth-telling level in the one-sender game is significantly higher.
39 On the basis of trust level, the trust level of receivers is 70% in the one-sender game, which is highest in all games. Then I discuss trust levels in two-sender game. When the receiver gets non-conflictive messages, the percentage that receivers trust messages from senders is exact 50% in the Sequential Game, and the percentage in the Simultaneous Game is 65%. When the receiver gets conflictive messages, trust levels are quite different with different senders and in different variations. In the Simultaneous Game, the receiver believes that 52.5% messages from sender 1 and 47.5% messages from sender 2 are truthful. The trust levels are very close. However, in the Sequential Game, the receiver believes that 64% messages from sender 1 and 36% messages from sender 2 are truthful. The trust levels are very different. The receiver trusts sender 1 with more frequency, which means that the receiver thinks messages from sender 2 are dependent on messages from sender 1.
When subjects play the Choice Game, the receiver should first make choices to play one-sender game and two-sender game. I divide the analysis into two parts. The first part is to compare one-sender game with the Simultaneous Game. The other part is to compare one-sender game with the Sequential Game. I find that receivers did not always choose games that brought them higher earnings. The most often choice was to choose one-sender game 3 periods and two-sender game 3 periods. This happened in both simultaneous groups and sequential groups. For receivers who have shown clear preferences towards the one-sender game and two-sender game, most receivers chose to play the two-sender game. This is more evident in sequential groups. As for truth-telling levels and trust levels, both decease in the Choice Game. This could be the influence of learning. Subjects learn that there are strategic communications, so senders lie more often and receivers believe that senders will transmit untruthful messages to them. Noting that there is an interval between first two games and the last Choice game and this may affect the validity of results in the Choice Game.
In this experiment, messages are more informational than theory would predict and receivers trust messages more often than predictions. It implies that subjects often fail to play equilibrium strategies. Previous literature has shown overcommunication in sender-receiver games (Cai & Wang (2006), Peeters et al. (2008), Sánchez-Pagés &
40 Vorsatz(2009)). Sánchez-Pagés & Marc Vorsatz (2007), Gurdal et al. (2013) also find overcommunication. They explain it with the agent quantal response equilibrium (AQRE). AQRE is a statistical theory of equilibrium, which is proposed by Mckelvey & Palfrey in 1995. They use this theory in games that are based on the concept that subjects play better strategies more often, compared with worse strategies, but they do not always play the best strategies. It is a model to explain experimental data from the perspective of bounded rationality. Besides, it requires the condition that the
equilibrium quantal response probabilities can represent the expected payoff to a strategy in equilibrium. Then it defines a quantal response equilibrium (QRE) as a reference point of the process and establish existence. Gurdal et al. also estimate their experiment with the logic-AQRE. They show that the rationality levels and the cost of lying are statistically different from zero. In addition, senders have higher expected payoffs in the Sequential Game, compared with the Simultaneous Game. However, receivers get less in the Sequential Game and become better off under the condition of non-conflicting messages. Combining these two articles together, logit-AQRE could offer an explanation for the experimental results.
My other findings, for example, messages in the Simultaneous Game are more informational than messages in the Sequential Game when there are two senders are also consistent with Gurdal et al. (2013) findings. This could be complained by competition between senders (Milgrom & Roberts (1986)). Milgrom & Roberts discuss that competition among interested parties (senders) who are competing in providing information will lead to the emergency of truth when information of interested parties is complete and verifiable. In addition, the “revert” behavior of sender 2, which is also found by Gurdal et al. (2013) is another important reason of this finding. Another finding that one-sender game is the most informational is also the same with findings in previous literature (Krishna & Morgan (2001b), Li (2008), Ishida & Shimizu (2012)). In my experiment, senders have the same interest
alignment, which satisfies the condition that two senders have like biases in literature. For receivers’ choices between one sender and two senders, some receivers show explicit preference to choose receivers, although they got less earnings. The potential reason could be that the receiver anticipates to get more informational messages from more senders. They are afraid to lose information if they play one-sender game.
41 To summarize, when senders have the same interest alignment and interest
preferences are distinct for senders and the receiver, there exist systematic deviations from the theoretical predictions. In short term, messages from one sender are more likely to be truthful. If receivers have to obtain information from two senders, they should ask senders to send messages simultaneously. In another word, receivers become better off in the Simultaneous Game, not the Sequential Game. Similarly, there is overtrust for the receiver, especially there is only one sender and two senders playing the Simultaneous Game. In long term, I have to consider the effect of learning and results become close to levels that model predicts. The limitations in this paper are that the sample is too small and this experiment was conducted by pens and paper. In further research, I can do the experiment with a larger sample to get more valid results. I also should intend to explore the influence of interest alignments. For instance, two senders have opposing interests.
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44
Appendix A. Proof
Following the proof of Gurdal et al. (2013), I first derive the best responses of the sender. The sender will maximize his payoff.
The best responses after the actual payoff table is observed can be derived from:
max GH, GI { * +[ J +𝛾$𝑝$+ * +(1 − 𝛾$)𝑝$+ * +𝛾$𝑝&+ J +(1 − 𝛾$)𝑝&+ J +𝛾&(1 − 𝑝$) + * +(1 − 𝛾&)(1 − 𝑝$)+ * +𝛾&(1 − 𝑝&)+ J +(1 − 𝛾&)(1 − 𝑝&)]} This maximization problem is equivalent to
max
GH, GI {2[𝑝$(𝛾$ − 𝛾&) + 𝑝&(𝛾&− 𝛾$)] + L +}
Hence, the best response correspondences for the sender is:
𝑝$ = [0, 1] 1 0
𝑖𝑓 𝛾$ > 𝛾& 𝑖𝑓 𝛾$ = 𝛾&
𝑖𝑓 𝛾$ < 𝛾& and 𝑝& =
1 [0, 1] 0 𝑖𝑓 𝛾$ < 𝛾& 𝑖𝑓 𝛾$ = 𝛾& 𝑖𝑓 𝛾$ > 𝛾&
Then, I calculate the best response of the receiver given the strategy (𝑝$, 𝑝&). Assume that the actual payoff table is table A. If the sender transmits message A, the
receiver’s belief that the sender reveals the actual payoff table is
𝜇$=G(OP$|RP$)S($)S(OP$) = S(OP$|RP$)S $ TS(OP$|RP&)S(&)G(OP$|RP$)S($) = GH GHTGI
If, on the other hand, the receiver obtains message B, the belief that table A will determine the final earnings is
𝜇&=G(OP&|RP$)S($)S(OP&) = S(OP&|RP$)S $ TS(OP&|RP&)S(&)G(OP&|RP$)S($) = (*U G*U GH H)T(*UGI)
