Cheap Talk Games with TwoSenders and Different Modes of Communication^{ †}
Abstract
:1. Introduction
2. The Model and the Theoretical Predictions
2.1. The Simultaneous Game
2.2. The Sequential Game
2.3. The Choice Game
2.4. Hypotheses
3. Experimental Design and Procedures
4. Experimental Results
4.1. Senders
4.2. Receivers
4.3. Observers
5. Logistic Analysis for the Choice Treatment
6. Discussion and Concluding Remarks
Author Contributions
Funding
Conflicts of Interest
Appendix A. Instructions
Table A  G1  G2  Receiver 

Action U  4.5  4.5  1 
Action D  0.5  0.5  9 
Table B  G1  G2  Receiver 
Action U  0.5  0.5  9 
Action D  4.5  4.5  1 
Appendix B. Proofs of Propositions
References
 Crawford, V.; Sobel, J. Strategic information transmission. Econometrica 1982, 50, 1431–1451. [Google Scholar] [CrossRef]
 Dickhaut, J.; McCabe, K.; Mukherji, A. An experimental study on strategic information transmission. Econ. Theory 1995, 6, 389–403. [Google Scholar] [CrossRef]
 Cai, H.; Wang, J. Overcommunication in strategic information transmission games. Games Econ. Behav. 2006, 56, 7–36. [Google Scholar] [CrossRef]
 SánchezPagés, S.; Vorsatz, M. An experimental study of truthtelling in a sender receiver game. Games Econ. Behav. 2007, 61, 86–112. [Google Scholar] [CrossRef] [Green Version]
 SánchezPagés, S.; Vorsatz, M. Enjoy the silence: An experiment on truthtelling. Exp. Econ. 2009, 12, 220–241. [Google Scholar] [CrossRef]
 Sutter, M. Deception through telling the truth?! Experimental evidence from individuals and teams. Econ. J. 2009, 12, 220–241. [Google Scholar] [CrossRef] [Green Version]
 Peeters, R.; Vorsatz, M.; Walz, M. Truth, trust, and sanctions: On institutional selection in sender–receiver games. Scand. J. Econ. 2013, 115, 508–548. [Google Scholar] [CrossRef] [Green Version]
 Gurdal, M.Y.; Ozdogan, A.; Saglam, I. Truthtelling and trust in sender–receiver games with intervention: An experimental study. Rev. Econ. Des. 2014, 18, 83–103. [Google Scholar] [CrossRef]
 Gilligan, T.; Krehbiel, K. Asymmetric information and legislative rules with a heterogeneous committee. Am. J. Political Sci. 1989, 33, 459–490. [Google Scholar] [CrossRef]
 Krishna, V.; Morgan, J. A model of expertise. Q. J. Econ. 2001, 116, 747–775. [Google Scholar] [CrossRef]
 Gick, W. Cheap Talk Equilibria—A Note on Two Senders; Mimeo, Harvard University: Cambridge, MA, USA, 2008. [Google Scholar]
 Li, M. Two (talking) heads are not better than one. Econ. Bull. 2008, 3, 1–8. [Google Scholar]
 Li, M. Advice from multiple experts: A comparison of simultaneous, sequential, and hierarchical communication. B.E. J. Theor. Econ. 2010, 10, 18. [Google Scholar] [CrossRef] [Green Version]
 AustenSmith, D. Information transmission in debate. Am. J. Political Sci. 1990, 34, 124–152. [Google Scholar] [CrossRef]
 AustenSmith, D. Credible debate equilibria. Soc. Choice Welf. 1990, 7, 75–93. [Google Scholar] [CrossRef]
 AustenSmith, D. Interested experts and policy advice: Multiple referrals under open rule. Games Econ. Behav. 1993, 5, 3–43. [Google Scholar] [CrossRef]
 Milgrom, P.; Roberts, J. Relying on information of interested parties. Rand J. Econ. 1986, 17, 18–32. [Google Scholar] [CrossRef]
 AustenSmith, D. Information acquisition and orthogonal argument, in Political Economy: Institutions Competition and Representation. In Proceedings of the Seventh International Symposium in Economic Theory and Econometrics; Barnet, W.A., Melvin, H.J., Schofield, N., Eds.; Cambridge University Press: Cambridge, UK; New York, NY, USA; Melbourne, Australia, 1993; pp. 407–436. [Google Scholar]
 Battaglini, M. Multiple referrals and multidimensional cheap talk. Econometrica 2002, 70, 1379–1401. [Google Scholar] [CrossRef] [Green Version]
 Battaglini, M. Policy advice with imperfectly informed experts. Adv. Theor. Econ. 2004, 4, 1. [Google Scholar] [CrossRef]
 Ambrus, A.; Takahashi, S. Multisender cheap talk with restricted state spaces. Theor. Econ. 2008, 3, 1–27. [Google Scholar]
 Lai, E.K.; Lim, W.; Wang, J.T.Y. Experimental Implementations and Robustness of Fully Revealing Equilibria in Multidimensional Cheap Talk; Mimeoy: Cambridge, MA, USA, 2011. [Google Scholar]
 Minozzi, W.; Woon, J. Competition and Exaggeration in Experimental Cheap Talk Games. 2018; unpublished manuscript. [Google Scholar]
 Vespa, E.; Wilson, A.J. Communication with multiple senders: An experiment. Quant. Econ. 2016, 7, 1–36. [Google Scholar] [CrossRef] [Green Version]
 Peeters, R.; Vorsatz, M.; Walz, M. Rewards in an experimental sender–receiver game. Econ. Lett. 2008, 101, 148–150. [Google Scholar] [CrossRef]
 Fischbacher, U. ZTree – Zurich toolbox for readymade economic experiments. Exp. Econ. 2007, 10, 171–178. [Google Scholar] [CrossRef] [Green Version]
1.  
2.  Also, in order to explain the observed differences in behavior of senders and receiver in simultaneous and sequential plays, we use a logit agent quantal response equilibrium (logitAQRE) model, following [7] and [8]. We present the maximum likelihood estimation results of this model in the online appendix at: http://isaglam.etu.edu.tr/BGOS2020OnlineAppendix.pdf. 
3.  For example, [9,10,11,12,13] among others extend the basic one sender and one receiver model in Crawford and Sobel [1] by allowing two perfectly informed senders. [14,15,16] consider the case with two imperfectly informed senders while [15,16] also analyze the effects of alternative communication modes, namely simultaneous and sequential transmission of information. A common feature of these extensions is that the policy space is unidimensional, while [17,18,19,20,21,22] consider multidimensional models of cheap talk. 
4.  
5.  An alternative behavioral explanation for excessive truthtelling is provided by SánchezPagés and Vorsatz [5] using their baseline and punishment games in [4] with a modification that the sender in the baseline game additionally has a costly option of remaining silent. They show that overcommunication in the baseline game can be attributed to lying aversion.

6.  Several papers have also studied the robustness of overcommunication phenomenon. For example, Peeters et al. [25] show that overcommunication of the sender disappears in the presence of rewards, whereas the trust by the receiver increases significantly. Gurdal et al. [8] analyze the robustness of excessive truthtelling and trust to the intervention of an honest regulator (equivalently to the presence of nonstrategic sender types). They show that excessive truthtelling and excessive trust are higher under intervention than under the absence of intervention.

7.  For instance, $p=\frac{3}{4}$, $q=\frac{3}{4}$ and ${r}_{AA}=\frac{1}{3}$, ${r}_{BB}=\frac{1}{2}$, ${r}_{AB}={r}_{BA}=\frac{1}{4}$ constitute an equilibrium. 
8.  We consider the observer role so as to check whether our subjects, when they are neither the receiver nor a sender, are able to make correct guesses about the outcomes of the games.

9.  Given that the subjects make choices in the same group for 36 periods, it may be a valid concern that anonymity may have been somewhat disregarded as the repetition could have an effect in the experiment even though the roles are randomly assigned in each period within a group. Since all treatments may be affected to some extent by the repetition, we believe that the differences in treatments are not due to the repetition effect. Moreover, our experimental results indicate that the behavior of the players are not significantly different than the theoretical predictions of the oneshot game in most of the cases (see Section 4). As the choice of running sessions with groups of 4 is essentially done for using nonparametric tests for the receiver behavior in the Choice Treatment, our estimation finding (Result 4 in Section 5) in that regard shows that the receiver values the information acquired in the previous period more than the information acquired in all other past periods in the Choice Treatment. These results imply that the learning effect due to repetition must be minimal.

10.  Looking at the 276 instances during the Choice Treatment, we see that the receivers preferred simultaneous messages in 152 cases (55%) and sequential messages in 124 cases (45%). 
11.  The theoretical predictions for the probabilities in the first and last rows are 1/2, whereas the theoretical prediction for the probability that the two senders’ messages are nonconflictive in simultaneous plays is ${p}_{A}{q}_{A}+(1{p}_{A})(1{q}_{A})\in [0,1]$. 
12.  The theoretical predictions for the probabilities in the first two rows and the last row are 1/2. The theoretical predictions for the probabilities in all the remaining rows are arbitrary in the interval $[0,1]$. To see this, one can check that the probability that sender 2 is truthful when sender 1 is truthful is ${p}_{A}{q}_{A}\left(A\right)+(1{p}_{B})(1{q}_{B}\left(B\right))$. Similarly, the probability that sender 2 is truthful when sender 1 lies is $(1{p}_{A}){q}_{A}\left(B\right)+{p}_{B}(1{q}_{B}\left(A\right))$. One can also check that the probability that senders are nonconflictive is ${p}_{A}{q}_{A}\left(A\right)+(1{p}_{A})(1{q}_{A}\left(B\right))$. 
13.  Also, the receiver is statistically significantly the least likely to select simultaneous play if the receiver earned the high payoff in a sequential play in the last period. Also, high ratio of nonconflicting messages in both the Simultaneous and the Sequential Treatment increases the likelihood of simultaneous choice in the Choice Treatment, while the estimated impact is larger in the Simultaneous Treatment than in the Sequential Treatment.

Table A  Sender 1  Sender 2  Receiver 

Action U  4.5  4.5  1 
Action D  0.5  0.5  9 
Table B  Sender 1  Sender 2  Receiver 
Action U  0.5  0.5  9 
Action D  4.5  4.5  1 
Simultaneous Treatment  Choice Treatment  

% Sender is truthful  54.0 *  49.0 * 
% Senders are nonconflictive  51.4 **  53.4 ** 
% Nonconflicting messages  58.2 *  50.2 * 
are correct  
N  23  23 
Sequential Treatment  Choice Treatment  

% Sender is truthful  53.3 *  52.3 * 
% Sender 1 is truthful  50.0 *  51.6 * 
% Sender 2 is truthful when  54.0 **  48.0 ** 
sender 1 is truthful  
% Sender 2 is truthful when  58.9 **  63.0 ** 
sender 1 lies  
% Senders are nonconflictive  46.4 **  45.9 ** 
% Nonconflicting messages  54.5 *  62.5 * 
are correct  
N  23  23 
Simultaneous Messages  Sequential Messages  

Simultaneous Treatment  59.2 *  Sequential Treatment  52.3 * 
Choice Treatment  50.7 *  Choice Treatment  48.6 * 
N  23  23 
Simultaneous Messages  Sequential Messages  

Simultaneous Treatment  56.5 *  Sequential Treatment  59.6 * 
Choice Treatment  61.3 *  Choice Treatment  55.1 * 
N  23  23 
Number of Times Simultaneous  Number of Subjects 

Messages Is Preferred  
0  22 
1  18 
2  22 
3  30 
% Guesses of Favorable  % Correct  

Outcome for the Receiver  Guesses  
Simultaneous Treatment  44.2 *  45.7 * 
Sequential Treatment  48.2 *  48.2 * 
Choice Treatment (Simultaneous)  59.1 *  51.2 * 
Choice Treatment (Sequential)  55.2 *  44.5 * 
N  23  23 
I  II  III  IV  V  VI  VII  

Simwon ${}^{\u2020}$ (1 if previous play was simultaneous and  2.43 **  1.81  1.14  2.27 *  1.72  1.12  
receiver earned high payoff)  (2.06)  (0.84)  (0.32)  (1.92)  (0.8)  (0.28)  
Seqwon ${}^{\u2020}$ (1 if previous play was sequential and  0.47 *  0.68  0.43 **  0.43 *  0.67  0.41 **  
receiver earned high payoff)  (−1.74)  (−0.71)  (−2.41)  (−1.93)  (−0.74)  (−2.44)  
Simlost ${}^{\u2020}$ (1 if previous play was simultaneous and  1.18  2.02  1.08  1.90  
receiver earned low payoff)  (0.42)  (1.33)  (0.21)  (1.24)  
Simwonnc ${}^{\u2020}$ (1 if previous play was simultaneous with  3.67 **  3.67 **  3.70 **  3.68 **  
nonconflicting messages and receiver earned high payoff)  (1.96)  (1.96)  (2.04)  (2.04)  
Seqwonnc ${}^{\u2020}$ (1 if previous play was sequential with  0.99  0.88  
nonconflicting messages and receiver earned high payoff)  (−0.01)  (−0.24)  
Simlostnc ${}^{\u2020}$ (1 if previous play was simultaneous with  0.72  0.70  
nonconflicting messages and receiver earned low payoff)  (−0.71)  (−0.77)  
Seqlostnc ${}^{\u2020}$ (1 if previous play was sequential with  2.05  2.10  
nonconflicting messages and receiver earned low payoff)  (1.33)  (1.45)  
Average payoff of the receivers over simultaneous plays,  0.97  0.95  0.90  0.91  
updated in the Choice Treatment  (−0.14)  (−0.27)  (−0.59)  (−0.51)  
Average payoff of the receivers over sequential plays,  1.26  1.27 *  1.26 *  1.26  
updated in the Choice Treatment  (1.47)  (1.76)  (1.67)  (1.61)  
Ratio of nonconflicting messages in the Simultaneous  8.88  2.88  2.59  2.53  5.26  4.73  4.68 
Treatment  (1.6)  (0.98)  (0.95)  (0.90)  (1.33)  (1.37)  (1.28) 
Ratio of nonconflicting messages in the Sequential  1.46  1.30  1.27  1.26  1.65  1.72  1.71 
Treatment  (0.33)  (0.26)  (0.22)  (0.22)  (0.49)  (0.5)  (0.50) 
Constant  0.11  0.54  0.40  0.64  0.14  0.13  0.19 
(−1.29)  (−0.76)  (−1.01)  (−0.59)  (−1.26)  (−1.21)  (−1.00)  
Pseudo R2  0.015  0.063  0.086  0.079  0.072  0.095  0.088 
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Bayindir, E.E.; Gurdal, M.Y.; Ozdogan, A.; Saglam, I. Cheap Talk Games with TwoSenders and Different Modes of Communication. Games 2020, 11, 18. https://doi.org/10.3390/g11020018
Bayindir EE, Gurdal MY, Ozdogan A, Saglam I. Cheap Talk Games with TwoSenders and Different Modes of Communication. Games. 2020; 11(2):18. https://doi.org/10.3390/g11020018
Chicago/Turabian StyleBayindir, Esra E., Mehmet Y. Gurdal, Ayca Ozdogan, and Ismail Saglam. 2020. "Cheap Talk Games with TwoSenders and Different Modes of Communication" Games 11, no. 2: 18. https://doi.org/10.3390/g11020018