# Valuable Cheap Talk and Equilibrium Selection

## Abstract

**:**

## 1. Introduction

## 2. Previous Literature

## 3. Motivation

## 4. Model

**G**with n players and finite action spaces ${S}_{i}$ for $i=1,\dots ,n$.13 Payoffs are given by ${u}_{i}$ for $i=1,\dots ,n$. It will be simplest to think of

**G**in normal form.

**G**is played exactly once, though

**G**itself may be a repeated game. Before this happens, there is a

**conversation**C(

**G**), defined as follows. Each player begins the pregame conversation with a totally mixed prior

**forecast**${\pi}_{i}={\pi}_{i}^{1}\in \mathsf{\Delta}\left({S}_{i}\right)$ about his or her behavior. The forecasts are common knowledge among all the players. At each round $t=1,2,3\dots $ of the conversation, player i announces ${m}_{i}^{t}\in {S}_{i}$. The announcements are made simultaneously by all players in each round.14

**G**, and define ${E}_{i}\subseteq {S}_{i}$ by:

**credible**. If ${m}_{i}^{1}$ was credible, then we define:

**appearance**is given by ${p}_{i}^{2}={m}_{i}^{1}$. If the initial announcement was not credible, then the forecast is not updated, and the appearance is undefined. Recursively, we now define ${m}_{i}^{t}$ to be credible when:

**G**are given by ${\mu}_{i}=\underset{j\ne i}{\times}{b}_{j}$.

**Definition**

**1.**

**acceptable equilibrium**(of

**G**) is a profile $\sigma \in \underset{i=1}{\stackrel{n}{\times}}\mathsf{\Delta}\left({S}_{i}\right)$ such that $\sigma =b$ for some belief vector b resulting from a convergent conversation starting at some prior forecasts π; the set of acceptable equilibria is denoted $AccE\left(G\right)$.

**Theorem**

**1.**

- 1.
- $NE\left(G\right)\subseteq AccE\left(G\right)$
- 2.
- $AccE\left(G\right)\subseteq \epsilon NE\left(G\right)$

**Proof.**

**Definition**

**2.**

**directly attainable**from ${\sigma}^{\prime}\in NE\left(G\right)$ by the coalition S if ${\sigma}_{s}$ is a Nash equilibrium in the induced game fixing all players outside of S to play as in ${\sigma}^{\prime}$, and if also $\forall i\notin S$, we have ${u}_{i}({\sigma}_{i},{\sigma}_{S},{\sigma}_{-i,S}^{\prime})>{u}_{i}({\sigma}_{i}^{\prime},{\sigma}_{S},{\sigma}_{-i,S}^{\prime})$.

**Definition**

**3.**

**attainable**from ${\sigma}^{\prime}\in NE\left(G\right)$ by the coalition S if there is a chain of equilibria, each directly attainable by S, leading from ${\sigma}^{\prime}$ to σ; if also, $\forall i\in S$${u}_{i}\left(\sigma \right)>{u}_{i}\left({\sigma}^{\prime}\right)$; and if finally, there is no similar such chain (for any coalition) leading away from σ.

**Definition**

**4.**

**G**is

**stably efficient**if nothing is attainable from it; the set of these equilibria is denoted $StEff\left(G\right)$.

**optimal**if it is not weakly dominated.

**Definition**

**5.**

**agreeable equilibrium**(of

**G**) is a profile $\sigma \in \stackrel{n}{\underset{i=1}{\times}}\mathsf{\Delta}\left({S}_{i}\right)$ such that $\sigma =b$ for some belief vector b resulting from a convergent optimal conversation starting at some prior forecasts π; the set of agreeable equilibria is denoted $AgrE\left(G\right)$.

**Theorem**

**2.**

- 1.
- $StEff\left(G\right)\subseteq AgrE\left(G\right)$
- 2.
- $AgrE\left(G\right)\subseteq \epsilon StEff\left(G\right)$

**Proof.**

## 5. Examples

**G**is a repeated game and the players have a full conversation between each stage, then optimal speech should lead to efficient outcomes all along the extensive form game tree, both on and off the equilibrium path. This gives rise to the difficult problem of finding renegotiation-proof equilibria18.

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Osborne, M.J. An Introduction to Game Theory; Oxford University Press: New York, NY, USA, 2004; Volume 3. [Google Scholar]
- Crawford, V.P.; Sobel, J. Strategic information transmission. Econom. J. Econom. Soc.
**1982**, 50, 1431–1451. [Google Scholar] [CrossRef] - Farrell, J. Cheap talk, coordination, and entry. RAND J. Econ.
**1987**, 18, 34–39. [Google Scholar] [CrossRef] - Farrell, J.; Gibbons, R. Cheap talk can matter in bargaining. J. Econ. Theory
**1989**, 48, 221–237. [Google Scholar] [CrossRef] [Green Version] - Forges, F. Universal mechanisms. Econom. J. Econom. Soc.
**1990**, 58, 1341–1364. [Google Scholar] [CrossRef] - Farrell, J. Meaning and credibility in cheap-talk games. Games Econ. Behav.
**1993**, 5, 514–531. [Google Scholar] [CrossRef] [Green Version] - Aumann, R.; Hart, S. Polite Talk Isn’t Cheap; Technical Report; Mimeo, Hebrew University of Jerusalem: Jerusalem, Israel, 1993. [Google Scholar]
- Blume, A.; Sobel, J. Communication-proof equilibria in cheap-talk games. J. Econ. Theory
**1995**, 65, 359–382. [Google Scholar] [CrossRef] - Farrell, J.; Rabin, M. Cheap talk. J. Econ. Perspect.
**1996**, 10, 103–118. [Google Scholar] [CrossRef] - Rabin, M. A model of pre-game communication. J. Econ. Theory
**1994**, 63, 370–391. [Google Scholar] [CrossRef] - Matsui, A. Cheap-talk and cooperation in a society. J. Econ. Theory
**1991**, 54, 245–258. [Google Scholar] [CrossRef] - Canning, D. Learning and efficiency in common interest signaling games. In The Dynamics of Norms; Bicchieri, C., Jeffrey, R.C., Skyrms, B., Eds.; Cambridge University Press: Cambridge, UK, 1997; pp. 67–85. [Google Scholar]
- Sandroni, A. Reciprocity and cooperation in repeated coordination games: The principled-player approach. Games Econ. Behav.
**2000**, 32, 157–182. [Google Scholar] [CrossRef] - Crawford, V.P.; Haller, H. Learning how to cooperate: Optimal play in repeated coordination games. Econom. J. Econom. Soc.
**1990**, 58, 571–595. [Google Scholar] [CrossRef] - Young, H.P. The evolution of conventions. Econom. J. Econom. Soc.
**1993**, 61, 57–84. [Google Scholar] [CrossRef] - Kalai, E.; Lehrer, E. Rational learning leads to Nash equilibrium. Econom. J. Econom. Soc.
**1993**, 61, 1019–1045. [Google Scholar] [CrossRef] - Cooper, R.; De Jong, D.V.; Forsythe, R.; Ross, T.W. Forward induction in coordination games. Econ. Lett.
**1992**, 40, 167–172. [Google Scholar] [CrossRef] - Brandts, J.; Cooper, D.J. It’s what you say, not what you pay: An experimental study of manager-employee relationships in overcoming coordination failure. J. Eur. Econ. Assoc.
**2007**, 5, 1223–1268. [Google Scholar] [CrossRef] - Cachon, G.P.; Camerer, C.F. Loss-avoidance and forward induction in experimental coordination games. Q. J. Econ.
**1996**, 111, 165–194. [Google Scholar] [CrossRef] [Green Version] - Boulu-Reshef, B.; Holt, C.A.; Rodgers, M.S.; Thomas-Hunt, M.C. The impact of leader communication on free-riding: An incentivized experiment with empowering and directive styles. Leadersh. Q.
**2020**, 31, 101351. [Google Scholar] [CrossRef] - Bernheim, B.D.; Peleg, B.; Whinston, M.D. Coalition-proof nash equilibria i. concepts. J. Econ. Theory
**1987**, 42, 1–12. [Google Scholar] [CrossRef] - Jnawali, K.; Morsky, B.; Bauch, C.T. Strategic Interactions in Antiviral Drug Use during an Influenza Pandemic. 2017. Available online: http://currents.plos.org/outbreaks/index.html%3Fp=65448.html (accessed on 3 August 2020).
- Aumann, C. Nash Equilibria are not Self-Enforcing. In Economic Decision Making: Games, Econometrics, and Optimization (Contributions in Honor of Jacques Drèze); Elsevier: Amsterdam, The Netherlands, 1990. [Google Scholar]
- Charness, G. Self-serving cheap talk: A test of Aumann’s conjecture. Games Econ. Behav.
**2000**, 33, 177–194. [Google Scholar] [CrossRef] - Miller, J.H.; Moser, S. Communication and coordination. Complexity
**2004**, 9, 31–40. [Google Scholar] [CrossRef] [Green Version] - Bergin, J.; MacLeod, W.B. Efficiency and renegotiation in repeated games. J. Econ. Theory
**1993**, 61, 42–73. [Google Scholar] [CrossRef]

1. | The limit is an $\epsilon $-Nash equilibrium. |

2. | This is discussed in further detail in Section 3. |

3. | The notion of efficiency used here is stable efficiency, a concept that is equivalent to Pareto efficiency in generic two person games. |

4. | The players do not have beliefs about the full strategies of their opponents, only ideas about what might actually occur in the game. Thus, the preplay forecasts are distributions over actions, not distributions over mixed strategies (which themselves are distributions over actions). This is not crucial to the conclusions reached. |

5. | It is not strictly necessary for the results that the priors be totally mixed. |

6. | The author performed the analysis under the seemingly weaker assumption that all that is known about the prior forecasts is that they place a certain minimum weight on each action, but the results carry over. Since this assumption adds complexity, but is no sounder in justification (Why can the entire distributions not be known if the minimum weights are?), it has been left out. |

7. | Recall that the average of multiple sets of actions is equivalent to a mixed strategy. |

8. | We assume that players only care about payoffs up to some arbitrarily small constant $\epsilon $, either because they cannot perceive finer differences or because they are indifferent over this range. |

9. | We make the standard assumptions on the action game so that a best response always exists. |

10. | In particular, continually starting over inhibits convergence, in which case, the player has no influence on the ultimate course of the discussion. This is never optimal, as shown below. |

11. | If no credible announcements were made after some finite stage, this is taken to mean that the limit does not exist. However, as above, we may assume that this does not occur. |

12. | Naturally, since full rationality is assumed, we could endlessly iterate the process, but there is no need. |

13. | The assumption of finiteness can be weakened. |

14. | Sequential announcements lead to a forced asymmetry regarding who speaks when. The effects of this generalized first-mover advantage are irrelevant for the present discussion. |

15. | Unless player i has only one possible credible announcement, as discussed in Section 3. |

16. | In particular, since the conversation converges, there must be some round after which nobody ever cleans their slate and starts over. |

17. | Consider, as one example, fictitious play in the rock-paper-scissors game. |

18. | See, for example, the survey paper by Bergin and MacLeod (1993) [26]. |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jamison, J.
Valuable Cheap Talk and Equilibrium Selection. *Games* **2020**, *11*, 34.
https://doi.org/10.3390/g11030034

**AMA Style**

Jamison J.
Valuable Cheap Talk and Equilibrium Selection. *Games*. 2020; 11(3):34.
https://doi.org/10.3390/g11030034

**Chicago/Turabian Style**

Jamison, Julian.
2020. "Valuable Cheap Talk and Equilibrium Selection" *Games* 11, no. 3: 34.
https://doi.org/10.3390/g11030034