# Cooperation between Emotional Players

## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Analysis

#### 3.1. The Finitely Repeated Interaction

**Proposition**

**1.**

**Proposition**

**2.**

**Proposition**

**3.**

#### 3.2. The Infinitely Repeated Interaction

#### 3.2.1. Grim Trigger

**Proposition**

**4.**

**Proposition**

**5.**

#### 3.2.2. Mutual Minmax

**Proposition**

**6.**

**Proposition**

**7.**

## 4. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**State transitions. The emotional players can transition between three states of mind and the transitions depend stochastically on the observed actions of the other player.

**Proposition**

**A1.**

## References

- Levine, D. Modeling altruism and spitefulness in experiments. Rev. Econ. Dyn.
**1998**, 1, 593–622. [Google Scholar] [CrossRef][Green Version] - Cox, J.; Friedman, D.; Gjerstad, S. A tractable model of reciprocity and fairness. Games Econ. Behav.
**2007**, 59, 17–45. [Google Scholar] [CrossRef][Green Version] - Cox, J.; Friedman, D.; Sadiraj, V. Revealed altruism. Econometrica
**2008**, 76, 31–69. [Google Scholar] [CrossRef] - Shapley, L.S. Stochastic games. Proc. Natl. Acad. Sci. USA
**1953**, 39, 1095–1100. [Google Scholar] [CrossRef] [PubMed] - Drouvelis, M.; Grosskopf, B. The effects of induced emotions on pro-social behaviour. J. Pub. Econ.
**2016**, 134, 1–8. [Google Scholar] [CrossRef][Green Version] - Capra, M.C. Mood-driven behavior in strategic interactions. Am. Econ. Rev.
**2004**, 94, 367–372. [Google Scholar] [CrossRef] - Ben-Shakhar, G.; Bornstein, G.; Hopfensitz, A.; Van Winden, F. Reciprocity and emotions in bargaining using physiological and self-report measures. J. Econ. Psychol.
**2007**, 28, 314–323. [Google Scholar] [CrossRef] - Kirchsteiger, G.; Rigotti, L.; Rustichini, A. Your morals might be your moods. J. Econ. Behav. Organ.
**2006**, 59, 155–172. [Google Scholar] [CrossRef] - Oechssler, J. Finitely repeated games with social preferences. Exp. Econ.
**2013**, 16, 222–231. [Google Scholar] [CrossRef][Green Version] - Ekman, P.; Friesen, W. Constants across cultures in the face and emotion. J. Pers. Soc. Psychol.
**1971**, 17, 124–129. [Google Scholar] [CrossRef][Green Version] - Izard, C.E. The Face of Emotion; Appleton Century Crofts: New York, NY, USA, 1971. [Google Scholar]
- Sorin, S. Stochastic games with incomplete information. In Stochastic Games and Applications, 1st ed.; Neyman, A., Sorin, S., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2003; pp. 375–395. [Google Scholar]
- Dutta, P. A folk theorem for stochastic games. J. Econ. Theory
**1995**, 66, 1–32. [Google Scholar] [CrossRef][Green Version] - Mailath, G.; Samuelson, L. Repeated Games and Reputations: Long-Run Relationships, 1st ed.; Oxford University Press: Oxford, UK, 2006. [Google Scholar]
- Frank, R.H. Passions Within Reason: The Strategic Role of the Emotions; W. W. Norton & Company: New York, NY, USA, 1988. [Google Scholar]
- Bernheim, D.; Braghieri, L.; Martinez-Marquina, A.; Zuckerman, D. A theory of chosen preferences. Unpublished.
- Steele, C. The psychology of self-affirmation: Sustaining the integrity of the self. Adv. Exp. Soc. Psychol.
**1988**, 21, 261–302. [Google Scholar] - Kunda, Z. The case for motivated reasoning. Psychol. Bull.
**1990**, 108, 480–498. [Google Scholar] [CrossRef] - Epley, N.; Gilovich, T. The mechanics of motivated reasoning. J. Econ. Perspect.
**2016**, 30, 133–140. [Google Scholar] [CrossRef] - Gino, F.; Norton, M.; Weber, R. Motivated Bayesians: Feeling moral while acting egoistically. J. Econ. Perspect.
**2016**, 30, 189–212. [Google Scholar] [CrossRef][Green Version] - Aumann, R.J. Survey of repeated games. In Essays in Game Theory and Mathematical Economics in Honor of Oskar Morgenstern, 1st ed.; Henn, R., Moeschlin, O., Eds.; Bibliograph Inst.: Mannheim, Germany, 1981; pp. 11–42. [Google Scholar]
- Rubinstein, A. Finite automata play the repeated prisoner’s dilemma. J. Econ. Theory
**1986**, 39, 83–96. [Google Scholar] [CrossRef] - Elster, J. Emotions and economic theory. J. Econ. Lit.
**1998**, 36, 47–74. [Google Scholar] - Geanakoplos, J.; Pearce, D.; Stacchetti, E. Psychological games and sequential rationality. Games Econ. Behav.
**1989**, 1, 60–79. [Google Scholar] [CrossRef][Green Version] - Battigalli, P.; Dufwenberg, M. Dynamic psychological games. J. Econ. Theory
**2009**, 144, 1–35. [Google Scholar] [CrossRef] - Charness, G.; Dufwenberg, M. Promises and partnership. Econometrica
**2006**, 74, 1579–1601. [Google Scholar] [CrossRef][Green Version] - Battigalli, P.; Dufwenberg, M. Guilt in games. Am. Econ. Rev.
**2007**, 97, 170–176. [Google Scholar] [CrossRef][Green Version] - Battigalli, P.; Dufwenberg, M.; Smith, A. Frustration, aggression, and anger in leader-follower games. Games Econ. Behav.
**2019**, 117, 15–39. [Google Scholar] [CrossRef] - Myerson, R.; Weibull, J. Tenable strategy blocks and settled equilibria. Econometrica
**2015**, 83, 943–976. [Google Scholar] [CrossRef][Green Version]

1. | |

2. | |

3. | Multiplying by $(1-\delta )$ as in (2) is a non-essential normalization, made to simplify calculations. |

4. | The transitions are initially assumed to be deterministic. An analysis of players with stochastic state transitions can be found in Appendix A. |

5. | An alternative is to let the player transition to the neutral state rather than to the hostile state. This would however not affect the analysis of the game. |

6. | The stage game is thus not a prisoner’s dilemma in the state $s=NF$. |

7. | The action profile $(D,C)$ is an unintuitive Nash equilibrium. It is an equilibrium because the players first receive their payoffs, both material and psychological, and then transition to the other state of mind. If the players would transition before receiving their payoffs, and thus evaluate the payoffs according to their new state, $(D,C)$ would not be an equilibrium. |

8. | The observant reader may note that the state $s=FF$, with two emotional players, also transforms the stage game. In short, the stage game of $s=FF$ is a prisoner’s delight if $c+\alpha c>b>d+\alpha d$, a stag hunt if $c+\alpha c>b$ and $d+\alpha d>b$, and a hawk-dove game if $c+\alpha c<b$ and $\alpha b>d+\alpha d$. |

9. | The following analysis relies on the folk theorem for stochastic games proved by Dutta [13]. |

10. | The values of $\gamma $, and $b,c,d$ determines which present values that are vertices in the convex hull. |

11. | This may seem unintuitive, but consider for example a husband and wife’s cooperation over household work. One of them fails to do the dishes, and they punish each other by letting the dishes pile up. After three days they are supposed to return to cooperation. However, at this point the punishment may even have exacerbated the hostile emotions. |

**Figure 1.**State transitions. The emotional player can transition between three states of mind and the transitions depend deterministically on the observed actions of the other player.

**Figure 2.**The set of feasible and individually rational utility vectors. One emotional player with three states of mind and one Homo oeconomicus play the game.

C | D | |
---|---|---|

C | $c,c$ | $0,b$ |

D | $b,0$ | $d,d$ |

C | D | |
---|---|---|

C | $c,c+\alpha c$ | $0,b$ |

D | $b,\alpha b$ | $d,d+\alpha d$ |

(a) $s=NF$ | ||

C | D | |

C | $c,c$ | $0,b$ |

D | $b,0$ | $d,d$ |

(b) $s=NN$ | ||

C | D | |

C | $c,c-\gamma c$ | $0,b$ |

D | $b,-\gamma b$ | $d,d-\gamma d$ |

(c) $s=NH$ |

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Andersson, L.
Cooperation between Emotional Players. *Games* **2020**, *11*, 45.
https://doi.org/10.3390/g11040045

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Andersson L.
Cooperation between Emotional Players. *Games*. 2020; 11(4):45.
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Andersson, Lina.
2020. "Cooperation between Emotional Players" *Games* 11, no. 4: 45.
https://doi.org/10.3390/g11040045