# Imitation and Local Interactions: Long Run Equilibrium Selection

## Abstract

**:**

## 1. Introduction

## 2. Related Literature

## 3. The Model

#### 3.1. Interaction

#### 3.2. Revision Opportunities

## 4. Unperturbed Dynamics

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

#### Absorbing States

**Lemma**

**1.**

**Lemma**

**2.**

## 5. Stochastic Analysis

**Lemma**

**3.**

**Lemma**

**4.**

**Lemma**

**5.**

**Proposition**

**1.**

**Proposition**

**2.**

**Lemma**

**6.**

- For$$\beta <\frac{2k-2}{2k-3}-\frac{3\alpha}{2k-3},$$$$R\left(\overline{A}\right)=2k+1.$$
- For$$\beta >\frac{2k-2}{2k-4}-\frac{4\alpha}{2k-4},$$$$R\left(\overline{A}\right)<2k.$$
- For$$\frac{2k-2}{2k-3}-\frac{3\alpha}{2k-3}<\beta <\frac{2k-2}{2k-4}-\frac{4\alpha}{2k-4},$$$$R\left(\overline{A}\right)=2k.$$

**Lemma**

**7.**

**Proposition**

**3.**

- $\overline{A}$ is the unique stochastically stable convention if$$\beta <\frac{2k-2}{2k-3}-\frac{3\alpha}{2k-3}.$$
- $\overline{A}$, $\overline{B}$ both belong to stochastically stable set if$$\frac{2k-2}{2k-3}-\frac{3\alpha}{2k-3}<\beta <\frac{2k-2}{2k-4}-\frac{4\alpha}{2k-4}.$$
- $\overline{B}$ is the unique stochastically stable convention if$$\beta >\frac{2k-2}{2k-4}-\frac{4\alpha}{2k-4}.$$

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A. Radius and Coradius

## Appendix B. Proofs

**Proof of Lemma**

**1.**

**Proof of Lemma**

**2.**

**Proof of Lemma**

**3.**

**Figure A1.**Example of a minimal cost path from $\overline{A}$ to $\overline{B}$. A agents are white, while B agents are black. In the example, $N=15$, ${D}_{A}^{min}=5$, and ${D}_{B}^{min}=4$. Step 8 $\in D\left(\overline{B}\right)$.

**Proof of Lemma**

**4.**

**Proof of Lemma**

**5.**

**Proof of Proposition**

**1.**

**Proof of Proposition**

**2.**

**Proof of Lemma**

**6.**

**Figure A3.**Example of a mixed group composed by $2k-1$ agents A and three B agents such that the first and the last one are A agents, and all the agents outside the group are B agents. $k=5$.

**Proof of Lemma**

**7.**

**Proof of Proposition**

**3.**

## References

- Harsanyi, J.C.; Selten, R. A General Theory of Equilibrium Selection in Games; MIT Press: Cambridge, MA, USA, 1988; Volume 1. [Google Scholar]
- Alós-Ferrer, C.; Weidenholzer, S. Imitation, local interactions, and efficiency. Econ. Lett.
**2006**, 93, 163–168. [Google Scholar] [CrossRef] - Kandori, M.; Mailath, G.J.; Rob, R. Learning, mutation, and long run equilibria in games. Econometrica
**1993**, 61, 29–56. [Google Scholar] [CrossRef] - Young, H.P. The evolution of conventions. Econometrica
**1993**, 61, 57–84. [Google Scholar] [CrossRef] - Bergin, J.; Lipman, B.L. Evolution with state-dependent mutations. Econometrica
**1996**, 64, 943–956. [Google Scholar] [CrossRef][Green Version] - Bilancini, E.; Boncinelli, L. The evolution of conventions under condition-dependent mistakes. Econ. Theory
**2019**, 69, 497–521. [Google Scholar] [CrossRef][Green Version] - Sawa, R.; Wu, J. Prospect dynamics and loss dominance. Games Econ. Behav.
**2018**, 112, 98–124. [Google Scholar] [CrossRef] - Nax, H.H.; Newton, J. Risk attitudes and risk dominance in the long run. Games Econ. Behav.
**2019**, 116, 179–184. [Google Scholar] [CrossRef] - Staudigl, M.; Weidenholzer, S. Constrained interactions and social coordination. J. Econ. Theory
**2014**, 152, 41–63. [Google Scholar] [CrossRef][Green Version] - Ellison, G. Learning, local interaction, and coordination. Econometrica
**1993**, 61, 1047–1071. [Google Scholar] [CrossRef][Green Version] - Blume, L.E. The statistical mechanics of strategic interaction. Games Econ. Behav.
**1993**, 5, 387–424. [Google Scholar] [CrossRef][Green Version] - Norman, T.W. Rapid evolution under inertia. Games Econ. Behav.
**2009**, 66, 865–879. [Google Scholar] [CrossRef][Green Version] - Jiang, G.; Weidenholzer, S. Local interactions under switching costs. Econ. Theory
**2017**, 64, 571–588. [Google Scholar] [CrossRef][Green Version] - Alós-Ferrer, C.; Weidenholzer, S. Contagion and efficiency. J. Econ. Theory
**2008**, 143, 251–274. [Google Scholar] [CrossRef] - Khan, A. Coordination under global random interaction and local imitation. Int. J. Game Theory
**2014**, 43, 721–745. [Google Scholar] [CrossRef][Green Version] - Cui, Z. More neighbors, more efficiency. J. Econ. Dyn. Control
**2014**, 40, 103–115. [Google Scholar] [CrossRef] - Ellison, G. Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution. Rev. Econ. Stud.
**2000**, 67, 17–45. [Google Scholar] [CrossRef][Green Version] - Alós-Ferrer, C.; Buckenmaier, J.; Farolfi, F. Imitation, network size, and efficiency. Netw. Sci.
**2020**, 9, 123–133. [Google Scholar] [CrossRef] - Alós-Ferrer, C.; Buckenmaier, J.; Farolfi, F. When Are Efficient Conventions Selected in Networks? J. Econ. Dyn. Control
**2021**, 124, 104074. [Google Scholar] [CrossRef] - Lim, W.; Neary, P.R. An experimental investigation of stochastic adjustment dynamics. Games Econ. Behav.
**2016**, 100, 208–219. [Google Scholar] [CrossRef] - Mäs, M.; Nax, H.H. A behavioral study of “noise” in coordination games. J. Econ. Theory
**2016**, 162, 195–208. [Google Scholar] [CrossRef][Green Version] - Bilancini, E.; Boncinelli, L.; Nax, H.H. What noise matters? Experimental evidence for stochastic deviations in social norms. J. Behav. Exp. Econ.
**2021**, 90, 101626. [Google Scholar] [CrossRef] - Hwang, S.H.; Lim, W.; Neary, P.; Newton, J. Conventional contracts, intentional behavior and logit choice: Equality without symmetry. Games Econ. Behav.
**2018**, 110, 273–294. [Google Scholar] [CrossRef][Green Version] - Newton, J. Evolutionary game theory: A renaissance. Games
**2018**, 9, 31. [Google Scholar] [CrossRef][Green Version] - Newton, J. Conventions under Heterogeneous Behavioural Rules. Rev. Econ. Stud.
**2020**, 0, 1–25. [Google Scholar]

**Figure 1.**Areas characterized by different dimension of minimal A groups. Areas on the top, lighter colors, are associated with greater values of minimal A group dimension.

**Figure 2.**Areas characterized by different dimension of minimal B groups. Areas on the top, darker colors, are associated with lower values of minimal B group dimension.

**Figure 3.**Example of a group composed by $2k$ agents A and three B agents such that the first and the last one of the group are A agents, and all the agents outside the group are B agents. $k=5$, A agents are white, and B agents are black.

**Figure 5.**In the blue area, the all-A monomorphic state is the unique long run equilibrium, when $N>2k(2k+1)$.

**Figure 6.**In the green area, the dimension of the minimal A group is equal to $2k$, while $2k+1$ is the dimension of the minimal B group.

**Figure 7.**Example of a mixed group composed by $2k$ agents B and three A agents such that the first and the last one of the group are B agents, and all the agents outside the group are A agents. $k=5$.

**Figure 8.**Example of a mixed group composed by $2k-1$ agents B and four A agents such that the first and the last one of the group are B agents, and all the agents outside the group are A agents. $k=5$.

**Figure 9.**In the red area, $\overline{B}$ is the unique long run equilibrium; in the blue area, $\overline{A}$ is the unique long run equilibrium; in the orange area, both $\overline{A}$ and $\overline{B}$ are long run equilibria.

**Figure 10.**In the red area, $\overline{B}$ is the unique long run equilibrium; in the blue area, $\overline{A}$ is the unique long run equilibrium; in the orange area, both $\overline{A}$ and $\overline{B}$ are long run equilibria.

A | B | |
---|---|---|

A | $a,a$ | $c,d$ |

B | $d,c$ | $b,b$ |

A | B | |
---|---|---|

A | 1 | 0 |

B | $\alpha $ | $\beta $ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vicario, E.
Imitation and Local Interactions: Long Run Equilibrium Selection. *Games* **2021**, *12*, 30.
https://doi.org/10.3390/g12020030

**AMA Style**

Vicario E.
Imitation and Local Interactions: Long Run Equilibrium Selection. *Games*. 2021; 12(2):30.
https://doi.org/10.3390/g12020030

**Chicago/Turabian Style**

Vicario, Eugenio.
2021. "Imitation and Local Interactions: Long Run Equilibrium Selection" *Games* 12, no. 2: 30.
https://doi.org/10.3390/g12020030