# Competing Conventions with Costly Information Acquisition

## Abstract

**:**

## 1. Introduction

## 2. The Model

**Lemma**

**1.**

**Lemma**

**2.**

## 3. Complete Information with Free Acquisition

#### 3.1. Unperturbed Dynamics

**Lemma**

**3.**

**Lemma**

**4.**

#### 3.2. Perturbed Dynamics

**Definition**

**1.**

**Theorem**

**1.**

## 4. Incomplete Information with Costly Acquisition

#### 4.1. Unperturbed Dynamics

**Lemma**

**5.**

#### 4.2. Perturbed Dynamics

#### 4.2.1. Low Cost

**Corollary**

**1.**

**Theorem**

**2.**

#### 4.2.2. High Cost

**Corollary**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

- If $N\left({\pi}_{B}-{\pi}_{A}\right)>{N}_{B}{\mathsf{\Pi}}_{B}-{N}_{A}{\pi}_{B}-{\mathsf{\Pi}}_{B}+{\pi}_{B}+{\mathsf{\Pi}}_{A}$, then $M{S}_{a}$ is uniquely stochastically stable.
- If $N\left({\pi}_{A}-{\pi}_{B}\right)>{N}_{A}{\mathsf{\Pi}}_{A}-{N}_{B}{\pi}_{A}-{\mathsf{\Pi}}_{A}+{\mathsf{\Pi}}_{B}+{\pi}_{A}$, then $M{S}_{b}$ is uniquely stochastically stable.
- If $min\left\{{N}_{A}{\mathsf{\Pi}}_{A}-{N}_{B}{\pi}_{A}+{\pi}_{A},{N}_{B}{\mathsf{\Pi}}_{B}-{N}_{A}{\pi}_{B}+{\pi}_{B}\right\}-{\mathsf{\Pi}}_{A}-{\mathsf{\Pi}}_{B}>N\left({\pi}_{A}+{\pi}_{B}\right)$, then $TS$ is uniquely stochastically stable.

- If $N\left({\pi}_{B}-{\pi}_{A}\right)>{N}_{B}({\mathsf{\Pi}}_{B}+{\pi}_{A})-{N}_{A}({\mathsf{\Pi}}_{A}+{\pi}_{B})+{\mathsf{\Pi}}_{A}-{\pi}_{A}+{\pi}_{B}-{\mathsf{\Pi}}_{B}$, then $M{S}_{a}$ is uniquely stochastically stable.
- If $N\left({\pi}_{A}-{\pi}_{B}\right)>{N}_{A}({\mathsf{\Pi}}_{A}+{\pi}_{B})-{N}_{B}({\mathsf{\Pi}}_{B}+{\pi}_{A})-{\mathsf{\Pi}}_{A}+{\pi}_{A}-{\pi}_{B}+{\mathsf{\Pi}}_{B}$, then $M{S}_{b}$ is uniquely stochastically stable.
- If $N\left({\pi}_{A}-{\pi}_{B}\right)={N}_{A}({\mathsf{\Pi}}_{A}+{\pi}_{B})-{N}_{B}({\mathsf{\Pi}}_{B}+{\pi}_{A})-{\mathsf{\Pi}}_{A}+{\pi}_{A}-{\pi}_{B}+{\mathsf{\Pi}}_{B}$, then both $M{S}_{a}$ and $M{S}_{b}$ are stochastically stable.

**Lemma**

**6.**

## 5. Discussion

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proofs

**Proof**

**of**

**Lemma**

**1.**

**Proof**

**of**

**Lemma**

**2.**

#### Appendix A.1. Proofs of Complete Information with Free Acquisition

**Proof**

**of**

**Lemma**

**3.**

**Proof**

**of**

**Lemma**

**4.**

**Lemma**

**A1.**

**Proof.**

**Corollary**

**A1.**

**Lemma**

**A2.**

**Proof.**

**Lemma**

**A3.**

**Proof.**

**Proof**

**of**

**Theorem**

**1.**

#### Appendix A.2. Proofs of Incomplete Information with Costly Acquisition

**Proof**

**of**

**Lemma**

**5.**

**Proof**

**of**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**4.**

**Proof**

**of**

**Lemma**

**6.**

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**Figure 1.**$P{S}_{b}=(0,0)$ is uniquely stochastically stable: $\frac{{\pi}_{B}}{{\pi}_{A}}<\frac{{N}_{B}}{{N}_{A}}$.

**Figure 2.**$P{S}_{a}=(10,5)$ is uniquely stochastically stable: $\frac{{\pi}_{B}}{{\pi}_{A}}>\frac{{N}_{B}}{{N}_{A}}$.

a | b | |
---|---|---|

a | ${\mathsf{\Pi}}_{A},{\mathsf{\Pi}}_{A}$ | $0,0$ |

b | $0,0$ | ${\pi}_{A},{\pi}_{A}$ |

a | b | |
---|---|---|

a | ${\pi}_{B},{\pi}_{B}$ | $0,0$ |

b | $0,0$ | ${\mathsf{\Pi}}_{B},{\mathsf{\Pi}}_{B}$ |

a | b | |
---|---|---|

a | ${\mathsf{\Pi}}_{A},{\pi}_{B}$ | $0,0$ |

b | $0,0$ | ${\pi}_{A},{\mathsf{\Pi}}_{B}$ |

State | Condition on Group Size and Payoffs | Conditions on c |
---|---|---|

$M{S}_{a}$ | none | none |

$M{S}_{b}$ | none | none |

$TS$ | $\frac{{\pi}_{B}}{{\mathsf{\Pi}}_{B}}<\frac{{N}_{B}-1}{{N}_{A}}$ | $c>max\left\{\frac{{N}_{B}}{N-1}{\pi}_{A},\frac{{N}_{A}}{N-1}{\pi}_{B}\right\}$ |

$P{S}_{b}$ | none | $c<\frac{{N}_{B}}{N-1}{\pi}_{A}$ |

$P{S}_{a}$ | (1) $\frac{{\pi}_{B}}{{\mathsf{\Pi}}_{B}}>\frac{{N}_{B}-1}{{N}_{A}}$ (2) $\frac{{\pi}_{B}}{{\mathsf{\Pi}}_{B}}<\frac{{N}_{B}-1}{{N}_{A}}$ | (1) $c<\frac{{N}_{B}-1}{N-1}{\mathsf{\Pi}}_{B}$ (2) $c<\frac{{N}_{A}}{N-1}{\pi}_{B}$ |

$(0,{N}_{A},{N}_{B},{N}_{B})$ | (1) $\frac{{\pi}_{A}}{{\mathsf{\Pi}}_{A}}<\frac{{N}_{B}}{{N}_{A}-1}$ (2) $\frac{{\pi}_{A}}{{\mathsf{\Pi}}_{A}}>\frac{{N}_{B}}{{N}_{A}-1}$ | (1) $c<\frac{{N}_{A}-1}{N-1}{\pi}_{A}$ (2) $c<\frac{{N}_{B}}{N-1}{\mathsf{\Pi}}_{A}$ |

$({N}_{A},0,0,{N}_{B})$ | none | $c<min\left\{\frac{{N}_{B}}{N-1}{\pi}_{A},\frac{{N}_{B}-1}{N-1}{\pi}_{B}\right\}$ |

$(0,{N}_{A},{N}_{B},0)$ | (1) $\frac{{\pi}_{A}}{{\mathsf{\Pi}}_{A}}<\frac{{N}_{B}}{{N}_{A}-1}$ and $\frac{{\pi}_{B}}{{\mathsf{\Pi}}_{B}}>\frac{{N}_{B}-1}{{N}_{A}}$ (2) $\frac{{\pi}_{A}}{{\mathsf{\Pi}}_{A}}>\frac{{N}_{B}}{{N}_{A}-1}$ and $\frac{{\pi}_{B}}{{\mathsf{\Pi}}_{B}}>\frac{{N}_{B}-1}{{N}_{A}}$ (3) $\frac{{\pi}_{A}}{{\mathsf{\Pi}}_{A}}<\frac{{N}_{B}}{{N}_{A}-1}$ and $\frac{{\pi}_{B}}{{\mathsf{\Pi}}_{B}}<\frac{{N}_{B}-1}{{N}_{A}}$ (4) $\frac{{\pi}_{A}}{{\mathsf{\Pi}}_{A}}>\frac{{N}_{B}}{{N}_{A}-1}$ and $\frac{{\pi}_{B}}{{\mathsf{\Pi}}_{B}}<\frac{{N}_{B}-1}{{N}_{A}}$ | (1) $c<min\left\{\frac{{N}_{A}-1}{N-1}{\pi}_{A},\frac{{N}_{B}-1}{N-1}{\mathsf{\Pi}}_{B}\right\}$ (2) $c<min\left\{\frac{{N}_{B}}{N-1}{\mathsf{\Pi}}_{A},\frac{{N}_{B}-1}{N-1}{\mathsf{\Pi}}_{B}\right\}$ (3) $c<min\left\{\frac{{N}_{A}-1}{N-1}{\pi}_{A},\frac{{N}_{A}}{N-1}{\pi}_{B}\right\}$ (4) $c<min\left\{\frac{{N}_{B}}{N-1}{\mathsf{\Pi}}_{A},\frac{{N}_{A}}{N-1}{\pi}_{B}\right\}$ |

$(0,0,0,{N}_{B})$ | none | $c<\frac{{N}_{B}-1}{N-1}{\pi}_{B}$ |

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Rozzi, R. Competing Conventions with Costly Information Acquisition. *Games* **2021**, *12*, 53.
https://doi.org/10.3390/g12030053

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Rozzi R. Competing Conventions with Costly Information Acquisition. *Games*. 2021; 12(3):53.
https://doi.org/10.3390/g12030053

**Chicago/Turabian Style**

Rozzi, Roberto. 2021. "Competing Conventions with Costly Information Acquisition" *Games* 12, no. 3: 53.
https://doi.org/10.3390/g12030053