# Exchange Networks with Stochastic Matching

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## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. Convergent Expectations

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

#### 2.2. Divergent Expectations

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

#### 2.3. Social Preferences

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

## 3. Simulations

#### 3.1. Convergent Expectations

**Result**

**1.**

#### 3.2. Divergent Expectations

**Result**

**2.**

#### 3.3. Social Preferences

**Result**

**3.**

## 4. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | The consolidation of partnership does not involve some form of obstinacy such as in [25]. Their paper shows that, in bargaining, outside options may cancel out the effects of obstinacy. |

2 | In his coconut model, [38] has provided an example of the unlikelihood of witnessing a frictionless decentralized coordination mechanism in labor markets. |

3 | The sum of probabilities equals 1, such that $\mathbb{P}\left[\right(u,v)\notin M(t\left)\right]=1-\mathbb{P}\left[\right(u,v)\in M(t\left)\right]$ when ${\sum}_{(u,k)\in M\left(t\right)}{[{x}_{u}\left(t\right)-{x}_{k}\left(t\right)]}^{\alpha}={[{x}_{u}\left(t\right)-{x}_{v}\left(t\right)]}^{\alpha}$. We know that ${x}_{k}\left(t\right)=0$ for any unmatched node $k\notin M\left(t\right)$. We observe that ${lim}_{{x}_{k}\left(t\right)\to 0}{\sum}_{(u,k)\in M\left(t\right)}{[{x}_{u}\left(t\right)-{x}_{k}\left(t\right)]}^{\alpha}{[{x}_{u}\left(t\right)-{x}_{v}\left(t\right)]}^{-\alpha}=1$, where ${\sum}_{(u,k)\in M\left(t\right)}{[{x}_{u}\left(t\right)-{x}_{k}\left(t\right)]}^{\alpha}=1/{[{x}_{u}\left(t\right)-{x}_{v}\left(t\right)]}^{-\alpha}$, which holds true for a very large population of players. Consequently, $\mathbb{P}\left[\right(u,v)\in M(t\left)\right]+\mathbb{P}\left[\right(u,v)\notin M(t\left)\right]=1$ in the presence of an atomistic-type market structure: |

4 | An outside option is the best alternative that a player can command if it withdraws from the bargaining process in a unilateral way ([8]). |

5 | Provided that only an actual match involves the surplus splitting between the players, the value into play in the negotiation, or ${w}_{u,v}\left(t\right)$, is weighted by the probability that players do not match. Actually, the expected value to be exchanged depends on the probability that it has not been the subject of previous trading: |

6 | Whereas, in a Nash form of game, the alternatives are given exogenously, the alternatives in the network bargaining game are given endogenously: |

7 | Decreasing the level of $\varphi $ increases the expected outcomes of both. |

8 | Hayek himself admitted that market mechanisms were based on bounded rationality ([64]). |

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**Figure 1.**Nash bargaining solutions (${x}_{u}^{\u2605}$, ${x}_{v}^{\u2605}$) with convergent expectations. The x-axis corresponds to the decay parameter ($\alpha $). The y-axis denotes Nash allocations (${x}_{u}^{\alpha},{x}_{u}^{\alpha}$).

**Figure 2.**Option values (${\beta}_{u}$, ${\beta}_{v}$) with convergent expectations. The x-axis corresponds to the decay parameter ($\alpha $). The y-axis denotes the values alternative to Nash allocations (${\beta}_{u}\left(\alpha \right),{\beta}_{v}\left(\alpha \right)$).

**Figure 3.**Convergence dynamics of Nash allocations (${x}_{u}^{\u2605}$, ${x}_{v}^{\u2605}$) for $\alpha =0.50$. The x-axis corresponds to the timeline (t). The y-axis denotes the evolution of Nash allocations as a function of time (${x}_{u}^{\u2605}\left(t\right)$, ${x}_{v}^{\u2605}\left(t\right)$).

**Figure 4.**Nash bargaining solution (${x}_{u}^{\u2605}$) with divergent expectations. The left-sided x-axis corresponds to the decay parameter of u (${\alpha}_{u}$). The right-sided y-axis is the decay parameter of v (${\alpha}_{v}$). The z-axis denotes Nash allocations (${x}_{u}^{\u2605}({\alpha}_{u},{\alpha}_{v})$).

**Figure 5.**Nash bargaining solution (${x}_{v}^{\u2605}$) with divergent expectations. The left-sided x-axis corresponds to the decay parameter of u (${\alpha}_{u}$). The right-sided y-axis is the decay parameter of v (${\alpha}_{v}$). The z-axis denotes Nash allocations (${x}_{v}^{\u2605}({\alpha}_{u},{\alpha}_{v})$).

**Figure 6.**Convergence dynamics of Nash allocations (${x}_{u}^{\u2605}$, ${x}_{v}^{\u2605}$) for ${\alpha}_{u}=0.60$ and ${\alpha}_{v}=0.70$. The x-axis corresponds to the timeline (t). The y-axis denotes the evolution of Nash allocations as a function of time (${x}_{u}^{\u2605}\left(t\right)$, ${x}_{v}^{\u2605}\left(t\right)$).

**Figure 7.**Nash bargaining solution (${x}_{u}^{\u2605}$) with social preferences. The left-sided x-axis corresponds to the share of u ($\varphi $). The right-sided y-axis is the decay parameter ($\alpha $). The z-axis denotes Nash allocations (${x}_{u}^{\u2605}(\varphi ,\alpha )$).

**Figure 8.**Nash bargaining solution (${x}_{v}^{\u2605}$) with social preferences. The left-sided x-axis corresponds to the share of v ($1-\varphi $). The right-sided y-axis is the decay parameter ($\alpha $). The z-axis denotes Nash allocations (${x}_{v}^{\u2605}(1-\varphi ,\alpha )$).

**Figure 9.**Convergence dynamics of Nash allocations (${x}_{u}^{\u2605}$, ${x}_{v}^{\u2605}$) for $\alpha =0.50$ and $\varphi =0.80$. The x-axis corresponds to the timeline (t). The y-axis denotes the evolution of Nash allocations as a function of time (${x}_{u}^{\u2605}\left(t\right)$, ${x}_{v}^{\u2605}\left(t\right)$).

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Dragicevic, A.Z.
Exchange Networks with Stochastic Matching. *Games* **2023**, *14*, 2.
https://doi.org/10.3390/g14010002

**AMA Style**

Dragicevic AZ.
Exchange Networks with Stochastic Matching. *Games*. 2023; 14(1):2.
https://doi.org/10.3390/g14010002

**Chicago/Turabian Style**

Dragicevic, Arnaud Zlatko.
2023. "Exchange Networks with Stochastic Matching" *Games* 14, no. 1: 2.
https://doi.org/10.3390/g14010002