# An Adaptive Model of Demand Adjustment in Weighted Majority Games

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Literature

## 3. The Model

#### 3.1. Aspirations in Cooperative Games

**Definition**

**1.**

**aspiration**if it is maximal ($x\left(S\right)\ge v\left(S\right)$ for all S) and feasible (for all i, there exists $S\ni i$ such that $x\left(S\right)\le v\left(S\right)$).

**Definition**

**2.**

**generating collection**$GC\left(x\right)=\{S:x(S)=v(S\left)\right\}$ is the set of coalitions that can satisfy the demands of their members.

**Definition**

**3.**

**partnered**if ${C}_{i}$ is nonempty for all i and for any i, j in N:

**Definition**

**4.**

**partnered**if $GC\left(x\right)$ is partnered.

**Definition**

**5.**

**separating**if ${C}_{i}\backslash {C}_{j}$ and ${C}_{j}\backslash {C}_{i}$ are both nonempty for any $i,j$.

**Definition**

**6.**

**separating**if $GC\left(x\right)$ is separating.

**Definition**

**7.**

**balanced**if x solves the problem

#### 3.2. The Basic Demand Adjustment Process

**Definition**

**8.**

**absorbing**if $\mathrm{\Psi}\left(\mathcal{A}\right)=\mathcal{A}.$ An absorbing set $\mathcal{A}$ is

**minimal**if no strict subset of $\mathcal{A}$ is absorbing.

**Definition**

**9.**

**absorbing set solution**is the union of all minimal absorbing sets.

**Lemma**

**1.**

- (i)
- If player i’s demand ${x}_{i}^{t-1}$ is not feasible, ${x}_{i}^{t}<{x}_{i}^{t-1}.$
- (ii)
- If player i’s demand ${x}_{i}^{t-1}$ is not maximal, ${x}_{i}^{t}>{x}_{i}^{t-1}.$

**Proof.**

- (i)
- Because ${max}_{S:i\in S}\{v\left(S\right)-{x}^{t-1}(S\backslash i)\}<{x}_{i}^{t-1}$ (given that ${x}_{i}^{t-1}$ is not feasible), it follows that ${x}_{i}^{t}<{x}_{i}^{t-1}$.
- (ii)
- Because ${max}_{S:i\in S}\{v\left(S\right)-{x}^{t-1}(S\backslash i)\}>{x}_{i}^{t-1}$ (given that ${x}_{i}^{t-1}$ is not maximal), it follows that ${x}_{i}^{t}>{x}_{i}^{t-1}$.

- (i)
- only one player adjusts at a time;
- (ii)
- a player will increase his demand if some coalition can support the larger demand, given the demands of others;
- (iii)
- a player will decrease his demand if no coalition can support his current demand, given the demands of others.

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Corollary**

**1.**

#### 3.3. The Demand Adjustment Process with Mutations

**Lemma**

**2.**

**Proof.**

**Definition**

**10.**

**stochastically stable**if it has a positive probability in the limit stationary distribution as ε goes to 0, that is, ${\mu}_{x}^{0}>0$.

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Definition**

**11.**

**locally stable**if (i) all states in $\mathcal{B}$ are in an absorbing set; (ii) for any $\mathcal{S}\subseteq \mathcal{B}$, after a mutation of one player to a higher demand the basic process converges to a state in $\mathcal{B}$; (iii) there is no proper subset of $\mathcal{B}$ that has this property.

**Proposition**

**3.**

## 4. Demand Adjustment in Weighted Majority Games

#### 4.1. Weighted Majority Games

**Definition**

**12.**

**constant-sum**if $v\left(S\right)+v(N\backslash S)=1$.

**Definition**

**13.**

**homogeneous**if all minimal winning coalitions have the same total weight q.

**Definition**

**14.**

**homogeneous game**.

#### 4.2. Aspirations in Weighted Majority Games

**Remark**

**1.**

**Example**

**1.**

#### 4.3. Symmetric Majority Games

**Proposition**

**4.**

**Proof.**

#### 4.4. Apex Games

**Proposition**

**5.**

**Proof.**

#### 4.5. Stochastic Stability in Other Weighted Majority Games

**Example**

**1**

**(continued).**

**Example**

**2.**

**Example**

**3.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Lemma 2

**Lemma**

**A1.**

**Proof.**

#### Appendix A.2. An Aspiration Vector That Is Balanced but Not Partnered

**Example**

**A1.**

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Montero, M.; Possajennikov, A.
An Adaptive Model of Demand Adjustment in Weighted Majority Games. *Games* **2022**, *13*, 5.
https://doi.org/10.3390/g13010005

**AMA Style**

Montero M, Possajennikov A.
An Adaptive Model of Demand Adjustment in Weighted Majority Games. *Games*. 2022; 13(1):5.
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**Chicago/Turabian Style**

Montero, Maria, and Alex Possajennikov.
2022. "An Adaptive Model of Demand Adjustment in Weighted Majority Games" *Games* 13, no. 1: 5.
https://doi.org/10.3390/g13010005