# The Unanimity Rule under a Two-Agent Fixed Sequential Order Voting

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## Abstract

**:**

## 1. Introduction

## 2. The Model

#### 2.1. Decision Problem

**Axiom**

**1.**

**Axiom**

**2.**

**Axiom**

**3.**

**Axiom**

**4.**

#### 2.2. Order Schemes

- Partial histories: histories in which either no proposal has been made or agreement has not been reached after finitely many proposals. These are formalised by the initial history ${h}^{0}$ and finite histories $({h}^{1},\cdots ,{h}^{n})$, where either ${h}^{n}={c}_{1}^{n}$ or ${h}^{n}=({c}_{1}^{n},{c}_{2}^{n})$ and ${c}_{1}^{n}\ne {c}_{2}^{n}$, respectively. Let H denote the set of partial histories.
- Terminal histories: histories in which either agreement is reached or disagreement is persistent. These are formalised by finite histories $({h}^{1},\cdots ,{h}^{n})$, where ${h}^{n}=({c}_{1}^{n},{c}_{2}^{n})$ and ${c}_{1}^{n}={c}_{2}^{n}$, and infinite histories, respectively. Let Z denote the set of terminal histories. Notice that each terminal history z induces a unique outcome $o\left(z\right)\in O$. This is obtained as follows: $\left(i\right)$ if z is finite, then $z={\left({h}^{t}\right)}_{t=1}^{n}$ with ${h}^{n}=({c}_{1}^{n},{c}_{2}^{n})$ and ${c}_{1}^{n}={c}_{2}^{n}$; clearly, $o\left(z\right)=(n,{c}_{1}^{n})$; $\left(ii\right)$ if z is infinite, then $o\left(z\right)=(\infty ,\varnothing ).$

- Fixed order scheme: The designated proposer is the same at every stage; that is, at every stage t, agent a makes a proposal and agent b decides whether to accept it or continue with the voting. This scheme is formalised by assuming that agent a’s and b’s information sets are, respectively:$$\begin{array}{cc}\hfill {H}_{a}^{1}& =\left(\right)open="\{"\; close="\}">({h}^{1},\cdots ,{h}^{n})\in H\left(\right)open="|"\; close>\phantom{\rule{0.166667em}{0ex}}{h}^{n}=({c}_{1}^{n},{c}_{2}^{n})\hfill & \cup \left\{{h}^{0}\right\},\end{array}\hfill {H}_{b}^{1}& =\left(\right)open="\{"\; close="\}">({h}^{1},\cdots ,{h}^{n})\in H\left(\right)open="|"\; close>\phantom{\rule{0.166667em}{0ex}}{h}^{n}={c}_{1}^{n}\hfill & .$$
- Switching order scheme: The designated proposer changes at every stage; that is, agent a (respectively, b) makes a proposal at each odd (respectively, even) stage t, and agent b (respectively, a) decides whether to accept agent a’s (respectively, b’s) proposal or to continue with the voting. In this case, we assumed that agent a’s and b’s information sets are, respectively:$$\begin{array}{cc}\hfill {H}_{a}^{2}& =\left(\right)open="\{"\; close="\}">({h}^{1},\cdots ,{h}^{n})\in H\left(\right)open="|"\; close>\begin{array}{cc}\hfill rl\left(i\right)& {h}^{n}=({c}_{1}^{n},{c}_{2}^{n})\mathrm{for}\mathrm{even}n,\\ \hfill \left(ii\right)& {h}^{n}={c}_{1}^{n}\mathrm{for}\mathrm{odd}n\end{array}\hfill & \cup \left\{{h}^{0}\right\},\end{array}\hfill {H}_{b}^{2}& =\left(\right)open="\{"\; close="\}">({h}^{1},\cdots ,{h}^{n})\in H\left(\right)open="|"\; close>\begin{array}{cc}\hfill rl\left(i\right)& {h}^{n}=({c}_{1}^{n},{c}_{2}^{n})\mathrm{for}\mathrm{odd}n,\\ \hfill \left(ii\right)& {h}^{n}={c}_{1}^{n}\mathrm{for}\mathrm{even}n\end{array}\hfill & .$$

**Example**

**1.**

**Definition**

**1**

## 3. The Strategic Impact of Order Schemes

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Note

1 | This requires an inductive construction. If $h\in {H}_{i}^{k}$, then let $z(s|h)=(h,\widehat{h})$, where: $(i)$${\widehat{h}}^{1}=({\widehat{c}}_{1}^{1},{\widehat{c}}_{2}^{1})$, consisting of ${c}_{1}^{1}={s}_{i}(h)$ and ${\widehat{c}}_{2}^{1}={s}_{-i}(h,{\widehat{c}}_{1}^{1})$; $(ii)$ for any $m\ge 2$, define inductively ${\widehat{h}}^{m}$ as $({\widehat{c}}_{1}^{m},{\widehat{c}}_{2}^{m})$, consisting of ${\widehat{c}}_{1}^{m}={s}_{i}(h,{\widehat{h}}^{1},\cdots ,{\widehat{h}}^{m-1})$ for the corresponding i, and ${\widehat{c}}_{2}^{m}={s}_{-i}(h,{\widehat{h}}^{1},\cdots ,{\widehat{h}}^{m-1},{\widehat{c}}_{1}^{m})$, until ${\widehat{c}}_{1}^{m}={\widehat{c}}_{2}^{m}$. |

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**MDPI and ACS Style**

Bánnikova, M.; Giménez-Gómez, J.-M.
The Unanimity Rule under a Two-Agent Fixed Sequential Order Voting. *Games* **2022**, *13*, 77.
https://doi.org/10.3390/g13060077

**AMA Style**

Bánnikova M, Giménez-Gómez J-M.
The Unanimity Rule under a Two-Agent Fixed Sequential Order Voting. *Games*. 2022; 13(6):77.
https://doi.org/10.3390/g13060077

**Chicago/Turabian Style**

Bánnikova, Marina, and José-Manuel Giménez-Gómez.
2022. "The Unanimity Rule under a Two-Agent Fixed Sequential Order Voting" *Games* 13, no. 6: 77.
https://doi.org/10.3390/g13060077