# Time-Preference Heterogeneity and Multiplicity of Equilibria in Two-Group Bargaining

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model and the Results

**Definition 1.**

**Lemma 1.**

- For all $i\in B$ and all ${h}_{j}\in {H}_{j}$, $j\in N$,${v}_{i}\left({h}_{j},{x}^{\prime}\right)=no$ if ${x}^{\prime}>{z}_{B}^{+}\left(\overline{x}\right)$; and for all ${h}_{j}\in {H}_{j}$, $j\in N$, and ${x}^{\prime}<{z}_{A}^{+}\left(\underline{x}\right)$ there is some $i\in \overline{A}$ such that ${v}_{i}^{*}\left({h}_{j},{x}^{\prime}\right)=no$.
- For all $i\in B$, $j\in \overline{A}$ and ${h}_{j}\in {H}_{j}$, ${v}_{i}^{*}\left({h}_{j},{x}^{\prime}\right)=yes$ for all ${x}^{\prime}<{z}_{B}^{+}\left(\underline{x}\right)$.
- For all $i\in \overline{A}$, $j\in B$ and ${h}_{j}\in {H}_{j}$, ${v}_{i}^{*}\left({h}_{j},{x}^{\prime}\right)=yes$ for all ${x}^{\prime}>{z}_{A}^{-}\left(\overline{x}\right)$.

**Proof.**

**Lemma 2.**

- If $\delta \ge \mu $ then ${x}_{i}^{*}\left({h}_{i}\right)\ge {z}_{B}^{+}\left(\underline{x}\right)$ for all $i\in A$ and any ${h}_{i}\in {H}_{i}$.
- If $\delta \le \mu $ then ${x}_{a}^{*}\left({h}_{a}\right)\ge {z}_{B}^{+}\left(\underline{x}\right)$ for all ${h}_{a}\in {H}_{a}$.
- ${x}_{i}^{*}\left({h}_{i}\right)\le {z}_{A}^{-}\left(\overline{x}\right)$ for all $i\in B$ and any ${h}_{i}\in {H}_{i}$.

**Proof.**

**Lemma 3.**

**Proof.**

**Lemma 4.**

**Proof.**

**Example 1.**

**Lemma 5.**

- ${v}_{a}^{*}\left({h}_{i}^{0},{x}^{\prime}\right)=no$ for all ${x}^{\prime}\in \left({x}_{i}^{*}\left({h}_{i}^{0}\right),{z}_{B}^{+}\left(\underline{x}\right)\right)$
- There is some ${x}^{\prime}\in \left({x}_{i}^{*}\left({h}_{i}^{0}\right),{z}_{B}^{+}\left(\underline{x}\right)\right)$ such that $x\left({h}^{\left(j\right)}|{\sigma}^{*}\right)\in \left[{y}_{a}\left(\underline{x}\right),min\left\{{y}_{i}\left({x}_{i}^{*}\left({h}_{i}^{0}\right)\right),{x}^{s}\right\}\right]$, where ${h}^{\left(j\right)}=\left({h}_{i}^{0},{x}^{\prime},j\right)\in {H}^{1}$ is the history that follows when some player $j\in N$ rejects ${x}^{\prime}$.

**Proof.**

**Lemma 6.**

**Proof.**

- ${\widehat{H}}^{1}=\{\left({h}_{j}^{0},{x}^{\prime},k\right):j\in A,{x}^{\prime}\in \left(\tilde{x},{z}_{B}^{+}\left(\underline{x}\right)\right],k\in N\}$
- ${\tilde{H}}^{1}=\{\left({h}_{j}^{0},{x}^{\prime},k\right):j\in A,{x}^{\prime}\notin \left(\tilde{x},{z}_{B}^{+}\left(\underline{x}\right)\right],k\in N\}$

- ${x}_{j}\left({h}_{j}^{0}\right)=\tilde{x}$ for all $j\in A$; ${x}_{j}\left({h}_{j}^{0}\right)={x}_{j}^{*}\left({h}_{j}^{0}\right)$ for all $j\in B$; and ${x}_{a}\left({h}_{a}^{0}\right)={x}_{a}^{*}\left({h}_{a}^{0}\right)$.
- ${v}_{j}\left({h}_{k}^{0},{x}^{\prime}\right)={v}_{j}^{*}\left({h}_{k}^{0},{x}^{\prime}\right)$ for all $j\in N$ and $k\notin A$; ${v}_{j}\left({h}_{k}^{0},{x}^{\prime}\right)=yes$ iff ${x}^{\prime}\le {z}_{B}^{+}\left(\underline{x}\right)$ for all $j\in B$ and $k\in A$; ${v}_{j}\left({h}_{k}^{0},{x}^{\prime}\right)=yes$ iff ${x}^{\prime}\ge \delta \underline{x}$ for all $j\in A$, $k\in A$; and ${v}_{a}\left({h}_{k}^{0},{x}^{\prime}\right)=yes$ iff ${x}^{\prime}\in \left[{z}_{A}^{-}\left(\underline{x}\right),\tilde{x}\right]\cup \left({z}_{B}^{+}\left(\underline{x}\right),1\right]$ for all $k\in A$.
- $\sigma \left({h}^{1}\right)$ is a strategy profile of mutually best responses starting at some ${h}^{1}\in {H}^{1}$ such that:
- −
- $x\left({h}^{1}|\sigma \right)={x}^{E}\in \left[{y}_{a}\left(\underline{x}\right),{y}_{i}\left({x}_{i}^{*}\left({h}_{i}^{0}\right)\right)\right]$ if ${h}^{1}\in {\widehat{H}}^{1}$
- −
- $x\left({h}^{1}|\sigma \right)=\underline{x}$ if ${h}^{1}\in {\tilde{H}}^{1}$
- −
- $x\left({h}^{1}|\sigma \right)=x\left({h}^{1}|{\sigma}^{*}\right)$ for all ${h}^{1}\in {H}^{1}\backslash \left({\widehat{H}}^{1}\cup {\tilde{H}}^{1}\right).$9

**Proposition 1.**

**Proof.**

- For all $h\in \widehat{H}\subset H$
- (a)
- ${x}_{i}\left({h}_{i}\right)\equiv {x}_{A}\in \left[{z}_{A}^{-}\left(\underline{x}\right),{z}_{B}^{+}\left(\underline{x}\right)\right)$ for $i\in A$; ${x}_{i}\left({h}_{i}\right)={z}_{A}^{-}\left(\underline{x}\right)$ for $i\in B$, and ${x}_{a}\left({h}_{a}\right)={z}_{B}^{+}\left(\underline{x}\right)$, such that$$\underline{x}=\frac{{n}_{B}{z}_{A}^{-}\left(\underline{x}\right)+{n}_{A}{x}_{A}+{z}_{B}^{+}\left(\underline{x}\right)}{n}$$
- (b)
- For all $i\in B$ and any $j\in N$, ${v}_{i}\left({h}_{j},{x}^{\prime}\right)=yes$ iff ${x}^{\prime}\le {z}_{B}^{+}\left(\underline{x}\right)$. For all $i\in A$ and any $j\in N$, ${v}_{i}\left({h}_{j},{x}^{\prime}\right)=yes$ iff $x\ge \delta \underline{x}$. For agent a, ${a}_{a}\left({h}_{j},x\right)=yes$ iff ${x}^{\prime}\ge {z}_{A}^{-}\left(\underline{x}\right)$ and $j\in B$, and ${v}_{a}\left({h}_{j},{x}^{\prime}\right)=yes$ iff ${x}^{\prime}\in \left[{z}_{A}^{-}\left(\underline{x}\right),{x}_{A}\right]\cup \left({z}_{B}^{+}\left(\underline{x}\right),1\right]$ and $j\in A$.
- (c)
- ${h}^{\prime}=\left({h}_{i},{x}^{\prime},j\right)\in \tilde{H}$, for all $j\in N$ if $i\in A$ and ${x}^{\prime}\in \left({x}_{A},{z}_{B}^{+}\left(\underline{x}\right)\right]$; and ${h}^{\prime}=\left({h}_{i},{x}^{\prime},j\right)\in \widehat{H}$ otherwise.

- For all $h\in \tilde{H}=H\backslash \widehat{H}$
- (a)
- ${x}_{i}\left({h}_{i}\right)={z}_{A}^{-}\left(z\right)=\mu z$ for all $i\in B$ and ${x}_{i}\left({h}_{i}\right)\equiv {x}_{A}^{\prime}$ for all $i\in \overline{A}$, such that$$z=\frac{{n}_{B}\mu z+\left({n}_{A}+1\right){x}_{A}^{\prime}}{n}\in \left[{y}_{a}\left(\underline{x}\right),min\{{y}_{A}\left(\underline{x}\right),{x}^{s}\}\right]$$
- (b)
- For all $i\in \overline{A}$, ${v}_{i}\left({h}_{j},{x}^{\prime}\right)=yes$ for all $j\in N$ iff ${x}^{\prime}\ge {\delta}_{i}z$; and for all $i\in B$, ${v}_{i}\left({h}_{j},{x}^{\prime}\right)=yes$ iff either $j\in \overline{A}$ and ${x}^{\prime}\le {x}_{A}^{\prime}$ or $j\in B$ and ${x}^{\prime}\le {z}_{B}^{+}\left(z\right)$.
- (c)
- ${h}^{\prime}\left({h}_{i},{x}^{\prime},j\right)\in \widehat{H}$ for all $j\in N$ if $i\in \overline{A}$ and ${x}^{\prime}>{x}_{A}^{\prime}$; ${h}^{\prime}\left({h}_{i},{x}^{\prime},j\right)\in \tilde{H}$ for all $j\in N$ if $i\in \overline{A}$ and ${x}^{\prime}\le {x}_{A}^{\prime}$; and ${h}^{\prime}\left({h}_{i},{x}^{\prime},j\right)\in \tilde{H}$ for all $j\in N$ if $i\in B$ and for all ${x}^{\prime}$.

**Proposition 2.**

**Proof.**

**Corollary 1.**

- $\delta \in \left(\frac{1}{n-{n}_{B}},\overline{\delta}\right]$ and $\mu \ge {\mu}_{2}\left(\delta ,n,{n}_{B}\right)$, or
- $\delta >\overline{\delta}$ and $\mu \ge {\mu}_{3}\left(\delta ,n,{n}_{B}\right).$

**Proof.**

## 3. An Illustrative Example

- The minimal no-delay SPE expected outcome $\underline{x}$ is not monotone in μ. For low values of μ such a relationship is negative, which might seem counter-intuitive. As noted in the previous section, this happens because when the costs of delaying an agreement for player a are reduced, their impatient partners make smaller proposals in order to prevent a delayed agreement induced from a’s rejection. Hence, although the proposals of agents in B increase in μ, those of agents in A are reduced. Since, in this example, ${n}_{B}$ is low relatively to ${n}_{A}$, the second effect dominates the first one.12 For large values of μ, agents in A make the same proposal as agents in B. Hence, only the first effect appears and therefore $\underline{x}={x}_{s}$ increases in μ. Worthy, as $\mu \to 1$ the minimal no-delay SPE expected outcome converges to the maximal no-delay SPE expected outcome.
- There is a discontinuity at $\mu ={\mu}_{1}$. The reason for this lies in the fact that in order to get multiplicity there must exist $\underline{x}\in \left[{x}_{s},{x}^{s}\right)$ with ${y}_{a}\left(\underline{x}\right)\le min\left\{{y}_{A}\left(\underline{x}\right),{x}^{s}\right\}$, which happens only when $\mu \ge {\mu}_{1}>\delta $. Moreover, when such an outcome exists then there is a (equilibrium) threat by agent a to reject proposals in $\left[{z}_{A}^{-}\left(\underline{x}\right),{z}_{B}^{+}\left(\underline{x}\right)\right]$, inducing a jump in the equilibrium proposals of agents in A, which is reflected in such a discontinuity.
- If $\mu \in \left[0.95498,0.99786\right)$ then $\underline{x}<{n}_{A}/\left(n-1\right)$, which is the unique no-delay SPE expected outcome when a is not included into negotiations. So, within this parameter range the members of A would prefer not to include agent a into negotiations rather than their least favorable equilibrium when a is added. This does not happen when $\mu >0.99786$ where including a makes all agents in A better off even in their least preferred no-delay SPE.
- If $\mu >0.99813$ then $\underline{x}>\left({n}_{A}+1\right)/n$, which is the unique no-delay SPE expected outcome when $\mu =\delta $. So, even if agents in A anticipate that their worst equilibrium will be played, they all benefit from such heterogeneity.

## 4. When a Is more Impatient

**Proposition 3.**

## 5. Final Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Proposition A1.**

**Proof.**

- For all $h\in \widehat{H}$: (i) ${h}^{\prime}=\left({h}_{i},{x}^{\prime},j\right)\in \widehat{H}$ if ${x}^{\prime}=0$, $j\in N$; (ii) ${h}^{\prime}=\left({h}_{i},{x}^{\prime},j\right)\in \widehat{H}$ if ${x}^{\prime}\ne 0$ and $j\in B$; and (iii) ${h}^{\prime}=\left({h}_{i},{x}^{\prime},j\right)\in \tilde{H}$ if ${x}^{\prime}\ne 0$ and $j\in \overline{A}$.
- For all $h\in \tilde{H}$: (i) ${h}^{\prime}=\left({h}_{i},{x}^{\prime},j\right)\in \tilde{H}$ if ${x}^{\prime}=1$, $j\in N$; (ii) ${h}^{\prime}=\left({h}_{i},{x}^{\prime},j\right)\in \tilde{H}$ if ${x}^{\prime}\ne 1$ and $j\in \overline{A}$; and (iii) ${h}^{\prime}=\left({h}_{i},{x}^{\prime},j\right)\in \widehat{H}$ if ${x}^{\prime}\ne 1$ and $j\in B$.

- For all $h\in \widehat{H}$:
- −
- ${x}_{i}\left({h}_{i}\right)=0$ for all $i\in N$
- −
- If $i\in \overline{A}$, then ${v}_{j}\left({h}_{i},{x}^{\prime}\right)=no$ iff ${x}^{\prime}\ne 0$ for the first responder is $j\in B$
- −
- If $i\in B$ then ${v}_{j}\left({h}_{i},{x}^{\prime}\right)=no$ iff ${x}^{\prime}\ne 0$ for the first responder is $j\in \overline{A}$

- For all $h\in \tilde{H}$:
- −
- ${x}_{i}\left({h}_{i}\right)=1$ for all $i\in N$
- −
- If $i\in \overline{A}$, then ${v}_{j}\left({h}_{i},{x}^{\prime}\right)=no$ iff ${x}^{\prime}\ne 1$ for the first responder is $j\in B$
- −
- If $i\in B$ then ${v}_{j}\left({h}_{i},{x}^{\prime}\right)=no$ iff ${x}^{\prime}\ne 1$ for the first responder is $j\in \overline{A}$

**Lemma A1.**

**Proof.**

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^{1}See their Proposition 5.1, in page 312.^{2}This intuition is not corroborated in experiments by [15], who show that the more impatient agents also have an impact on the final outcome.^{4}Without imposing such a restriction, any policy $x\in \left[0,1\right]$ can be sustained as a no-delay SPE expected outcome when ${\delta}_{i}\ge 1/2$ for all $i\in N$. This highlights the importance of the this assumption in [7] to obtain uniqueness when all agents have the same time preference. We prove this statement in the Appendix.^{5}When no confusion arises, we omit the parameters $\left(N,\delta ,\mu \right)$.^{6}Lemma 1.2 refers to Statement 2 in Lemma 1. We use this referencing throughout.^{7}In Section 5, the results for $\delta >\mu $ are presented.^{8}We thank a referee for pointing out this remark.^{9}We know that such strategy profiles do exist, as $\underline{x}<{x}^{s}$ (by assumption), ${x}^{E}$ (by Lemma 5) and ${\sigma}^{*}$ itself are no-delay SPE expected outcomes.^{10}The exact inequality to obtain a positive (negative) relationship is ${n}_{B}\ge \left(\le \right)\frac{\left({n}_{A}+1\right){n}_{A}\delta}{2\left({n}_{A}+1\right)\mu +{n}_{A}\delta +{\mu}^{2}}$.^{11}In the example, we use some terms (as $\overline{\delta},{\mu}_{1},{\mu}_{2},{\mu}_{3}$ or $\overline{\mu}$) defined in the proofs of Proposition 2 and Corollary 1.^{12}Changing the sizes of the groups in the example by considering ${n}_{B}=7$, and replicating the calculations, it can be easily checked that the first effect dominates the second one and that $\widehat{x}$ is increasing in μ; thus $\underline{x}$ would be monotone.^{13}It is immediate that such a ${x}_{A}$ must exist.^{14}As there exists ${\sigma}^{*}\in E$ yielding $x\left({h}^{0}|{\sigma}^{*}\right)=\underline{x}$, where ${y}_{a}\left(\underline{x}\right)<{y}_{i}\left({x}_{i}^{*}\left({h}_{i}^{0}\right)\right)$ for all $i\in A$, such a values ${x}_{A}$ and ${x}_{B}$ must exist.

**Figure 1.**Pairs $(\delta ,\mu )$ yielding multiple SPE expected outcomes when $n=11$ and ${n}_{B}=5$.

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Cardona, D.; Rubí-Barceló, A. Time-Preference Heterogeneity and Multiplicity of Equilibria in Two-Group Bargaining. *Games* **2016**, *7*, 12.
https://doi.org/10.3390/g7020012

**AMA Style**

Cardona D, Rubí-Barceló A. Time-Preference Heterogeneity and Multiplicity of Equilibria in Two-Group Bargaining. *Games*. 2016; 7(2):12.
https://doi.org/10.3390/g7020012

**Chicago/Turabian Style**

Cardona, Daniel, and Antoni Rubí-Barceló. 2016. "Time-Preference Heterogeneity and Multiplicity of Equilibria in Two-Group Bargaining" *Games* 7, no. 2: 12.
https://doi.org/10.3390/g7020012