A folk theorem for the one-dimensional spatial bargaining model

Abstract

We show that in the one-dimensional bargaining model based on the protocol of Baron and Ferejohn (Am Polit Sci Rev 83:1181–1206, 1989), if voting is simultaneous, publicly observed, and no agent has the power to unilaterally impose a choice, then arbitrary policies can be supported by subgame perfect equilibria in stage-undominated voting strategies when agents are patient. Moreover, in the model with a bad status quo, arbitrary outcomes can be supported for arbitrary positive discount factors. We formulate sufficient conditions for supporting arbitrary alternatives in terms of a system of equations. The system has a straightforward solution in the model with no discounting, and after verifying non-singularity of this system, we use the implicit function theorem to derive the folk theorem when agents are sufficiently patient.

Proof of Lemma 1

Lemma 1

Let $\varvec{\delta }\in (0,1)^{n}$. If $\mathbf Y \in \Omega ^{n}$ is plausible at $\varvec{\delta }$, then $\times _{i \in N} \text{ int } Y_{i}\subseteq X(\varvec{\delta })$.

Proof

Let $\varvec{\delta }\in [0,1)^{n}$ and let $\mathbf Y \in \Omega ^{n}$ be plausible at $\varvec{\delta }$. Let $(y_{1},\ldots , y_{n})\in \times _{i\in N} (\text{ int }Y_{i})$. It suffices to show that there exists $\sigma \in \Sigma (\varvec{\delta })$ such that for each $i\in N$, agent $i$ proposes $y_{i}$ in the first period, and it is accepted. Our equilibrium construction relies on the definition of three mappings, denoted $\alpha ^{j}_{i}$, $\beta ^{k}_{i}$, $\gamma ^{k}_{i}$ from alternatives to alternatives for each $i,j,k \in N$.

To define the mappings $\alpha ^{j}_{i}$, note that for every $j\in N$, strict quasi-concavity of $u_{j}$ implies that the minimum of $u_{j}$ in $Y_{j}$ is attained at a boundary point of the interval. Thus, for all $x\in \text{ int }Y_{j}$, we have $u_{j}(x) > \min _{z\in Y_{j}}u_{j}(z)$, and since $\mathbf {Y}$ is plausible, the inequality $u_{j}(x) > \delta _{j}\sum _{i\in N}\rho _{i}\min _{z_{i}\in Y_{i}}u_{j}(z_{i})$ then holds. Continuity of $u_{j}$ then implies that for each $i,j \in N$, we can define a mapping $\alpha ^{j}_{i}:\text{ int }Y_{j}\rightarrow \text{ int }Y_{i}$ such that for each $x\in \text{ int } Y_{j}$ and each $k\in N$, we have $\alpha ^{j}_{i}(x) \ne \hat{x}_{k}$, and so that in addition we have

$$\begin{aligned} u_{j}(x)&> \delta _{j}\sum _{i\in N}\rho _{i}u_{j}\left( \alpha ^{j}_{i}(x)\right) . \end{aligned}$$

(4)

Note that if agent $j$ compares having $x$ as the immediate outcome to rejection and having each $i$ propose and pass $\alpha ^{j}_{i}(x)$ in the next period, then the agent strictly prefers the former to the latter. In our equilibrium construction, we understand $\alpha ^{j}_{i}(x)$ to be an alternative proposed by agent $i$ if agent $j$ was supposed to propose $x$ in the preceding period and agent $j$’s proposal is not accepted; in equilibrium, this happens only when agent $j$ deviates from proposing $x$, so $\alpha ^{j}_{i}(x)$ acts to punish such deviations.

For each $i,k\in N$, define the mapping $\beta ^{k}_{i} :\text{ int }Y_{i} \rightarrow X$ by

$$\begin{aligned} \beta ^{k}_{i}(x)&= \left\{ \begin{array}{ll} \frac{x+\min \{\hat{x}_{k},\,\max Y_{i}\}}{2} &{} \quad \text{ if } \,x\le \hat{x}_{k},\\ \frac{x+\max \{\hat{x}_{k},\,\min Y_{i}\}}{2} &{} \quad \text{ if } \,x>\hat{x}_{k}.\end{array}\right. \end{aligned}$$

(5)

Thus, if $x < \hat{x}_{k}$, for example, then $\beta ^{k}_{i}(x)$ is located between $x$ and $\hat{x}_{k}$, so that $u_{k}\left( \beta ^{k}_{i}(x)\right) >u_{k}(x)$. We understand $\beta ^{k}_{i}(x)$ to be the alternative proposed by agent $i$ to reward agent $k$ if the proposer in the previous period proposed the correct alternative, if this was rejected by all agents but agent $k$, and if $x$ would be the current proposal dictated by $\alpha ^{j}_{i}$; this essentially moves the current proposal toward the ideal point of agent $k$, while remaining within the set $Y_{i}$ for the current proposer.

Define the mapping $\gamma ^{k}_{i}:\text{ int }Y_{i} \rightarrow X$ by

$$\begin{aligned} \gamma ^{k}_{i}(x)&= \left\{ \begin{array}{ll} \frac{x+\min Y_{i}}{2} &{} \quad \text{ if } \,x\le \hat{x}_{k},\\ \frac{x+\max Y_{i}}{2} &{} \quad \text{ if } \,x>\hat{x}_{k}.\end{array}\right. \end{aligned}$$

(6)

Thus, if $x < \hat{x}_{k}$, for example, then $\gamma ^{k}_{i}(x)$ is located between $\min Y_{i}$ and $x$, so that $u_{k}\left( \gamma ^{k}_{i}(x)\right) <u_{k}(x)$. We understand $\gamma ^{k}_{i}(x)$ to be the alternative proposed by agent $i$ to punish agent $k$ if the proposer in the previous period proposed the wrong alternative, if this was rejected by all agents but agent $k$, and if $x$ would be the current proposal dictated by $\alpha ^{j}_{i}$; this essentially moves the current proposal away from the ideal point of agent $k$, while remaining within the set $Y_{i}$ for the current proposer.

To construct an equilibrium $\sigma$ such that each agent $i$ proposes $y_{i}$ in the first period and it is accepted, we define proposal and voting strategies as follows. For each $i\in N$, we let $\mathbf {b}^{i}=(b^{i}_{1},\ldots ,b^{i}_{n})\in \{a,r\}^{n}$ denote the voting profile such that $b^{i}_{i}=a$ and $b^{i}_{j}=r$ for all $j\in N\setminus \{i\}$. Now, for each $i\in N$, define the proposal strategy $p_{i} :H^{pi}\rightarrow X$ so that for each $t=1,2,\dots$ and each $h_{t}\in H^{pi}_{t}$,

• 1. $p_{i}(h_{1})=y_{i}$,

• 2. if $t=2,3\dots$ and for some $h_{t-1}\in H^{pj}_{t-1}$, $x\in X$, and $\mathbf {b}\in B$, we have $h_{t}=(h_{t-1},x,\mathbf {b},i)$, then

  $$\begin{aligned} p_{i}(h_{t})=\left\{ \begin{array}{ll} \alpha ^{j}_{i}\left( p_{j}(h_{t-1})\right) &{} \quad \text{ if } \,\, \mathbf {b}\ne \mathbf {b}^{k} \text{ for } \text{ all } \,k\in N,\\ \beta ^{k}_{i}\left( \alpha ^{j}_{i}\left( p_{j}(h_{t-1})\right) \right) &{} \quad \text{ if } \,\, \mathbf {b}=\mathbf {b}^{k} \text{ for } \text{ some }\,k\in N \text{ and } \,x=p_{j}(h_{t-1}),\\ \gamma ^{k}_{i}\left( \alpha ^{j}_{i}\left( p_{j}(h_{t-1})\right) \right) &{} \quad \text{ if } \,\, \mathbf {b}=\mathbf {b}^{k} \text{ for } \text{ some } \,k\in N \text{ and } x\ne p_{j}(h_{t-1}).\end{array}\right. \end{aligned}$$

In particular, if agent $j$ is supposed to propose $p_{j}(h_{t-1})$ in period $t-1$, if the agent’s proposal is rejected, and if there is not exactly one vote to accept, then agent $i$ proposes $\alpha ^{j}_{i}\left( p_{j}(h_{t-1})\right)$ to punish $j$. If agent $j$ proposes the correct alternative and exactly one agent, say $k$, accepts $j$’s proposal, then $i$ rewards $k$ with $\beta ^{k}_{i}\left( \alpha ^{j}_{i}\left( p_{j}(h_{t-1})\right) \right)$. And if agent $j$ proposes the incorrect alternative and exactly one agent $k$ accepts it, then $i$ punishes $k$ with $\gamma ^{k}_{i}(\alpha ^{j}_{i}(p_{j}(h_{t-1})))$. The effectiveness of these proposal strategies stems from the fact that by our specification of voting strategies, below, agent $j$’s proposal is rejected if and only if the agent proposes $x \ne p_{j}(h_{t-1})$, in which case it is rejected by a unanimous vote.

For each $k \in N$, define the voting strategy $v_{k} :H^{v}\rightarrow \{a,r\}$ so that for every $j\in N$, every $h\in H^{pj}$, and every $x\in X$,

$$\begin{aligned} v_{k}(h,x)=\left\{ \begin{array}{ll} a &{} \quad \text{ if } \, x=p_{j}(h),\\ r &{} \quad \text{ if } \, x\ne p_{j}(h).\end{array}\right. \end{aligned}$$

That is, if the correct alternative $p_{j}(h)$ is proposed then all agents accept, and otherwise all agents reject.

Let $\sigma =(p_{i},v_{i})_{i\in N}$. Since each agent $i$ proposes $y_{i}$ initially and all agents accept the proposal, it remains to be shown that $\sigma \in \Sigma (\varvec{\delta })$. Consider any $j\in N$ and any $h\in H^{pj}$. Note that by construction, $p_{j}(h)\in \text{ int }Y_{j}$. By the construction of voting strategies, $p_{j}(h)$ passes if proposed, implying $U^{\sigma }_{j}(h,p_{j}(h)|\varvec{\delta })= \delta ^{\tau (h)-1}_{j}u_{j}(p_{j}(h))$. Now consider any $x\ne p_{j}(h)$. If agent $j$ deviates and proposes $x$ at $h$, then the outcome of voting is $v(h,x)=(r,\ldots ,r)$ and so, in the next period, each $i\in N$ proposes $\alpha ^{j}_{i}(p_{j}(h))$, which passes by a unanimous vote. Thus, we have

$$\begin{aligned} U^{\sigma }_{j}(h,x|\varvec{\delta })&= \delta ^{\tau (h)}_{j}\sum _{i\in N}\rho _{i}u_{j}\left( \alpha ^{j}_{i}\left( p_{j}(h)\right) \right) . \end{aligned}$$

Then, by Eq. (4), we have

$$\begin{aligned} U^{\sigma }_{j}\left( h,p_{j}(h)|\varvec{\delta }\right)&> U^{\sigma }_{j}(h,x|\varvec{\delta }), \end{aligned}$$

so the deviation is not profitable.

Next, consider any $h\in H^{v}$, and write $h=(h',x)$ for some $h'\in H^{pj}$ and $x\in X$, and consider any agent $k \in N$. First, suppose $x=p_{j}(h')$. Then the outcome of voting is $v(h)=(a,\dots ,a)$. Since $N\in \mathcal {D}$ and $N\setminus \{k\}\in \mathcal {D}$, we have

$$\begin{aligned} U^{\sigma }_{k}(h,(v_{k}(h),v_{-k}(h))|\varvec{\delta })&= U^{\sigma }_{k}\left( h,\left( b_{k},v_{-k}(h)\right) |\varvec{\delta }\right) \,\, = \,\, u_{k}(x) \end{aligned}$$

for every $b_{k}\in \{a,r\}$. Second, suppose $x\ne p_{j}(h')$. Then $v(h)=(r,\ldots ,r)$. Note that $\emptyset \notin \mathcal {D}$ and $\{k\}\notin \mathcal {D}$. By construction of proposal strategies, we have

$$\begin{aligned} U^{\sigma }_{k}\left( h,(v_{k}(h),v_{-k}(h))|\varvec{\delta }\right)&= \delta ^{\tau (h)}_{k} \sum _{i}\rho _{i}u_{k}\left( \alpha ^{j}_{i}\left( p_{j}(h')\right) \right) , \end{aligned}$$

and

$$\begin{aligned} U^{\sigma }_{k}\left( h,(a,v_{-k}(h))|\varvec{\delta }\right)&= \delta ^{\tau (h)}_{k}\sum _{i}\rho _{i}u_{k}\left( \gamma ^{k}_{i}\left( \alpha ^{j}_{i} \left( p_{j}(h')\right) \right) \right) . \end{aligned}$$

Recall that for every $k\in N$, we have $\alpha ^{j}_{i}\left( p_{j}(h')\right) \ne \hat{x}_{k}$, so (6) implies $u_{k}\left( \alpha ^{j}_{i}\left( p_{j}(h')\right) \right) >u_{k}\left( \gamma ^{k}_{i}\left( \alpha ^{j}_{i} \left( p_{j}(h')\right) \right) \right)$. Thus,

$$\begin{aligned} U^{\sigma }_{k} \left( h,\left( v_{k}(h),v_{-k}(h)\right) |\varvec{\delta }\right) > U^{\sigma }_{k}\left( h,\left( a,v_{-k}(h)\right) |\varvec{\delta }\right) , \end{aligned}$$

so again no agent has a profitable deviation. Therefore, $\sigma$ is a subgame perfect equilibrium.

Continuing, we must show that $\sigma$ is in stage-undominated voting strategies. Let $\mathbf {b}_{-k}^{k}$ be the profile of ballots for all $i\in N\setminus \{k\}$ such that they all choose $r$. In case $x \ne p_{j}(h')$, we have $v_{k}(h)=r$. In the previous paragraph, we have already shown that

$$\begin{aligned} U^{\sigma }_{k}\left( h, \left( v_{k}(h),\mathbf {b}_{-k}^{k}\right) |\varvec{\delta }\right) > U^{\sigma }_{k}\left( h, \left( a,\mathbf {b}_{-k}^{k}\right) |\varvec{\delta }\right) , \end{aligned}$$

so agent $k$’s acceptance leads to a strictly lower payoff. In case $x=p_{j}(h^{\prime })$, we have $v_{k}(h)=a$. Then, by construction, we have

$$\begin{aligned} U^{\sigma }_{k}\left( h,(v_{k}(h),\mathbf {b}_{-k}^{k}\right) |\varvec{\delta }) = \delta ^{\tau (h)}_{k}\sum _{i}\rho _{i}u_{k}\left( \beta ^{k}_{i}\left( \alpha ^{j}_{i}(x)\right) \right) , \end{aligned}$$

and

$$\begin{aligned} U^{\sigma }_{k}\left( h, \left( r,\mathbf {b}_{-k}^{k}\right) |\varvec{\delta }\right) = \delta ^{\tau (h)}_{k}\sum _{i}\rho _{i}u_{k}\left( \alpha ^{j}_{i}(x)\right) . \end{aligned}$$

Recall that for every $i\in N$, we have $\alpha ^{j}_{i}(x)\ne \hat{x}_{k}$, so (5) implies $u_{k}\left( \beta ^{k}_{i}\left( \alpha ^{j}_{i}(x)\right) \right) >u_{k}\left( \alpha ^{j}_{i}(x)\right)$. Thus,

$$\begin{aligned} U^{\sigma }_{k}\left( h,(v_{k}(h),\mathbf {b}_{-k}\right) |\varvec{\delta })&> U^{\sigma }_{k}\left( h,(r,\mathbf {b}_{-k})|\varvec{\delta }\right) , \end{aligned}$$

establishing that $\sigma$ is in stage-undominated voting strategies. We conclude that $\sigma \in \Sigma (\varvec{\delta })$, as required. $\square$