A bargaining model of endogenous procedures

Abstract

This paper endogenizes policymaking procedures in a multilateral bargaining framework. A procedure specifies players’ proposal power in bargaining over one-dimensional policies. In procedural bargaining players internalize the procedures’ effects on subsequent policy bargaining. In policy bargaining players’ utilities are continuous, strictly concave, and order-restricted. The paper provides equilibrium characterization, existence, and uniqueness results for this two-tier bargaining model. Although the procedural choice set is multidimensional, sequentially rational procedures feature “limited power sharing” and admit a total order. In equilibrium, endogenous procedures and policies are strategic complements.

Appendices

Appendix A: Mathematical proofs

Lemma 1

In equilibrium, a policy proposal is approved if and only if it is acceptable to the core player. That is, $A^{u}({\varvec{\rho }})=A_{k}^{u}({\varvec{\rho }})$, for all ${\varvec{\rho }} \in \Delta ^{n}.$

Proof of Lemma 1

Given a policy bargaining strategy profile ${\varvec{\sigma }}^{u}( {\varvec{\rho }}) =[ \zeta _{i}^{u}( {\varvec{\rho }}), A_{i}^{u}( {\varvec{\rho }}) ] _{i\in N},$ and a measurable set $Z\subseteq [ 0,1]$, define the following probability measure:

$$\begin{aligned} \pi ^{u}( {\varvec{\rho }}) ( Z)&=( 1-\gamma ) \mathbb {I}\big \{ p^{D}\in Z\big \} \nonumber \\&\quad \,+\,\gamma \frac{\sum \nolimits _{j\in N}\rho _{j}\big \{ \zeta _{j}^{u}( {\varvec{\rho }}) \big [ Z\cap A^{u}({\varvec{\rho }}) \big ]+\mathbb {I}\big \{ p^{D}\in Z\big \} ( 1-\gamma ) \zeta _{j}^{u}( {\varvec{\rho }}) \big [ [ 0,1] {\setminus } A^{u}( {\varvec{\rho }}) \big ] \big \} }{1-\gamma \sum \nolimits _{j\in N}\rho _{j}\zeta _{j}^{u}( {\varvec{\rho }}) \big [[0,1] {\setminus } A^{u}( {\varvec{\rho }}) \big ] } \end{aligned}$$

(14)

where $\mathbb {I} \{ p^{D}\in Z \}$ is an indicator function. By Eq. (4), we have $\int u_{i}( p) \pi ^{u}( {\varvec{\rho }}) ( dp) =( 1-\gamma ) u_{i}(p^{D})+\gamma U_{i}({\varvec{\rho }}|{\varvec{\sigma }}^{u}),$ for all $i\in N.$

Take a policy $p\in A_{k}^{u}({\varvec{\rho }}).$ That means $u_{k}(p)\ge ( 1-\gamma ) u_{k}(p^{D})+\gamma U_{k}({\varvec{\rho }}|{\varvec{\sigma }} ^{u}).$ Thus the core player weakly prefers p to $\pi ^{u}( {\varvec{\rho }}).$ Since all players vote between the same two policy lotteries, by order-restriction of preferences over policy lotteries, either $\underline{C}_{k}=\{ i\in N\mid {\tilde{p}}_{i}\le {\tilde{p}} _{k}\}$ or $\overline{C}_{k}=\{ i\in N\mid {\tilde{p}}_{i}\ge {\tilde{p}}_{k}\}$ weakly prefer p to $\pi ^{u}( {\varvec{\rho }} )$. We claim that $\underline{C}_{k},\overline{C}_{k}\in \mathcal {D}$ and thus $p\in A^{u}({\varvec{\rho }}),$ establishing that $A_{k}^{u}({\varvec{\rho }})\subseteq A^{u}({\varvec{\rho }}).$ Suppose $\underline{C}_{k}\notin \mathcal {D}$; then, because $\mathcal {D}$ is strong, $\overline{C} _{k+1}=N{\setminus } \underline{C}_{k}\in \mathcal {D}.$ Thus, ${\tilde{p}}_{k+1}$ beats ${\tilde{p}}_{k}$, since $u_{i}({\tilde{p}}_{k+1})>u_{i}({\tilde{p}}_{k})$ for all $i\ge k+1,$ that is, $i\in \overline{C}_{k+1}$, contradicting that ${\tilde{p}}_{k}\in K.$ The argument for $\overline{C}_{k}\in \mathcal {D}$ is analogous. Now take a policy $p\in A^{u}({\varvec{\rho }})$ and suppose that $p\notin A_{k}^{u}({\varvec{\rho }}).$ That is, $( 1-\gamma ) u_{k}(p^{D})+\gamma U_{k}({\varvec{\rho }}|{\varvec{\sigma }}^{u})>u_{k}(p).$ Since the core player strictly prefers $\pi ^{u}( {\varvec{\rho }})$ to p, by order-restriction of preferences over policy lotteries, either there exists $i\ge k$ such that $\underline{C}_{i}$ strictly prefer $\pi ^{u}( {\varvec{\rho }})$ to p while its complement $\overline{C} _{i+1}$ weakly prefer p to $\pi ^{u}( {\varvec{\rho }}),$ or there exists $i\le k$ such that $\overline{C}_{i}$ strictly prefer $\pi ^{u}( {\varvec{\rho }})$ to p while its complement $\underline{C} _{i-1}$ weakly prefer p to $\pi ^{u}( {\varvec{\rho }})$. Assume the former; the argument for the second case is analogous. Because $\mathcal { D}$ is monotonic, $\underline{C}_{k}\subseteq \underline{C}_{i}$ and $\underline{C}_{k}\in \mathcal {D}$ imply $\underline{C}_{i}\in \mathcal {D}$, and since $\mathcal {D}$ is proper, $\overline{C}_{i+1}=N{\setminus } \underline{ C}_{i}\notin \mathcal {D}$. The same is true for $\overline{C}_{i+2}, \overline{C}_{i+3},\ldots ,\overline{C}_{n}$, implying that there is no decisive coalition that weakly prefers p to $\pi ^{u}( {\varvec{\rho }})$, so $p\notin A^{u}({\varvec{\rho }}),$ a contradiction. This establishes that $A^{u}({\varvec{\rho }})\subseteq A_{k}^{u}({\varvec{\rho }}).$ $\square$

Proposition 1

Given any policymaking procedure ${\varvec{\rho }} \in \Delta ^{n}$, a unique policy bargaining equilibrium exists, and it is no-delay and in pure strategies.

Proof of Proposition 1

The existence of a no-delay equilibrium follows from Theorem 1 in Banks and Duggan (2006a) since policy utilities $u_{i}$ satisfy their conditions of continuity, concavity, and LSWP. Since $[ 0,1] \subseteq \mathbb {R },$ Theorem 2 in the same paper implies that every no-delay equilibrium is in pure strategies. We show that a no-delay equilibrium must be unique and there are no equilibria with delay. Since in the rest of the proof all policy proposal and voting strategies are conditional on the same procedure ${\varvec{\rho }}$, we simplify notation by temporarily suppressing this dependence.

By Lemma 1, $A^{u}=A_{k}^{u}.$ Thus, in a pure-strategy no-delay policy equilibrium $\mathbf {s}^{u}$ the collective acceptance set is $A^{u}=\{ p\in [ 0,1] \mid u_{k}(p)\ge ( 1-\gamma ) u_{k}( p^{D}) +\gamma U_{k}(\mathbf {s}^{u})\},$ and the stationary bargaining value is $U_{k}(\mathbf {s}^{u})=\sum _{i\in N}\rho _{i}u_{k}( z_{i}^{u})$. By the strict concavity of $u_{k},$ this set is a compact interval of the form $A^{u}=[ \underline{ p},\overline{p}] \ni {\tilde{p}}_{k}$ where $\underline{p},\overline{p}$ are defined implicitly by:

$$\begin{aligned} u_{k}(\underline{p})= & {} u_{k}(\overline{p})=( 1-\gamma ) u_{k}( p^{D}) \nonumber \\&+\,\gamma \left[ u_{k}(\underline{p})\sum _{{\tilde{p}}_{i}< \underline{p}}\rho _{i}+\sum _{\underline{p}\le {\tilde{p}}_{i}\le \overline{p }}\rho _{i}u_{k}( {\tilde{p}}_{i}) +u_{k}(\overline{p})\sum _{\tilde{ p}_{i}>\overline{p}}\rho _{i}\right] \end{aligned}$$

(15)

Suppose $\mathbf {s}^{I}\ne \mathbf {s}^{II}$ are two pure-strategy no-delay equilibria with $A^{u}( \mathbf {s}^{I}) \subset A^{u}( \mathbf {s}^{II}).$ If $\gamma =0,$ then $u_{k}(\underline{p} ^{I})=u_{k}(\underline{p}^{II})=u_{k}( p^{D}),$ and therefore $A^{u}( \mathbf {s}^{I}) =A^{u}( \mathbf {s}^{II}),$ contradiction. If $\gamma >0,$ then $u_{k}(\underline{p}^{I})>u_{k}( \underline{p}^{II})$ and so, by Eq. (15), $U_{k}(\mathbf {s} ^{I})>U_{k}(\mathbf {s}^{II})$. Also, $u_{k}(\underline{p}^{I})-u_{k}( \underline{p}^{II})=\gamma [ U_{k}(\mathbf {s}^{I})-U_{k}(\mathbf {s} ^{II})] <U_{k}(\mathbf {s}^{I})-U_{k}(\mathbf {s}^{II}),$ since $\gamma <1.$ By no-delay, $U_{k}(\mathbf {s})=u_{k}(\underline{p})\sum _{{\tilde{p}}_{i}< \underline{p}}\rho _{i}+\sum _{\underline{p}\le {\tilde{p}}_{i}\le \overline{p }}\rho _{i}u_{k}( {\tilde{p}}_{i}) +u_{k}(\overline{p})\sum _{\tilde{ p}_{i}>\overline{p}}\rho _{i}$ and so:

$$\begin{aligned}&U_{k}(\mathbf {s}^{I})-U_{k}(\mathbf {s}^{II}) =\big [ u_{k}(\underline{p} ^{I})-u_{k}(\underline{p}^{II})\big ] \sum _{{\tilde{p}}_{i}<\underline{p} ^{II}}\rho _{i}\nonumber \\&\quad +\sum _{\underline{p}^{II}\le {\tilde{p}}_{i}\le \underline{p} ^{I}}\big [ u_{k}(\underline{p}^{I})-u_{k}( {\tilde{p}}_{i}) \big ] \rho _{i} \nonumber \\&+\sum _{\overline{p}^{I}\le {\tilde{p}}_{i}\le \overline{p}^{II}}\big [ u_{k}(\overline{p}^{I})-u_{k}( {\tilde{p}}_{i}) \big ] \rho _{i}+ \big [ u_{k}(\overline{p}^{I})-u_{k}(\overline{p}^{II})\big ] \sum _{\tilde{p }_{i}>\overline{p}^{II}}\rho _{i} \nonumber \\\le & {} u_{k}(\underline{p}^{I})-u_{k}(\underline{p}^{II})=u_{k}(\overline{p} ^{I})-u_{k}(\overline{p}^{II}) \end{aligned}$$

(16)

since $u_{k}(\underline{p}^{II})\le u_{k}( {\tilde{p}}_{i})$ for $\underline{p}^{II}\le {\tilde{p}}_{i}\le \underline{p}^{I}\le {\tilde{p}} _{k},$ and $u_{k}(\overline{p}^{II})\le u_{k}( {\tilde{p}}_{i})$ for ${\tilde{p}}_{k}\le \overline{p}^{I}\le {\tilde{p}}_{i}\le \overline{p} ^{II}$. Contradiction. Therefore $\mathbf {s}^{I}=\mathbf {s}^{II},$ so a pure-strategy no-delay equilibrium is unique.

Suppose there exists a pure-strategy policy equilibrium with delay, denoted $\mathbf {s}^{u}.$ A proposal with delay occurs because a recognized player is strictly better off moving into the next bargaining round than proposing any policy in the collective acceptance set $A^{u}$: there exists $i\in N$ such that $\rho _{i}>0$ and $( 1-\gamma ) u_{i}(p^{D})+\gamma U_{i}(\mathbf {s}^{u})>\sup \{ u_{i}(p)\mid p\in A^{u} \}.$ Let $\pi ^{u}$ be the continuation policy lottery, see Eq. (14), and $\mu ( \pi ^{u}) \equiv \int p\pi ^{u}( dp)$ its mean. By the strict concavity of each $u_{j}$ we have $u_{j}[ \mu ( \pi ^{u}) ] \ge \int u_{j}( p) \pi ^{u}( dp) =( 1-\gamma ) u_{j}(p^{D})+\gamma U_{j}(\mathbf {s}^{u})$ for all $j\in N,$ including i, where equality occurs only when $\pi ^{u}( \{ p^{D} \} ) =1.$ But this implies that $\mu ( \pi ^{u}) \in A^{u}$ and so $\sup \{ u_{i}(p)\mid p\in A^{u} \} \ge u_{i}[ \mu ( \pi ^{u}) ].$ This inequality together with the the delay condition implies that $\left( 1-\gamma \right) u_{i}(p^{D})+\gamma U_{i}(\mathbf {s}^{u})>u_{i}[ \mu ( \pi ^{u}) ],$ which contradicts concavity of $u_{i}$. Thus, there can be no equilibrium with delay. $\square$

Lemma 2

In equilibrium, a procedural proposal is approved if and only if it is acceptable to the core player. That is, $A^{U}=A_{k}^{U}.$

Proof of Lemma 2

Given a strategy profile $( {\varvec{\sigma }}^{U},{\varvec{\sigma }} ^{u}) =( \sigma _{i}^{U},{\sigma } _{i}^{u}) _{i\in N},$ and a measurable set $Z\subseteq \Delta ^{n}$, define the following probability measure in procedural space:

$$\begin{aligned} \psi ( Z)= & {} ( 1-\delta ) \mathbb {I}\big \{ {\varvec{\rho }}^{D}\in Z\big \}\nonumber \\&+\,\delta \frac{\sum \nolimits _{j\in N}\rho _{0j}\big \{ \zeta _{j}^{U}(Z\cap A^{U})+\mathbb {I}\big \{ {\varvec{\rho }}^{D}\in Z\big \} ( 1-\delta ) \zeta _{j}^{U}(\Delta ^{n}{\setminus } A^{U})\big \} }{1-\delta \sum \nolimits _{j\in N}\rho _{0j}\zeta _{j}^{U}(\Delta ^{n}{\setminus } A^{U})} \end{aligned}$$

(17)

where $\mathbb {I}\{ {\varvec{\rho }}^{D}\in Z\}$ is an indicator function. Then the continuation policy lottery in procedural bargaining is:

$$\begin{aligned} \pi ^{U}=\int \pi _{{\varvec{\rho }}}\psi ( d{\varvec{\rho }}) \end{aligned}$$

(18)

where $\pi _{{\varvec{\rho }}}$ was defined in Eq. (6), implying $\int U_{i}( {\varvec{\rho }}|{\varvec{\sigma }}^{u}) \psi ( d {\varvec{\rho }}) = \int \int u_{i}( p) \pi _{{\varvec{\rho }} }( dp) \psi ( d{\varvec{\rho }}) =\int u_{i}( p) \pi ^{U}( dp).$ By Eq. (3), we have $\int U_{i}( {\varvec{\rho }}|{\varvec{\sigma }}^{u}) \psi ( d{\varvec{\rho }} ) =(1-\delta )U_{i}({\varvec{\rho }}^{D})+\delta \mathcal {U}_{i}( {\varvec{\sigma }}^{U},{\varvec{\sigma }}^{u}),$ for all $i\in N,$ where $U_{i}( {\varvec{\rho }}|{\varvec{\sigma }}^{u})$ is defined recursively by Eq. (4). The strategy of proof then follows that for Lemma 1 in showing first that if ${\varvec{\rho }}\in A_{k}^{U}$ then ${\varvec{\rho }}\in A^{U},$ and second, if ${\varvec{\rho }}\in A^{U}$ and ${\varvec{\rho }}\notin A_{k}^{U}$ then no decisive coalition weakly prefers the procedure ${\varvec{\rho }}$ to the procedural lottery $\psi ,$ or equivalently, the policy lottery $\pi _{{\varvec{\rho }}}$ to the policy lottery $\pi ^{U}$. $\square$

Lemma 3

Given a no-delay, pure-strategy, policy strategy profile $\mathbf {s}^{u}$: (i) ${\varvec{\rho }}$ effective and ${\varvec{\rho }}\in \arg \max \{ U_{i}( z\mathbf {|s} ^{u}) \mid z\in A_{k}^{U} \}$ for some $i\in N,$ imply that ${\varvec{\rho }}$ is LPS. (ii) Within the subset of LPS procedures the reduced-form utility functions $v_{i}$ are continuous and strictly quasi-concave on their domain $Z^{v}=[ \underline{p}_{k}^{D},\overline{p}_{k}^{D}] \cap [ {\tilde{p}}_{1}, {\tilde{p}}_{n}].$

Proof of Lemma 3

(i) If the proposer is $i=k$ then, by Lemmas 1 and 2, he proposes ${\varvec{\rho }} =\mathbf {e}_{k},$ which is LPS and yields $p={\tilde{p}}_{k}$ with probability one. Assume without loss of generality that $i>k.$ Suppose player i proposes procedure ${\varvec{\rho }}$, and denote by $A^{u}({\varvec{\rho }})= [ \underline{p}( {\varvec{\rho }}) ,\overline{p}( {\varvec{\rho }}) ]$ the corresponding policy collective acceptance set; see Eq. (14). We claim that ${\varvec{\rho }}$ effective and ${\varvec{\rho }}\in \arg \max \{ U_{i}( z\mathbf {|s}^{u}) \mid z\in A_{k}^{U} \}$ imply that $\rho _{j}=0$ for $j\notin \{ k,\ldots ,i \}.$

Suppose, first, that $\rho _{j}>0$ for some $j>i.$ Because ${\varvec{\rho }}$ is effective, $z_{j}^{u}=\min \{ {\tilde{p}}_{j},\overline{p}( {\varvec{\rho }}) \} >{\tilde{p}}_{i}.$ Consider the alternative procedure ${\varvec{\rho }}^{\epsilon }$ that reallocates proposal power from j to i :  $\rho _{j}^{\epsilon }=\rho _{j}-\epsilon _{i},\rho _{i}^{\epsilon }=\rho _{i}+\epsilon _{i}$, with all other components unchanged. Pick $\epsilon _{i}>0$ such that $\epsilon _{i}<\rho _{j}$ and $\rho _{i}+\epsilon _{i}<1.$ Then ${\varvec{\rho }}^{\epsilon }$ makes both i and k strictly better off by shifting weight from $z_{j}^{u}$ to ${\tilde{p}} _{i}$ and/or reducing $z_{h}^{u},$ for $h>i,$ and increasing $z_{h}^{u},$ for $h<k;$ see Eq. (15). Thus, ${\varvec{\rho }}$ cannot be sequentially rational for player i.

Suppose, second, that $\rho _{j}>0$ for some $j<k.$ If $\overline{p}( {\varvec{\rho }}) \le {\tilde{p}}_{i},$ consider the alternative procedure ${\varvec{\rho }}^{\epsilon }$ that reallocates proposal power from j to k and i,  and leaves the core player k indifferent: $\rho _{j}^{\epsilon }=\rho _{j}-\epsilon _{k}-\epsilon _{i},\rho _{k}^{\epsilon }=\rho _{k}+\epsilon _{k},\rho _{i}^{\epsilon }=\rho _{i}+\epsilon _{i}$, with all other components unchanged. Pick $\epsilon _{k},\epsilon _{i}>0$ such that $0<\epsilon _{k}+\epsilon _{i}<\rho _{j},\rho _{k}+\epsilon _{k}<1,\rho _{i}+\epsilon _{i}<1,$ and $U_{k}\left( {\varvec{\rho }}^{\epsilon }\right) =U_{k}( {\varvec{\rho }}) .$ Then $u_{k}[ \underline{p }({\varvec{\rho }})] =u_{k}[ \overline{p}({\varvec{\rho }})] =( 1-\gamma ) u_{k}(p^{D})+\gamma U_{k}( {\varvec{\rho }} )$ and $\gamma <1$ imply $\underline{p}( {\varvec{\rho }} ^{\epsilon }) =\underline{p}( {\varvec{\rho }})$ and $\overline{p}({\varvec{\rho }}^{\epsilon })=\overline{p}({\varvec{\rho }}).$ By no-delay, $U_{k}( {\varvec{\rho }}^{\epsilon }) =U_{k}( {\varvec{\rho }})$ implies $\epsilon _{k} \{ u_{k}({\tilde{p}} _{k})-u_{k}[ \max \{ {\tilde{p}}_{j},\underline{p}( {\varvec{\rho }}) \} ] \} -\epsilon _{i} \{ u_{k}[ \max \{ {\tilde{p}}_{j},\underline{p}( {\varvec{\rho }}) \} ] -u_{k}[ \overline{p}({\varvec{\rho }})] \} =0.$ Note that ${\varvec{\rho }}^{\epsilon }$ makes the proposer i strictly better off because it shifts weight from $z_{j}^{u}=\max \{ {\tilde{p}}_{j},\underline{p}( {\varvec{\rho }}) \} <{\tilde{p}} _{k}$ to ${\tilde{p}}_{k}$ and $\overline{p}({\varvec{\rho }}).$ If $\overline{p} ( {\varvec{\rho }}) >{\tilde{p}}_{i},$ consider the alternative procedure ${\varvec{\rho }}^{\epsilon }$ that reallocates proposal power from j to k :  $\rho _{j}^{\epsilon }=\rho _{j}-\epsilon _{k},\rho _{k}^{\epsilon }=\rho _{k}+\epsilon _{k}$, with all other components unchanged. Pick $\epsilon _{k}>0$ such that $\epsilon _{k}<\rho _{j},\rho _{k}+\epsilon _{k}<1,$ and $\overline{p}( {\varvec{\rho }}^{\epsilon }) >{\tilde{p}}_{i}.$ Then, ${\varvec{\rho }}^{\epsilon }$ makes both i and k strictly better off by shifting weight from $z_{j}^{u}$ to ${\tilde{p}} _{k}$ and/or reducing $z_{h}^{u},$ for $h>i,$ and increasing $z_{h}^{u},$ for $h<k;$ see Eq. (15). Thus, in either case, ${\varvec{\rho }}$ cannot be sequentially rational for player i.

Now suppose player i’s utility-maximizing effective procedure ${\varvec{\rho }}$ from the core player’s procedural acceptance set $A_{k}^{U}$ is not LPS, i.e., there are players l, c, r such that ${\tilde{p}}_{k}\le {\tilde{p}} _{l}<{\tilde{p}}_{c}<{\tilde{p}}_{r}\le {\tilde{p}}_{i}$, and $\rho _{l},\rho _{r}>0$ are effective proposal probabilities. As shown above no players below k and above i can have positive proposal power under ${\varvec{\rho }}$. Note that ${\varvec{\rho }}$ effective implies ${\tilde{p}}_{c}<\overline{p} ( {\varvec{\rho }}),$ since otherwise $\rho _{r}$ can be transferred to player $r-1\ge c$ without affecting equilibrium policy outcomes. We show that ${\varvec{\rho }}$ cannot be utility-maximizing for player i by identifying another procedure ${\varvec{\rho }}^{\epsilon }\in A_{k}^{U}$ that reallocates proposal power from all other players to c,  leaves the core player k indifferent, and makes i strictly better off.

Define ${\varvec{\rho }}^{\epsilon }$ as the following reshuffling among ${\varvec{\rho }}$’s components: $\rho _{j}^{\epsilon }=\rho _{j}-\epsilon _{l},$ for all $j<c$ with $\rho _{j}>0$, $\rho _{j}^{\epsilon }=\rho _{j}-\epsilon _{r},$ for all $j>c$ with $\rho _{j}>0,$ and $\rho _{c}^{\epsilon }=\rho _{c}+n_{l}\epsilon _{l}+n_{r}\epsilon _{r}$, where $n_{l}$ and $n_{r}$ are the number of players with positive proposal power to the left, respectively to the right, of player c. Pick $\epsilon _{l},\epsilon _{r}>0$ such that $U_{k}( {\varvec{\rho }}^{\epsilon } \mathbf {|s}^{u}) =U_{k}( {\varvec{\rho }} \mathbf {|s}^{u}),$ and $\epsilon _{l}<\rho _{j}$ for all $j<c$ with $\rho _{j}>0,$ and $\epsilon _{r}<\rho _{j}$ for all $j>c$ with $\rho _{j}>0,$ and $\rho _{c}+n_{l}\epsilon _{l}+n_{r}\epsilon _{r}<1.$ This is possible because $U_{k}$ is continuous in ${\varvec{\rho }}$ so the solution $( \epsilon _{l},\epsilon _{r})$ is continuous at (0, 0). The decisiveness of the core player in policy voting under ${\varvec{\rho }}$ and ${\varvec{\rho }}^{\epsilon }$ implies $\overline{p}({\varvec{\rho }}^{\epsilon })=\overline{p}({\varvec{\rho }})$ and $\underline{p}( {\varvec{\rho }} ^{\epsilon }) =\underline{p}( {\varvec{\rho }})$. Thus, the support of $\pi _{{\varvec{\rho }}^{\epsilon }}$ remains the same as the support of $\pi _{{\varvec{\rho }}},$ in particular ${\tilde{p}}_{c}$ belongs in the support of both policy lotteries. Now notice that by no-delay of policy strategies, $U_{k}( {\varvec{\rho }}^{\epsilon }\mathbf {|s}^{u}) =U_{k}( {\varvec{\rho }} \mathbf {|s}^{u})$ implies $\sum _{j\in N}\rho _{j}^{\epsilon }u_{k}( z_{j}^{u}) =\sum _{j\in N}\rho _{j}u_{k}( z_{j}^{u}).$ By cancelling terms, we get $\epsilon _{l}\sum _{j<c}u_{k}( z_{j}^{u}) +\epsilon _{r}\sum _{j>c}u_{k}( z_{j}^{u}) -( n_{l}\epsilon _{l}+n_{r}\epsilon _{r}) u_{k}( z_{c}^{u}) =0$ leading to:

$$\begin{aligned} u_{k}( z_{c}^{u}) =\frac{\epsilon _{l}\sum _{j<c}u_{k}( z_{j}^{u}) +\epsilon _{r}\sum _{j>c}u_{k}( z_{j}^{u}) }{ n_{l}\epsilon _{l}+n_{r}\epsilon _{r}}<u_{k}\left( \frac{\epsilon _{l}\sum _{j<c}z_{j}^{u}+\epsilon _{r}\sum _{j>c}z_{j}^{u}}{n_{l}\epsilon _{l}+n_{r}\epsilon _{r}}\right) \end{aligned}$$

(19)

where the inequality follows from the strict concavity of $u_{k}.$ Then, because $u_{k}$ is strictly decreasing to the right of ${\tilde{p}}_{k},$ it also follows that $z_{c}^{u}>\frac{\epsilon _{l}\sum _{j<c}z_{j}^{u}+\epsilon _{r}\sum _{j>c}z_{j}^{u}}{n_{l}\epsilon _{l}+n_{r}\epsilon _{r}}.$ This immediately implies that the mean policy under $\pi _{{\varvec{\rho }} ^{\epsilon }}$ is larger than the mean policy under $\pi _{{\varvec{\rho }}},$ or $\sum _{j\in N}\rho _{j}^{\epsilon }z_{j}^{u}>\sum _{j\in N}\rho _{j}z_{j}^{u}$, equivalent to $\epsilon _{l}\sum _{j<c}z_{j}^{u}+\epsilon _{r}\sum _{j>c}z_{j}^{u}-( n_{l}\epsilon _{l}+n_{r}\epsilon _{r}) z_{c}^{u}>0.$

By construction, the cdf’s of the policy lotteries $\pi _{{\varvec{\rho }}}$ and $\pi _{{\varvec{\rho }}^{\epsilon }}$ feature single crossing at ${\tilde{p}} _{c}.$ Since $\pi _{{\varvec{\rho }}^{\epsilon }}$ also has higher mean, it follows that $\pi _{{\varvec{\rho }}^{\epsilon }}$ second-order stochastically dominates $\pi _{{\varvec{\rho }}};$ see Whitmore and Findlay (1978, Sec. 2.16). Because $u_{i}$ is stricty concave and strictly increasing to the left of ${\tilde{p}}_{i}$ second-order stochastic dominance implies that player i strictly prefers $\pi _{{\varvec{\rho }}^{\epsilon }}$ to $\pi _{ {\varvec{\rho }}},$ and thus ${\varvec{\rho }}^{\epsilon }$ to ${\varvec{\rho }},$ or $U_{i}( {\varvec{\rho }}^{\epsilon }\mathbf {|s}^{u}) >U_{i}( {\varvec{\rho }} \mathbf {|s}^{u}).$ Since both ${{\varvec{\rho }},{\varvec{\rho }} } ^{\epsilon }\in A_{k}^{U},$ then ${\varvec{\rho }}\notin \arg \max \{ U_{i}( z\mathbf {|s}^{u}) \mid z\in A_{k}^{U} \}.$ Contradiction. Therefore, ${\varvec{\rho }}$ has to be LPS.

(ii) Consider an LPS procedure ${\varvec{\rho }}$. This means that for some $j,j+1\in N,$ $\rho _{j},\rho _{j+1}$ are effective proposal probabilities with $\rho _{j}+\rho _{j+1}=1$. Denote the policy collective acceptance set induced by this procedure by $[ \underline{p}( {\varvec{\rho }} ),\overline{p}( {\varvec{\rho }}) ].$ Let ${\hat{p}} ( {\varvec{\rho }}) =\rho _{j}{\tilde{p}}_{j}+\rho _{j+1}{\tilde{p}} _{j+1}$ and $v_{i}[ {\hat{p}}( {\varvec{\rho }}) ] =U_{i}( {\varvec{\rho }})=\rho _{j}u_{i}(z_{j}^{u})+\rho _{j+1}u_{i}(z_{j+1}^{u}),$ by the no-delay property of equilibrium policy strategies. Then, $v_{i}$ is continuous in ${\hat{p}}$ because $u_{i}$ is continuous in p and $z_{j}^{u},z_{j+1}^{u}$ are continuous in ${\varvec{\rho }}$. Take any player $i\in N$ and suppose, without loss of generality, that $i\ge k.$ We claim that $v_{i}$ is strictly increasing in ${\hat{p}}$, when ${\hat{p}}<{\tilde{p}} _{i},$ and strictly decreasing in ${\hat{p}}$, when ${\hat{p}}>{\tilde{p}}_{i},$ on the set $[ \underline{p}_{k}^{D},\overline{p}_{k}^{D}] \cap [ {\tilde{p}}_{1},{\tilde{p}}_{n}],$ therefore strictly quasi-concave in ${\hat{p}}$ on that set; $\underline{p}_{k}^{D},\overline{p} _{k}^{D}$ were defined in footnote 14. Note that $[ \underline{p}( {\varvec{\rho }}),\overline{p}( {\varvec{\rho }}) ] \subseteq [ \underline{p}_{k}^{D},\overline{p}_{k}^{D}].$ We claim that $[ \underline{p}( {\varvec{\rho }}),\overline{p} ( {\varvec{\rho }}) ]$ shrinks as ${\hat{p}}$ increases below ${\tilde{p}}_{k},$ and expands as ${\hat{p}}$ increases above ${\tilde{p}}_{k}.$

To see this, note that $u_{k}(\underline{p})=u_{k}(\overline{p})=( 1-\gamma ) u_{k}( p^{D}) +\gamma [ \rho _{j}u_{k}(z_{j}^{u})+\rho _{j+1}u_{k}(z_{j+1}^{u})].$ The support of the policy lottery induced by procedure ${\varvec{\rho }}$, namely $\{ z_{j}^{u},z_{j+1}^{u} \},$ is either $\{ {\tilde{p}}_{j} \},$ or $\{ {\tilde{p}}_{j},\overline{p}( {\varvec{\rho }}) \} ,$ or $\{ {\tilde{p}}_{j},{\tilde{p}}_{j+1} \},$ or $\{ {\tilde{p}}_{j+1} \},$ if $j,j+1\ge k,$ and either $\{ {\tilde{p}}_{j} \},$ or $\{ {\tilde{p}}_{j},{\tilde{p}} _{j+1} \},$ or $\{ \underline{p}( {\varvec{\rho }}), {\tilde{p}}_{j+1} \},$ or $\{ {\tilde{p}}_{j+1} \} ,$ if $j,j+1\le k.$ Thus, as ${\hat{p}}$ increases below ${\tilde{p}}_{k}$, by assigning greater proposal power to players closer to k,  both $u_{k}( \underline{p}),u_{k}(\overline{p})$ go up, which means that $\underline{p} ( {\varvec{\rho }}),\overline{p}( {\varvec{\rho }})$ get closer to ${\tilde{p}}_{k}\in [ \underline{p}( {\varvec{\rho }}),\overline{p}( {\varvec{\rho }}) ];$ an analogous argument shows that $[ \underline{p}( {\varvec{\rho }}),\overline{p} ( {\varvec{\rho }}) ]$ expands as ${\hat{p}}$ increases above ${\tilde{p}}_{k}.$ Increasing ${\hat{p}}$ below ${\tilde{p}}_{i}$ thus increases either the weight in the policy lottery $\pi _{{\varvec{\rho }}}$ on a policy closer to ${\tilde{p}}_{i},$ or shifts the support of the policy lottery $\pi _{{\varvec{\rho }}}$ closer to ${\tilde{p}}_{i}$, or both. The new policy lottery thus first order stochastically dominates the old policy lottery. Since $u_{i}$ is strictly increasing to the left of ${\tilde{p}}_{i},$ in either case $v_{i}$ increases. Analogously, $v_{i}$ is strictly decreasing in ${\hat{p}}$ above ${\tilde{p}}_{i}.$ $\square$

Proposition 2

An equilibrium of the full game exists. Any equilibrium leads to the same distribution of policy outcomes, i.e., equilibrium behavior features outcome-uniqueness. In equilibrium, procedural proposals are no-delay and in pure strategies, and give effective proposal power to at most two players who are adjacent to each other.

Proof of Proposition 2

We first prove existence of a no-delay equilibrium in the one-dimensional reduced-form game specified by $\langle Z^{v},{\varvec{\rho }}_{0},\mathcal {D},\delta ,v_{i},U_{i}({\varvec{\rho }} ^{D}),V_{i} \rangle$, and show it must be in pure strategies. Then we show that a no-delay equilibrium is unique and that there are no equilibria with delay in the reduced-form game.

Let ${{\varvec{\zeta }}}^{v}$ denote a profile of mixed proposal strategies in the reduced-form game. For all $i\in N$ define individual acceptance sets as $A_{i}^{v}({\varvec{\zeta }}^{v})= \{ z\in Z^{v}\mid v_{i}( z) \ge (1-\delta )U_{i}({\varvec{\rho }}^{D})+\delta f_{i}({\varvec{\zeta }} ^{v}) \},$ where by no delay, $f_{i}({\varvec{\zeta }}^{v})=\sum _{j\in N}\rho _{0j}\int v_{i}(z)\zeta _{j}^{v}(dz)$. Because $v_{i}( {\tilde{p}} _{i}) =U_{i}( \mathbf {e}_{i}) \ge U_{i}({\varvec{\rho }})$ for all ${\varvec{\rho }} \in \Delta ^{n},$ and $v_{i}( {\tilde{p}} _{i}) \ge v_{i}({\hat{p}})$ for all ${\hat{p}}\in Z^{v}$, each $A_{i}^{v}( {\varvec{\zeta }}^{v})$ is a non-empty set. Because $v_{i}$ is continuous and strictly quasi-concave on the compact set $Z^{v},$ by Lemma 3(ii), each $A_{i}^{v}({\varvec{\zeta }}^{v})$ is a compact and convex set, as well as a continuous correspondence. Applying the proof strategy in Lemmas 1 and 2, voting strategies imply $A^{v}({\varvec{\zeta }}^{v})=A_{k}^{v}({\varvec{\zeta }} ^{v})$. Thus, $A^{v}({\varvec{\zeta }}^{v})$ is a non-empty set, and a compact-valued, convex-valued, continuous correspondence. For all $i\in N$ define player i’s best response correspondence: $M_{i}({\varvec{\zeta }} ^{v})=\arg \max \{ v_{i}(z)\mid z\in A^{v}({\varvec{\zeta }} ^{v}) \}.$ Because, by Lemma 3(ii), $v_{i}$ is strictly quasi-concave, and $A^{v}$ has the properties mentioned above, the Theorem of the Maximum implies that $M_{i}$ is a continuous function.

Let $M=\times _{i\in N}M_{i}$ denote the joint best-response function. The domain of M is the cartesian product of sets of probability distributions over the compact convex set $Z^{v},$ which is thus itself compact and convex. Since M is itself continuous on a compact convex set, Brower’s Theorem implies that it has a fixed point. Because each $M_{i}$ maps into degenerate probability distributions on $Z^{v}$, this fixed point must be a vector of points each belonging to $Z^{v}.$ Denote it by $\mathbf {z} ^{v}=( z_{1}^{v},\ldots ,z_{n}^{v}) .$ We claim that $\mathbf {s}^{v}= [ ( z_{1}^{v},A_{1}^{v}(\mathbf {z}^{v})),\ldots ,( z_{n}^{v},A_{n}^{v}(\mathbf {z}^{v})) ]$ is a no-delay equilibrium of the reduced-form procedural bargaining game. Weak dominance is satisfied by construction, since $A_{i}^{v}(\mathbf {z}^{v})$ only contains choice alternatives weakly preferred to delay. For sequential rationality, we need to show that $\sup \{ v_{i}(z)\mid z\in A^{v} \} \ge (1-\delta )U_{i}({\varvec{\rho }}^{D})+\delta \sum _{j\in N}\rho _{0j}v_{i}(z_{j}^{v}),$ for all $i\in N$ with $\rho _{0i}>0.$

Denote by $[ \underline{{\hat{p}}},\overline{{\hat{p}}}]$ the range of procedural indices the core player weakly prefers to the continuation value. Since $v_{k}$ is strictly quasi-concave, the range is defined by:

$$\begin{aligned} v_{k}(\underline{{\hat{p}}})=v_{k}(\overline{{\hat{p}}})=(1-\delta )U_{k}( {\varvec{\rho }}^{D})+\delta \sum _{j\in N}\rho _{0j}v_{k}(z_{j}^{v})=\int u_{k}(p)\pi ^{U}(dp) \end{aligned}$$

(20)

where $\pi ^{U}$ denotes the policy lottery induced by ${\varvec{\rho }}^{D}$ and $( z_{j}^{v}) _{j\in N}$; see Eq. (18). Since $A^{v}=A_{k}^{v},$ we have $A^{v}=[ \underline{{\hat{p}}},\overline{ {\hat{p}}}].$ Denote by $\underline{{\varvec{\rho }}},\overline{{\varvec{\rho }}}$ their corresponding LPS procedures. If $\pi ^{U}$ is degenerate, yielding a single policy outcome p, it has to be the ideal point of a player. Then $p\in A_{i}^{v}$ for all $i\in N,$ and so $p\in A^{v}.$ All players may then propose $z_{i}^{v}=p$ which gives utility equal to delay. In that follows we assume that $\pi ^{U}$ is nondegenerate.

Suppose, without loss of generality, that ${\tilde{p}}_{k}\le \mu ( \pi ^{U}).$ Then, $v_{i}({\tilde{p}}_{k})=U_{i}( \mathbf {e}_{k}) =u_{i}( {\tilde{p}}_{k}) \ge u_{i}[ \mu ( \pi ^{U}) ] >\int u_{i}( p) \pi ^{U}( dp),$ for all $i\le k$. This, together with ${\tilde{p}}_{k}\in A^{v}=[ \underline{{\hat{p}}},\overline{{\hat{p}}}],$ gives sequential rationality for these players, since they would propose $z_{i}^{v}={\tilde{p}} _{k}$ rather than delay.

For $i>k$ and $\mu ( \pi ^{U}) \le \min supp( \pi _{ \overline{{\varvec{\rho }}}})$ let ${\tilde{p}}_{l}=\max \{ \tilde{p }_{i}\mid {\tilde{p}}_{i}\le \mu ( \pi ^{U}) \}$ and ${\tilde{p}}_{h}=\min \{ {\tilde{p}}_{i}\mid \mu ( \pi ^{U}) \le {\tilde{p}}_{i}\le \min supp( \pi _{\overline{{\varvec{\rho }}} }) \}$. Then ${\tilde{p}}_{l},{\tilde{p}}_{h}\le \overline{{\hat{p}} },$ so ${\tilde{p}}_{l},{\tilde{p}}_{h}\in A^{v},$ and $v_{i}({\tilde{p}} _{l})=U_{i}( \mathbf {e}_{l}) =u_{i}( {\tilde{p}}_{l}) \ge u_{i}[ \mu ( \pi ^{U}) ] >\int u_{i}( p) \pi ^{U}( dp),$ for all i with $k<i\le l$, which gives sequential rationality for these players, since they would propose $z_{i}^{v}={\tilde{p}}_{l}$ rather than delay. And $v_{i}({\tilde{p}} _{h})=U_{i}( \mathbf {e}_{h}) =u_{i}( {\tilde{p}}_{h}) \ge u_{i}[ \mu ( \pi ^{U}) ] >\int u_{i}( p) \pi ^{U}( dp),$ for all $i\ge h,$ which gives sequential rationality for these players, since they would propose $z_{i}^{v}={\tilde{p}}_{h}$ rather than delay.

Note that $\mu ( \pi ^{U}) <\max supp( \pi _{\overline{ {\varvec{\rho }}}}),$ since otherwise ${\tilde{p}}_{k}\le \min supp( \pi _{\overline{{\varvec{\rho }}}}) \le \max supp( \pi _{\overline{{\varvec{\rho }}}})$ $\le \mu ( \pi ^{U}),$ implying that $v_{k}(\overline{{\hat{p}}})=U_{k}(\overline{{\varvec{\rho }}} )\ge u_{k}[ \mu ( \pi ^{U}) ] >\int u_{k}(p)\pi ^{U}( dp),$ a contradiction to (20). Therefore, it remains to show sequential rationality for $i>k$ and $\mu ( \pi ^{U}) \in ( \min supp( \pi _{\overline{{\varvec{\rho }}} }),\max supp( \pi _{\overline{{\varvec{\rho }}}}) ).$ We show that $\pi _{\overline{{\varvec{\rho }}}}$ second-order stochastically dominates $\pi ^{U}$. If that is true, because rightmost player n ’s utility function is strictly increasing and strictly concave on $[ 0,{\tilde{p}}_{n}]$, and no policy proposals can exceed ${\tilde{p}}_{n},$ he prefers $\pi _{\overline{{\varvec{\rho }}}}$ over $\pi ^{U}.$ Since the core player k is indifferent between $\pi _{\overline{ {\varvec{\rho }}}}$ over $\pi ^{U},$ by Eq. (20), by order restriction of preferences over policy lotteries, all $i>k$ at least weakly prefer $\pi _{\overline{{\varvec{\rho }}}}$ over $\pi ^{U},$ which is equivalent to stating $v_{i}(\overline{{\hat{p}}})\ge (1-\delta )U_{i}( {\varvec{\rho }}^{D})+\delta \sum _{j\in N}\rho _{0j}v_{i}(z_{j}^{v}),$ for all $i>k;$ then, $\overline{{\hat{p}}}\in A^{v}$ implies sequential rationality for all $i>k,$ since they would propose $z_{i}^{v}=\overline{{\hat{p}}}$ rather than delay.

We now show that, when $\mu ( \pi ^{U}) \in ( \min supp( \pi _{\overline{{\varvec{\rho }}}}),\max supp( \pi _{ \overline{{\varvec{\rho }}}}) ),$ the binary policy lottery $\pi _{\overline{{\varvec{\rho }}}}$ second-order stochastically dominates $\pi ^{U}.$ The claim follows from two observations: (i) if $p\in supp( \pi ^{U})$ then $p\notin ( \min supp( \pi _{\overline{{\varvec{\rho }}}}),\max supp( \pi _{\overline{{\varvec{\rho }}}}) ),$ implying single-crossing, and (ii) $\mu ( \pi _{\overline{ {\varvec{\rho }}}}) \ge \mu ( \pi ^{U}).$

To see (i), consider the support of $\pi ^{U}.$ The pure proposal strategies $z_{j}^{v}$ may be in $A^{v}=[ \underline{{\hat{p}}},\overline{{\hat{p}}} ]$ or not. If $z_{j}^{v}\in [ \underline{{\hat{p}}},\overline{\hat{ p}}]$ then, by Lemma 3(i), its associated policy lottery is binary with support either the same as $\pi _{\overline{{\varvec{\rho }}}}$ or $\pi _{ \underline{{\varvec{\rho }}}},$ or its support is the singleton of some player’s ideal policy ${\tilde{p}}_{i}\in [ \underline{{\hat{p}}},\overline{ {\hat{p}}}].$ If $z_{j}^{v}\notin [ \underline{{\hat{p}}},\overline{ {\hat{p}}}]$ then its associated policy lottery is $\pi ^{U}.$ This puts positive probability $1-\delta$ on $supp( \pi _{{\varvec{\rho }} ^{D}}).$ Since $U_{k}(\overline{{\varvec{\rho }}})=(1-\delta )U_{k}( {\varvec{\rho }}^{D})+\delta \sum _{j\in N}\rho _{0j}v_{k}(z_{j}^{v}),$ by Eq. (20), and since $v_{k}(z_{j}^{v})\ge (1-\delta )U_{k}( {\varvec{\rho }}^{D})+\delta \sum _{j\in N}\rho _{0j}v_{k}(z_{j}^{v})$ for all j,  then $v_{k}(z_{j}^{v})\ge U_{k}({\varvec{\rho }}^{D}),$ and so $U_{k}( \overline{{\varvec{\rho }}})\ge U_{k}({\varvec{\rho }}^{D})$. Using Eq. (15), $u_{k}[ \overline{p}(\overline{{\varvec{\rho }}})] =( 1-\gamma ) u_{k}( p^{D}) +\gamma U_{k}(\overline{ {\varvec{\rho }}})\ge ( 1-\gamma ) u_{k}( p^{D}) +\gamma U_{k}({\varvec{\rho }}^{D})=u_{k}[ \overline{p}({\varvec{\rho }} ^{D})],$ and therefore $\max supp( \pi _{\overline{{\varvec{\rho }} }}) =\overline{p}(\overline{{\varvec{\rho }}})\le \overline{p}({\varvec{\rho }}^{D}).$ Thus any policy in $supp( \pi _{{\varvec{\rho }} ^{D}})$ above ${\tilde{p}}_{k}$ and strictly below $\overline{p}( {\varvec{\rho }}^{D})$ has to be the ideal policy of some player. Since $\pi _{ \overline{{\varvec{\rho }}}}$ is a binary lottery, the implication of (i) is that the cdf of $\pi _{\overline{{\varvec{\rho }}}}$ single-crosses the cdf of $\pi ^{U}$ from below, i.e., there exists $p^{*}$ such that cdf($\pi _{ \overline{{\varvec{\rho }}}}$)<cdf($\pi ^{U}$), for $p<p^{*},$ and cdf($\pi _{\overline{{\varvec{\rho }}}}$)$\ge$cdf($\pi ^{U}$), for $p\ge p^{*}$. That is the case here, since if cdf($\pi _{\overline{{\varvec{\rho }}}}$)<cdf($\pi ^{U}$) on $[ \min supp( \pi _{\overline{{\varvec{\rho }}} }),\max supp( \pi _{\overline{{\varvec{\rho }}}}) ),$ then $p^{*}=\max supp( \pi _{\overline{{\varvec{\rho }}}});$ if however cdf($\pi _{\overline{{\varvec{\rho }}}}$)$\ge$cdf($\pi ^{U}$) on $[ \min supp( \pi _{\overline{{\varvec{\rho }}}}),\max supp( \pi _{\overline{{\varvec{\rho }}}}) ),$ then $p^{*}=\min supp( \pi _{\overline{{\varvec{\rho }}}}).$

To see (ii), suppose $\mu ( \pi _{\overline{{\varvec{\rho }}}}) <\mu ( \pi ^{U}).$ Then, $\pi _{\overline{{\varvec{\rho }}}}$ would first-order stochastically dominate, for the core player, the binary policy lottery $\pi ^{\prime }$ with the same support as $\pi _{\overline{ {\varvec{\rho }}}}$ but mean $\mu ( \pi ^{U}),$ which can be obtained, because $\mu ( \pi ^{U}) \in ( \min supp( \pi _{\overline{{\varvec{\rho }}}}),\max supp( \pi _{\overline{{\varvec{\rho }}}}) ),$ by shifting probability mass toward the higher support point. However, because of single-crossing between the cdfs of the equal-mean distributions $\pi ^{\prime }$ and $\pi ^{U},$ we have that $\pi ^{\prime }$ second-order stochastically dominates $\pi ^{U}$ for all players, including the core player, since policy utilities are strictly concave. Thus, the core player strictly prefers $\pi _{\overline{{\varvec{\rho }}}}$ to $\pi ^{U},$ a contradiction to (20). Now, putting (i) and (ii) together, single-crossing between the cdfs of $\pi _{\overline{ {\varvec{\rho }}}}$ and $\pi ^{U},$ and $\mu ( \pi _{\overline{{\varvec{\rho }}}}) \ge \mu ( \pi ^{U})$, imply that $\pi _{ \overline{{\varvec{\rho }}}}$ second-order stochastically dominates $\pi ^{U};$ see Whitmore and Findlay (1978, Sec. 2.16).

Uniqueness of a no-delay equilibrium in reduced-form procedural bargaining over the one-dimensional index ${\hat{p}}$ can be shown using the same strategy used for policy bargaining over the one-dimensional policy p in Proposition 1; see Eq. (16). The outcome-uniqueness of equilibria for procedural bargaining over ${\varvec{\rho }}$ then follows from the uniqueness of equilibria for bargaining over LPS procedures since (a) every LPS procedure, i.e., each ${\hat{p}},$ is associated with a unique binary (or degenerate) policy lottery, and (b) such a policy lottery and LPS procedure can be associated with multiple non-LPS procedures by reallocating proposal power from the more extreme effective proposer in the LPS procedure to even more extreme proposers with ideal policies outside the policy collective acceptance set.

To show that procedural bargaining cannot feature delay, consider an equilibrium strategy profile $( {\varvec{\sigma }}^{U}\mathbf {,s} ^{u})$ with pure policy strategies, by Proposition 1, where some procedural proposals are not in the procedural collective acceptance set, $\sum _{i\in N}\rho _{0i}\zeta _{i}^{U}(A^{U})<1.$ Denote by $\underline{ {\varvec{\rho }}},\overline{{\varvec{\rho }}}$ the LPS procedures that leave the core player indifferent between approving them and continuing bargaining, $U_{k}(\underline{{\varvec{\rho }}})=U_{k}(\overline{{\varvec{\rho }}})=(1-\delta )U_{k}({\varvec{\rho }}^{D})+\delta \mathcal {U}_{k}( {\varvec{\sigma }}^{U} \mathbf {,s}^{u}).$ As before, let $\pi ^{U}$ denote the policy lottery induced by the continuation subgame; see Eq. (18). Then, one can follow the same strategy used above for proving the sequential rationality of $\mathbf {z}^{v}$ to show that $\pi _{\underline{{\varvec{\rho }} }}$ and $\pi _{\overline{{\varvec{\rho }}}}$ second-order stochastically dominate $\pi ^{U}$. Since $\underline{{\varvec{\rho }}},\overline{{\varvec{\rho }}}\in A_{k}^{U},$ and $A_{k}^{U}=A^{U}$ by Lemma 2, all players have the LPS choice alternatives $\underline{{\varvec{\rho }}},\overline{{\varvec{\rho }}}$ in $A^{U}$ one of which does at least as well for them as delay. $\square$

Proposition 3

The policy outcome bounds are located between the ideal policies of the marginal effective proposers.

$$\begin{aligned}{}[ {\tilde{p}}_{L+1},{\tilde{p}}_{H-1}] \subseteq [ z_{\min }^{u},z_{\max }^{u}] \subseteq [ {\tilde{p}}_{L},{\tilde{p}}_{H} ]. \end{aligned}$$

(21)

Proof of Proposition 3

We claim that ${\tilde{p}}_{H-1}\le z_{\max }^{u}\le {\tilde{p}}_{H}.$ The argument for why ${\tilde{p}}_{L}\le z_{\min }^{u}\le {\tilde{p}}_{L+1}$ is analogous. By the policy equilibrium characterization in Eq. (5) and the LPS property proven in Proposition 2, we have that:

$$\begin{aligned} z_{\max }^{u}=\max _{i\in N}z_{i}^{u}( \overline{{\varvec{\rho }}}) =z_{H}^{u}\left( \overline{{\varvec{\rho }}}\right) =\arg \max \left\{ u_{H}(z)\mid z\in A_{k}^{u}\left( \overline{{\varvec{\rho }}}\right) \right\} \end{aligned}$$

(22)

Since $A_{k}^{u}( \overline{{\varvec{\rho }}}) =[ \underline{p} ( \overline{{\varvec{\rho }}}),\overline{p}( \overline{ {\varvec{\rho }}}) ]$ player H will propose either $\overline{p} ( \overline{{\varvec{\rho }}})$, if $\overline{p}( \overline{ {\varvec{\rho }}}) <{\tilde{p}}_{H},$ or ${\tilde{p}}_{H}$ if $\overline{p} ( \overline{{\varvec{\rho }}}) \ge {\tilde{p}}_{H}.$ Therefore,

$$\begin{aligned} z_{H}^{u}\left( \overline{{\varvec{\rho }}}\right) =\min \left\{ {\tilde{p}} _{H},\overline{p}\left( \overline{{\varvec{\rho }}}\right) \right\} . \end{aligned}$$

(23)

By the effectiveness of player H’s proposal power, $\overline{p}( \overline{{\varvec{\rho }}}) \ge {\tilde{p}}_{H-1}.$ Therefore, we have ${\tilde{p}}_{H-1}\le \min \{ {\tilde{p}}_{H},\overline{p}( \overline{ {\varvec{\rho }}}) \} \le {\tilde{p}}_{H},$ and the stated claim follows. $\square$

Proposition 4

Equilibrium procedure and policy are strategic complements: (i) policy moderation is associated with procedural moderation: $\overline{p}( \overline{{\varvec{\rho }}}) -\underline{p}( \underline{{\varvec{\rho }}})$ and $\overline{{\hat{p}}}-\underline{{\hat{p}}}$ are decreasing in $\gamma ,\delta ,$ and (ii) policy extremism is associated with procedural extremism: $\overline{p}( \overline{{\varvec{\rho }}}) -\underline{ p}( \underline{{\varvec{\rho }}})$ and $\overline{{\hat{p}} }-\underline{{\hat{p}}}$ are increasing in $\vert p^{D}-\tilde{ p}_{k} \vert$.

Proof of Proposition 4

We first show that the procedural default’s value to the core player $U_{k}( {\varvec{\rho }}^{D})$ is increasing in $\gamma$ and decreasing in $\vert p^{D}-{\tilde{p}}_{k} \vert .$ By no-delay in policy bargaining, see Proposition 1, $u_{k}(z_{i}^{u})\ge ( 1-\gamma ) u_{k}(p^{D})+\gamma \sum _{j\in N}\rho _{j}^{D}u_{k}(z_{j}^{u}),$ for all proposers $i\in N.$ Averaging over all i,  using weights $\rho _{i}^{D},$ we have $\sum _{i\in N}\rho _{i}^{D}u_{k}(z_{i}^{u})\ge ( 1-\gamma ) u_{k}(p^{D})+\gamma \sum _{j\in N}\rho _{j}^{D}u_{k}(z_{j}^{u}),$ and since $\gamma <1,$ it immediately follows that $u_{k}(p^{D})\le \sum _{i\in N}\rho _{i}^{D}u_{k}(z_{i}^{u})=\sum _{{\tilde{p}}_{i}\in [ \underline{p}( {\varvec{\rho }}^{D}),\overline{p}({\varvec{\rho }}^{D})] }\rho _{i}^{D}u_{k}({\tilde{p}}_{i})+u_{k}[ \overline{p}({\varvec{\rho }}^{D}) ] \sum _{{\tilde{p}}_{i}\notin [ \underline{p}({\varvec{\rho }}^{D}), \overline{p}({\varvec{\rho }}^{D})] }\rho _{i}^{D}\le u_{k}({\tilde{p}} _{i}),$ for all ${\tilde{p}}_{i}\in [ \underline{p}({\varvec{\rho }}^{D}), \overline{p}({\varvec{\rho }}^{D})].$ By Eq. (15) we have that:

$$\begin{aligned} u_{k}[ \overline{p}({\varvec{\rho }}^{D})]= & {} \frac{1}{1-\gamma \mathop {\textstyle \sum }\nolimits _{{\tilde{p}}_{i}\notin [ \underline{p}({\varvec{\rho }}^{D}), \overline{p}({\varvec{\rho }}^{D})] }\rho _{i}^{D}}\nonumber \\&\times \left[ \left( 1-\gamma \right) u_{k}(p^{D})+\gamma \mathop {\textstyle \sum }\limits _{{\tilde{p}}_{i}\in [ \underline{p}({\varvec{\rho }}^{D}),\overline{p}({\varvec{\rho }}^{D})] }\rho _{i}^{D}u_{k}({\tilde{p}}_{i})\right] \end{aligned}$$

(24)

namely $u_{k}[ \overline{p}({\varvec{\rho }}^{D})]$ is an average of $u_{k}(p^{D})$ and $[ u_{k}({\tilde{p}}_{i})] _{{\tilde{p}}_{i}\in [ \underline{p}({\varvec{\rho }}^{D}),\overline{p}({\varvec{\rho }}^{D}) ] }.$ Since $u_{k}(p^{D})\le u_{k}({\tilde{p}}_{i}),$ for all ${\tilde{p}} _{i}\in [ \underline{p}({\varvec{\rho }}^{D}),\overline{p}({\varvec{\rho }} ^{D})],$ we have that $u_{k}[ \overline{p}({\varvec{\rho }}^{D}) ]$ is increasing in $\gamma ,$ and also that $u_{k}[ \overline{p} ({\varvec{\rho }}^{D})]$ is increasing in $u_{k}(p^{D}),$ which is strictly decreasing in $\vert p^{D}-{\tilde{p}}_{k} \vert .$ Since $U_{k}({\varvec{\rho }}^{D})=\sum _{{\tilde{p}}_{i}\in [ \underline{p}( {\varvec{\rho }}^{D}),\overline{p}({\varvec{\rho }}^{D})] }\rho _{i}^{D}u_{k}({\tilde{p}}_{i})+u_{k}[ \overline{p}({\varvec{\rho }}^{D}) ] \sum _{{\tilde{p}}_{i}\notin [ \underline{p}({\varvec{\rho }}^{D}), \overline{p}({\varvec{\rho }}^{D})] }\rho _{i}^{D},$ it is increasing in $u_{k}[ \overline{p}({\varvec{\rho }}^{D})],$ and therefore increasing in $\gamma$ and decreasing in $\vert p^{D}-{\tilde{p}} _{k} \vert .$

Let $\overline{p}( \overline{{\varvec{\rho }}}),\underline{p} ( \underline{{\varvec{\rho }}})$ denote the equilibrium policy bounds and $\overline{{\hat{p}}}=\overline{\rho }_{H-1}{\tilde{p}}_{H-1}+ \overline{\rho }_{H}{\tilde{p}}_{H},\underline{{\hat{p}}}=\underline{\rho }_{L} {\tilde{p}}_{L}+\underline{\rho }_{L+1}{\tilde{p}}_{L+1}$ the equilibrium procedural index bounds, where $( \overline{\rho }_{H-1},\overline{\rho }_{H}),( \underline{\rho }_{L},\underline{\rho }_{L+1})$ are effective recognition probabilities. We now show that $\overline{p}$ and $\overline{{\hat{p}}}$ are decreasing in $\gamma ,\delta$ and increasing in $\vert p^{D}-{\tilde{p}}_{k} \vert .$ An symmetric argument can be used to show that $\underline{p}$ and $\underline{{\hat{p}}}$ are increasing in $\gamma ,\delta$ and decreasing in $\vert p^{D}-{\tilde{p}} _{k} \vert .$ By Propositions 1 and 2, the equilibrium policy and procedure can be characterized by the following no-delay conditions.

$$\begin{aligned} u_{k}[ \overline{p}( \overline{{\varvec{\rho }}}) ]= & {} \left( 1-\gamma \right) u_{k}(p^{D})+\gamma \left[ \overline{\rho } _{H-1}u_{k}({\tilde{p}}_{H-1})+\overline{\rho }_{H}u_{k}(\min \left\{ \tilde{p }_{H},\overline{p}\left( \overline{{\varvec{\rho }}}\right) \right\} )\right] \nonumber \\\end{aligned}$$

(25)

$$\begin{aligned} v_{k}(\overline{{\hat{p}}})= & {} \left( 1-\delta \right) U_{k}({\varvec{\rho }} ^{D})+\delta \left\{ \sum _{{\tilde{p}}_{i}\in \left[ \underline{{\hat{p}}}, \overline{{\hat{p}}}\right] }\rho _{0i}v_{k}({\tilde{p}}_{i})+v_{k}(\overline{ {\hat{p}}})\sum _{{\tilde{p}}_{i}\notin \left[ \underline{{\hat{p}}},\overline{\hat{ p}}\right] }\rho _{0i}\right\} \end{aligned}$$

(26)

In the above equations note that $u_{k}(p^{D})\le \overline{\rho } _{H-1}u_{k}({\tilde{p}}_{H-1})+\overline{\rho }_{H}u_{k}(\min \{ \tilde{p }_{H},\overline{p} \} )\le u_{k}({\tilde{p}}_{H-1}),$ and $U_{k}( {\varvec{\rho }}^{D})\le \sum _{{\tilde{p}}_{i}\in [ \underline{{\hat{p}}}, \overline{{\hat{p}}}] }\rho _{0i}v_{k}({\tilde{p}}_{i})+v_{k}(\overline{ {\hat{p}}})\sum _{{\tilde{p}}_{i}\notin [ \underline{{\hat{p}}},\overline{\hat{ p}}] }\rho _{0i}\le v_{k}({\tilde{p}}_{i})$ for ${\tilde{p}}_{i}\in [ \underline{{\hat{p}}},\overline{{\hat{p}}}].$ Solving for $u_{k}[ \overline{p}( \overline{{\varvec{\rho }}}) ]$ and $v_{k}( \overline{{\hat{p}}}),$ we have, in the case $\overline{p}( \overline{ {\varvec{\rho }}}) <{\tilde{p}}_{H}$:

$$\begin{aligned} u_{k}[ \overline{p}( \overline{{\varvec{\rho }}}) ]= & {} \frac{1}{1-\gamma ( 1-\overline{\rho }_{H-1}) }\left[ \left( 1-\gamma \right) u_{k}(p^{D})+\gamma \overline{\rho }_{H-1}u_{k}({\tilde{p}} _{H-1})\right] \end{aligned}$$

(27)

$$\begin{aligned} v_{k}(\overline{{\hat{p}}})= & {} \frac{1}{1-\delta \mathop {\textstyle \sum }\nolimits _{{\tilde{p}} _{i}\notin [ \underline{{\hat{p}}},\overline{{\hat{p}}}] }\rho _{0i}} \left[ \left( 1-\delta \right) U_{k}({\varvec{\rho }}^{D})+\delta \mathop {\textstyle \sum }\limits _{{\tilde{p}}_{i}\in [ \underline{{\hat{p}}},\overline{{\hat{p}}} ] }\rho _{0i}v_{k}({\tilde{p}}_{i})\right] \end{aligned}$$

(28)

By Eq. (24), an increase in $\gamma$ increases $U_{k}( {\varvec{\rho }}^{D}),$ which, by Eq. (28), increases $v_{k}( \overline{{\hat{p}}}).$ Since $v_{k}$ is strictly decreasing to the right of ${\tilde{p}}_{k},$ it follows that $\overline{{\hat{p}}}$ decreases in $\gamma .$ This also means $\overline{\rho }_{H-1}$ is increasing in $\gamma ,$ since $\overline{{\hat{p}}}=\overline{\rho }_{H-1}{\tilde{p}}_{H-1}+( 1-\overline{ \rho }_{H-1}) {\tilde{p}}_{H}.$ Thus, $\gamma$ affects $\overline{p} ( \overline{{\varvec{\rho }}})$ in Eq. (27), and in Eq. (25) for the case ${\tilde{p}}_{H}\le \overline{p} ( \overline{{\varvec{\rho }}})$, both directly and through $\overline{\rho }_{h}.$ Since $u_{k}(p^{D})\le u_{k}({\tilde{p}}_{H-1})<u_{k}( {\tilde{p}}_{H})$ and $\frac{\gamma \overline{\rho }_{H-1}}{1-\gamma ( 1- \overline{\rho }_{H-1}) }$ is strictly increasing in both $\gamma$ and $\overline{\rho }_{H-1},$ then $u_{k}(\overline{p})$ is increasing in $\gamma$ and $\overline{\rho }_{H-1}.$ Since $u_{k}$ is strictly decreasing to the right of ${\tilde{p}}_{k}$, it follows that $\overline{p}( \overline{ {\varvec{\rho }}})$ is decreasing in $\gamma .$ Analogous arguments imply that $\overline{p}( \overline{{\varvec{\rho }}}),\overline{ {\hat{p}}}$ are decreasing in $\delta$ and increasing in $\vert p^{D}- {\tilde{p}}_{k} \vert .$ $\square$

Appendix B: Detailed solution for Example 3

Example 3 solution

In policy bargaining $\gamma =0$ and so proposers use take-it-or-leave-it strategies toward the core player $k=2,$ as in the Romer–Rosenthal proposer-pivot model. Since $p^{D}=0$ and ${\tilde{p}}_{k}={\tilde{p}}_{2}=0.3$, the policy collective acceptance set is $A^{u}=A_{k}^{u}=[ 0,0.6].$ This implies the equilibrium policy strategies are: $z_{1}^{u}=0.1,z_{2}^{u}=0.3,z_{3}^{u}=0.4,z_{4}^{u}=z_{5}^{u}=0.6$ and $U_{k}({\varvec{\rho }}^{D})=-\frac{1}{5}[ (0.2)^{2}+(0.0)^{2}+(0.1)^{2}+(0.3)^{2}+(0.3)^{2}] =-0.046.$ Note that $U_{k}(\mathbf {e}_{2})=0>U_{k}(\mathbf {e}_{3})=-0.01>U_{k}({\varvec{\rho }} ^{D})$. Thus, $z_{2}^{U}=\mathbf {e}_{2},z_{3}^{U}=\mathbf {e}_{3}.$ By Proposition 2’s characterization of equilibrium procedures, player 1 will share effective proposal power with player 2,  the core player. Denote this procedure by ${\varvec{\rho }}_{1}=( \rho _{1},1-\rho _{1},0,0,0).$ Because players 4 and 5 make the same policy proposal, player 5 cannot have effective procedural power. Thus, his procedural proposal cannot give effective proposal power to himself. Then $z_{4}^{U}$ and $z_{5}^{U}\$have to be outcome-equivalent procedures with some reshuffling of proposal power between players 4 and 5. Denote the LPS-equivalent of player 4’s procedural proposal by ${\varvec{\rho }} _{4}=( 0,0,1-\rho _{4},\rho _{4},0)$. Because procedural proposals are no-delay:

$$\begin{aligned} U_{k}({\varvec{\rho }}_{1})=U_{k}({\varvec{\rho }}_{4})=( 1-\delta ) U_{k}({\varvec{\rho }}^{D})+\delta \sum _{i\in N}\rho _{0i}U_{k}(z_{i}^{U}) \end{aligned}$$

(29)

where LPS implies that $U_{k}({\varvec{\rho }}_{1})=\rho _{1}u_{k}(z_{1}^{u})+( 1-\rho _{1}) u_{k}(z_{2}^{u})=-\rho _{1}( 0.2) ^{2}-( 1-\rho _{1}) ( 0) ^{2}$ and $U_{k}({\varvec{\rho }}_{4})=( 1-\rho _{4}) u_{k}(z_{3}^{u})+\rho _{4}u_{k}(z_{4}^{u})=-( 1-\rho _{4}) ( 0.1) ^{2}-\rho _{4}( 0.3) ^{2}.$ The two equations in (29) can then be solved for the two unknowns $\rho _{1},\rho _{4}$, yielding $\rho _{1}=0.439,\rho _{4}=0.094.$ These imply the procedural proposal strategies for players 1, 4,  and 5 are $z_{1}^{U}=( 0.439,0.561,0,0,0),z_{4}^{U}=( 0,0,0.906,\rho _{4},0.094-\rho _{4}),z_{5}^{U}=( 0,0,0.906,0.094-\rho _{5},\rho _{5}),$ where $0\le \rho _{4},\rho _{5}\le 0.094.$ Notice that because $\gamma =0,$ policy strategies are constant across procedures. With a positive policy discount factor, the solution would be considerably more intensive computationally, since it has to include the policy bargaining equilibrium conditions as well. $\square$