Non-bossiness

Abstract

An allocation rule is “non-bossy” if whenever a change in an agent’s preferences does not bring about a change in his assignment, then it does not bring about a change in anybody’s assignment. We discuss the multiple interpretations that have been proposed for this property. We question their validity, arguing that in most cases, non-bossiness either says too little or that it says too much. We also make a case against the property. We propose as its main justification the technical help that it often provides in structuring classes of rules, making characterizations more manageable.

Appendices

Appendix A

This appendix contains lists of models that should help understand the role played by non-bossiness in relating strategy-proofness and group strategy-proofness.

(i)  Models for which non-bossiness (or a version of it) and strategy-proofness together imply group strategy-proofness (or its weak form) are identified by Barberà and Jackson (1995), Serizawa (1996), Pápai (2000a, (2000b), Rhee (2011), Manjunath (2014), and Long (2016).

(ii)  Models for which a related implication holds that involves some other requirements are identified by Barberà et al. (1997), Klaus (2001a, (2001b), and Mutuswami (2005).

(iii)  Models for which non-bossiness and strategy-proofness together do not imply group strategy-proofness are identified by Miyagawa (1997), Pápai (2003), Alcalde-Unzu and Molis (2011), Manjunath (2012), Jaramillo and Manjunath (2012), Hashimoto and Saitoh (2012), Ehlers (2014) and Morimoto et al. (2013).

Appendix B

In this appendix we support our claim that there is no logical relation between localness and non-bossiness. We omit the formal definition of the Walrasian rule. It is clearly local. However, there are domains on which it is bossy.

Claim 1

The Walrasian rule violates non-bossiness on the quasi-linear domain.

To prove this, we consider the following example, in the description of which $U(R_i,x_i)$ designates the upper contour set of the relation $R_i$ at $x_i$. Here, by a quasi-linear relation we mean a relation such that if two bundles are indifferent, then so are the two bundles obtained from them by adding the same quantity of a particular good. Let ${\varvec{\mathcal {R}}}_{{{\varvec{ql}}}}$ denote the class of continuous, monotonic, convex, and quasi-linear preferences for which the particular good is good 1. An economy is a pair $(\omega , R)$ of a profile $\omega \equiv (\omega _i)_{i \in N}$ of endowments and a profile $R\equiv (R_i)_{i \in N}\in \mathcal {R}^N_{ql}$. Let W denote the Walrasian rule. To make our point, we need not vary the endowment profile and therefore we do not list it as an argument of W.

Proof

Let $\ell =2$, $N\equiv \{1,2,3\}$, $\omega _1= \omega _2 =\omega _3 \equiv (5,5)$, and preferences $R \in \mathcal {R}^N_{ql}$ be such that $U(R_1, x_1)$ is uniquely supported by the line of slope $-1$ at each point $x_1$ of ordinate 5, $U(R_2, x_2)$ is supported by any line of slope between $-1$ and $-\frac{1}{2}$ at each point $x_2$ of ordinate 3, $U(R_3, x_3)$ is supported by any line of slope between $-1$ and $-\frac{1}{2}$ at each point $x_3$ of ordinate 7. Let $y\equiv ((5,5), (7,3),(3,7))$. We have $\{y\} =W(R)$. Now, let $R'_1\in \mathcal {R}_{ql}$ be such that $U(R'_1, x_1)$ is uniquely supported by the line of slope $-\frac{1}{2}$ at each point $x_1$ of ordinate 5. Let $y'_1\equiv y_1$, $y'_2\equiv (9, 3)$,  $y'_3\equiv (1,7)$, and $y'\equiv (y'_1, y'_2, y'_3)$. Then, $\{y'\} =W(R'_1, R_2,R_3)$. Agent 1’s assignment has not changed, but the other agents’ assignments have. $\square$

Agents 2 and 3’s preferences in the example used to prove Claim 1 are not smooth: they have non-degenerate cones of lines of support at each point on the horizontal lines of ordinates 3 and 5 respectively. The relevance of smoothness in guaranteeing non-bossiness is discussed by Satterthwaite and Sonnenschein (1981).

Non-bossiness does not imply localness. The proof if by means of an example. It concerns the fair division problem when preferences belong to the domain ${\varvec{\mathcal {R}}}_{{{\varvec{cl}}}}$ of continuous, monotonic, and strictly convex preferences.

Example 1

Let r be a point in the simplex of commodity space, and let ${{\varvec{E}}}^{{\varvec{r}}}$ be the rule that selects, for each economy, the efficient allocation x—under our assumptions, it is unique—such that, for some $\lambda \in \mathbb {R}_+$, each agent $i \in N$ is indifferent between $x_i$ and $\lambda r$.

The egalitarian-equivalence correspondence (Pazner and Schmeidler 1978) selects for each economy each allocation x such that there is a reference bundle $x_0$ that each agent finds indifferent to his assignment. The rules $\{E\}_{r \in \Delta ^{\ell -1}}$ are canonical selections from the Pareto–and–egalitarian-equivalence correspondence.

Claim 2

For each $r \in \Delta ^{\ell -1}$, the rule $E^r$ is non local and non-bossy.

Proof

It is obvious that each $E^r$ violates localness. Now, let $R \in \mathcal {R}^N_{cl}$ and $x\equiv E^r(R)$ with associated parameter $\lambda$. Let $i \in N$. Let $R'_i \in \mathcal {R}_{cl}$ and $x'\equiv E^r(R'_i, R_{-i})$. Suppose that $x'_i = x_i$. Let $\lambda '$ be the parameter associated with $(R'_i, R_{-i})$. We claim that $\lambda '=\lambda$. Indeed, if $\lambda '>\lambda$, each agent $j \in N{\setminus } \{i\}$ is better off than he was initially. This means that in R, $x'$ Pareto dominates x. If $\lambda '<\lambda$, in $(R'_i, R_{-i})$, x Pareto dominates $x'$. In each case, we obtain a contradiction to the fact that $E^r$ is a selection from the Pareto correspondence. If $\lambda '=\lambda$, for each $j \in N{\setminus } \{i\}$, $x'_j = x_j$. Altogether, $x' = x$. $\square$

Because of its importance to public economics, let us also discuss the Lindahl correspondence, again omitting the formal definitions. (The only difference with the Walrasian correspondence is that agents face individualized prices.) It is clearly local. Also, there are interesting preference domains on which it is single-valued and non-bossy. One such domain is when (i) preferences are strictly convex and the public good is “strictly normal”, in the sense that an increase in the individualized price an agent faces leads to an increase in the public good component of the bundle at which he maximizes his preferences in his budget set, and (ii) the technology is linear. Then, there is a unique Lindahl allocation. Now, if agent i’s preferences change but his assignment does not, the public good level does not change. For an agent’s maximizing bundle on his new budget set to have the same public good component, his budget set should be the same. He faces the same individualized prices. So then, the allocation remains a Lindahl allocation.

Appendix C

The individual-endowments-lower-bound correspondence selects all the allocations that each agent finds at least as desirable as his endowment.

Claim 3

Any subsolution of the Pareto–and–individual-endowments-lower-bound correspondence that is local is a subsolution of the Walrasian correspondence.

Proof

Let $\varphi$ be a subsolution of the Pareto correspondence. Let $R \in \mathcal {R}^N$, $x \in \varphi (R)$, and suppose that $x \notin W(R)$. Then, at the supporting prices p—they exist because $\varphi$ is a subsolution of the Pareto correspondence—there is $i \in N$ such that $px_i <p\omega _i$. Then, let $R'_i \in \mathcal {R}$ be such that $U(R'_i,x_i)$ still admits p as supporting prices, but $\omega _i \mathrel {P'_i} x_i$.³⁹ By localness, $x\in \varphi (R'_i,R_{-i})$. However, in $(R'_i,R_{-i})$, the individual-endowments lower bound is violated for agent i at x. $\square$