Strategy-proof choice with monotonic additive preferences

doi:10.1016/j.geb.2020.12.001

Games and Economic Behavior

Volume 126, March 2021, Pages 94-99

https://doi.org/10.1016/j.geb.2020.12.001 Get rights and content

Abstract

We describe the class of strategy-proof mechanisms for choosing sets of objects when preferences are additive and monotonic. Keywords: strategy-proofness; additive preferences; decomposability; participatory budgeting.

Introduction

We study strategy-proof mechanisms for choosing a set of objects from a given collection of feasible sets. We assume that the objects are public in nature, independent of each other, and valuable to all participants in the mechanism. Under these assumptions, a participant's preferences are naturally representable by a measure, i.e., a monotonic additive set function.

The interpretation of the term “feasible object” depends on the considered application. Let us be briefly discuss three examples.

(1) Participatory budgeting (Cabannes (2004), Shah (2007), Ganuza and Baiocchi (2012)) is an increasingly popular practice whereby a local government allows its residents to determine how an exogenously given fraction of its budget should be spent. Here the objects are the (indivisible) public projects contemplated by the members of the community (renovating a sport center, building a new school, and so on); and a subset of projects is feasible if the sum of the associated costs does not exceed the fraction of the budget that was set aside. The assumption that preferences are representable by an additive set function makes sense if the projects are independent.

From a theoretical viewpoint, this is a multi-agent version of the well-known knapsack problem. The literature explores the issues of Pareto optimality (Kellerer et al. (2004)), welfare maximization and fairness (see among others Benade et al. (2017), and Aziz et al. (2018)), and computational complexity (see for instance Fluschnik et al. (2019)). The issue of strategic manipulability remains relatively little studied.¹

Multiple budget constraints (arising from the need to use multiple scarce resources to complete the public projects) are considered in Fain et al. (2018). For a comprehensive survey of other theoretical extensions and variants of the participatory budgeting model (including alternative assumptions on preferences), see Aziz and Shah (2020).

(2) In a second interpretation of our model, the objects are candidates to be recruited to fill a number of unrelated positions. Here the feasibility constraints may take the form of budget constraints, upper bounds, lower bounds, or diversity requirements.

(3) A third interpretation arises in the context of uncertainty. The objects are the possible states of nature and the problem consists in choosing an event, i.e., a subset of states of nature.

As an illustration, consider the problem of allocating a given amount of public funds to develop a vaccine against a new infectious disease. A number of pharmaceutical firms are competing for the funds; only one firm can receive the funds and no firm can succeed without funding. If funded, a firm may either succeed or fail to develop the vaccine, and it is fully characterized by the particular event (subset of states of nature) where it succeeds. A group of experts must decide which firm to fund. Under our assumptions, the problem amounts to selecting an event. According to the standard subjective-expected-utility paradigm, each expert evaluates a firm based on her subjective probability (of the event) that the firm succeeds: the expert's preferences are represented by a measure over the possible events.²

These three examples show that it is important to study mechanisms whose range may be constrained in different ways. For any collection of feasible sets, the theorem we shall prove describes the class of strategy-proof mechanisms whose range coincides with that collection.

To illustrate our result, suppose there are three agents and the set of relevant objects is $X = {a, b, c, a^{'}, b^{'}}$ . Let agent 1 select which of the sets ${a, b}, {a, c}, {b, c}$ will be part of the selected set, and let the agents decide by majority vote which of ${a^{'}}$ or ${b^{'}}$ will be included as well. The selected set is the union of these two separate choices. If preferences are additive, it is easy to see that this mechanism is strategy-proof. Note that not all subsets of X can be chosen: the range of the mechanism contains the six sets ${a, b, a^{'}}$ , ${a, c, a^{'}}$ , ${b, c, a^{'}}$ , ${a, b, b^{'}}$ , ${a, c, b^{'}}$ , ${b, c, b^{'}}$ . Observe also that (i) no set in the range is a strict subset of another, (ii) every set in the range is the union of one of the sets ${a, b}, {a, c}, {b, c}$ and one of the sets ${a^{'}}, {b^{'}}$ , (iii) the choice between ${a, b}, {a, c}, {b, c}$ is dictatorial whereas the choice between ${a^{'}}$ and ${b^{'}}$ is majority-based. As we shall see, all strategy-proof mechanisms defined for monotonic additive preferences have a structure similar to the one in this example.

Our work belongs to the literature on strategy-proofness in contexts where the set of social alternatives has a Cartesian product structure and preferences are separable or additive. To see the connection, note that the collection of all subsets of a finite set of objects ${1, . . ., m}$ may be identified with the Cartesian product ${0, 1}^{m}$ . Early works in the field, such as Barberà et al. (1991) and Le Breton and Sen (1999), describe the unconstrained strategy-proof mechanisms — those whose range is the entire set of alternatives.

Subsequent contributions explore the consequences of imposing feasibility constraints on the range of the mechanism. Barberà et al. (2005) characterize all the strategy-proof mechanisms for choosing sets of objects (or, equivalently, alternatives in the cube ${0, 1}^{m}$ ) when preferences are either additive or separable. Reffgen and Svensson (2012) generalize the analysis to the case where alternatives form an arbitrary finite product set.

Compared to this strand of work, the specificity of the current note lies in the assumption that preferences are monotonic. In the three examples discussed above, and in most economic applications, the assumption of monotonic preferences is natural and salient. It is therefore useful to understand its implications for the class of strategy-proof mechanisms. Although the structure of strategy-proof mechanisms may in general be very sensitive to the restrictions imposed on individual preferences,³ it turns out that restricting attention to the domain of monotonic additive preferences does not substantially alter the characterization offered in Theorem 1 of Barberà et al. (2005).

Section snippets

Setup

Let $N = {1, . . ., n}$ , $n \geq 2$ , be the set of agents and let $X = {1, . . ., m}$ , $m \geq 2$ , be a set of objects. The set of (social) alternatives is $X : = {A : \emptyset \neq A \subseteq X}$ . We emphasize that each alternative is a set of objects. Throughout this note, ⊆ denotes inclusion and ⊂ is reserved for strict inclusion.

Agent i's preference $P_{i}$ is a binary relation on $X$ which is complete (for all $A, B \in X$ , $[A \neq B] \Rightarrow [A P_{i} B or B P_{i} A]$ ), transitive (for all $A, B, C \in X$ , $[A P_{i} B and B P_{i} C] \Rightarrow [A P_{i} C])$ , and asymmetric (there do not exist $A, B \in X$ such that $A P_{i} B$ and $B P$

Characterization

For any nonempty set of alternatives $A \subseteq X$ , a decomposition of $A$ is a collection ${A_{1}, . . ., A_{L}}$ such that $(i) A_{l} \subseteq X for l = 1, . . ., L, (ii) A_{l} \cap A_{l^{'}} = \emptyset for all A_{l} \in A_{l}, A_{l^{'}} \in A_{l^{'}}, l \neq l^{'}, (iii) A = {⋃_{l = 1}^{L} A_{l} : (A_{1}, . . ., A_{L}) \in A_{1} \times . . . \times A_{L}} .$ The sets $A_{1}, . . ., A_{L}$ are components of $A$ . A decomposition of $A$ is maximal if there is no other decomposition ${A_{1}^{'}, . . ., A_{L^{'}}^{'}}$ of $A$ such that $L^{'} > L$ . Barberà et al. (2005) prove that every set $A \subseteq X$ has a unique maximal decomposition; see also Svensson and Torstensson (2008).

For example, if $X = {a, b, c$

Proof

We derive Theorem 2 from Theorem 1. Let $f : P_{m a}^{N} \to X$ be an SCF and let ${A_{1}, . . ., A_{L}}$ be the maximal decomposition of $R_{f}$ . It is easy to check that f is strategy-proof if conditions (i), (ii), and (iii) hold. Conversely, suppose that f is strategy-proof.

Step 1. The range $R_{f}$ is a clutter.

Substep 1.1. The SCF f is unanimous on its range: for all $A \in R_{f}$ and all $P \in P_{m a}^{N}$ , $[τ (P_{i}; R_{f}) = A for all i \in N] \Rightarrow [f (P) = A]$ .

The argument is standard. Fix $A \in R_{f}$ and $P \in P_{m a}^{N}$ ; and suppose that $τ (P_{i}; R_{f}) = A$ for all $i \in N$ . Since $A \in R_{f}$ , there

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^☆: We thank S. Horan, two referees, and an associate editor for useful feedback.

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NoteStrategy-proof choice with monotonic additive preferences☆

Abstract

Introduction

Section snippets

Setup

Characterization

Proof

J. Econ. Theory

Proportionally representative participatory budgeting: axioms and algorithms

Participatory budgeting: models and approaches

Strategyproof choice of social acts

Am. Econ. Rev.

Voting by committees

Econometrica

Preference elicitation for participatory budgeting

Truthful and near-optimal mechanisms for welfare maximization in multi-winner elections

Participatory budgeting: a significant contribution to participatory democracy

Environ. Urban.

Note
Strategy-proof choice with monotonic additive preferences☆