Elsevier

Journal of Mathematical Economics

Volume 61, December 2015, Pages 21-33
Journal of Mathematical Economics

Decomposing random mechanisms

https://doi.org/10.1016/j.jmateco.2015.06.002Get rights and content

Abstract

Random mechanisms have been used in real-life situations for reasons such as fairness. Voting and matching are two examples of such situations. We investigate whether the desirable properties of a random mechanism survive decomposition of the mechanism as a lottery over deterministic mechanisms that also hold such properties. To this end, we represent properties of mechanisms–such as ordinal strategy-proofness or individual rationality–using linear constraints. Using the theory of totally unimodular matrices from combinatorial integer programming, we show that total unimodularity is a sufficient condition for the decomposability of linear constraints on random mechanisms. As two illustrative examples we show that individual rationality is totally unimodular in general, and that strategy-proofness is totally unimodular in some individual choice models. We also introduce a second, more constructive approach to decomposition problems, and prove that feasibility, strategy-proofness, and unanimity, with and without anonymity, are decomposable in non-dictatorial single-peaked voting domains. Just importantly, we establish that strategy-proofness is not decomposable in some natural problems.

Introduction

Random mechanisms are frequently used in sustaining fairness among market participants. For example, admission to public schools through school choice in the US (cf.  Abdulkadiroğlu and Sönmez, 2003) is administered in many districts through centralized random mechanisms that use random tie-breakers. Some voting and social-choice environments also use random mechanisms. Jury selection, draft lotteries, and ballot positioning are further examples (cf.  Fishburn, 1984). Other examples include voting in Olympic figure skating competitions and the election method of military leaders (known as doges) in Venice (used for more than 500 years; cf.  Lines, 1986). Some random mechanisms are designed directly to use a lottery over predetermined deterministic mechanisms, as in school choice. Another approach in random mechanism design uses probabilistic assignment over outcomes for each situation rather than deterministic mechanisms in the support of the random mechanism. Competitive equilibrium from equal incomes of Hylland and Zeckhauser (1979), the probabilistic serial mechanism of Bogomolnaia and Moulin (2001) for object allocation, and maximal lottery methods (cf.  Kreweras, 1965, Fishburn, 1984) for voting are some examples of this approach.

Random mechanisms correspond to the full range of possible mechanisms. From the point of view of mechanism design, they cannot be neglected in the search for the best mechanism to implement a desired goal. On the other hand, many market design situations require transparency of the mechanism. Randomness of a mechanism is often a source of additional complexity in explaining and educating the agents who will participate in its implementation. Although simple tie-breakers can easily be explained to the participants in certain situations (e.g., in school choice), more complex random mechanism implementation often hinges on the condition that we can implement a deterministic mechanism to represent the random mechanism. For this reason, the market designer may want to resolve the uncertainty regarding the mechanism as soon as possible, before the participants’ private information is collected. Thus, the representability of a random mechanism as a randomization over deterministic mechanisms that also have the same properties could be crucial to the success of the design.

When a property is transferable through decomposition, it holds both ex ante, i.e., before the uncertainty regarding the mechanism is resolved, and ex post, i.e., after this uncertainty is resolved. In this case, the mechanism is more robust and is not affected by the market participants’ access to information regarding the resolution of the uncertainty in the mechanism. For example, if dominant-strategy incentive-compatibility (or strategy-proofness) is decomposable, then it is best for an agent to reveal his preferences truthfully regardless of if all he knows is that a stochastically strategy-proof mechanism will be implemented or if he knows exactly, after the lottery is resolved, which strategy-proof deterministic mechanism will be implemented. If such a decomposition goes through, this deterministic mechanism, in many cases, can be explained more transparently to the participants.1

The goal of this paper is to narrow the gap between our understanding of random and deterministic mechanisms in ordinal environments. Although we have a good understanding of which properties of deterministic mechanisms are preserved when we randomize over deterministic mechanisms, the other direction remains quite unclear. Exploring the possibility (or impossibility) of decomposition of a property will show whether, without loss of generality, we can focus on lotteries over deterministic mechanisms in mechanism design.

We adopt two approaches in determining the decomposability of properties of random mechanisms. We start with formulating a simple sufficient condition and then use a constructive approach for more complex properties where this first approach is inconclusive.

First, we reformulate a useful approach to mechanism design that has been used in combinatorial integer programming in various applications. We show how to analyze which properties of a random mechanism are decomposable by employing totally unimodular (TUM) decomposition (cf. Theorem 1). In this way, we contribute to the growing literature on new approaches to mechanism design using linear programming tools, which have recently found their way into mainstream economics (see Vohra, 2011). Using these methods, we show that every individually rational random mechanism is a lottery over individually rational deterministic mechanisms in a variety of environments including object allocation, social choice, and matching (cf. Theorem 2). Strategy-proofness with and without individual rationality constraints are also TUM in certain models. We give an example of an individual choice model where strategy-proofness is TUM and hence decomposable (cf. Theorem 3).23

Surprisingly, we find a counterexample in which even with a single agent, in the universal house allocation or voting domains, strategy-proofness is not decomposable and hence not TUM (cf. Theorem 4). On the other hand, together with other properties, strategy-proofness can still be decomposable in these domains.

Moreover, TUM decomposability is sometimes too strong. Even though a property is not TUM, it could still be decomposable. For example, it is straightforward to show that in the single-peaked voting domain (and hence in the universal domain), strategy-proofness, unanimity, and feasibility taken together are not TUM.45 Despite this fact, we prove that they are decomposable (cf. Theorem 5). In proving this result, we employ a constructive approach, which requires the knowledge of the characterization of deterministic mechanisms that carry the same properties as the random mechanism. Using this information, we construct a lottery over deterministic mechanisms with the required properties that induces a given random mechanism. Moreover, we prove that strategy-proofness is decomposable for tops-only mechanisms (i.e., when the mechanism outcome relies only on the reported top choices of the agents) in a single-peaked voting domain and unanimity is not needed for this result as an additional property (cf. Theorem 6).

As a corollary to the proof of the decomposability of strategy-proofness and unanimity in a single-peaked voting domain, we also establish that anonymity, unanimity, strategy-proofness, and feasibility are jointly decomposable (cf. Theorem 7).6

A forerunner to our work, Gibbard (1977) studied the decomposition of strategy-proofness in voting when all strict preference rankings are admissible, i.e., in the universal social-choice domain. In this model, he showed that any unanimous and strategy-proof random mechanism is a randomization over unanimous and strategy-proof deterministic mechanisms. Such deterministic mechanisms are known to be dictatorships (cf.  Gibbard, 1973, Satterthwaite, 1975). The question of whether such a decomposition is possible in restricted domains in which there are non-dictatorial unanimous and strategy-proof deterministic mechanisms has remained open. Using our tools, we answer it in the affirmative in the single-peaked voting domain. Deterministic strategy-proof and unanimous mechanisms in this domain were characterized by Moulin (1980) and have been studied intensively ever since. It turns out that strategy-proofness and unanimity, with and without anonymity, are decomposable even though they are not TUM.7 This result is surprising given the observation by Ehlers et al. (2002) that some strategy-proof and unanimous mechanisms cannot be decomposed in the same domain as a randomization over the particular subset of strategy-proof and unanimous deterministic mechanisms that they study. This paper’s main contribution is the characterization of strategy-proof and unanimous random mechanisms in the single-peaked preference voting model. Unlike our approach, they come up with a random mechanism class that is not defined as a probability distribution over deterministic strategy-proof and unanimous mechanisms. Hence, our result also implies that the Ehlers et al. (2002) class is equivalent to probability distributions over the Moulin (1980)’s unanimous subclass.

We introduce the decomposition tools in a unified model of many economic environments. In our model, there are a finite number of agents and a finite number of social and personalized outcomes. Agents have preferences over personal outcomes. The model encompasses voting, public goods provision, assignment of discrete goods with and without transfers, assignment of divisible goods, matching, coalition formation, and network formation. Each of these environments corresponds in the unified model to a set of conditions on what outcomes are feasible and a condition on the class of allowable ordinary preference profiles. The feasibility condition allows us to include both standard strict-preference voting problems (everybody obtains the same outcome) and object allocation (everybody obtains a different outcome). The preference domain condition allows us to include both environments without transfers (all preference profiles over outcomes are allowed) and environments with transfers.

In this unified model we primarily study ordinal mechanisms, that is, mechanisms whose message space consists of ordinal preference rankings over sure outcomes.8 To draw on combinatorial integer programming, we represent the random mechanism as a vector of probabilities indexed by agents, agents’ outcomes, and agents’ preference profiles. We show that the feasibility of the mechanism (e.g., the sum of the probabilistic outcomes sum up to 1, or constraints implying this end) along with certain properties can be represented as a TUM matrix whose rows are indexed by agents, agents’ outcomes, and agents’ preference profiles, and columns correspond to these constraints. A matrix is TUM if all of its square submatrices have determinants equal to 1, 0, or 1. We represent the feasibility constraints by columns corresponding to each preference profile separately, over every pair of agent and agent’s outcome. Certain properties, like individual rationality constraints, are also separable across preference profiles. However, properties such as strategy-proofness are defined over preference profile pairs as well as agents and their outcomes. Provided the structure of feasibility constraints of the environment is also TUM, we show that TUM feasibility constraints and the individual-rationality constraint may be jointly represented by a TUM matrix. We also show that strategy-proofness and simple feasibility can be represented by a TUM constraint matrix in certain environments. Finally, we rely on the result of Hoffman and Kruskal (1956) to show that the real-valued vector that codes a random mechanism satisfying the constraint represented by the TUM matrix is equal to a probability-weighted sum of integer-valued vectors. Each of these integer-valued vectors represents a deterministic mechanism that satisfies the same feasibility constraints and properties.

A forerunner to our study is Sethuraman et al. (2003, STV from now on). They used the theory of combinatorial integer programming to represent and decompose Arrovian social choice functions through linear constraints. The work of Budish et al. (2013, BCKM from now on) is also related to ours in that they use combinatorial integer programming to study decomposition problems related to matching. In contrast to our paper, both of these papers only study the question of whether we can decompose a particular random allocation into a randomization over deterministic outcomes while preserving constraints. This is an important question: the outcome of a random mechanism is a matrix of marginal probabilities that need to be implemented through feasible deterministic outcomes. They restrict their attention to constraints expressible as an unweighted sum of probabilistic decision variables and allocation probabilities, respectively. Our setup allows richer, integer-weighted constraints on implementation of random allocations. Our constraint language is rich enough to study mechanism design, and, for instance, to express and study constraints such as strategy-proofness constraints that are not expressible in the language of STV and BCKM.9

Our paper is also related to Peters et al. (2014), Chatterji et al. (2012), Picot and Sen (2012), and Chatterji et al. (2014). Their and our papers are independent.10 The closest to ours among these papers is Peters et al. (2014). They show that in the single-peaked domain, every strategy-proof and unanimous random mechanism is a lottery over such deterministic mechanisms; their proof technique relies on the Farkas Lemma and is different from our approach for the proof of Theorem 6. Chatterji et al. (2012) prove a decomposability property in lexicographic product domains; Picot and Sen (2012) prove it for the case of two social alternatives; and Chatterji et al. (2014) show that the decomposability of strategy-proofness and unanimity does not hold in general.

Section snippets

Motivating examples

We start by giving some examples of random mechanisms and the types of constraints and properties that are decomposable. Our results will generalize these examples.

We start with simple feasibility constraints of summation type over random outcomes, which our model covers.

Example 1

Consider an environment in which a random assignment gives each agent a probability of assignment for each object such that each object and each agent cannot be given a total assignment probability greater than 1. This

Environments

Let I be a finite set of agents. Let Oi be a finite set of sure personal outcomes of agent iI. For instance, the sure outcome might be a social choice, voting result, an object the agent was assigned, or the assigned object and the price paid. Let O×iIOi be the set of feasible profiles of sure social outcomes. We will use also the term environment to refer to O. We will derive our strongest positive results for voting problems where O1==O|I| and O={(o1,,o|I|)×iIOio1==o|I|}. Although by

Totally unimodular decomposition

We define the incidence matrix of a constraint C as a matrix whose rows are indexed by (,i,o) and columns indexed by the first two coordinates (C,C) of elementary constraints in C. Without loss of generality, we assume that constraint C contains at most one elementary constraint with a given pair of the first two coordinates (C,C). The value of a cell is 0 if it does not belong to CC, +1 if it belongs to C, and 1 if it belongs to C. We thus impose asymmetric roles on sets C and C even

A constructive approach to decomposition of strategy-proofness

In this section, we employ a constructive method to show the decomposability of strategy-proofness along with other desirable properties. We start by showing that strategy-proofness and feasibility together are not decomposable in general:

Theorem 4

For universal voting and house allocation problem models, strategy-proofness is not decomposable.

Proof of Theorem 4

The proof is through a counterexample. Suppose we have a single agent I={i} and three outcomes O=Oi={a,b,c}. With a single agent, feasibility constraints are the

Conclusion

In this paper, we study which desirable properties of a random mechanism survive decomposition of the mechanism as a lottery over deterministic mechanisms that also hold such properties. When desirable properties survive decomposition, we can focus our mechanism design efforts on deterministic mechanisms. We represented properties of random mechanisms as linear constraints, and, using combinatorial integer programming, we studied a sufficient condition, the total unimodularity of the

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    For their comments, we would like to thank Andy Atkeson, Christian Hellwig, Matthias Koeppe, Hervé Moulin, Aaron Roth, Arunava Sen, William Thomson, Rakesh Vohra, William Zame, and audiences at UCLA in May 2010 and at Northwestern Matching Conference in February 2011.

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