On Nash Implementability in Allotment Economies under Domain Restrictions with Indifference

1 Introduction

In games and social choice theories, many domain restrictions have been used in order to obtain positive results. In most cases, these results were obtained when special conditions were imposed on preference profiles. For example, limiting the preference of agents to linear orders or allowing some very limited indifference classes simplifies the expression of the conditions considerably, as well as the proofs of results based on restricting the domains of social choice rules. Among these conditions, the well-known notions are single-peaked preferences that were introduced by Black (1948) and single-dipped preferences that were provided by Inada (1964). In addition to these properties, other important conditions have been imposed on preference profiles and they are useful tools that obtain interesting results. ^[1]

Nevertheless, many authors have considered that some of the conditions imposed on preferences profiles are very strong. Thus, they have tried to relax these conditions to wider classes. For example, the fact that they have only one maximal element in a single-peaked domain, without admitting the indifference of at least one other alternative, is not always considered natural. Hence, many enlargements to this domain have been explored to allow either many best indifferent elements, which are now known as single-plateaued domains, or many worst indifferent elements, which were introduced by Cantala (2004) and they are referred to as outside options. ^[2] We also cite the extension of the single-dipped preferences domain to a wider class of preferences where agents can express their indifference among multiple worst alternatives, termed single-trouthed preferences, or multiple best alternatives, which were introduced by Manjunath (2011) and Aragón and Caramuta (2011).

The motivation for this article is to study the implications of these extensions in the implementation literature that has a close connection with social choice theory. Before explaining the problematic and providing our contribution in this direction, we start by giving a brief definition of implementation theory. This theory studies the problem of a society that chooses a social choice correspondence (SCC) to represent the social welfare of individuals or desired outcomes and the planner confronts the difficulty of how to implement this rule. This occurs because the planner does not know the exact preferences of individuals regarding these outcomes. Thus, in order to participate with a lower cost (e. g., in the case of the construction of public goods), the agents may falsify their preferences. To ensure that they disclose their true preferences, the planner will organize a non-cooperative game among these individuals. In a given solution context, when the set of equilibrium outcomes of this game coincides with the set of socially desired alternatives, we can say that an SCC is implementable in this given solution context.

To achieve this objective, some conditions should be imposed on the SCCs that the planner hopes to implement. Thus, Maskin (1977, 1999) was the first to show that there is an important connection between the implementability of SCCs and a property now known as Maskin monotonicity. He proved that this requirement is a necessary condition for an SCC to be Nash implementable, and it becomes sufficient together with an additional property called no-veto power. Since this latter requirement fails to be satisfied for many examples of Maskin monotonic SCCs in economic and political applications like, among others, the core correspondence, the individually rational correspondence in voting problems and assignment of indivisible goods, the stable rule in matching problems, and the no-envy correspondence, the individually rational correspondence from equal division in private good economies, many studies have proposed either alternative conditions or different mechanisms to overcome this deficiency. For instance, Moore and Repullo (1990), Dutta and Sen (1991), Sjöström (1991) and Danilov (1992) provided full characterizations whereas many other authors proposed either necessary conditions, such as Ziad (1997, 1998), sufficient conditions, such as Yamato (1992) and Doghmi and Ziad (2008a), or different mechanisms like the one recently proposed by Doğan (2015) to implement the no-envy correspondence on general domains of economies.

In relation to the standard domain restrictions with private values of single-peaked and single-dipped preferences, Thomson (1990, 2010), Doghmi and Ziad (2008b), and Doghmi (2013a) tried to apply some of these theoretical results to implement many examples of solutions of the problem of fair division. On single-peaked domain, Thomson (1990, 2010) proved that only the Pareto correspondence can be implemented by Maskin’s theorem (1999). To implement the no-envy correspondence, the individually rational correspondence from equal division and their intersection, he applied Yamato’s theorem (1992). For the implementability of the intersections of the Pareto correspondence with the no-envy correspondence, and with the individually rational correspondence from equal division, these tools do not work and hence he appealed to Sjöström’s algorithm (1991). On single-peaked and single-dipped domains, Doghmi and Ziad (2008b), and Doghmi (2013a) applied the results they developed in Doghmi and Ziad (2008a) and they examined the implementability of all these correspondences in an easy way. They proved that only Maskin monotonicity is a necessary and sufficient condition to implement these unanimous solutions.

In these domains of private good economies when we allow different types of indifference between alternatives, the properties used in these previous findings do not always apply, and they are only a few studies which are interested in this topic. To our knowledge, only the recent works of Doghmi (2013b) and Doghmi and Ziad (2013a, 2013b, 2015) are concerned with this issue. They studied Nash-implementability of SCCs when preferences profile are single-plateaued, and single-dipped with best indifferent allocations. Thus, they extended the results of Doghmi and Ziad (2008b), Thomson (1990, 2010), and Doghmi (2013a) on single-peaked and single-dipped domains in considering preference profiles which admit multiple best indifferent elements. However, these studies do not cover preference profiles which contain multiple worst indifferent elements.

Our objective in this article is to fill this gap and to contribute to the understanding of the implications of implementability in these domains of economies which admit different types of indifference between alternatives. We examine in particular the following domains.

1. Single-peakedness with worst indifferent allocations. This class of preferences extends the standard concept of single-peaked preferences by allowing multiple worst indifferent alternatives. It was introduced by Cantala (2004)^[3] and explored recently by Klaus and Bochet (2013) to study the relationship between monotonicity and strategy-proofness.

2. Single-plateauedness with worst indifferent allocations. This class of preferences contains two types of indifference between alternatives: multiple best indifferent elements and multiple worst indifferent elements when generalizing the well-known domains of single-peaked and single-plateaued preferences.

3. Single-troughedness. This class of preferences is a natural enlargement of the domain of single-dipped preferences, which requires that each agent is indifferent among several worst alternatives. These worst alternatives are called the “trough”. More recently, Bossert and Peters (2014) examined the notion of single-troughedness in a choice-theoretic setting. ^[4] This domain of preferences has also been used by Manjunath (2011) to characterize all rules that satisfy unanimity and strategy-proofness properties.

4. Single-troughedness with best indifferent allocations. This domain is an enlargement of the domain of single-dipped preferences with best indifferent allocations, which was explored recently by Aragón and Caramuta (2011) and Doghmi (2013b), but also an extension of the single-troughed domain examined by Bossert and Peters (2014), and by Manjunath (2011).

In this work we give a full characterization of Nash-implementability of SCCs on these four domains of economies. We introduce a weak variant of no-veto power, namely I∗-weak no-veto power, and we prove that this property together with unanimity and a stronger version of Maskin monotonicity, termed I-monotonicity, developed in Doghmi and Ziad (2015), are sufficient for Nash-implementability in general environments. Using this result, which differs from the previous findings, we show that on the above preference domains any SCC that has full range is Nash implementable if and only if it satisfies Maskin monotonicity. To derive this characterization, we first prove that the new property of I∗-weak no-veto power is implied by unanimity. In this auxiliary result, the role of our new requirement is central given that the different variants of no-veto power in the previous literature do not work. Second, we demonstrate that Doghmi and Ziad’s property of I-monotonicity continue, together with the full range requirement, to involve the property of unanimity in the preference domains with different types of indifference under consideration here. Moreover, I-monotonicity becomes equivalent to Maskin monotonicity. Given this full characterization, we inspect the implementability of various well-known SCCs. Firstly, we show that the monotonic solutions in the unrestricted domains, which violate no-veto power, like the no-envy correspondence, the individually rational correspondence from equal division and their intersection, are Nash implementable in our setup. Hence, we deduce that these solutions that are implementable in the domains of strict preferences persist and keep their implementability when we allow different types of indifference. Secondly, we give examples of correspondences that are not monotonic in the unrestricted domains but monotonic in our context, like the strong Pareto indifferent correspondence and its intersections with the no-envy correspondence, with the individually rational correspondence from equal division, and with the Pareto correspondence and therefore Nash implementable. Finally, we provide examples of correspondences which do not satisfy Maskin monotonicity in general domain, like the Pareto correspondence and its intersections with the no-envy correspondence and with the individually rational correspondence from equal division, and we prove that these correspondences continue to violate this property in our setup, and hence they are not Nash implementable. Thus, we conclude that these correspondences, which are Nash implementable in some domains of strict preferences like in single-peaked domains, lose their implementability when we extend the preference domains to allow for indifferences.

The rest of this paper is organized as follows. In Section 2, we introduce notations and definitions for the general setup. In Section 3, we introduce a weaker version of no-veto power and we provide a sufficient condition for Nash implementation in general environment. In Section 4, we present our main results on preference domains with private values and indifference; we discuss the challenges for these domains compared to the previous findings, we give a full characterization for Nash implementation, and we study the implementability of various well-known SCCs. In Section 5, we provide a conclusion.

2 Notations and Definitions

Let A be a set of alternatives, and let N={1,...,n} be a set of individuals, with generic element i. Each individual i is characterized by a preference relation Ri defined over A, which is a complete, and transitive relation in some class ℜi of admissible preference relations. Let ℜ=ℜ1×...×ℜn. An element R=(R1,...,Rn)∈ℜ is a preference profile. The relation Ri indicates the individual i’s preference. For a,b∈A, the notation aRib means that the individual i weakly prefers a to b. The asymmetrical and symmetrical parts of Ri are denoted respectively by Pi and ∼i.

A social choice correspondence (SCC) F is a multi-valued mapping from ℜ into 2A∖{∅}, that associates with every R a nonempty subset of A. For all Ri∈ℜi and all a∈A, the lower contour set for agent i at alternative a is denoted by: L(a,Ri)={b∈A | aRib}. The strict lower contour set and the indifference lower contour set are denoted respectively by LS(a,Ri)={b∈A | aPib} and LI(a,Ri)={b∈L(a,Ri) | a∼ib}.

A mechanism (or a game form) is given by Γ=(S,g) where S=Πi∈NSi; Si denotes the strategy set of the agent i and g is a function from S to A. The elements of S are denoted by s=(s1,s2,...,sn)=(si,s−i), where s−i=(s1,...,si−1,si+1,...,sn). When s∈S and bi∈Si,(bi,s−i)=(s1,...,si−i,bi,si+1,...,sn) is obtained after replacing si by bi, and g(Si,s−i) is the set of results which agent i can obtain when the other agents choose s−i from S−i=Πj∈N,j≠iSj.

A Nash equilibrium of the game (Γ,R) is a vector of strategies s∈S such that for any i, g(s)Rig(bi,s−i) for all bi∈Si, i. e. when the other players choose s−i, the player i cannot deviate from si. Given N(S,g,R) the set of Nash equilibria of the game (Γ,R), a mechanism Γ=(S,g) implements a SCC F in Nash equilibria if for all R∈ℜ,F(R)=g(N(S,g,R)). We say that a SCC F is implementable in Nash equilibria if there is a mechanism which implements it in these equilibria.

A SCC F satisfies unanimity if for any a∈A and any R∈ℜ, if for any i∈N, L(a,Ri)=A, then a∈F(R).

A central theorem of the implementatability of SCCs is provided by Maskin (1977/1999). It is based on the following properties:

Monotonicity: A SCC F satisfies monotonicity if for all R,R′∈ℜ, for any a∈F(R), if for any i∈N, L(a,Ri)⊆L(a,Ri′), then a∈F(R′).

No-veto power: A SCC F satisfies no-veto power if for each i, each R∈ℜ, and each a∈A, if L(a,Rj)=A for all j∈N∖{i}, then a∈F(R).

Maskin (1977/1999) proved that any Nash implementable correspondence must satisfy Maskin monotonicity and, when the number of agents is at least three, any SCC that satisfies Maskin monotonicity and no-veto power is Nash implementable.

3 A Weak Property of No-veto Power for Sufficiency

Recently, Doghmi and Ziad (2015) provided a full characterization for the implementability of SCCs in private good economies with single-plateaued preferences. They showed that any SCC that has full range is Nash implementable if and only if it satisfies Maskin monotonicity. To derive this result, they provided a new sufficient condition in general environment. Nevertheless, this result of general setup does not apply to domains of economies which admit multiple worst indifferent alternatives. To fill this gap and generalize these results for different types of indifference, we introduce a weak version of no-veto power that covers a large domain of preferences with indifference. To define this property, we start with the following definition.

For any agenti’s preferenceRi, any alternativea∈F(R)for some singleton “operator”{o}∈LI(a,Ri)witho≠a, the indifferent options subset is the subsetI(a,o,Ri)={b∈A∖{a,o}s.t. a ∼i b ∼i o}. If|LI(a,Ri)|≥3, thenI(a,o,Ri)≠∅, otherwiseI(a,o,Ri)=∅.

Now, we introduce the weak property of no-veto power that we call I∗-weak no-veto power. Before describing this property, we give the following key notation. The arguments of the minimum of a set of alternatives A at an individual i’s preference Ri, denoted Argmin(Ri,A), are the set of the bottom-ranked alternatives for an agent i at a preference profile R; i. e., Argmin(Ri,A)={a| for all b∈A:bRia}.

An SCCFsatisfiesI∗-weak no-veto power if for eachi, eachR∈ℜ, eacha∈F(R), and eachb∈A, if for eachR′∈ℜ, b∈LS(a,Ri)∪I(a,o,Ri)−argmin(Ri′,A)⊆L(b,Ri′)andL(b,Rj′)=Afor allj∈N∖{i}, for someo∈LI(a,Ri)∖{a}, thenb∈F(R′).

Loosely speaking, the property of I∗-weak no-veto power means that if an alternative a is socially chosen in a profile R and, for an agent i, if an alternative b is an element of the union of the subset of the alternatives which are ranked strictly below a and the subset of the indifferent options, in excluding all alternatives of i that are bottom-ranked in a new profile R′, improves its ranking in Ri′ and becomes top-ranked for all j≠i in Rj′, then it must be socially chosen in R′. To illustrate this property, we give the following example.

Let F(R)={a,b} and F(R′)={a}. In this example, the I∗-weak no-veto power condition is satisfied. In profile R′ we have L(c,Ri=2,3′)=A, and in profile R, we have two optimal alternatives: {a,b}⊆F(R). For b∈F(R), we have for player 1 I(b,o=d,R1)={c,e}, I(b,o=c,R1)={d,e} or I(b,o=e,R1)={c,d}. For I(b,o=d,R1)={c,e}, we have c∉LS(b,R1)∪I(b,o=d,R1)−argmin(R1′,A)={e}/⊆L(c,R1′)={c}. For I(b,o=c,R1)={d,e}, we have c∉LS(b,R1)∪I(b,o=c,R1)−argmin(R1′,A)={d,e}/⊆L(c,R1′)={c}. For I(b,o=e,R1)={c,d}, we have c∉LS(b,R1)∪I(b,o=e,R1)−argmin(R1′,A)={d}/⊆L(c,R1′)={c}. Now, for a∈F(R), we have I(a,o,R1)={∅} and c∉LS(a,R1)∪I(a,o,R1)−argmin(R1′,A)={b,d,e}/⊆L(c,R1′)={c}; thus c∉F(R′).

I∗-weak no-veto power lies between the standard properties of no-veto power and unanimity, it is implied by the former, but it has no logical relationship with the latter, as we show in the following observation.

Observation 1: i) TheI∗-weak no-veto power property is implied by no-veto power, but the converse does not always hold. ii) There is no logical relationship betweenI∗-weak no-veto power and unanimity.

Proof. For the part (i), it is trivial that no-veto power implies I∗-weak no-veto power. For the converse, it follows from Example 1 that I∗-weak no-veto power is satisfied, but no-veto power is not.

Concerning the part (ii), to prove that I∗-weak no- veto power does not imply the unanimity condition, it suffices to have an alternative b ranked at the top and strictly preferred to an optimal outcome a∈F(R), in this case I∗-weak no-veto power is satisfied, but unanimity is not. For the converse, we just replace in Example 1 the preference order for agent 1 at profile R′; we make aPi′dPi′c∼i′ ePb′ instead of aPi′dPi′ePi′bPc′. In this case, it is easy to see that the unanimity property is satisfied, but I∗-weak no-veto power is not. We have b∈F(R), I(b,o=d,R1)={c,e}, c∈LS(b,R1)∪I(b,o=d,R1)−Argmin(R1′,A)={c,e}⊆L(c,R1′)={b,c,e} and L(c,Ri=2,3′)=A, but c∉F(R′). Q.E.D

The second property, called I-monotonicity, that we consider for sufficiency has been introduced recently by Doghmi and Ziad (2015). An SCC F satisfies I-monotonicity if for all R,R′∈ℜ, for any a∈F(R), if for any i∈N, LS(a,Ri)∪I(a,o,Ri)∪{a}⊆L(a,Ri′) for some o∈LI(a,Ri)∖{a}, then a∈F(R′). This property can be equivalently reformulated as follows.

An SCC F satisfies I-monotonicity if for all R,R′∈ℜ, for any a∈F(R), if for any i∈N, [LS(a,Ri)∪I(a,o,Ri)−argmin(Ri′,A)]∪{a}⊆L(a,Ri′) for some o∈LI(a,Ri)∖{a}, then a∈F(R′).

I-monotonicity says that if an alternative a is selected for some profile of preferences R, and if for all agents, the alternatives produced by the union of the subset of those that are ranked strictly below a and the subset of indifferent options, in excluding all bottom-ranked alternatives in a new profile R′ (including a) remain ranked below a (in large sense) in R′, then the alternative a must be chosen for the new profile R′.

The next result provides a sufficient condition for an SCC to be Nash implementable. We shall use this result, which differs from the previous findings in the literature, as a principle tool to prove the main result in the next section.

Proof. See appendix.

4 Main Result on Preference Domains with Private Values and Indifference

In this section, we present a model of allotment economies in preference domains with private values and different types of indifference, which generalize single-peaked and single-dipped preference domains. We show that I∗-weak no-veto power is implied by unanimity and I-monotonicity with full range property imply the two properties of I∗-weak no-veto power and unanimity. Thus, we conclude that I-monotonicity with the full range property form a sufficient condition for Nash implementation, and moreover I-monotonicity becomes equivalent to Maskin monotonicity, which imply that an SCC with full range is Nash implementable if and only if it satisfies Maskin monotonicity. We use this result to study the implementability of some well-known correspondences, such as the Pareto correspondence, the no-envy correspondence, the individually rational correspondence from equal division, the strong Pareto indifferent correspondence and some of their intersections.