The objective of implementation theory is to study, in a rigorous manner, the relationship between outcomes in a society and the institutions under which those outcomes arise. The outcomes represent the production and allocation of public and private goods and services in a society where each agent has preferences over these outcomes. The institutions are the set of rules under which the allocation is decided and applied (property rights, voting laws and voting rules, constitutions, contract law, etc.). The relationship between institutions and outcomes can be viewed from a perspective of design, and, in this form, is called the theory of mechanism design. The “planner” or the mechanism designer (representative of state, a local collectivity, a region, manager of a firm, etc.) has a goal in mind—to maximize an objective function called social choice correspondence (or rule). This correspondence represents the social objectives that the society or its representatives want to achieve. A difficulty arises in the asymmetry of information between planner and agents. This asymmetric information results because the planner does not know the exact preferences of agents. If, for example, the “options” or outcomes are public goods, the agents will state false preferences in order to participate with lower costs, and once the public property is constructed, they can use it like everyone. In order for the agents to reveal their true preferences, the planner will implement a mechanism (non-cooperative game). It is a way of collecting messages and making a decision; it represents the communication and decision aspects of the organization. This mechanism is a list of messages or “strategy” spaces, and an outcome function mapping messages into allocations, where the strategies of players depend essentially on the preference profiles and the options (or alternatives) set. A social choice correspondence (SCC) is implementable in a given solution concept if the payment with this solution of the game corresponds to the socially desired alternative and vice versa. The planner, thus, hopes to, via this game, get the agents to reveal their sincere preferences.

To characterize the class of the SCCs which are implementable, some conditions on these correspondences should be imposed. Thus, Maskin [1], as the first to identify an important relationship between implementability and a condition known now as Maskin monotonicity, demonstrated that any implementable SCC in Nash equilibria must satisfy this condition. However, this condition alone is not sufficient, and so Maskin [1] gave an additional condition called no-veto power, thus proving that in a society where at least three agents exist, an SCC is implementable if the conditions of monotonicity and no-veto power hold.

Nevertheless, the problem of implementation theory is that there are important correspondences in economic, social, and political sciences which do not satisfy no-veto power1. Thus, many authors like Moore and Repullo [10], Dutta and Sen [11], Sjöström [12], Danilov [13], Yamato [14], and Ziad [15,16], tried to address this problem by providing alternative conditions.

Thomson [17,18] applied a certain number of conditions for these previous theoretical results to implement solutions of the problem of fair allocation with single-peaked preferences. He showed that Maskin’s theorem is silent about the implementability of many solutions which can be implemented by the Yamato’s condition of strong monotonicity. However, he found that this latter condition is not stable under intersection and so he appealed to the algorithm of Sjöm [12]. More recently, Doghmi and Ziad [19,20] provided new sufficient conditions and showed that many solutions can be implemented on the single-peaked preferences domain in an easier way compared to that of the previous conditions in the literature.

This domain of single-peaked preferences has been explored in social choice theory since the work of Black [21]. In parallel, an inverse configuration to this domain, calledsingle-dipped preferences, has also been explored in this theory and in many economic applications, but it has never been explored in relation to the problem of implementation theory.

Thus, in this paper, we examine the implementability of solution to the fair division problems when agents have single-dipped preferences. This type of preference requires that each agent has a unique bad alternative. Assuming that there is an amount Ω ∈ ℝ₊₊ of a certain infinitely divisible good that is to be allocated among a set of n agents. The preference of each agent i is represented by a continuous and single-dipped preference relation R_i over [0, Ω] as illustrated in Figure 1.

This type of preference has been introduced by Inada [22] to study public good situations. It is also explored for private goods areas by Klaus et al. [23] to characterize two classes of division rules that satisfy Pareto optimality, strategy-proofness, and replacement-domination proprieties. Klaus [24,25] provided two allocation models with single-dipped preferences. In the first model, she considered the problem of allocating an infinitely divisible commodity among agents. She characterized the class of rules that satisfy a whole selection of proprieties. In the second model, she studied the allocation of an indivisible object to a group of agents. Similar to the first model, she characterized the class of rules that satisfy some conditions. Ehlers [26] studied the problem of allocating an infinitely divisible endowment among a group of agents; he characterized a class of Pareto optimal and coalitionally strategy-proof allocation rules and studied some proprieties of fairness. Manjunath [27] explored the problem of locating a single public good when preferences of agents are single-dipped, thus providing a characterization of all rules satisfying unanimity and strategy-proofness properties. More recently, Barberà et al. [28] were interested in the characterization of the family of all individual and group strategy-proof rules on the domain of single-dipped preferences. They examined the implications of these strategy-proofness requirements on the maximum size of the rules’ range.

In this paper, we study the Nash implementation in an allocation problem with single-dipped preferences. We prove that, with at least three agents, any solution of the problem of fair division can be implemented in Nash equilibria if—and only if—it satisfies Maskin monotonicity. Thus, via this result, the no-veto power condition is dispensed of in this area, and so we implement some well-known correspondences which satisfy Maskin monotonicity but violate the condition of no-veto power.

The rest of this paper is organized as follows. In Section 2, we introduce notations and definitions. In Section 3, we state and prove our main result in a domain of the allocation problem with single-dipped preferences, and we justify this result providing some well-known correspondences which satisfy Maskin monotonicity and violate the no-veto power condition. Section 4 provides our concluding remarks.

Before defining our domain of applications in Section 3, we provide in this section general notations and definitions in order to present the key concepts of implementation theory. Let A be a setof alternatives, and let N = {1, ..., n} be a set of individuals, with generic element i. Each individual i is characterized by a preference relation R_i defined over A, which is a complete, transitive, and reflexive relation in some class ℜ_i of admissible preference relations. Let ℜ = ℜ₁ × ... ×ℜ_n. An element R = (R₁, ..., R_n) ∈ ℜ is a preference profile. The relation R_i indicates the individual’s i preference. For a, b ∈ A, the notation aR_ib means that the individual i weakly prefers a to b. The asymmetrical and symmetrical parts of R_i are noted respectively by P_i and ∼_i.

A social choice correspondence (SCC) F is a mapping from ℜ into 2^A \{∅}, that associates with every R a non-empty subset of A. For all ℜ_i ∈ R_i and all a ∈ A, the lower contour set for agent i at alternative a is noted by: L(a, R_i) = {b ∈ A | aR_ib}. The strict lower contour set and the indifference lower contour set are noted respectively by LS(a, R_i) = {b ∈ A | aP_ib} and LI(a, R_i) = {b ∈ A | a ∼i b}.

A mechanism (or form game) is given by Γ = (S, g) where S = Π_i∈N S_i; S_i 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, ..., s_n) = (s_i, s_−i), where s_−i = (s₁, ..., s_i−1, s_i+1, ..., s_n). When s ∈ S and b_i ∈ S_i, (b_i, s_−i) = (s₁, ..., s_i−i,b_i, s_i+1, ..., s_n) is obtained after replacing s_i by b_i, and g(S_i, s_−i) is the set of results which agent i can obtain when the other agents choose s_−i from S_−i = Π_{j ∈ N, j ≠ i}S_j.

A Nash equilibrium of the game (Γ,R) is a vector of strategies s ∈ S such that for any i, g(s)R_ig(b_i, s_−i) for all b_i ∈ S_i, i.e. when the other players choose s_−i, the player i cannot deviate from s_i. Given N(g, R, S) the set of Nash equilibria of the game (Γ,R), a mechanism Γ = (S, g) implements an SCC F in Nash equilibria if for all R ∈ ℜ,F (R) = g(N(g, R, S)).

We say that an 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, R_i) = A, then a ∈ F (R).

Maskin [1] introduced the following conditions on F to characterize the SCCs that are implementable in Nash equilibria.

Monotonicity: This condition stipulates that if an alternative a is socially chosen in a profile R, and if the alternatives ranked below a for all agents remain ranked below it (in the large sense) in a new profile R′, then the alternative a must be socially chosen in R′.

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

Maskin [1] proved that any Nash implementable correspondence must satisfy Maskin monotonicity. However, this condition alone is not sufficient. For this, Maskin [1] gave the following additional condition.

No-veto power: This condition requires that if an alternative is ranked at the top for all agents except one, then this alternative must be socially chosen.

A SCC F satisfies no-veto power if for i, R ∈ ℜ, and a ∈ A, if L(a, R_j) = A for all j ∈ N\{i}, then a ∈ F (R).

Maskin [1] showed that, with at least three players, any social choice correspondence satisfies Maskin monotonicity, and the no-veto power is thus Nash implementable.

Doghmi and Ziad [19] reexamined this result by introducing the following new sufficient conditions.

Strict monotonicity: It is a strong version of Maskin monotonicity. A SCC F satisfies strict monotonicity if for all R, R′ ∈ ℜ, for any a ∈ F (R), if for any i ∈ N, LS(a, R_i) ∪{a}⊆ L(a, R_i’), then a ∈ F (R’).

Strict weak no-veto power. It is a weak version of no-veto power. It lies between no-veto power and unanimity. A SCC F satisfies strict weak no-veto power if for i, R ∈ ℜ, and a ∈ F (R), if for R′ ∈ ℜ, b ∈ LS(a, R_i) ⊆ L(b, R′_i) and L(b, R′_j) = A for all j ∈ N\{i}, then b ∈ F (R′).

Doghmi and Ziad [19] showed that, in addition to unanimity, strict monotonicity and strict weak no-veto power are sufficient for an SCC F to be implementable.

There is an amount Ω ∈ ℝ₊₊ of a certain infinitely divisible good that is to be allocated among a set N = {1, ..., n} of n agents. The preference of each agent i ∈ N is represented by a continuous and single-dipped preference relation R_i over [0, Ω] (the asymmetrical part is written P_i and the symmetrical part ∼_i). For all x_i, y_i ∈ [0, Ω],x_iR_iy_i means that, for the agent i, to consume a share x_i is as good as to consume the quantity y_i. A feasible allocation for the economy (R, Ω) is a vector x ≡ (x_i)_{i ∈ N} ∈ ℝ₊ⁿ such that Σ_{i ∈ N} x_i = Ω and X is the set of the feasible allocations. We note that the feasible allocations set is X = [0, Ω] × ... × [0, Ω]. Thus, L(x, R_i) = X is equivalent to L(x_i, R_i) = [0, Ω]. For the set L(x, R_i) = X, xR_iy for all y ∈ X implies that x_iR_iy_i. Thus, the agents’ preferences are defined over individual consumption spaces, not over allocation space. Then the proprieties of implementation theory, presented in general setup in Section 2, become as follows: A SCC F satisfies monotonicity if for all R, R′ ∈ ℜ, for any x ∈ F (R), if for any i ∈ N, L(x_i, R_i) ⊆ L(x_i, R′_i), then x ∈ F (R′). A SCC F satisfies strict monotonicity if for all R, R′ ∈ ℜ, for any x ∈ F (R), if for any i ∈ N, LS(x_i, R_i) ∪{x_i}⊆ L(x_i, R′_i), then x ∈ F (R′). A SCC F satisfies no-veto power if for i, R ∈ ℜ, and x ∈ X, if L(x_j, R_j) = [0, Ω] for all j ∈ N\{i}, then x ∈ F (R). A SCC F satisfies strict weak no-veto power if for i, R ∈ ℜ, x, y ∈ X, and x ∈ F (R), if for R′ ∈ ℜ, y_i ∈ LS(x_i, R_i) ⊆ L(y_i, R′_i) and L(y_j, R′) = [0, Ω] for all j ∈ N\{i}, then y ∈ F (R′). A SCC F satisfies unanimity if for any x ∈ X and any R ∈ ℜ, if for any i ∈ N, L(x_i, R_i) = [0, Ω], then x ∈ F (R). We note that the free disposability of the good is not assumed.

A preference relation R_i is single-dipped if there is a number d(R_i) ∈ [0, Ω] such that for all x_i, y_i ∈ [0, Ω] if x_i < y_i ≤ d(R_i) or d(R_i) ≤ y_i < x_i, then x_iP_iy_i. We call d(R_i) the dip of R_i.

The class of all single-dipped preference relations is represented by ℜ_sd ⊆ℜ. For R ∈ ℜ_sd, let d(R) = (d(R₁), ..., d(R_n)) be the profile of dips. A single-dipped preference relation R_i ∈ ℜ_sdi is described by the function r_i : [0, Ω] → [0, Ω] which is defined as follows: r_i(x_i) is the consumption of the agent i on the other side of the dip which is indifferent to x_i (if it exists); otherwise, it is 0 or Ω. Formally, if x_i ≤ d(R_i), then, r_i(x_i) ≥ d(R_i) and x_i ∼_i r_i(x_i) if such a number exists or r_i(x_i) = Ω otherwise; if x_i ≥ d(R_i), then, r_i(x_i) ≤ d(R_i) and x_i ∼_i r_i(x_i) if such a number exists or r_i(x_i) = 0 otherwise.

Let us introduce some well-known correspondences.

No-Envy correspondence, NE, (Foley [29]). This correspondence selects the feasible allocations for which each agent prefers his own share than the shares of the other agents. It is defined as follows: Let R ∈ ℜ_sd, NE(R) = {x ∈ X if x_iR_ix_j for each pair {i, j}⊆ N}.

Individually Rational Correspondence from Equal Division, I_ed: This correspondence selects the feasible allocations for which each agent prefers his own share rather than the average one. It is defined as follows: Let R ∈ ℜ_sd, I_ed(R) = {x ∈ X : x_iR_i(Ω/n) for all i ∈ N}.

Pareto correspondence, P : This solution selects the feasible allocations which are not weakly dominated by another allocation for all agents and not strictly dominated for at least one agent. It is defined as follows: Let R ∈ ℜ_sd, P (R) = {x ∈ X : ∄x′ ∈ X such that for all i ∈ N, x′_iR_ix_i, and for some i ∈ N, x′_iP_ix_i}.

We have examined the Nash implementation in an allocation problem with single-dipped preferences. We have shown that, with at least three agents, an SCC is implementable in Nash equilibrium if, and only if, it is Maskin monotonic. We have showed that in this area, some well-known SCCs satisfy Maskin monotonicity, but they violate no-veto power, and so Maskin’s theorem is silent about its implementability. Through our result, the no-veto power condition is no longer required and hence these correspondences are Nash implementable.

For future research, the Nash implementation can be examined in an extended class of preferences called single-troughed preferences. This domain of individual preferences is a natural enlargement of the domain of single-dipped preferences. It requires that each agent is indifferent among several bad alternatives. It has been used recently by Manjunath [27] to characterize all rules satisfying unanimity and strategy-proofness properties.