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In this paper, we study the Nash implementation in an allocation problem with single-dipped preferences. We show that, with at least three agents, Maskin monotonicity is necessary and sufficient for implementation. We examine the implementability of various social choice correspondences (SCCs) in this environment, and prove that some well-known SCCs are Maskin monotonic ( but they do not satisfy no-veto power) and hence Nash implementable.

The objective of implementation theory is to study, in a rigorous manner, the

To characterize the class of the SCCs which are implementable, some conditions on these correspondences should be imposed. Thus, Maskin [

Nevertheless, the problem of implementation theory is that there are important correspondences in economic, social, and political sciences which do not satisfy no-veto power

Thomson [

This domain of single-peaked preferences has been explored in social choice theory since the work of Black [

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 _{i}

Single-dipped preferences.

This type of preference has been introduced by Inada [

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

Before defining our domain of applications in _{i}_{i}_{1} × ... ×_{n}_{1}, ..., _{n}_{i}_{i}b_{i}_{i}_{i}

A social choice correspondence (SCC) ^{A }\{∅}, that associates with every _{i}_{i}_{i}_{i}b_{i}_{i}b_{i}

A mechanism (or form game) is given by Γ = (_{i∈N} _{i}_{i}_{1},_{2}, ..., _{n}_{i}_{−i}_{−i}_{1}, ..., _{i−1}_{i+1}_{n}_{i}_{i}_{i}_{−i}) = (s_{1}, ..., _{i−i}_{i}_{i+1}_{n}_{i}_{i}_{i}_{−i}) is the set of results which agent _{−i}_{−i}_{j ∈ N, j ≠ i}_{j}

A Nash equilibrium of the game (Γ,_{i}g_{i}_{−i}) for all _{i}_{i}_{−i}, the player _{i}

We say that an SCC

A SCC _{i}

Maskin [

A SCC _{i}_{i}), then

Maskin [

A SCC _{j}

Maskin [

Doghmi and Ziad [

_{i}_{i}’), then

_{i}_{i}) and _{j}) =

Doghmi and Ziad [

There is an amount Ω ∈ ℝ_{++} of a certain infinitely divisible good that is to be allocated among a set _{i}_{i}_{i}). For all _{i}_{i}_{i}R_{i}y_{i}_{i}_{i}_{i}_{i ∈ N} ∈ ℝ_{+}^{n} such that Σ_{i ∈ N} _{i}_{i}_{i}_{i}_{i}_{i}y_{i}R_{i}y_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}), then _{j}_{j}_{i}_{i}_{i}_{i}_{i}) and _{j}_{i}_{i}

A preference relation _{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}P_{i}y_{i}_{i}_{i}

The class of all single-dipped preference relations is represented by ℜ_{sd}_{sd}_{1}_{n}_{i}_{sd}_{i }is described by the function _{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i} _{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i} _{i}_{i}_{i}_{i}

Let us introduce some well-known correspondences.

_{sd}_{i}R_{i}x_{j}

_{ed}_{sd}_{ed}_{i}R_{i}

_{sd}_{i}R_{i}x_{i}_{i}_{i}x_{i}

As mentioned in

_{sd}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}

_{sd}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}

_{sd}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}

According to Lemmas 1 and 2, we have the following corollary:

In the following, we show that strict monotonicity, alone, is sufficient for Nash implementation when preferences are single-dipped.

Now, we show that strict monotonicity is not only sufficient, but is also necessary, as long as it becomes equivalent to Maskin monotonicity.

_{sd}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}′ (_{i}_{i}_{i}_{i}_{i}) ∀i. Thus, we have the inclusion of Maskin monotonicity. Q.E.D.

Through propositions 1 and 2, we complete the proof of the following theorem which is the main result of the paper.

To justify this result, we provide in the next subsection a list of solutions which satisfy Maskin monotonicity, and violate the condition of no-veto power.

In this subsection, we check whether the no-envy correspondence, the individually rational correspondence from equal division, (_{ed}_{ed}

In the considered examples of SCCs, we begin by examining the implementability of the no-envy correspondence. We provide the following proposition.

As the no-envy correspondence, the individually rational correspondence from the equal division also satisfies Maskin monotonicity.

The proofs of Propositions 3 and 4 are omitted. This is because the no-envy correspondence and the individually rational correspondence from equal division satisfy Maskin monotonicity in the general environment and so it is obvious that this condition is checked for the restricted domain of an allocation problem with single-dipped preferences.

By stability under the intersection of Maskin monotonicity, we give the next proposition.

According to Doghmi and Ziad [

_{sd}_{sd}_{i}R_{i}x_{i}_{i}_{i}x_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}P_{i}y_{i}_{j}P_{j}y_{j}_{k}P_{k}x_{k}_{i}_{i}_{i}_{i}_{j}_{j}_{j}_{k}_{k}_{i}P_{i}x_{i}_{j}_{j} _{j}_{k}P_{k}x_{k}

Through Proposition 3 and Lemma 3, we complete the proof of the following proposition.

Now, we examine the intersection of the Pareto correspondence with the individually rational correspondence from equal division. We provide the following lemma.

_{sd}_{ed}_{ed}_{sd}_{ed}_{i}R_{i}x_{i}_{i}P_{i}x_{i}_{i}x_{i}_{i}_{i}_{i}_{i}_{k}_{j}_{i}_{i}_{j}_{j}_{k}_{i}_{i}_{k}_{i}P_{i}y_{i}_{j}P_{j}y_{j}_{k}P_{k}x_{k}_{i}x_{i}_{i}_{j}_{j}_{k}_{j}_{i}P_{i}x_{i}_{j}_{j} _{j}_{k}_{j}_{k}_{k}_{k}P_{k}x_{k}

Through Proposition 4 and Lemma 4, we complete the proof of the following proposition.

We check whether the monotonic correspondences satisfy this additional condition for sufficiency or not. We provide the following proposition.

_{ed}

_{sd}_{1}_{2}_{i = 1,2},_{i = 1,2} = [0, Ω]. _{i = 1,2}_{3}_{3}, hence

For the individually rational correspondence from equal division, we have from _{i = 1,2},_{i = 1,2} = [0, Ω], but _{3}_{3}. Hence _{ed}_{ed}

We conclude that, in the domain of the allocation problem with single-dipped preferences, Maskin’s theorem is silent about the implementability of the monotonic correspondences under consideration. By Theorem 1, all these monotonic correspondences are Nash implementable.

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.

The _{ed}_{ed}

For future research, the Nash implementation can be examined in an extended class of preferences called

I am indebted to two anonymous referees of the journal for remarkably insightful and detailed comments that greatly improved this paper.

Also there are well-known correspondences that satisfy neither Maskin monotonicity nor no-veto power. Thus, literature on implementation theory has recently emerged (Matsushima [

The Pareto correspondence satisfies the unanimity condition in the domains of the allocation problem with single-dipped and single-plateaued preferences, and so it is Nash implementable with partially honest agents in these domains. For more details, see Doghmi and Ziad [