Strategy Proof Package Assignment Of Lease

There are \(n\ge 2\) agents and \(m\ge 2\) objects. We denote the set of agents by \(N\equiv \{1,\dots ,n\}\) and the set of objects by \(M\equiv \{1,\dots ,m\}\). Let \(\mathcal {M}\) be the power set of M. With abuse of notation, for each \(a\in M\), we may write a to mean \(\{a\}\). Each agent receives a subset of M and pays some amount of money. Thus, the agents’ common consumption set is \(\mathcal {M}\times \mathbb {R}\) and a generic (consumption)bundle for agent i is a pair \(z_i=(A_i,t_i)\in \mathcal {M}\times \mathbb {R}\). Let \(\varvec{0}\equiv (\emptyset ,0)\).

Each agent i has a complete and transitive preference relation \(R_i\) over \(\mathcal {M}\times \mathbb {R}\). Let \(P_i\) and \(I_i\) be the strict and indifference relations associated with \(R_i\). A typical class of preferences is denoted by \(\mathcal {R}\). We call \(\mathcal {R}^n\) a domain. The following are standard conditions of preferences.

Money monotonicity: For each \(A_i\in \mathcal {M}\) and each pair \(t_i,t'_i\in \mathbb {R}\) with \(t_i<t'_i\), \((A_i,t_i)\mathrel {P_i}(A_i,t'_i)\).

First object monotonicity: For each \((\{a\},t_i)\in \mathcal {M}\times \mathbb {R}\), \((\{a\},t_i)\mathrel {P_i}(\emptyset ,t_i)\).

Possibility of compensation: For each \((A_i,t_i)\in \mathcal {M}\times \mathbb {R}\) and each \(A'_i\in \mathcal {M}\), there are \(t'_i,t''_i\in \mathbb {R}\) such that \((A_i,t_i)\mathrel {R_i}(A'_i,t'_i)\) and \((A'_i,t''_i)\mathrel {R_i}(A_i,t_i)\).

Continuity: For each \(z_i\in \mathcal {M}\times \mathbb {R}\), the upper contour set at \(z_i\), \(UC_i(z_i)\equiv \{z'_i\in \mathcal {M}\times \mathbb {R}:z'_i\mathrel {R_i}z_i\}\), and the lower contour set at \(z_i\), \(LC_i(z_i)\equiv \{z'_i\in \mathcal {M}\times \mathbb {R}:z_i\mathrel {R_i}z'_i\}\), are both closed.

Free disposal: For each \((A_i,t_i)\in \mathcal {M}\times \mathbb {R}\) and each \(A'_i\in \mathcal {M}\) with \(A'_i\subseteq A_i\), \((A_i,t_i)\mathrel {R_i}(A'_i,t_i)\).

Definition 1

A preference relation is classical if it satisfies money monotonicity, first object monotonicity, possibility of compensation, and continuity.

Let \(\mathcal {R}^C\) be the class of classical preferences. We call \((\mathcal {R}^{C})^n\) the classical domain. Let \(\mathcal {R}^C_+\) be the class of classical preferences satisfying free disposal. Obviously, \(\mathcal {R}^C_+\subsetneq \mathcal {R}^C\).

Lemma 1 holds for classical preferences. The proof is relegated to the Appendix.

Lemma 1

Let \(R_i\in \mathcal {R}^C\) and \(A_i,A'_i\in \mathcal {M}\). There is a continuous function \(V_i(A'_i;(A_i,\cdot )):\mathbb {R}\rightarrow \mathbb {R}\) such that for each \(t_i\in \mathbb {R}\), \((A'_i,V_i(A'_i;(A_i,t_i)))\mathrel {I_i}(A_i,t_i)\).

For each \(R_i\in \mathcal {R}^C\), each \(z_i\in \mathcal {M} \times \mathbb {R}\), and each \(A_i\in \mathcal {M}\), we call \(V_i(A_i;z_i)\) the valuation of \(A_i\)at\(z_i\)for\(R_i\). By money monotonicity, for each \(R_i\in \mathcal {R}^C\) and each pair \((A_i,t_i),(A'_i,t'_i)\in \mathcal {M}\times \mathbb {R}\), \((A_i,t_i)\mathrel {R_i}(A'_i,t'_i)\) if and only if \(V_i(A'_i;(A_i,t_i))\le t'_i\).

Definition 2

A preference relation \(R_i\in \mathcal {R}^C\) is quasi-linear if for each pair \((A_i,t_i),(A_i',t'_i)\in \mathcal {M}\times \mathbb {R}\) and each \(t''_i\in \mathbb {R}\), \((A_i,t_i)\mathrel {I_i}(A_i',t'_i)\) implies \((A_i,t_i+t''_i)\mathrel {I_i}(A_i',t'_i+t''_i)\).

Let \(\mathcal {R}^Q\) be the class of quasi-linear preferences. We call \((\mathcal {R}^Q)^n\) the quasi-linear domain. Obviously, \(\mathcal {R}^Q\subsetneq \mathcal {R}^C\).

Remark 1

Let \(R_i\in \mathcal {R}^Q\). Then,
  1. (i)

    there is a valuation function\(v_i:\mathcal {M}\rightarrow \mathbb {R}_{+}\) such that \(v_i(\emptyset )=0\), and for each pair \((A_i,t_i),(A_i',t'_i)\in \mathcal {M}\times \mathbb {R}\), \((A_i,t_i)\mathrel {R_i}(A_i',t'_i)\) if and only if \(v_i(A_i')-t'_i\le v_i(A_i)-t_i\), and

  2. (ii)

    for each \((A_i,t_i)\in \mathcal {M}\times \mathbb {R}\) and each \(A'_i\in \mathcal {M}\), \(\textit{V}_i(A'_i;(A_i,t_i))-t_i=v_i(A'_i)-v_i(A_i)\).

Now we define important classes of preferences. The following property formalizes the notion that given a payment level, an agent desires to consume at most one object.

Definition 3

A preference relation \(R_i\in \mathcal {R}^C\) satisfies the unit-demand property if for each \((A_i,t_i)\in \mathcal {M}\times \mathbb {R}\) with \(|A_i|>1\), there is \(a\in A_i\) such that \((a,t_i)\mathrel {R_i}(A_i,t_i)\).7\(^{\text {,}}\)8

The condition means that given a payment level, for each set of objects, the most preferred one in the set is at least as good as the set itself. Note that it is possible that when an agent with a unit-demand preference relation receives an object and his payment is fixed, an additional object makes him better off. However, this occurs only when he prefers the additional object to the original one. Figure 1 illustrates a unit-demand preference relation.

Let \(\mathcal {R}^U\) be the class of unit-demand preferences. We call \((\mathcal {R}^U)^n\) the unit-demand domain. Obviously, \(\mathcal {R}^U\subsetneq \mathcal {R}^C\).

We also consider a property that formalizes the notion that given a payment level, an agent desires to consume several objects.

Definition 4

A preference relation \(R_i\in \mathcal {R}^C\) satisfies the multi-demand property if for each \((\{a\},t_i)\in \mathcal {M}\times \mathbb {R}\), there is \(A_i\in \mathcal {M}\) such that \(a\in A_i\) and \((A_i,t_i)\mathrel {P_i}(\{a\},t_i)\).

The condition says that given a payment level, when an agent receives an object, receiving some additional object(s) makes him better off. Note that given a payment level, even if an agent has a multi-demand preference relation, when he receives a set consisting of several objects, he may find it worse than each object in the set. Figure 2 illustrates a multi-demand preference relation.

Let \(\mathcal {R}^M\) be the class of multi-demand preferences. We call \((\mathcal {R}^M)^n\) the multi-demand domain. The following are examples of preferences satisfying the multi-demand property.

Example 1:

k-object-demand preferences. Given \(k\in \{1,\dots , m\}\), a preference relation \(R_i\in \mathcal {R}^C\) satisfies the k-object-demand property if (i) for each \((A_i,t_i)\in \mathcal {M}\times \mathbb {R}\) with \(|A_i|<k\), and each \(a\in M{\setminus } A_i\), \((A_i\cup \{a\},t_i)\mathrel {P_i}(A_i,t_i)\), and (ii) for each \((A_i,t_i)\in \mathcal {M}\times \mathbb {R}\) with \(|A_i|\ge k\), there is \(A'_i\subseteq A_i\) with \(|A'_i|= k\) such that \((A'_i,t_i)\mathrel {R_i}(A_i,t_i)\).9 Clearly, for each \(k\in \{2,\dots , m\}\), preferences satisfying the k-object-demand property satisfy the multi-demand property.

Example 2:

Substitutes and complements. Suppose that the set of objects are divided into two non-empty sets K and L, and agent i with a preference relation \(R_i\) views objects a and b as substitutes if both a and b are in the same set, and as complements if a and b are in different sets. For example, objects in K can be pens and objects in L can be notebooks. Formally, \(R_i\) satisfies the following property: For each \(A_i\in \mathcal {M}\) with \(|A_i|> 1\) and each \(t_i\in \mathbb {R}\), if \(A_i\subseteq K\) or \(A_i\subseteq L\), then there is \(a\in A_i\) such that \((A_i,t_i)\mathrel {I_i}(a,t_i)\), and otherwise, for each \(a\in A_i\), \((A_i,t_i)\mathrel {P_i}(a,t_i)\). Clearly, this preference relation \(R_i\) satisfies the multi-demand property.

Some preferences in \(\mathcal {R}^C\) violate both of the unit-demand property and the multi-demand property.

Example 3:

(Fig.3). A preference relation violating the unit-demand property and the multi-demand property. Let \(R_i\in \mathcal {R}^C\) be such that for each \(a\in M\) and each \(t_i\in \mathbb {R}\), \(V_i(a;(\emptyset ,t_i))=t_i+5\), and for each \(A_i\in \mathcal {M}\) with \(|A_i|>1\), and each \(t_i\in \mathbb {R}\),

$$\begin{aligned} V_i(A_i;(\emptyset ,t_i))= {\left\{ \begin{array}{ll} t_i+5&{}\text {if }t_i\ge -5,\\ \frac{1}{2}(t_i+5)&{}\text {otherwise}. \end{array}\right. } \end{aligned}$$

Then, for each pair \(a,b\in M\) and each \(t_i\in \mathbb {R}\) with \(t_i<-5\), \(V_i(\{a,b\};(\emptyset ,t_i))=\frac{1}{2}(t_i+5)>t_i+5=V_i(a;(\emptyset , t_i))=V_i(b;(\emptyset ,t_i))\), and thus, we have \((\{a,b\}, t_i+5)\mathrel {P_i}(a,t_i+5)\mathrel {I_i}(b,t_i+5)\). Thus, \(R_i\) does not satisfy the unit-demand property. Moreover, for each \(a\in M\), each \(A_i\in \mathcal {M}\) with \(a\in A_i\), and each \(t_i\in \mathbb {R}\) with \(t_i\ge -5\), \(V_i(A_i;(\emptyset ,t_i))=t_i+5=V_i(a;(\emptyset ,t_i))\), and thus, we have \((A_i,t_i+5)\mathrel {I_i}(a,t_i+5)\). Thus, \(R_i\) does not satisfy the multi-demand property.

An object allocation is an n-tuple \(A\equiv (A_1,\ldots ,A_{n})\in \mathcal {M}^n\) such that \(A_i\cap A_j=\emptyset \) for each \(i,j\in N\) with \(i\ne j\). We denote the set of object allocations by \(\mathcal {A}\). A (feasible) allocation is an n-tuple \(z\equiv (z_1,\dots ,z_{n})\equiv ((A_1,t_1),\dots ,(A_{n},t_{n}))\in (\mathcal {M}\times \mathbb {R})^n\) such that \((A_1,\dots ,A_{n})\in \mathcal {A}\). We denote the set of feasible allocations by Z. Given \(z\in Z\), we denote the object allocation and the agents’ payments at z by \(A\equiv (A_1,\dots ,A_n)\) and \(t\equiv (t_1\dots , t_n)\), respectively, and we also write \(z=(A,t)\).

A preference profile is an n-tuple \(R\equiv (R_1,\ldots R_n)\in \mathcal {R}^n\). Given \(R\in \mathcal {R}^n\) and \(i\in N\), let \(R_{-i}\equiv (R_j)_{j\ne i}\).

An allocation rule, or simply a rule on \(\mathcal {R}^n\) is a function \(f: \mathcal {R}^n\rightarrow Z\). Given a rule f and \(R\in \mathcal {R}^n \), we denote the bundle assigned to agent i by \(f_i(R)\) and we write \(f_i(R)=(A_i(R),t_i(R))\).

Now, we introduce standard properties of rules. The efficiency notion here takes the planner’s preferences into account and assumes that he is only interested in his revenue. Formally, an allocation \(z\equiv ((A_i,t_i))_{i\in N}\in Z\) is (Pareto-)efficient for \(R\in \mathcal {R}^n\) if there is no feasible allocation \(z'\equiv ((A_i',t'_i))_{i\in N}\in Z\) such that \((\text {i})\text { for each }i\in N,\;z_i'\mathrel {R_i}z_i\text {, }(\text {ii})\text { for some }j\in N,z_j'\mathrel {P_i}z_j, \text { and }(\text {iii})\sum _{i\in N}t'_i\ge \sum _{i\in N}t_i.\)

The first property states that for each preference profile, a rule chooses an efficient allocation.

Efficiency: For each \(R\in \mathcal {R}^n\), f(R) is efficient for R.

Remark 2

By money monotonicity and Lemma 1, the efficiency of allocation z is equivalent to the property that there is no allocation \(z'\equiv ((A'_i,t'_i))_{i\in N}\in Z\) such that

(i\('\)) for each \(i\in N\), \(z_i'\mathrel {I_i}z_i\), and (ii\('\)) \(\sum _{i\in N}t'_i>\sum _{i\in N}t_i\).

The second property states that no agent benefits from misrepresenting his preferences.

Strategy-proofness: For each \(R\in \mathcal {R}^n\), each \(i\in N\), and each \(R'_i\in \mathcal {R}\), \(f_i(R)\,R_i \,f_i(R'_i,R_{-i})\).

The third property states that an agent is never assigned a bundle that makes him worse off than he would be if he had received no object and paid nothing.

Individual rationality: For each \(R\in \mathcal {R}^n\) and each \(i\in N\), \(f_i(R)\mathrel {R_i}\varvec{0}\).

The fourth property states that the payment of each agent is always nonnegative.

No subsidy: For each \(R\in \mathcal {R}^n\) and each \( i\in N\), \(t_i(R)\ge 0\).

The final property is a weaker variant of the fourth: If an agent receives no object, his payment is nonnegative.

No subsidy for losers: For each \(R\in \mathcal {R}^n\) and each \(i\in N\), if \(A_i(R)=\emptyset \), \(t_i(R)\ge 0\).

Почему? - рассердился Беккер. - У меня его уже нет, - сказала она виноватым тоном.  - Я его продала.

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