# 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

- (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 - (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}

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 \((A_i,t_i)\mathrel {P_i}(\{a\},t_i)\).

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 \(t_i\in \mathbb {R}\),

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

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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