Allocation of an indivisible object on the full preference domain: axiomatic characterizations
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Abstract
We study the problem of allocating an indivisible object to one of several agents on the full preference domain when monetary transfers are not allowed. Our main requirement is strategyproofness. The other properties we seek are Pareto optimality, nondictatorship, and nonbossiness. We provide characterizations of strategyproof rules that satisfy Pareto optimality and nonbossiness, nondictatorship and nonbossiness, and Pareto optimality and nondictatorship. As a consequence of these characterizations, we show that a strategyproof rule cannot satisfy Pareto optimality, nondictatorship, and nonbossiness simultaneously.
Keywords
Matching Indivisible object Strategyproofness Pareto optimality Full preference domainJEL Classification
C78 D711 Introduction
We consider the following problem. A central agency allocates an indivisible object to one of several agents (e.g., a task to employees, a house to applicants) where monetary transfers are not possible. We study this allocation problem on the full preference domain—in particular, an agent may prefer to receive the object, prefer not to receive it, or be indifferent. In practice, agents are often indifferent when they do not know the quality or features of an object. In such situations, it may be costly to restrict the full preference domain to the strict one by treating indifferent agents as if they prefer to receive the object or prefer not to. For example, an agent is indifferent, and he is considered as preferring the object on the strict preference domain, and he receives it. When there are agents who actually prefer to receive the object, allocating it to an indifferent agent results in an efficiency loss. Similarly, if the indifferent agent is considered as not preferring the object on the strict preference domain, an efficient rule requires that the object is not allocated to this agent. This requirement introduces an artificial constraint for an efficient rule, which unnecessarily restricts the set of efficient rules when there are agents who actually prefer not to receive the object. Erdil and Ergin (2008) establish this result by showing that random tiebreaking in school choice districts in the United States adversely affects the welfare of students by introducing artificial constraints.
In this paper, our main requirement is strategyproofness, which means that no agent can gain from reporting false preference. This strategic robustness property has been widely used to guarantee that agents reveal their true preferences.^{1} In addition to strategyproofness, we are interested in Pareto optimality, nondictatorship, and nonbossiness. Pareto optimality is a standard efficiency notion used in almost all economic applications, and nondictatorship and nonbossiness are fairness properties. A rule is Pareto optimal if there is no other rule that provides weakly better outcomes for all agents and strictly better outcomes for some agents. A rule is nondictatorial if there is no agent who receives the object whenever he prefers to do so and does not receive it whenever he prefers not to. A rule is nonbossy if there is no agent who can change the outcome of another agent without changing his own outcome.
We first show, interestingly, that Pareto optimality implies strategyproofness (Proposition 1). We prove that a strategyproof and nonbossy rule has a ranking function (Proposition 2). We then characterize strategyproof rules as follows. A Pareto optimal and nonbossy rule is dictatorial (Proposition 3), a nondictatorial and nonbossy rule is suboptimal (Proposition 4), and a Pareto optimal and nondictatorial rule is bossy (Proposition 5). As a result of these characterizations, we show that a strategyproof rule cannot satisfy Pareto optimality, nondictatorship, and nonbossiness simultaneously (Proposition 6).
2 Model and analysis
There is an indivisible object and a set \(N=\{1,2,\ldots ,n\}\) of finite agents, where \(n\ge 2\). The indivisible object can be allocated to an agent or may not be allocated at all. If the object is allocated to agent \(i\in N\), the outcome of the allocation \( x=i\); if the object is not allocated at all, the outcome of the allocation \(x=0\). Then, the set of possible outcomes is \(N_{0}=\{0,1,2,\ldots ,n\}\). An agent \(i\in N\) may or may not receive the object, so the allocation of agent i in the outcome x is \(x_{i}\in \{0,i\}\).
The preference of agent \(i\in N\) is \(R_{i}\), where \(R_{i}\) is reflexive for each agent i, and the strict component of \(R_{i}\) is \(P_{i}\). If an agent prefers to receive the object, we denote it by \(R_{i}=R_{i}^{+}\); if he is indifferent, we denote it by \(R_{i}=R_{i}^{0}\); and if he prefers not to receive the object, we denote it by \(R_{i}=R_{i}^{}\). The set of preference profiles for agent \(i \in N\) is \({\mathfrak {R}}_{{\mathfrak {i}}}\in \{R_{i}^{+},R_{i}^{0},R_{i}^{}\}\), where \({\mathfrak {R}}=\times _{i\in N}{\mathfrak {R}}_{{\mathfrak {i}}}\). For notational convenience and ease of distinguishing preference types, we define a binary relationship (a strict partial order) “\(\vartriangleright \)” such that for \(i\in N\) and \(R_{i}\in \) \({\mathfrak {R}}_{i}\), we have \(R_{i}^{+}\vartriangleright R_{i}^{0}\vartriangleright R_{i}^{}\). For example, \(R_i^+ \vartriangleright \ {\tilde{R}}_{i}\) means \({\tilde{R}}_{i} \in \{R_{i}^{0},R_{i}^{}\}\); and \({\tilde{R}}_{i} \vartriangleright \ R_{i}^\) means \({\tilde{R}}_{i} \in \{R_{i}^{+}, R_{i}^{0}\}\).
If all agents prefer to receive the object, we denote it by \(R^{+}\), where \(R^{+}=(R_{1}^{+},R_{2}^{+} \ldots ,R_{n}^{+})\); if all agents are indifferent, we denote it by \(R^{0}\), where \(R^{0}=(R_{1}^{0},R_{2}^{0} \ldots ,R_{n}^{0})\); and if all agents prefer not to receive the object, we denote it by \(R^{}\), where \(R^{}=(R_{1}^{},R_{2}^{} \ldots ,R_{n}^{})\). Moreover, when an agent is indifferent or prefers not to receive the object, we denote it by \(R_{i}=R_{i}^{\lnot }\), where \(R_{i}^{\lnot }\in \{R_{i}^{0},R_{i}^{}\}\). A rule \(f:{\mathfrak {R}} \longrightarrow N_{0}\) is a function that associates a feasible outcome to each preference profile. The outcome of agent i by rule f under the preference profile R is \(f_{i}(R)\), i.e., when \(f(R)=x\), \(f_{i}(R)=x_{i}\).
We next define the properties studied in the paper. First, strategyproofness requires that no agent ever gains by misrepresenting his true preferences. For all \(i\in N\), \({\tilde{R}}_{i}\in {\mathfrak {R}}_{i}\) and \(R\in {\mathfrak {R}}\), a rule f is strategyproof if \(f_{i}(R)R_{i}f_{i}({\tilde{R_{i}}},R_{i})\). If a rule is not strategyproof, then it is manipulable. For some \( i\in N\), \({\tilde{R}}_{i}\in {\mathfrak {R}}_{i}\) and \(R\in {\mathfrak {R}}\), a rule f is manipulable and agent i can manipulate it at R if \( f_{i}({\tilde{R}}_{i},R_{i})P_{i}f_{i}(R)\). Second, a rule is Pareto optimal if there is no other rule that provides weakly better outcomes for all agents and strictly better outcomes for some agents. Formally, a rule f is Pareto optimal if for all \(R\in {\mathfrak {R}}\), there is no \(x\in N_{0}\) such that \(x_{i}R_{i}f_{i}(R)\) for all \(i\in N\), and \(x_{j}P_{j}f_{j}(R)\) for some \(j\in N\). Third, dictatorship requires that there is an agent who receives the object whenever he prefers to receive, and does not receive it whenever he prefers not to. Formally, a rule f is dictatorial if there exists \(i\in N\) such that whenever \(R_{i}=R_{i}^{+}\), \(f(R)=i\), and whenever \(R_{i}=R_{i}^{}\), \(f_{i}(R)=0\). In this case, agent i is a dictator with respect to f. A rule f is nondictatorial if there is no dictator agent with respect to f. Fourth, bossiness requires that there is an agent who can change another agent’s outcome without changing his own outcome. Formally, a rule f is bossy if there exist \(i,j\in N\), \({\tilde{R}}_{i}\in {\mathfrak {R}}_{i}\), and \(R\in {\mathfrak {R}}\) such that \(f_{i}(R)=f_{i}({\tilde{R}}_{i},R_{i})\) and \(f_{j}(R)\ne f_{j}({\tilde{R}}_{i},R_{i})\), and in this case, agent i becomes bossy with agent j at R. If a rule f is not bossy, it is nonbossy. Finally, full range requires that every possible outcome occurs at least at one preference profile. Formally, a rule f is full range if for all \(x\in N_{0}\), there exists \(R\in {\mathfrak {R}}\) such that \(f(R)=x\).
The following proposition shows, interestingly, that Pareto optimality implies strategyproofness.
Proposition 1
If a rule f is Pareto optimal, then f is strategyproof.
Proof
Suppose that f is Pareto optimal but not strategyproof. Then, there exists an agent \(i\in N\), \({\tilde{R}}_{i}\in {\mathfrak {R}}_{i}\), and \(R\in {\mathfrak {R}}\) such that \(f_{i}({\tilde{R}}_{i},R_{i})P_{i}f_{i}(R)\). If \(R_i=R_i^0\), agent i cannot manipulate f because he cannot be better off. If \(R_{i}= R_{i}^{}\), we need \(f(R)=i\), which can be improved with \(f_i(R)=0\), and this contradicts the Pareto optimality of f. If \(R_i=R_i^+\), for f to be manipulable, we need \(f_i(R)=0\), \(f({\tilde{R}}_{i},R_{i})=i\). Pareto optimality of f requires that there exists \(j\in N\) such that \(R_{j}=R^+_j\) and \(f(R)=j\). Then, \(f({\tilde{R}}_{i},R_{i})=i\) can be Pareto improved by \(f({\tilde{R}}_{i},R_{i})=j\), contradiction.\(\square \)
Pápai (2001) proves that strategyproofness, nonbossiness, and full range are sufficient for Pareto optimality on the strict preference domain. The following example shows that this result does not hold on the full preference domain.
Example 1
Let \(N=\{1,2\}\). Let the rule f be such that \( f(R_{1}^{},R_{2}^{+})=2\), \(f(R_{1}^{},R_{2}^{0})=2\), \( f(R_{1}^{},R_{2}^{})=0\) and for all \(R\in {\mathfrak {R}}{\setminus } \{(R_{1}^{},R_{2}^{+}),(R_{1}^{},R_{2}^{0}),(R_{1}^{},R_{2}^{})\}\), \( f(R)=1\). Agent 1 receives the object whenever he prefers to do so (or he is indifferent) and he does not receive it whenever he does not prefer it. Then, agent 1 does not need to manipulate. When agent 2 prefers to receive the object, he either gets it or cannot get it because agent 1 gets it. Moreover, agent 2 does not receive the object whenever he prefers not to. Thus, f is strategyproof. Whenever agent 1 reports \(R_{1}^{0}\) or \( R_{1}^{+}\), he receives the object, and hence, he cannot be bossy with agent 2, and agent 2 cannot be bossy with agent 1. Then, f is nonbossy. All possible outcomes (\(x=0,1,2\)) occur at least at one preference profile, so f is full range. Yet, f is not Pareto optimal because the outcome can be improved by letting \(f(R_{1}^{0},R_{2}^{+})=2\) instead of \(f(R_{1}^{0},R_{2}^{+})=1\).
Example 1 shows that strategyproofness, nonbossiness, and full range are not sufficient for Pareto optimality on the full preference domain because the object is assigned to an indifferent agent even though another agent prefers to receive it. We define this case formally as follows. A rule f is forceful if there exist \(i,j\in N\) and \(R\in {\mathfrak {R}}\) such that \(R_{i}=R_{i}^{+},R_{j}=R_{j}^{0}\), and \(f(R)=j\). If f is not forceful, it is nonforceful. In Lemma 2, we show that strategyproofness, nonbossiness, full range, and nonforcefulness are sufficient for Pareto optimality on the full preference domain.
Before showing Lemma 2, we define two more properties and derive a necessary and sufficient condition for Pareto optimality. A rule f is wasteful if there exists an agent \(i\in N\) such that \(R_{i}=R_{i}^{+}\) and \(R\in {\mathfrak {R}}\), and \(f(R)=0\); i.e., f is wasteful if there is an agent who prefers to receive the object, but the object is not allocated at all. If a rule f is not wasteful, it is nonwasteful. An agent \(i\in N\) is a discard agent with respect to f if there exists \(R_{i}\in {\mathfrak {R}}_{i}\) such that \(f(R_{i}^{},R_{i})=i\); i.e., if an agent prefers not to receive the object but receives it, he is a discard agent. Lemma 1 proves that nonforcefulness, nonwastefulness, and containing no discard agent are necessary and sufficient for Pareto optimality.
Lemma 1
A rule f is Pareto optimal if and only if f is nonforceful, nonwasteful, and contains no discard agent.
Proof
 (a)
As it is obvious, we omit this proof, but it is available from the author upon request.
 (b)
If a rule f is nonforceful, nonwasteful, and contains no discard agents, then f is Pareto optimal. Suppose to the contrary that f is not Pareto optimal. Then, there exists \(x\in N_{0}\) and \(R \in {\mathfrak {R}}\) such that \(x_{i}R_{i}f_{i}(R)\) for all \(i\in N\), and \(x_{j}P_{j}f_{j}(R)\) for (at least) an agent \(j\in N{\setminus } \left\{ i\right\} \) . There are three cases: (i) \(R_j=R_j^0\), (ii) \(R_j=R_j^\), and (iii) \(R_j=R_j^+\). (i): \(R_j=R_j^0\), agent j cannot be strictly better off. (ii): \(R_j=R_j^\) and \(f(R)=j\), then \(x_j=0\) is such that \(x_{j}P_{j}f_{j}(R)\) and \(x_{i}R_{i}f_{i}(R)\) for all \(i\in N{\setminus }\{j\}\). However, \(R_j=R_j^\) and \(f(R)=j\) means that j is a discard agent, contradiction. (iii): \(R_j=R_j^+\) and \(f_j(R)=0\). Then \(x=j\) is such that \(x_{j}P_{j}f_{j}(R)\) and \(x_{i}R_{i}f_{i}(R)\) for all \(i\in N{\setminus }\{j\}\) if a) \(f(R)=0\) or b) \(f(R)=i\), where \(R_i=R_i^0\) or c) \(f(R)=i\), where \(R_i=R_i^\). a) \(R_j=R_j^+\) and \(f(R)=0\) contradicts nonwastefulness of f. b) \(R_j=R_j^+\) and \(f(R)=i\), where \(R_i=R_i^0\) contradicts nonforcefulness of f. Case c) \(f(R)=i\), where \(R_i=R_i^\) contradicts not containing a discard agent. \(\square \)
Lemma 2
If a rule f is strategyproof, nonbossy, full range, and nonforceful, f is Pareto optimal.
Proof
 (i)(wastefulness): For some \(i\in N\), there is \(R\in {\mathfrak {R}}\) such that \(R_{i}=R_{i}^{+}\) and \(f(R)=0\). Let \(R_{1}=R_{1}^{+}\), without loss of generality. Since f is full range, there exists \({\tilde{R}}\in \mathfrak {R}\) such that \( f({\tilde{R}})=1\). Then, consider the sequence of profiles given below:Since f is nonbossy, when \(R_{j}\) changes to \({\tilde{R}}_{j}\), either the outcome does not change or by strategyproofness, \({\tilde{R}}_{j}\ne R_{j}^{}\), \(R_{j}\ne R_{j}^{+}\) and \(f(R_{1},\ldots ,R_{j1},{\tilde{R}}_{j},\ldots , {\tilde{R}}_{n})=j\) for \(j\in \{1,\ldots ,n\}\). First, \({\tilde{R}}_{j}\ne R_{j}^{}\) because otherwise, j can manipulate by reporting \(R_{j}\) when his actual preference is \({\tilde{R}}_{j}\). Second, \(R_{j}\ne R_{j}^{+}\) because otherwise, j can manipulate by reporting \({\tilde{R}}_{j}\) when his actual preference is \(R_{j}\). In the last step, when \(R_{1}\) is replaced by \({\tilde{R}} _{1}\), we know that the outcome changes to \(f({\tilde{R}}_{1},{\tilde{R}}_{2} \ldots ,{\tilde{R}}_{n})=f({\tilde{R}})=1\). Then \({\tilde{R}} _{1}\) should be different than \(R_{1}\). Because \(R_{1}=R_{1}^{+}\), agent 1 can manipulate by reporting \({\tilde{R}} _{1}\) when his actual preference is \(R_{1}\). Thus, this contradicts strategyproofness of f.$$\begin{aligned} (R_{1},\ldots ,R_{n})\rightarrow & {} (R_{1},R_{2},\ldots ,\tilde{R}_{n})\rightarrow \cdots \rightarrow (R_{1},\ldots ,R_{j1},{\tilde{R}}_{j},\ldots ,{\tilde{R}}_{n})\rightarrow \cdots \\\rightarrow & {} (\tilde{R}_{1},\ldots ,{\tilde{R}}_{n}). \end{aligned}$$
 (ii)
(discard agent) Let agent 1 be a discard agent without loss of generality, i.e., there exists \({\tilde{R}}\in {\mathfrak {R}}\) such that \(f({\tilde{R}})=1\) and \( {\tilde{R}}_{1}=R_{1}^{}\). Since f is full range, there exists \(R\in {\mathfrak {R}}\) such that \(f(R)=0\). Then, by the argument above, we can establish that \({\tilde{R}}_{j}\ne R_{j}^{}\). Using the same reasoning, when \(R_{1}\) changes to \({\tilde{R}}_{1}\) in the last step, \({\tilde{R}}_{1}\ne R_{1}^{}\), which is a contradiction. \(\square \)
3 Characterizations of strategyproof rules
In this section, we characterize strategyproof rules that satisfy (i) Pareto optimality and nonbossiness, (ii) nondictatorship and nonbossiness, and (iii) Pareto optimality and nondictatorship. We start by defining new terms. First, an agentpreference pair \(\left( i,R_{i}\right) \) is a pair where agent \(i\in N\) has a preference \(R_{i}\in {\mathfrak {R}}_{i}\), and \(\left( i,R_{i}\right) \) is an element of \(\varPi =N\times {\mathfrak {R}}_{i}\), where \(\varPi \) is the set of agentpreference pairs. Moreover, \(\varPi _{0}=\varPi \cup \left\{ 0\right\} \), where \(\left\{ 0\right\} \) refers to the case in which the object is not assigned at all. Second, a rule f has a complete hierarchy if there exists an injective ranking function \(r:\varPi _{0}\rightarrow \mathbb {N}\) such that the following property holds: for all \(j\in N{\setminus } \left\{ i\right\} \) and \(R\in {\mathfrak {R}}\), if \(r\left( i,R_{i}\right) > r(j,R_{j})\) and \(r\left( i,R_{i}\right) > r\left( 0\right) \), then \(f\left( R\right) =i\); and if \(r\left( 0\right) > r\left( k,R_{k}\right) \) for all \(k\in N,\) then \(f\left( R\right) =0\). The intuition of the ranking function is as follows. An agent receives the object when the ranking of his agentpreference pair is higher than the ranking of other agents’ agentpreference pairs and it is higher than the ranking of the case where the object is not allocated at all. Moreover, if the ranking of the case where the object is not allocated at all is higher than the ranking of all agents’ agentpreference pairs, the object is not allocated at all. Third, we define hierarchical choice function in the context of the full preference domain as follows.
Definition 1
 (a)
For all \(i\in N,\) if \(R_{i}\vartriangleright {\tilde{R}}_{i}\) and \(r\left( i,R_{i}\right) > r\left( 0\right) \), there exists no \(j\in N{\setminus } \left\{ i\right\} \) such that \(r(i,{\tilde{R}}_{i})> r(j,R_{j})> r\left( i,R_{i}\right) \).
 (b)
For all \(i\in N,\) if \(r(i,{\tilde{R}}_{i})> r\left( 0\right) > r\left( i,R_{i}\right) ,\) then \({\tilde{R}}_{i}\vartriangleright R_{i}\).
We next present a remark that is used in the proof of Proposition 2.
Remark 1
 (a)
Let f be a nonbossy rule, \(i,j\in N\), and \(R\in {\mathfrak {R}} .\) If \(f\left( R\right) =i,\) then for all \({\tilde{R}}\in {\mathfrak {R}}\), \( f_{j}(R_{\left\{ i,j\right\} },{\tilde{R}}_{\left\{ i,j\right\} })=0\).
 (b)
Let f be a strategyproof and nonbossy rule, \(i,j\in N\), \(R_{i}\in {\mathfrak {R}}_{i}\), and \(R_{j}\in {\mathfrak {R}}_{j}\). Suppose that \(\left( i,R_{i}\right) \curlyeqsucc (j,{\tilde{R}}_{j})\). Then, for all \({\tilde{R}}_{i}\in {\mathfrak {R}}_{i}\) such that \({\tilde{R}}_{i}\vartriangleright R_{i}\) (if any), \((i,{\tilde{R}}_{i})\curlyeqsucc (j,{\tilde{R}}_{j})\). Moreover, for all \({\tilde{R}}_{j}\in {\mathfrak {R}}_{j}\) such that \(R_{j}\vartriangleright {\tilde{R}}_{j}\) (if any), \(\left( i,R_{i}\right) \curlyeqsucc (j,{\tilde{R}}_{j})\).
 (c)
Let f be a strategyproof and nonbossy rule. For \(i\in N\) and \(R\in {\mathfrak {R}}\), if \(f\left( R\right) =0\), then for all \({\tilde{R}} _{i}\in {\mathfrak {R}}_{i}\) such that \(R_{i}\vartriangleright {\tilde{R}}_{i}\) (if any), \(f({\tilde{R}}_{i},R_{i})=0\). Moreover, if \(f\left( R\right) =i\), then for all \({\tilde{R}}_{i}\in {\mathfrak {R}}_{i}\) such that \({\tilde{R}}_{i}\vartriangleright R_{i}\) (if any), \(f({\tilde{R}}_{i},R_{i})=i\).
The proof of Remark 1 is available upon request from the author. The following proposition characterizes strategyproof and nonbossy rules.
Proposition 2
A rule f is strategyproof and nonbossy if and only if f is a HCF.
Proof
 (a)
A HCF f with a ranking function r is strategyproof and nonbossy. Suppose to the contrary that there is an agent \(i\in N\) that can manipulate f at R. \(R_{i}\not =R_{i}^{0}\) because an indifferent agent cannot manipulate f. Then, there are two possible manipulations: either \(R_{i}=R_{i}^{+}\) or \(R_{i}=R_{i}^{}\). If \(R_{i}=R_{i}^{+}\) and \(f(R)=i\), since agent i achieves the best outcome, he cannot manipulate f. If \(R_{i}=R_{i}^{+}\), \(f_{i}\left( R\right) =0\), and \(r(i,R_{i}^{+})> r\left( 0\right) \), then there is an agent \(j\in N\ \)such that \(f(R)=j\). Then, by definition of complete hierarchy, \(r(j,R_{j})> r(i,R_{i}^{+})\). By the first condition of HCF, \(r(j,R_{j})> r(i,R_{i}^{})\) and \(r(j,R_{j})> r(i,R_{i}^{0})\). If \(r\left( 0\right) > r(i,R_{i}^{+})\), by the second condition of HCF, \(r(0)> r(i,R_{i}^{})\) and \(r\left( 0\right) > r(i,R_{i}^{0})\). Either case, agent i cannot receive the object by reporting a false preference. If \(R_{i}=R_{i}^{}\ \)and \(f_{i}(R)=0\), since this is the best outcome, he cannot manipulate f. If \(R_{i}=R_{i}^{}\ \)and \(f(R)=i\), for all \(j\in N\), \(r(i,R_{i}^{})> r(j,R_{j})\). Then, by the first condition of HCF, \(r(i,R_{i}^{0})> r(j,R_{j})\) and \(r(i,R_{i}^{+})> r(j,R_{j})\). Thus, agent i retains the object even if he reports his preference as \(R_{i}^{0}\) or \(R_{i}^{+}.\) This contradicts the manipulability of f. Thus, f is strategyproof. Suppose that f is bossy. Then, there exist \(i,j\in N\) such that agent i is bossy with agent j, i.e., there exists \(R\in \) \({\mathfrak {R}}\) such that \(f(R)=j\) , \(f_{j}(\widetilde{R}_{i},R_{i})=0\), and \(f_{i}(\widetilde{R} _{i},R_{i})=0\). \(f(R)=j\) implies that \(r(j,R_{j})> r\left( 0\right) \) and \(r(j,R_{j})> r\left( k,R_{k}\right) \) for all \(k\in N{\backslash } \left\{ j\right\} \). If \(f_{j}(\widetilde{R}_{i},R_{i})=0\), we have \(r(j,R_{j})> r\left( k,R_{k}\right) \) for all \(k\in N{\backslash } \left\{ i,j\right\} \) and we have \(r(i,{\tilde{R}}_{i})> r\left( j,R_{j}\right) \). Thus, we have \(r(i,{\tilde{R}}_{i})>r(j,R_{j})>r \left( 0\right) \) and \(r(i,{\tilde{R}}_{i})>r(j,R_{j})>r\left( k,R_{k}\right) \) for all \(k\in N{\backslash } \left\{ i\right\} \), and hence we have \(f(\widetilde{R}_{i},R_{i})=i\). However, \(f(\widetilde{R}_{i},R_{i})=i\) contradicts \(f_{i}(\widetilde{R}_{i},R_{i})=0\) (i.e., the bossiness of f), so f is nonbossy.
 (b)A strategyproof and nonbossy f is a HCF. We construct a ranking function r as follows:
 Initialization: start with \(\varPi =N\times {\mathfrak {R}}_{i}\) and \(\varPi _{0}=\varPi \cup \left\{ 0\right\} .\) LetInitially, \(R^{\max }=R^{+}\) and \(k=0.\) Rank the agentpreference pairs as follows:$$\begin{aligned} R_{i}^{\max }=\left\{ \text {for all }{\tilde{R}}_{i}\in {\mathfrak {R}} _{i}{\backslash } \left\{ R_{i}\right\} , R_{i}\in {\mathfrak {R}}_{{\mathfrak {i}}}\mid R_{i}\vartriangleright {\tilde{R}}_{i}\right\} \text { and }R^{\max }=\left( R_{i}^{\max }\right) _{i=1}^{n}\text {.} \end{aligned}$$

Step 1: for all \(R\in {\mathfrak {R}}\), if \(f(R^{\max })=0\), \(f\left( R\right) =0.\) Let \(r\left( 0\right) =3n+1\,k\), and rank the remaining agentpreference pairs in \(\varPi _{0}\) arbitrarily from 1 to \( 3nk\) and stop. Otherwise, go to step 2.

Step 2: for some agent \(i\in N\), if \(f(R^{\max })=i\), let \(r\left( i,R_{i}^{\max }\right) =3n+1k\) and \(k\leftarrow k+1.\) Let \({\mathfrak {R}} _{i}\leftarrow {\mathfrak {R}}_{i}\diagdown \left\{ R_{i}\right\} \) and \(\varPi _{0}\leftarrow \varPi _{0}{\backslash } \left\{ \left( i,R_{i}^{\max }\right) \right\} \). If \({\mathfrak {R}}_{i}=\emptyset ,\) let \(r\left( 0\right) =3n+1\,k \), and rank the remaining agentpreference pairs in \(\varPi _{0}\) arbitrarily from 1 to \(3nk\) and stop. Otherwise, let \(\mathfrak { R\leftarrow }\times _{i\in N}{\mathfrak {R}}_{i}\), recalculate \(R^{\max }\) and return to step 1.

(ii) The ranking function r first ranks \((i,R_{i}^{\max })\) before other possible agentpreference pairs of \(i\in N\) and the algorithm that generates the ranking function r guarantees the second condition of HCF. Moreover, since \((i,R_{i}^{\max })\) is ranked before other agentpreference pairs with smaller preferences of \(i\in N\), the first condition of HCF is guaranteed. Note that the first condition does not say anything about the agentpreference pairs with rank lower than \(r\left( 0\right) ,\) so ranking such pairs arbitrarily does not violate it. Thus, f is a HCF with the ranking function r. \(\square \)
This result is important as it not only characterizes the strategyproof and nonbossy rules but also helps characterize strategyproof rules that are Pareto optimal and nonbossy and that are nondictatorial and nonbossy.
3.1 Pareto optimal and nonbossy rules
In this section, we characterize strategyproof, Pareto optimal, and nonbossy rules; and we start by defining new terms. An agent i is the top agent with respect to f if \(f(R^{+})=i\), i.e., the top agent receives the object when all agents prefer to receive it. We next define two special cases of a HCF as follows.
Definition 2
 (i)
A rule f is a Serial Dictatorship if it has a complete hierarchy with \(r:\varPi _{0}\rightarrow \mathbb {N}\) that satisfies (a) for all \(i\in N\), \(r(i,R_{i}^{+})> r\left( 0\right) > r(i,R_{i}^{})\) and (b) for all \(i\in N\) and \(j\in N{\backslash } \left\{ i\right\} \), \(r(i,R_{i}^{+})> r(j,R_{j}^{0})\).
 (ii)
A Constrained HCF is a HCF with ranking function \(r:\varPi _{0}\rightarrow \mathbb {N}\) that satisfies one of the two conditions: (a) there is no top agent or (b) there is a top agent \(i \in N\), but there exists \(R\in {\mathfrak {R}}\) such that \(r(i,R_{i}^{})> r\left( 0\right) \) and \(r(i,R_{i}^{})> r(j,R_{j})\) for all \(j\in N{\backslash } \left\{ i\right\} \).
Definition 2(i) means that a Serial Dictatorship has a ranking function r and it requires that the rule is nonwasteful and nonforceful with no discard agent. Definition 2(ii) means that a Constrained HCF has a ranking function r and it requires that either there is no top agent or there is a top agent but he is also a discard agent. The following proposition characterizes Pareto optimal and nonbossy rules.
Proposition 3
A rule is Pareto optimal and nonbossy if and only if it is a Serial Dictatorship.
Proof
Let f be a Serial Dictatorship. Since a Serial Dictatorship is a special case of a HCF, f is a HCF. By Proposition 2, f is strategyproof and nonbossy. Definition 2(i) guarantees that f is nonwasteful and nonforceful with no discard agent. Then, by Lemma 1, f is Pareto optimal.
Let f be a Pareto optimal and nonbossy rule. As Proposition 1 shows, Pareto optimality implies strategyproofness. Then, f is strategyproof and nonbossy, so it is a HCF by Proposition 2. Then, it has a complete hierarchy with a ranking function \(r:\varPi _{0}\rightarrow \mathbb {N}\). By Lemma 1, Pareto optimality implies that f is nonwasteful and nonforceful with no discard agent. Being nonwasteful with no discard agent is guaranteed by part (a) of Definition 2(i), and being nonforceful is guaranteed by part (b) of Definition 2(i).\(\square \)
We next provide an example for Pareto optimal and nonbossy rules as follows.
Example 2
Let the rule f be such that \(f(R)=\begin{Bmatrix} \min \left\{ i\in NR_{i}=R_{i}^{+}\right\}&\text {if }R\ne R^{\lnot }\\ 0&\text {if }R=R^{\lnot } \end{Bmatrix}\) for all \(R\in {\mathfrak {R}}.\) The object is not awarded to agents who are indifferent and who prefer not to receive it. Then, f is nonforceful and contains no discard agent. Moreover, if there is an agent who prefers to receive the object, the object does not remain unassigned, so f is nonwasteful. Thus, f is Pareto optimal by Lemma 1. When an agent changes his preference, either his own outcome changes, or his outcome stays the same but other agents’ outcomes also stay the same. Hence, f is nonbossy. The first agent obtains the object whenever he prefers to do so, and he does not obtain it whenever he prefers not to, and hence the first agent is a dictator. Thus, f is dictatorial.
The following result is a corollary to Proposition 3 along with Lemma 2.
Corollary 1
A rule is strategyproof, nonbossy, full range, and nonforceful if and only if it is a Serial Dictatorship.
3.2 Nondictatorial and nonbossy rules
In this section, we characterize strategyproof, nondictatorial, and nonbossy rules as follows.
Proposition 4
A rule f is strategyproof, nondictatorial, and nonbossy if and only if it is a Constrained HCF.
Proof
Let f be a Constrained HCF. Since f is a special case of HCF, it is strategyproof and nonbossy by Proposition 2. Moreover, f satisfies one of the following requirements of being a Constrained HCF: (i) there is no top agent or (ii) there is a top agent i, but there exists \(R\in {\mathfrak {R}}\) such that \(r(i,R_{i}^{})>r\left( 0\right) \) and \(r(i,R_{i}^{})>r(j,R_{j})\) for all \(j\in N{\backslash } \left\{ i\right\} \). If the first requirement holds, then there is no agent who receives the object whenever he prefers to do so because \(f(R^+)=0\). If the second requirement holds, there is a top agent i, who is the only candidate for being a dictator, but he is a discard agent, so he cannot be a dictator. Thus, f is nondictatorial.
Let f be a strategyproof, nondictatorial, and nonbossy rule. By Proposition 2, f is a HCF with a ranking function \(r:\varPi _{0}\rightarrow \mathbb {N}\). Since f is nondictatorial, either there is no agent who receives the object whenever he prefers to do so, or such an agent exists but he receives the object even if he does not want to. If the former case occurs, for all \(i\in N\), there exists \(R\in {\mathfrak {R}}\) such that \(r\left( 0\right) >r\left( i,R_{i}^+\right) \) or \(r\left( j,R_j \right) >r\left( i,R_{i}^+\right) \) for some \(j \in N {\setminus } \{i\}\). Suppose to the contrary that there is a top agent i, i.e., \(f(R^+)=i\). Then, in any case, \(f_i(R_i^+,R_{i})=0.\) However, this means that there exists an agent \(k \in N {\setminus } \{i\}\) such that when agent k changes his preference from \(R_{k}^+\) to \(R_{k}\), agent i loses the object although he prefers to receive it. If k is the one that receives the object, f is not strategyproof. If k does not receive the object either, f is bossy. Thus, there is a contradiction. Hence, there cannot exist any top agent. This condition satisfies the first requirement of a Constrained HCF. Now suppose that there is an agent who receives the object whenever he prefers to do so but he receives the object even if he does not want to. Then, there exists \(R\in {\mathfrak {R}}\) such that \(r(i,R_{i}^{})>r\left( 0\right) \) and \(r(i,R_{i}^{})>r(j,R_{j})\) for all \(j\in N{\backslash } \left\{ i\right\} \). This condition satisfies the second requirement of a Constrained HCF. Thus, f is a Constrained HCF. \(\square \)
We next provide an example for strategyproof, nondictatorial, and nonbossy rules as follows.
Example 3
Let \(f(R)=0\) for all \(R\in {\mathfrak {R}}\) and \(R_i=R_i^+\). The object is not assigned, so no agent can manipulate f, and hence f is strategyproof. The object is not awarded at any profile, so there is no agent who obtains the object whenever he prefers to do so. Thus, f is nondictatorial. Since the object is not assigned at all, there is no agent who can change the outcome of another agent, and hence f is nonbossy. Because f does not assign the object even though agent i prefers to receive it, f is wasteful, so f is not Pareto optimal by Lemma 1.
3.3 Pareto optimal and nondictatorial rules
In this section, we characterize strategyproof, Pareto optimal, and nondictatorial rules, and start by defining new terms in the context of the full preference domain. First, a rule is constrained bossy if there exist two agents \(i,j\in N\) and preference profiles \(R,\widetilde{R}\in {\mathfrak {R}}\) such that \(R_{j}=R_{j}^{+}\), \(f(R)=j\), and \(f_{i}(\widetilde{R}_{i},R_{i})=f_{j}( \widetilde{R}_{i},R_{i})=0.\) In this case, agent i is constrained bossy with agent j. Second, recall that if \(f(R^{+})=i\), agent i is the top agent with respect to f. Then, if there is an agent who is constrained bossy with the top agent, this rule is a topbossy rule. Third, we define TopBossy Choice Function as follows.
Definition 3
If f is a topbossy rule in which there is no discard agent, and f is nonforceful and nonwasteful, then f is a TopBossy Choice Function.
The following proposition characterizes Pareto optimal and nondictatorial rules.
Proposition 5
A rule f is Pareto optimal and nondictatorial if and only if it is a TopBossy Choice Function.
Proof
Let f be a TopBossy Choice Function. By definition, f is nonforceful, nonwasteful, and contains no discard agent. Then f is Pareto optimal by Lemma 1. Suppose to the contrary that f is dictatorial. Then, there exists a dictator agent \(j\in N\) who receives the object whenever he prefers to do so, i.e., the top agent. Since f is topbossy, there is an agent \(i\in N\) such that i is constrained bossy with j, i.e., there exist \(R\in {\mathfrak {R}}\) and \(\widetilde{R}_{i}\in {\mathfrak {R}}_{i}\) such that \(R_{j}=R_{j}^{+}\), \(f(R)=j\), and \(f_{i}(\widetilde{R}_{i},R_{i})=f_{j}(\widetilde{R}_{i},R_{i})=0\). Since \(R_{j}=R_{j}^{+}\) and \(f_{j}(\widetilde{R}_{i},R_{i})=0\), agent j cannot obtain the object although he prefers to do so. Thus, agent j is not a dictator, contradiction.
Let f be Pareto optimal and nondictatorial. Pareto optimality of f implies that f is nonforceful, nonwasteful, and contains no discard agent by Lemma 1. Suppose to the contrary that f is not topbossy. By Lemma 1, Pareto optimality implies that the object is assigned at \(R^{+}\). Thus, there exists a top agent \(i\in N\) such that \(f(R^{+})=i\). Pareto optimality also implies strategyproofness of f by Proposition 1. Since f is strategyproof and not topbossy, \(f(R_{i}^{+},R_{i})=i\). Also, by Lemma 1, Pareto optimality requires that there is no discard agent, i.e., \(f_{i}(R_{i}^{},R_{i})=0\). Then, agent i receives the object whenever he prefers to do so, and does not receive it whenever he prefers not to. Thus, agent i is a dictator, contradiction.\(\square \)
We next provide an example for Pareto optimal and nondictatorial rules.
Example 4
As a result of Propositions 3, 4, and 5, we obtain the following impossibility result.
Proposition 6
No strategyproof rule satisfies Pareto optimality, nondictatorship, and nonbossiness simultaneously.
Proposition 6 is a direct result of Proposition 3 (which shows that a strategyproof, Pareto optimal, and nonbossy rule is dictatorial), Proposition 4 (which shows that a strategyproof, nondictatorial, and nonbossy rule is not Pareto optimal), and Proposition 5 (which shows that a strategyproof, Pareto optimal, and nondictatorial rule is (constrained) bossy).
Lemma 3 shows that the impossibility result is tight except when there are only two agents, Pareto optimality implies dictatorship.
Lemma 3
If \(n=2\) and a rule f is Pareto optimal, then f is dictatorial.
Proof
Suppose that \(n=2\) and f is Pareto optimal. Suppose to the contrary that f is nondictatorial. By Lemma 1, Pareto optimality implies that f is nonwasteful and nonforceful with no discard agent. Because of nonwastefulness, \(f(R^{+})>0\), and suppose that \(f(R^{+})=1\) without loss of generality. Since there is no dictator, there must be a case such that \(f(R_{1}^{+},R_{2}^{\lnot })\ne 1\). Then, either \(f(R_{1}^{+},R_{2}^{\lnot })=0\) or \(f(R_{1}^{+},R_{2}^{\lnot })=2\) holds. However, \(f(R_{1}^{+},R_{2}^{\lnot })=0\) contradicts nonwastefulness. If \(f(R_{1}^{+},R_{2}^{\lnot })=2\), then either \(f(R_{1}^{+},R_{2}^{})=2\) or \(f(R_{1}^{+},R_{2}^{0})=2\). If \(f(R_{1}^{+},R_{2}^{})=2\), agent 2 is a discard agent; and if \(f(R_{1}^{+},R_{2}^{0})=2\), f is forceful. Thus, all three cases contradict the Pareto optimality of f. Therefore, when there are two agents, Pareto optimality implies dictatorship.\(\square \)
4 Related literature
This paper belongs to the literature on the allocation problem of an indivisible object. This problem goes back to Glazer and Ma (1989) who consider allocating a prize to the agent who values it most without monetary transfers, and Perry and Reny (1999) and Olszewski (2003) generalize the results of Glazer and Ma (1989). Pápai (2001) considers the allocation problem of an indivisible object without monetary transfers, and shows the impossibility of finding a strategyproof rule that satisfies Pareto optimality, nondictatorship, and nonbossiness simultaneously. While Pápai (2001) restricts attention to the strict preference domain, we allow agents to be indifferent, and generalize the results of Pápai (2001) to the full preference domain.
The allocation problem of an indivisible object has been studied by allowing for monetary transfers as well, see for example, Tadenuma and Thomson (1993, 1995) and Ohseto (1999). Moreover, Ohseto (2000) proves the impossibility of finding a strategyproof and Pareto optimal rule, Fujinaka and Sakai (2009) analyze whether possible manipulations (i.e., the absence of strategyproofness) can have a serious impact on the outcome of agents, and in a recent study, Athanasiou (2013) characterizes strategyproof rules satisfying other properties. This paper is also related to the literature on the allocation problem of multiple indivisible objects without monetary transfers (e.g., Pápai 2000; Ehlers and Klaus 2007, and Kesten and Yazici 2012) and to the literature on the impartial allocation of a prize (e.g., Holzman and Moulin 2013).
Although agents are often indifferent in practice, most of the prior literature restricts attention to the strict preference domain for simplicity, except for several studies that consider the full preference domain. For instance, Bogomolnaia et al. (2005) show that all efficient allocations of indivisible objects can be generated using serially dictatorial rules; Katta and Sethuraman (2006) prove that strategyproofness is incompatible with efficiency on the full preference domain; Larsson and Svensson (2006) generalize strategyproof voting rules to the full preference domain; Yilmaz (2009) characterizes individually rational, efficient, and fair rules on the full preference domain; AlcaldeUnzu and Molis (2011) and Jaramillo and Manjunath (2012) generalize top trading cycles mechanism to the full preference domain; and Athanassoglou and Sethuraman (2011) prove that individual rationality, efficiency, and strategyproofness are incompatible on the full preference domain.
Footnotes
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