Object reallocation problems under single-peaked preferences: two characterizations of the crawler

Abstract

In object reallocation problems, if preferences are strict but otherwise unrestricted, TTC is the leading rule: It is the only rule satisfying efficiency, the endowment lower bound, and strategy-proofness. However, on the subdomain of single-peaked preferences, Bade (J Econ Theory 180:81–99, 2019) defines a new rule, the “crawler”, which also satisfies these properties, and in fact enjoys a stronger strategic property. We identify additional interesting properties that the crawler satisfies, and provide two characterizations of this rule. The first characterization is based on the endowment lower bound and two invariance properties, “adjacent-endowment-swapping invariance” and “separability”. The second characterization is based on the endowment lower bound, strategy-proofness, adjacent-endowment-swapping invariance, and another invariance property, “non-bossiness”.

Appendices

Appendices

Let $N \in {\mathcal {N}}$ and $\bigcup _{i \in N} \omega _i \equiv O \in {\mathcal {O}}$. Unless explicitly specified, without loss of generality, suppose that $N = \{1,\ldots ,n\}$ and $O = \{o_1,\ldots ,o_n\}$.

Proofs for Sect. 3

1.1 Proof of Proposition 1

Let $\varphi$ satisfy efficiency, the endowment lower bound, and strategy-proofness.

Let ${\mathcal {D}}^N \subset {\mathcal {P}}^N$ and $o_m,o_{m+1} \in O$ be such that for each $P \in {\mathcal {D}}^N$ and each $i~\in ~N$, $X_i(O) \in \{o_m,o_{m+1}\}$. Without loss of generality, let $\omega \in {\mathcal {X}}^N$ be such that $\omega ~=~(o_1,\ldots ,o_n)$. We first have the following fact:

Fact 1

For each $P \in {\mathcal {D}}^N$,

$$\begin{aligned} TTC(P,\omega ) = CR(P,\omega ). \end{aligned}$$

The proof is by induction on n.

Base step Suppose that $n = 2$. There are four possible preference profiles.

$$\begin{aligned} \begin{array}{llll} P_1:o_1,~o_2 &{}\quad P'_1:o_2,~o_1&{}\quad {\widehat{P}}_1:o_1,~o_2&{}\quad {\tilde{P}}_1:o_2,~o_1 \\ P_2:o_1,~o_2 &{}\quad P'_2:o_1,~o_2&{}\quad {\widehat{P}}_2:o_2,~o_1&{}\quad {\tilde{P}}_2:o_2,~o_1 \end{array} \end{aligned}$$

Because $\varphi$ satisfies efficiency and the endowment lower bound,

$$\begin{aligned} \begin{array}{ll} \varphi (P,\omega ) = TTC(P,\omega ) = (o_1,o_2)&{}\quad \varphi (P',\omega ) = TTC(P',\omega ) = (o_2,o_1)\\ \varphi ({\widehat{P}},\omega ) = TTC({\widehat{P}},\omega ) = (o_1,o_2) &{}\quad \varphi ({\tilde{P}},\omega ) = TTC({\tilde{P}},\omega ) = (o_1,o_2). \end{array} \end{aligned}$$

Induction hypothesis Let $k \ge 3$. Suppose that for each $n \le k - 1$, $\varphi = TTC$.

Induction step Suppose that $n = k$. Let $P \in {\mathcal {D}}^N$. Suppose that either (i) $X_m(O) = o_m$ or (ii) $X_{m+1}(O) = o_{m+1}$. Without loss of generality, suppose that (i) holds. Because $\varphi$ satisfies the endowment lower bound, $\varphi _m(P,\omega ) = o_m$. Then by the induction hypothesis together with Fact 1,

$$\begin{aligned} \varphi (P,\omega ) = CR(P,\omega ) = TTC(P,\omega ). \end{aligned}$$

Suppose that $X_m(O) = o_{m+1}$ and $X_{m+1}(O) = o_m$.

Let $P'_m,P'_{m+1} \in {\mathcal {D}}$ be such that

$$\begin{aligned}&P'_m: o_{m+1}, o_m, \ldots \\&P'_{m+1}: o_m, o_{m+1}, \ldots . \end{aligned}$$

By the endowment lower bound,

$$\begin{aligned}&\varphi _m\left( P'_{\{m,m+1\}},P_{-\{m,m+1\}},\omega \right) \in \{o_m,o_{m+1}\}\\&\quad \text {and}~\varphi _{m+1}\left( P'_{\{m,m+1\}},P_{-\{m,m+1\}},\omega \right) \in \{o_m,o_{m+1}\}. \end{aligned}$$

By efficiency,

$$\begin{aligned} \varphi _m\left( P'_{\{m,m+1\}},P_{-\{m,m+1\}},\omega \right) = o_{m+1} ~\text {and}~\varphi _{m+1}\left( P'_{\{m,m+1\}},P_{-\{m,m+1\}},\omega \right) = o_m. \end{aligned}$$

By strategy-proofness,

$$\begin{aligned} \varphi _m\left( P,\omega \right) = o_{m+1} ~\text {and}~\varphi _{m+1}\left( P,\omega \right) = o_m. \end{aligned}$$

Suppose that agents m and $m+1$ leave with their assignments. Then there are $n~-~2$ agents and each remaining agent has single-peaked preferences over the remaining objects. Moreover, there are two adjacent objects in $O \setminus \{o_m,o_{m+1}\}$ such that the most preferred object for each remaining agent in the remaining objects is one of them. Then by the induction hypothesis,

$$\begin{aligned} \varphi (P,\omega ) = TTC(P,\omega ). \end{aligned}$$

Finally by Fact 1,

$$\begin{aligned} \varphi (P,\omega ) = CR(P,\omega ) = TTC(P,\omega ). \end{aligned}$$

1.2 Proof of Lemma 2

Let $(P,\omega ) \in {\mathcal {P}}^N \times {\mathcal {X}}^N$, $i \in N$, and $P'_i \in {\mathcal {P}}$ be such that $CR_i(P,\omega ) = CR_i(P'_i,P_{-i},\omega )$. Then for each pair $j,k \in N \backslash \{i\}$, agent j receives his assignment and leaves with it earlier than agent k when the crawler is applied to $(P,\omega )$ if and only if they leave with same order when the crawler is applied to $(P'_i,P_{-i},\omega )$. Hence, $CR(P,\omega ) = CR(P'_i,P_{-i},\omega )$.

1.3 Proof of Proposition 2

By definition, group strategy-proofness implies both strategy-proofness and non-bossiness. We show the converse. Let $(P,\omega ) \in {\mathcal {P}}^N \in {\mathcal {X}}^N$, $N' \equiv \{i_1,\ldots ,i_{n'}\} \subseteq N$, and $P'_{N'} \in {\mathcal {P}}^{N'}$ be such that for each $i \in N'$,

$$\begin{aligned} \varphi _i(P'_{N'},P_{-N'},\omega )~R_i~\varphi _i(P_{N'},P_{-N'},\omega ). \end{aligned}$$

For each $i \in N'$, let ${\widehat{P}}_i \in {\mathcal {P}}$ be such that

$$\begin{aligned} {\widehat{P}}_i:\varphi _i(P'_{N'},P_{-N'},\omega ),\ldots ,~\text {and}~U({\widehat{P}}_i,\varphi _i(P,\omega )) \subseteq U(P_i,\varphi _i(P,\omega )). \end{aligned}$$

(1)

For each $i \in N$, because $\varphi _i(P'_{N'},P_{-N'},\omega )~R_i~\varphi _i(P_{N'},P_{-N'},\omega )$, we have $\varphi _i(P'_{N'},P_{-N},\omega ) \in U(P_i,\varphi _i(P,\omega ))$. Hence, preferences that satisfy (1) exist (Fig. 2).

Fig. 2

In Fig. 2a, b, $U({\widehat{P}}_i,\varphi _i(P,\omega )) \subseteq U(P_i,\varphi _i(P,\omega ))$. On the other hand, in Fig. 2c, $U({\widehat{P}}_i,\varphi (P,\omega )) \supset U(P_i,\varphi _i(P,\omega ))$

First, we show that $\varphi ({\widehat{P}}_{N'},P_{-N'},\omega ) = \varphi (P_{N'},P_{-N'},\omega )$.

Begin with P and suppose that agent $i_1$’s preferences change to ${\widehat{P}}_{i_1}$. By strategy-proofness,

$$\begin{aligned}&\varphi _{i_1}(P_{i_1},P_{-i_1},\omega )~R_i~\varphi _{i_1} ({\widehat{P}}_{i_1},P_{-i_1},\omega ) \end{aligned}$$

(2)

$$\begin{aligned}&\varphi _{i_1}({\widehat{P}}_{i_1},P_{-i_1},\omega ) ~{\widehat{R}}_{i_1}~\varphi _{i_1}(P_{i_1,},P_{-i_1},\omega ). \end{aligned}$$

(3)

Then (2) implies that

$$\begin{aligned} \varphi _{i_1}({\widehat{P}}_{i_1},P_{-i_1},\omega ) \notin U^*(P_i,\varphi _{i_1}(P,\omega )). \end{aligned}$$

On the other hand, (3) implies that

$$\begin{aligned} \varphi _{i_1}({\widehat{P}}_{i_1},P_{-i_1},\omega ) \in U({\widehat{P}},\varphi _{i_1}(P,\omega )). \end{aligned}$$

Because

$$\begin{aligned} U({\widehat{P}}_{i_1},\varphi _{i_1}(P,\omega )) \subseteq U(P_{i_1},\varphi _{i_1}(P,\omega )), \end{aligned}$$

we have

$$\begin{aligned} \varphi _{i_1}({\widehat{P}}_{i_1},P_{-i_1},\omega ) = \varphi _{i_1}(P,\omega ). \end{aligned}$$

By non-bossiness,

$$\begin{aligned} \varphi ({\widehat{P}}_{i_1},P_{-i_1},\omega ) = \varphi (P,\omega ). \end{aligned}$$

Consider $({\widehat{P}}_{i_1},P_{-i_1})$ and suppose that agent $i_2$’s preferences change to ${\widehat{P}}_{i_2}$. By strategy-proofness,

$$\begin{aligned}&\varphi _{i_2}({\widehat{P}}_{i_1},P_{i_2},P_{-\{i_1,i_2\}},\omega )~ R_{i_2}~\varphi _{i_2}({\widehat{P}}_{i_1},{\widehat{P} }_{i_2},P_{-\{i_1,i_2\}},\omega )~ \end{aligned}$$

(4)

$$\begin{aligned}&\varphi _{i_2}({\widehat{P}}_{i_1},{\widehat{P}}_{i_2},P_{-\{i_1,i_2\}},\omega )~ {\widehat{R}}_{i_2}~ \varphi _{i_2}({\widehat{P}}_{i_1},P_{i_2}, P_{-\{i_1,i_2\}},\omega ). \end{aligned}$$

(5)

Then (4) implies that

$$\begin{aligned} \varphi _{i_2}({\widehat{P}}_{i_1},{\widehat{P}}_{i_2},P_{-\{i_1,i_2\}}, \omega ) \notin U^*(P_{i_2},\varphi _{i_2}({\widehat{P}}_{i_1},P_{i_2}, P_{-\{i_1,i_2\}},\omega )) = U^*(P_{i_2},\varphi _{i_2}(P,\omega )). \end{aligned}$$

On the other hand, (5) implies that

$$\begin{aligned} \varphi _{i_2}({\widehat{P}}_{i_1},{\widehat{P}}_{i_2}, P_{-\{i_1,i_2\}},\omega ) \in U({\widehat{P}}_{i_2}, \varphi _{i_2}({\widehat{P}}_{i_1},P_{i_2},P_{-\{i_1,i_2\}},\omega )) = U({\widehat{P}}_{i_2},\varphi _{i_2}(P,\omega )). \end{aligned}$$

Because

$$\begin{aligned} U({\widehat{P}}_{i_2},\varphi _{i_2}(P,\omega )) \subseteq U(P_{i_2},\varphi _{i_2}(P,\omega )), \end{aligned}$$

we have

$$\begin{aligned} \varphi _{i_2}({\widehat{P}}_{\{i_1,i_2\}},P_{-\{i_1,i_2\}},\omega ) = \varphi _{i_2}(P,\omega ). \end{aligned}$$

By non-bossiness,

$$\begin{aligned} \varphi ({\widehat{P}}_{\{i_1,i_2\}},P_{-\{i_1,i_2\}},\omega ) = \varphi (P,\omega ). \end{aligned}$$

Repeating this argument for each $i \in \{i_3,\ldots ,i_{n'}\}$, we have

$$\begin{aligned} \varphi ({\widehat{P}}_{N'},P_{-N'},\omega ) = \varphi (P_{N'},P_{-N'},\omega ). \end{aligned}$$

Second, we show that $\varphi (P'_{N'},P_{-N'},\omega ) = \varphi ({\widehat{P}}_{N'},P_{-N'},\omega )$.

Begin with $(P'_{N'},P_{-N'})$ and suppose that agent $i_1$’s preferences change to ${\widehat{P}}_{i_1}$. Because for each $o \in O$,

$$\begin{aligned} \varphi _{i_1}(P'_{N'},P_{-N'},\omega )~{\widehat{R}}_{i_1}~o, \end{aligned}$$

by strategy-proofness,

$$\begin{aligned} \varphi _{i_1}({\widehat{P}}_{i_1},P'_{N'\backslash \{i_1\}},P_{-N'},\omega ) = \varphi _{i_1}(P'_{N'},P_{-N'},\omega ). \end{aligned}$$

By non-bossiness,

$$\begin{aligned} \varphi ({\widehat{P}}_{i_1},P'_{N'\backslash \{i_1\}},P_{-N'},\omega ) = \varphi (P'_{N'},P_{-N'},\omega ). \end{aligned}$$

Consider $({\widehat{P}}_{i_1},P'_{N' \backslash \{i_1\}},P_{-N'})$ and suppose that agent $i_2$’s preferences change to ${\widehat{P}}_{i_2}$. Because for each $o \in O$,

$$\begin{aligned} \varphi _{i_2}({\widehat{P}}_{i_1},P'_{N'\backslash \{i_1\}},P_{-N'},\omega ) = \varphi _{i_2}(P'_{N'},P_{-N'},\omega )~{\widehat{R}}_{i_2}~o, \end{aligned}$$

by strategy-proofness,

$$\begin{aligned} \varphi _{i_2}({\widehat{P}}_{\{i_1,i_2\}},P'_{N' \backslash \{i_1,i_2\}},P_{-N'},\omega ) = \varphi _{i_2}({\widehat{P}}_{i_1},P'_{N'\backslash \{i_1\}},P_{-N'},\omega ) = \varphi _{i_2}(P'_{N'},P_{-N'},\omega ). \end{aligned}$$

By non-bossiness,

$$\begin{aligned} \varphi ({\widehat{P}}_{\{i_1,i_2\}},P'_{N' \backslash \{i_1,i_2\}},P_{-N'},\omega ) = \varphi ({\widehat{P}}_{i_1},P'_{N'\backslash \{i_1\}},P_{-N'},\omega ) = \varphi (P'_{N'},P_{-N'},\omega ). \end{aligned}$$

Repeating this argument for each $i \in \{i_3,\ldots ,i_{n'}\}$, we have

$$\begin{aligned} \varphi ({\widehat{P}}_{N'},P_{-N'},\omega ) = \varphi (P'_{N'},P_{-N},\omega ). \end{aligned}$$

Overall,

$$\begin{aligned} \varphi (P'_{N'},P_{-N'},\omega ) = \varphi ({\widehat{P}}_{N'},P_{-N'},\omega ) = \varphi (P,\omega ). \end{aligned}$$

Thus, $\varphi$ is group strategy-proof.

1.4 Proof of Lemma 3

Let $(P,\omega ) \in {\mathcal {P}}^N \times {\mathcal {X}}^N$ and $i,i+1 \in N$ be such that $\omega _i \prec p(P_i)$ and $p(P_{i+1}) \prec \omega _{i+1}$. Suppose that agent $i+1$ receives his assignment $CR_{i+1}(P,\omega )$ at Step t when the crawler is applied to $(P,\omega )$. Then by definition of the crawler, the ownership of agent i is shifted to $\omega _{i+1}$ at Step t.

Suppose that agents i and $i+1$ swap their endowments. First, until Step t, the underlying procedure remains the same as above when the crawler is applied to $(P,\omega ^{i,i+1})$. Hence, $CR_{i+1}(P,\omega ^{i,i+1}) = CR_{i+1}(P,\omega )$. Moreover, after agent $i+1$ receives his assignment, the problem becomes the same as the problem of Step $t+1$ when the crawler is applied to $(P,\omega )$.

1.5 Proof of Theorem 1

We have seen that the crawler satisfies the endowment lower bound and adjacent endowment-swapping invariance. Moreover, it is clear that it satisfies separability.

Let $\varphi$ be a rule satisfying the properties listed in the theorem. Let $\omega \in {\mathcal {X}}^N$. Without loss of generality, suppose that $\omega = (o_1,o_2,\ldots ,o_n)$. We prove by induction on n that $\varphi$ is the crawler.

Base step Suppose that $n = 2$. There are four possible preference profiles.

$$\begin{aligned} \begin{array}{llll} P_1:o_1,~o_2 &{}\quad P'_1:o_2,~o_1&{}\quad {\widehat{P}}_1:o_1,~o_2&{}\quad {\tilde{P}}_1:o_2,~o_1 \\ P_2:o_1,~o_2 &{}\quad P'_2:o_1,~o_2&{}\quad {\widehat{P}}_2:o_2,~o_1&{}\quad {\tilde{P}}_2:o_2,~o_1 \end{array} \end{aligned}$$

Note that

$$\begin{aligned}&CR(P,\omega ) = (o_1,o_2),~CR(P',\omega ) = (o_2,o_1),~CR({\widehat{P}},\omega ) = (o_1,o_2),\\&\quad \text {and}~CR({\tilde{P}},\omega ) = (o_1,o_2). \end{aligned}$$

By the endowment lower bound,

$$\begin{aligned}&\varphi (P,\omega ) = (o_1,o_2),~\varphi (P',\omega ) \in \{(o_1,o_2),(o_2,o_1)\},~ \varphi ({\widehat{P}},\omega ) = (o_1,o_2),\\&\quad \text {and}~\varphi ({\tilde{P}},\omega ) = (o_1,o_2). \end{aligned}$$

Consider $(P',\omega )$. Suppose that agents 1 and 2 swap their endowments. Then by the endowment lower bound,

$$\begin{aligned} \varphi (P',\omega ^{1,2}) = (o_2,o_1). \end{aligned}$$

Hence, by adjacent endowment-swapping invariance,

$$\begin{aligned} \varphi (P,\omega ) = (o_2,o_1). \end{aligned}$$

Therefore, $\varphi = CR$.

Induction hypothesis Let $k \ge 3$. Suppose that for each $n \le k-1$, $\varphi = CR$.

Induction step Suppose that $n = k$. Let $P \in {\mathcal {P}}^N$. Let $i \in N$ and $P^0_i \in {\mathcal {P}}(\omega _i)$. Suppose that $P_i = P^0_{i}$, i.e., his most preferred object is his endowment. By the endowment lower bound,

$$\begin{aligned} \varphi _i(P^0_i,P_{-i},\omega ) = \omega _i. \end{aligned}$$

By separability,

$$\begin{aligned} \varphi (r_{N \backslash \{i\}}(P^0_i,P_{-i},\omega )) = \varphi _{N \backslash \{i\}}(P_i^0,P_{-i},\omega ). \end{aligned}$$

(6)

There are $k-1$ agents in $r_{N \backslash \{i\}}(P,\omega )$. Hence, by the induction hypothesis,

$$\begin{aligned} \varphi (r_{N \backslash \{i\}}(P^0_i,P_{-i},\omega )) = CR(r_{N \backslash \{i\}}(P^0_i,P_{-i},\omega )). \end{aligned}$$

Moreover, because the crawler meets the endowment lower bound,

$$\begin{aligned} \varphi (P^0_i,P_{-i},\omega ) = CR(P^0_i,P_{-i},\omega ). \end{aligned}$$

(7)

Let $i^* \in N$ be the first agent who is assigned an object when the crawler is applied to $(P,\omega )$. Let $N' \equiv \{i_1,i_2,\ldots ,i_m\} \subseteq N$, where $i_m \equiv i^*$, be such that $CR_{i^*}(P,\omega ) = \omega _{i_1} = p(P_{i^*})$, $o_{i_1}~\prec ~o_{i_2}~\prec ~\cdots ~\prec ~o_{i_m}$, and for each $k \in \{1,\ldots ,m-1\}$, agents $i_k$ and $i_{k+1}$ are adjacent. Let $(P,\omega ^{N'})~\in ~{\mathcal {P}}^N \times {\mathcal {X}}^N$ be the problem that differs from $(P,\omega )$ only in that the agents in $N'$ have exchanged their endowments as follows: For each $k \in \{1,\ldots ,m\}$, agent k owns object $o_{i_{k+1}}$ (agent $i_m$ owns object $o_1$). Because $p(P_{i^*}) = o_1$ and agent $i^*$ owns it at $(P,\omega ^{N'})$, by (7),

$$\begin{aligned} \varphi (P,\omega ^{N'}) = CR(P,\omega ^{N'}). \end{aligned}$$

(8)

Let $N^{''} = N' \setminus \{i_1\}$. Let $(P,\omega ^{N^{''}})~\in ~{\mathcal {P}}^N \times {\mathcal {X}}^N$ be the problem that differs from $(P,\omega )$ only in that the agents in $N^{''}$ have exchanged their endowments as follows: For each $k~\in ~\{2,\ldots ,m\}$, agent k owns object $o_{k+1}$ (agent $i_m$ owns $o_2$). Hence, agent $i_1$ owns $o_1$ and agent $i_m$ owns $o_2$. Then

• $o_1 \prec p(P_{i_1})$;

• $p(P_{i^*}) \prec o_2$;

• $o_1 \prec o_2$; and

• there is no $o \in O$ such that $o_1 \prec o \prec o_2$.

By adjacent endowment-swapping invariance,

$$\begin{aligned} \varphi (P,\omega ^{N^{''}}) = \varphi (P,\omega ^{N'}). \end{aligned}$$

Applying this argument repeatedly,

$$\begin{aligned} \varphi (P,\omega ) = \varphi (P,\omega ^{i^*-1,i^*}) = \cdots = \varphi (P,\omega ^{N'}). \end{aligned}$$

(9)

Because

$$\begin{aligned} CR(P,\omega ^{N'}) = CR(P,\omega ), \end{aligned}$$

together with (8) and (9), we have

$$\begin{aligned} \varphi (P,\omega ) = CR(P,\omega ). \end{aligned}$$

1.6 Proof of Theorem 2

We have seen that the crawler satisfies the endowment lower bound, strategy-proofness, non-bossiness, and adjacent endowment-swapping invariance.

We show the converse. The proof is almost the same as that of Theorem 1. The only difference is that we derive (6) from strategy-proofness and non-bossiness. Let $\varphi$ be a rule satisfying the properties. Let $\omega \in {\mathcal {X}}^N$. Without loss of generality, suppose that $\omega = (o_1,o_2,\ldots ,o_n)$. We prove by induction on n that $\varphi$ is the crawler.

Base step Suppose that $n = 2$. There are four possible preference profiles.

$$\begin{aligned} \begin{array}{llll} P_1:o_1,~o_2 &{}\quad P'_1:o_2,~o_1&{}\quad {\widehat{P}}_1:o_1,~o_2&{}\quad {\tilde{P}}_1:o_2,~o_1 \\ P_2:o_1,~o_2 &{}\quad P'_2:o_1,~o_2&{}\quad {\widehat{P}}_2:o_2,~o_1&{}\quad {\tilde{P}}_2:o_2,~o_1 \end{array} \end{aligned}$$

Note that

$$\begin{aligned}&CR(P,\omega ) = (o_1,o_2),~CR(P',\omega ) = (o_2,o_1),~CR({\widehat{P}},\omega ) = (o_1,o_2),\\&\quad \text {and}~CR({\tilde{P}},\omega ) = (o_1,o_2). \end{aligned}$$

By the endowment lower bound,

$$\begin{aligned}&\varphi (P,\omega ) = (o_1,o_2),~\varphi (P',\omega ) \in \{(o_1,o_2),(o_2,o_1)\},~ \varphi ({\widehat{P}},\omega ) = (o_1,o_2),\\&\quad \text {and}~\varphi ({\tilde{P}},\omega ) = (o_1,o_2). \end{aligned}$$

Consider $(P',\omega )$. Suppose that agents 1 and 2 swap their endowments. Then by the endowment lower bound,

$$\begin{aligned} \varphi (P',\omega ^{1,2}) = (o_2,o_1). \end{aligned}$$

Hence, by adjacent endowment-swapping invariance,

$$\begin{aligned} \varphi (P,\omega ) = (o_2,o_1). \end{aligned}$$

Therefore, $\varphi = CR$.

Induction hypothesis Let $k \ge 3$. Suppose that for each $n \le k-1$, $\varphi = CR$.

Induction step Suppose that Let $n = k$. Let $P \in {\mathcal {P}}^N$. Let $i \in N$ and $P^0_i~\in ~{\mathcal {P}}(\omega _i)$. Suppose that $P_i = P^0_i$, i.e., his most preferred object is his endowment. By the endowment lower bound,

$$\begin{aligned} \varphi _i(P^0_i,P_{-i},\omega ) = \omega _i. \end{aligned}$$

Let $j \in N \setminus \{i\}$ and $P'_j \in {\mathcal {P}}$ be such that for each pair $k,l \in N \backslash \{i\}$,

$$\begin{aligned} o_k~P_j~o_l~\iff ~o_k~P'_j~o_l. \end{aligned}$$

By strategy-proofness,

$$\begin{aligned} \varphi _j(P^0_i,P'_j,P_{-\{i,j\}},\omega ) = \varphi _j(P^0_i,P_j,P_{-\{i,j\}},\omega ). \end{aligned}$$

By non-bossiness,

$$\begin{aligned} \varphi (P^0_i,P'_j,P_{-\{i,j\}},\omega ) = \varphi (P^0_i,P_j,P_{-\{i,j\}},\omega ). \end{aligned}$$

This implies that

$$\begin{aligned} \varphi \left( r_{N \backslash \{i\}}(P^0_i,P_{-i},\omega )\right) = \varphi _{N \backslash \{i\}}(P^0_i,P_{-i},\omega ). \end{aligned}$$

There are $k-1$ agents in $r_{N \backslash \{i\}}(P^0_i,P_{-i})$. Hence, by the induction hypothesis,

$$\begin{aligned} \varphi \left( r_{N \backslash \{i\}}(P^0_i,P_{-i},\omega )\right) = CR\left( r_{N \backslash \{i\}}(P^0_i,P_{-i},\omega )\right) . \end{aligned}$$

Because the crawler meets the endowment lower bound,

$$\begin{aligned} \varphi (P^0_i,P_{-i},\omega ) = CR(P^0_i,P_{-i},\omega ). \end{aligned}$$

The rest of the proof is the same as that of Theorem 1.

Relation between the properties in Sect. 3

Figure 3 shows how the properties in Theorem 1 are related.

Fig. 3

An illustration of the relation between the properties in Theorem 1

Example 5

Let $N' \in {\mathcal {N}}$ be such that $|N'| \ge 3$.

Let $\varphi$ be defined by setting for each $(P,\omega )~\in ~{\mathcal {P}}^{N'}~\times ~{\mathcal {X}}^{N'}$, $\varphi (P,\omega ) = \omega$, and for each $N \in {\mathcal {N}}$ such that $N \ne N'$ and each $(P,\omega ) \in {\mathcal {P}}^N \times {\mathcal {X}}^N$, $\varphi (P,\omega ) = CR(P,\omega )$. Because the crawler meets the endowment lower bound, so does $\varphi$.

Let $N' = \{1,2,3\}$ and $\omega \in {\mathcal {X}}^{N'}$. Let $P' \in {\mathcal {P}}^{N'}$ be such that

$$\begin{aligned} P'_1&:\omega _1,~\omega _2,\omega _3~\ldots , \\ P'_2&:\omega _3,~\omega _2,\ldots \\ P'_3&:\omega _2,~\omega _3,\ldots . \end{aligned}$$

Then $\varphi (P',\omega ) = (\omega _1,\omega _2,\omega _3)$. Because

$$\begin{aligned}&\omega _2 \prec p(P'_2),~p(P'_3) \prec \omega _3,~\text {and}~\varphi (P',\omega ^{2,3}) = (\omega _1,\omega _3,\omega _2) \ne (\omega _1,\omega _2,\omega _3)\\&= \varphi (P',\omega ), \end{aligned}$$

$\varphi$ is not adjacent-endowment-swapping invariant. Because

$$\begin{aligned} \varphi _1(P',\omega ) = \omega _1~\text {and}~\varphi (r_{\{2,3\}}(P',\omega )) = (\omega _3,\omega _2) \ne (\omega _2,\omega _3) = \varphi _{\{2,3\}}(P',\omega ), \end{aligned}$$

$\varphi$ is not separable.

Example 6

Let $N' = \{1,2,3\}$. Let $\varphi$ be defined by setting for each $(P,\omega ) \in {\mathcal {P}}^{N'} \times {\mathcal {X}}^{N'}$,

$$\begin{aligned}&\varphi (P,\omega )\\&= {\left\{ \begin{array}{ll} \left( \omega _3,\omega _2,\omega _1\right) &{}\,\, \text {if }\omega _2\text { is between }\omega _1\text { and } \omega _3\text {, and}~p(P_1) = p(P_2) = p(P_3) = \omega _2 \\ CR(P,\omega ) &{}\,\,\text {otherwise}, \end{array}\right. } \end{aligned}$$

and for each $N \in {\mathcal {N}}$ such that $N \ne N'$ and each $(P,\omega ) \in {\mathcal {P}}^N \times {\mathcal {X}}^N$, $\varphi (P,\omega ) = CR(P,\omega )$.

By the same logic as in Example 3, $\varphi$ is adjacent-endowment-swapping invariant.

Consider the set $N'$ of agents. Let $\omega \in {\mathcal {X}}^{N'}$. Let $P' \in {\mathcal {P}}^{N'}$ be such that

$$\begin{aligned} P'_1&:\omega _2,~\omega _1,\ldots ,\\ P'_2&:\omega _2,\ldots , \\ P'_3&:\omega _2,~\omega _3,\ldots . \end{aligned}$$

Then $\varphi (P',\omega ) = (\omega _3,\omega _2,\omega _1)$. Because

$$\begin{aligned} \omega _1~P'_1~\omega _3 = \varphi _1(P',\omega )~\text {and}~\omega _3~P'_3~\omega _1 = \varphi _3(P',\omega ), \end{aligned}$$

$\varphi$ does not meet the endowment lower bound. Moreover, because

$$\begin{aligned} \varphi _2(P',\omega ) = \omega _2~\text {and}~\varphi (r_{\{1,3\}}(P',\omega )) = (\omega _1,\omega _3) \ne (\omega _3,\omega _1) = \varphi _{\{1,3\}}(P',\omega ), \end{aligned}$$

$\varphi$ is not separable.

Example 7

For each $N \in {\mathcal {N}}$, each $(P,\omega ) \in {\mathcal {P}}^N \times {\mathcal {X}}^N$, and each $i \in N$, let $\varphi _i(P) = \omega _{i+1}$, where the agent whose endowment is the rightmost in the order is assigned the leftmost object.

Because for each $N \in {\mathcal {N}}$, each $(P,\omega ) \in {\mathcal {P}}^N \times {\mathcal {X}}^N$, and each $i \in N$, $\varphi _i(P,\omega ) \ne \omega _i$, $\varphi$ is separable.

Let $N \in {\mathcal {N}}$. Let $(P,\omega ) \in {\mathcal {P}}^N \times {\mathcal {X}}^N$ and $i \in N$ be such that agent i’s endowment is not the rightmost object and $p(P_i) \prec \omega _i$. Because $p(P_i) \prec \omega _i \prec \omega _{i+1} = \varphi _i(P,\omega )$, we have $\omega _i ~P_i~\varphi _i(P,\omega )$. Hence, $\varphi$ does not meet the endowment lower bound.

Let $(P,\omega ) \in {\mathcal {P}}^N \times {\mathcal {X}}^N$ and $i,i+1 \in N$ be such that agent $i+1$’s endowment is not the rightmost object, $\omega _i \prec p(P_i)$ and $p(P_{i+1}) \prec \omega _{i+1}$. Because

$$\begin{aligned} \varphi _i(P,\omega ^{i,i+1}) = \omega _{i+2} \ne \omega _{i+1} = \varphi _i(P,\omega ), \end{aligned}$$

$\varphi$ is not adjacent-endowment-swapping invariant.

Figure 4 shows how the properties in Theorem 2 are related. In some of the examples below, we refer to problems consisting of a specific agent set. For those examples, for any other agent set, we apply the crawler.

Fig. 4

An illustration of the relation between the various examples and different sets of properties in Theorem 2

Example 8

Let $N \in {\mathcal {N}}$. Let $\varphi$ be defined by setting for each $(P,\omega ) \in {\mathcal {P}}^N \times {\mathcal {X}}^N$,

$$\begin{aligned} \varphi (P,\omega ) = {\left\{ \begin{array}{ll} \omega &{}\quad \text {if there is}~i \in N~\text {such that}~p(P_i) = \omega _i, \\ CR(P,\omega )&{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

Because the crawler satisfies the endowment lower bound, so does $\varphi$. No agent would benefit by pretending that he most prefers his endowment. Moreover, if an agent’s most preferred object is his endowment and he is truthful, he receives his endowment. Hence, $\varphi$ is strategy-proof.

Let $N \in {\mathcal {N}}$ and $\omega \in {\mathcal {X}}^N$. Let ${\mathcal {D}}^N \subset {\mathcal {P}}^N$ be such that for each $P' \in {\mathcal {D}}^N$, there is $i \in N$ such that $p(P'_i) \ne \omega _i$ and $\varphi _i(P',\omega ) = \omega _i$. Also, for each $P^0_i \in {\mathcal {P}}(\omega _i)$, we have $\varphi _i(P^0_i,P'_{-i},\omega ) = \omega _i$. However, there are $P' \in {\mathcal {D}}^N$, $i \in N$, and $P^0_i \in {\mathcal {P}}(\omega _i)$ such that $\varphi (P^0_i,P'_{-i},\omega ) \ne \varphi (P'_i,P'_{-i},\omega )$. Hence, $\varphi$ violates non-bossiness.

Let $\omega \in {\mathcal {X}}^N$. Let $P' \in {\mathcal {P}}^N$ be such that there is $i \in N$ such that

$$\begin{aligned} P'_i:~\omega _i,\ldots . \end{aligned}$$

Moreover, suppose that there are $j,j+1 \in N$ such that $\omega _j \prec p(P'_j)$ and $p(P'_{j+1}) \prec \omega _{j+1}$. Then $\varphi _{\{j,j+1\}}(P',\omega ) = (\omega _j,\omega _{j+1})$. Because

$$\begin{aligned} \varphi _{\{j,j+1\}}(P,\omega ^{j,j+1}) = (\omega _{j+1},\omega _{j}) \ne (\omega _j,\omega _{j+1}) = \varphi _{\{j,j+1\}}(P',\omega ), \end{aligned}$$

$\varphi$ is not adjacent-endowment-swapping invariant.

Example 9

Let $n \in \{2,\ldots ,{\overline{n}}\}$ and $N = \{1,2,\ldots ,n\} \in {\mathcal {N}}$. Let $\varphi$ be defined by setting for each $(P,\omega )~\in ~{\mathcal {P}}^N~\times ~{\mathcal {X}}^N$,

$$\begin{aligned} \varphi (P,\omega ) = {\left\{ \begin{array}{ll} (p(P_1),p(P_2),\ldots ,p(P_n))&{}\quad \text {if for each}~i \in N,~p(P_i) \ne \omega _i~\text {and} \\ &{}\quad \text {for each pair}~i,j \in N,~p(P_i) \ne p(P_j) \\ \omega &{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

It is clear that $\varphi$ satisfies the endowment lower bound. If for each $i \in N$, $p(P_i) \ne \omega _i$, we have $\varphi (P,\omega ) \ne \omega$. This implies that $\varphi$ is non-bossy.

Let $\omega \in {\mathcal {X}}^N$. Let $P' \in {\mathcal {P}}^N$ be such that there are $i,j \in N$ such that $p(P'_i) = p(P'_j)$. Moreover, suppose that there are $k,k+1 \in N$ such that $\omega _k \prec p(P'_k)$ and $p(P'_{k+1}) \prec \omega _{k+1}$. Then $\varphi _{\{k,k+1\}}(P',\omega )~=~(\omega _k,\omega _{k+1})$. Because

$$\begin{aligned} \varphi _{\{k,k+1\}}(P,\omega ^{k,k+1}) = (\omega _{k+1},\omega _k) \ne (\omega _k,\omega _{k+1}) = \varphi _{\{\omega _k,\omega _{k+1}\}}(P',\omega ), \end{aligned}$$

$\varphi$ is not adjacent endowment-swapping invariant.

Example 10

Let $N \in {\mathcal {N}}$. Let $\varphi$ be defined by setting for each $(P,\omega ) \in {\mathcal {P}}^N \times {\mathcal {X}}^N$, $\varphi (P,\omega ) = TTC^*(P,\omega )$, where at each step, no agent points to his endowment unless he is the only remaining agent. The only difference between $\varphi$ and TTC is at the step when an agent points to himself when applying TTC. Also, $\varphi$ assigns an agent his endowment only if he is the only agent present. This implies that being assigned his endowment is independent of his preferences. Then because TTC is strategy-proof, so is $\varphi$. Because TTC is non-bossy, so is $\varphi$.

It is trivial that $\varphi$ does not satisfy the endowment lower bound. Because TTC is not adjacent-endowment-swapping invariant, neither is $\varphi$.

Example 11

Let $N' = \{1,2,3,4\}$. Let $\varphi$ be defined by setting for each $(P,\omega ) \in {\mathcal {P}}^{N'} \times {\mathcal {X}}^{N'}$,

$$\begin{aligned} \varphi (P,\omega ) = {\left\{ \begin{array}{ll} TTC(P,\omega )&{}\quad \begin{array}{l} \text {if }\omega _2\text { and }\omega _3\text { are between }\omega _1\text { and }\omega _4\text { and} \\ (p(P_2),p(R_3),p(P_4)) = (\omega _2, \omega _3,\omega _2) \end{array}\\ \omega &{}\quad \begin{array}{l} \text {if }\omega _2\text { and }\omega _3\text { are between }\omega _1\text { and }\omega _4,\\ (p(P_2),p(P_3)) = (\omega _2,\omega _3)~\text {and}~p(P_4) \ne \omega _2 \end{array}\\ CR(P,\omega )&{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

Because both TTC and the crawler satisfy the endowment lower bound, so does $\varphi$. Consider the set $N'$ of agents. If $p(P_2)~=~\omega _2$ and $p(P_3) = \omega _3$, neither one of agents 2 and 3 would benefit from swapping his endowment with either of his neighbor. Also, the crawler is adjacent-endowment-swapping invariant. Hence, so is $\varphi$.

Let $\omega \in {\mathcal {X}}^{N'}$. Let $P' \in {\mathcal {P}}^{N'}$ be such that

$$\begin{aligned} P'_1&:\omega _4,~\omega _3,~\omega _2,~\omega _1,\ldots , \\ P'_2&:\omega _2,\ldots , \\ P'_3&:\omega _3,\ldots , \\ P'_4&:\omega _1,~\omega _2,~\omega _3,~\omega _4,\ldots . \end{aligned}$$

Then $\varphi (P',\omega ) = (\omega _1,\omega _2,\omega _3,\omega _4)$.

Let ${\widehat{P}}_4 \in {\mathcal {P}}$ be such that

$$\begin{aligned} {\widehat{P}}_4&:\omega _2,~\omega _1,\ldots . \end{aligned}$$

Then $\varphi ({\widehat{P}}_4,P'_{\{1,2,3\}},\omega ) = (\omega _4,\omega _2,\omega _3,\omega _1)$. Because

$$\begin{aligned} \varphi _4({\widehat{P}}_4,P'_{\{1,2,3\}},\omega ) = \omega _1~P'_4~\omega _4 = \varphi _4(P'_4,P'_{\{1,2,3\}},\omega ), \end{aligned}$$

$\varphi$ is not strategy-proof.

Let ${\widehat{P}}_2 \in {\mathcal {P}}$ be such that

$$\begin{aligned} {\widehat{P}}_2:\omega _3,~\omega _2,\ldots . \end{aligned}$$

Then $\varphi ({\widehat{P}}_2,P'_{\{1,3,4\}},\omega ) = (\omega _4,\omega _2,\omega _3,\omega _1)$. Because

$$\begin{aligned}&\varphi _2({\widehat{P}}_2,P'_{\{1,3,4\}},\omega ) = \omega _2 = \varphi _2(P'_2,P'_{\{1,3,4\}},\omega )~\text {and}\\&\varphi _{\{1,4\}}({\widehat{P}}_2,P'_{\{1,3,4\}},\omega ) = (\omega _4,\omega _1) \ne (\omega _1,\omega _4) = \varphi _{\{1,4\}}(P'_2,P'_{\{1,3,4\}},\omega ), \end{aligned}$$

$\varphi$ violates non-bossiness.

Example 12

Let $N' = \{1,2,3\}$ and denote $\bigcup _{i \in N'} \omega _i = \{o_1,o_2,o_3\}$. Let $f = (1,2,3,\ldots )$ and $f' = (1,3,2,\ldots )$. Let $\varphi$ be defined by setting for each $(P,\omega ) \in {\mathcal {P}}^{N'} \times {\mathcal {X}}^{N'}$,²⁰

$$\begin{aligned} \varphi (P,\omega ) = {\left\{ \begin{array}{ll} SP^f(P,\omega )&{}\quad \text {if}~P_1:o_2,~o_3,~o_1,\ldots , \\ SP^{f'}(P,\omega )&{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

Consider the set $N'$ of agents. Agent 1 is always assigned an object first. Hence, $\varphi$ is strategy-proof for him. Moreover, all sequential priority rules are strategy-proof. Because the ordering of agents depends only on agent 1’s preferences, $\varphi$ is strategy-proof. All sequential priority rules are adjacent-endowment-swapping invariant. Because the ordering of agents does not depend on the endowment profile, $\varphi$ is adjacent-endowment-swapping invariant. Because none of the sequential priority rules meets the endowment lower bound, neither does $\varphi$.

Let $\omega \in {\mathcal {X}}^{N'}$ and let $P' \in {\mathcal {P}}^{N'}$ be such that

$$\begin{aligned} P'_1&:o_2,~o_3,~o_1,\ldots , \\ P'_2&:o_1,~o_2,~o_3,\ldots , \\ P'_3&:o_1,~o_2,~o_3,\ldots . \end{aligned}$$

Let ${\widehat{P}}_1 \in {\mathcal {P}}$ be such that

$$\begin{aligned} {\widehat{P}}_1&: o_2,~o_1,\ldots . \end{aligned}$$

Then

$$\begin{aligned} \varphi (P'_1,P'_{\{2,3\}},\omega )&= (o_2,o_1,o_3), \\ \varphi ({\widehat{P}}_1,P'_{\{2,3\}},\omega )&= (o_2,o_3,o_1). \end{aligned}$$

Hence, $\varphi$ violates non-bossiness.

Example 13

Let $N' = \{1,2,3\}$ and denote $\bigcup _{i \in N'} \omega _i = \{o_1,o_2,o_3\}$. Let $\varphi$ be defined by setting for each $(P,\omega ) \in {\mathcal {P}}^{N'} \times {\mathcal {X}}^{N'}$,

$$\begin{aligned} \varphi (P,\omega ) = {\left\{ \begin{array}{ll} (o_3,o_2,o_1)&{}\quad \text {if}~p(P_1) = p(P_2) = p(P_3) = p(P_4) = o_2 \\ (o_1,o_3,o_2)&{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

Consider the set $N'$ of agents. It is trivial that $\varphi$ is non-bossy. Because $\varphi$ is independent of the endowment profile, it is adjacent-endowment-swapping invariant.

Let $\omega = (o_1,o_2,o_3)$. Let $P' \in {\mathcal {P}}^{N'}$ be such that

$$\begin{aligned} P'_1&:o_2,~o_1,~o_3,\ldots , \\ P'_2&:o_2,\ldots , \\ P'_3&:o_2,~o_3,~o_1,\ldots . \end{aligned}$$

Then $\varphi (P',\omega ) = (o_3,o_2,o_1)$. Because $o_1~P'_1~o_3 = \varphi _1(P',\omega )$, $\varphi$ does not satisfy the endowment lower bound.

Let ${\widehat{P}}_3 \in {\mathcal {P}}$ be such that

$$\begin{aligned} {\widehat{P}}_3&:o_1,~o_2,~o_3,\ldots . \end{aligned}$$

Then $\varphi ({\widehat{P}}_3,P'_{\{1,2\}},\omega ) = (o_1,o_3,o_2)$. Because

$$\begin{aligned} \varphi _3({\widehat{P}}_3,P'_{\{1,2\}},\omega ) = o_2~P'_3~o_1 = \varphi _3(P'_3,P'_{\{1,2\}},\omega ), \end{aligned}$$

$\varphi$ is not strategy-proof.