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Deferred acceptance is minimally manipulable

Abstract

This paper shows that the deferred acceptance mechanism (DA) cannot be improved upon in terms of manipulability without compromising stability. A conflict between manipulability and fairness is also identified. Stable mechanisms that minimize the set of individuals who match with their least preferred achievable mate are shown to be maximally manipulable among the stable mechanisms. These mechanisms are also more manipulable than DA. A similar conflict between fairness and manipulability is identified in the case of the median stable mechanisms.

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Notes

  1. 1.

    In particular, comparisons according to these partial orders differ from comparisons based on the occurrences of truthful Nash equilibria (see Section 2). Although I discuss all partial orders introduced by Pathak and Sönmez (2013) (which includes a partial order based on truthful Nash equilibria), I focus on the partial order the authors introduce in Section III of their paper, for reasons that are made clear in Section 2 of the present paper.

  2. 2.

    See, for example, Chen et al. (2016) who show that no two stable mechanisms can be compared in the sense of a third manipulability partial order also proposed by Pathak and Sönmez (2013).

  3. 3.

    An achievable mate is a mate that the individual matches with under some stable matching.

  4. 4.

    Beside the two papers already cited, see, for example, Aleskerov and Kurbanov (1999), Maus et al. (2007), Andersson et al. (2014), Fujinaka and Wakayama (2012), Barberà and Gerber (2017), and Decerf and Van der Linden (2016).

  5. 5.

    For previous papers on fair stable matchings, see Knuth (1997), Irving et al. (1987), Teo and Sethuraman (1998), and Klaus and Klijn (2006).

  6. 6.

    Similar profiles can also be constructed when \(\#W \ne \#M\), see footnote 19.

  7. 7.

    As Chen et al. (2016, Theorem 3) shows, \(\succeq ^{PS-I}\) is the ordinal equivalent of the partial order introduced in Pathak and Sönmez (2013, Section IV.), hence the label “\(PS-I\)”.

  8. 8.

    In general, \(\succeq ^{PS-NE}\) is also a sub-relation of \(\succeq ^{PS}\), whereas \(\succeq ^{AM}\) and \(\succeq ^{PS-NE}\) need not be logically connected. However, on the set of stable (one-to-one) mechanisms, we have \(\succeq ^{PS-I} \Rightarrow \succeq ^{PS} \Rightarrow \succeq ^{AM} \Rightarrow \succeq ^{PS-NE}\), where the first implication follows trivially from the fact that \(\succeq ^{PS-I}\) is unable to order any two stable mechanisms (Chen et al. 2016), and the last implication follows equally trivially from the fact that \(\succeq ^{PS-NE}\) is indifferent between any two mechanisms (Proposition 2).

  9. 9.

    In other words, when the mechanism changes from A to \(DA^W\), the increase in manipulability for men is “quantitatively” identical to the decrease for women, with each men able to match with one more woman as part of beneficial manipulations, and each woman able to match with one less man as part of beneficial manipulations. However, the decrease is “qualitatively” significant for woman (for whom it removes any room for beneficial manipulations), whereas the increase is somewhat insignificant for men (who now have one more option for manipulation in an otherwise already highly manipulable game).

  10. 10.

    Recall that minimal and maximal manipulability properties are implicitly defined with respect to the class of stable mechanisms.

  11. 11.

    Recall that manipulability properties that do not refer to either PS or AM hold for both partial orders.

  12. 12.

    Recall that, as explained above, Latin Square profiles can be constructed even when \(\#M = \#W > 3\) If \(\#M \ne \#W\), simply let all individuals other than \(\{w_1,\dots ,w_{\min \{\#M,\#W\}},m_1,\dots ,m_{\min \{\#M,\#W\}}\}\) rank being self-matched first.

  13. 13.

    For the case of minimal PS-manipulability, Theorem 1.(i) can be viewed as a consequence of Pathak and Sönmez (2013, Theorem 2).

  14. 14.

    Nevertheless, minimally AM-manipulable mechanisms different from DA and minimally PS-manipulable mechanisms different from DA can be shown to exist. Whether any of these mechanisms is of interest is left as an open question.

  15. 15.

    A mechanism would be trivially minimally manipulable if it cannot be compared with any other stable mechanism (see footnote 2).

  16. 16.

    It can be shown that \(\lim _{h\rightarrow \infty } \left( 1-\frac{1}{h}\right) ^{h!} = 0\).

  17. 17.

    Although the minimum regret and the miniworst criteria are similar in spirit, they differ in many ways. For example, the miniworst criterion does not ascribe a cardinal meaning to the rank of a mate. The two criteria are not logically related; neither criterion implies the other.

  18. 18.

    This does not follow directly from Proposition 7. In general, it is possible for a mechanism to be minimally manipulable but fail to be less manipulable than a maximally manipulable mechanism. See the example in the proof of Proposition 8 for the case of PS-manipulability.

  19. 19.

    For \(\#M = \#W = h >3\), consider any profile with \(R_{w_1}:m_1~ m_2 \dots \), \(R_{w_2}:m_2~ m_3 \dots \), \(\dots \), \(R_{w_h}:m_h~ m_1 \dots \) and \(R_{m_1}:w_h~ w_1 \dots \), \(R_{m_2}:w_1 ~w_2 \dots \), \(\dots \), \(R_{m_h}:m_{h-1}~ m_h \dots \). If \(\#M \ne \#W\), simply let all individuals other than \(\{w_1,\dots ,w_{\max \{\#M,\#W\}},m_1,\dots ,m_{\max \{\#M,\#W\}}\}\) rank being self-matched first.

  20. 20.

    Experiments suggest that in decentralized markets, subjects more often select the median stable matching than any other stable matching (Echenique and Yariv 2012).

  21. 21.

    To be precise, only such mechanisms andDA can possibly be minimally AM-manipulable. The addition of DA is reflected by the second term in the bound of Propositions 5.(ii) and 6.(i).

  22. 22.

    Strictly speaking, if \(\#W \ne \#M\), this is only true of the individuals who have acceptable mates. But by the construction explained in footnote 12, other students always match with themselves in all stable matchings, and therefore have no impact on manipulability comparisons.

  23. 23.

    In the iterated procedure leading to the formation of class \({\mathcal {A}}^*\), pairs formed of a individual \(j \in \{w_1, w_2, w_3, m_1, m_2 m_3\}\) and a preference \(\dot{R}_j\) are never considered because given profiles \(R^*\) and \(R^{**}\), a mechanism \(A \in {\mathcal {A}}\) can never satisfy both (A.28) and (A.29) for such pairs.

  24. 24.

    It can however be showed that no stable mechanism is less manipulable for colleges than college-proposing DA (Pathak and Sönmez 2013, Theorem 2).

  25. 25.

    For generalizations of the median stable matching to many-to-one problems, see Klaus and Klijn (2006) and Sethuraman et al. (2006).

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Correspondence to Martin Van der Linden.

Additional information

I am grateful to Tommy Andersson, Benoit Decerf, Paul Edelman, Eun Jeong Heo, Greg Leo, Jordi Massó, John Weymark and Myrna Wooders for helpful discussions and comments. I also want to thank the participants to the Vanderbilt Mechanism Design Conference (2016) for useful questions and remarks. Special thanks go to Patrick Prosser and his java code for finding stable matchings (http://www.dcs.gla.ac.uk/~pat/roommates/distribution/), which has been helpful in constructing some of the examples in this paper. This work is partially supported by National Science Foundation grant IIS-1526860.

Appendices

Appendix

A Omitted proofs

Proof of Proposition 3

In the definition of \(\succeq ^{PS}\), (7) is equivalent to

$$\begin{aligned} \begin{aligned}&\{i\in N~|~i \text { can}{} not \text { manipulate } A \text { given profile } R \} \\&\quad \supseteq \{i\in N~|~i \text { can}{} not \text { manipulate } B \text { given profile } R \}. \end{aligned} \end{aligned}$$

Similarly, in the definition of \(\succeq ^{AM}\), (10) is equivalent to

$$\begin{aligned} \begin{aligned}&\{i\in N~|~i \text { can}{} not \text { manipulate } A \text { given preference } R_* \} \\&\quad \supseteq \{i\in N~|~i \text { can}{} not \text { manipulate } B \text { given preference } R_* \}. \end{aligned} \end{aligned}$$

The proposition then follows directly from Lemma 2. \(\square \)

Proof of Proposition 4

I provide a proof for minimal PS-manipulability. The proof for maximal PS-manipulability is analogous.

Necessity. I prove the contrapositive. Suppose that \(A \succ ^{PS} B\). By Proposition 3, this implies that for some \(R^* \in {\mathcal {D}}\),

$$\begin{aligned} \begin{aligned}&\{i\in N~|~A_i(R^*) = f_i^{R^*} \} \subset \{i\in N~|~B_i(R^*) = f_i^{R^*} \}. \end{aligned} \end{aligned}$$

But because B is stable, \(B_i(R^*)\) is stable with respect to \(R^*\). Thus, there exists a profile (namely \(R^*\)) and a stable matching (namely \(B_i(R^*)\)) satisfy (14).

Sufficiency. I again prove the contrapositive. Suppose that \(\mu ^*\) is a stable matching satisfying (14) for some profile \(R^*\). Consider mechanism B constructed from A by setting \(B(R) = A(R)\) for all \(R \in {\mathcal {D}}\) with \(R\ne R^*\) and \(B(R^*) = \mu ^*\). Clearly, for all \(R \in {\mathcal {D}} \text { with } R\ne R^*\),

$$\begin{aligned} \{i\in N~|~A_i(R) = f_i^R \} = \{i\in N~|~B_i(R) = f_i^R \}. \end{aligned}$$
(A.1)

Also, by (14) and because \(B_i(R^*)=\mu ^*_i\),

$$\begin{aligned} \{i\in N~|~A_i(R^*) = f_i^{R^*} \} \subset \{i\in N~|~B_i(R^*)= f_i^{R^*} \}. \end{aligned}$$
(A.2)

By Proposition 3, (A.1) and (A.2) imply that \(B \succeq ^{PS} A\), but that the converse is not true. Hence, by definition, \(B\succ ^{PS} A\) and so A is not minimally PS-manipulable. \(\square \)

Proof of Theorem 1.(i)

I provide a proof for \(DA^W\) and minimal manipulability only. The proof for \(DA^M\) is analogous.

\({\varvec{\succeq ^{PS}.}}\) By Proposition 4, the PS part of (i) holds provided that there does not exist a profile \(R^*\) and a stable matching \(\mu ^*\) such that

$$\begin{aligned} \big \{i\in N~|~DA^W_i(R^*) = f_i^{R^*} \big \} \subset \big \{i\in N~|~\mu ^*_i = f_i^{R^*}\big \}. \end{aligned}$$
(A.3)

In order to derive a contradiction, suppose that there exists a stable matching \(\mu ^*\) and a preference profile \(R^*\) satisfying (A.3). By Lemma 1,

$$\begin{aligned} W \subseteq \big \{i\in N~|~DA^W_i(R^*) = f_i^{R^*} \big \}. \end{aligned}$$
(A.4)

Together, (A.3) and (A.4) imply \( W \subset \big \{i\in N~|~\mu ^*_i = f_i^{R^*}\big \}.\) But (A.3) implies \(\mu ^* \ne DA^W(R^*)\), which in turn implies that there exists a woman \(w^* \in W\) for whom \(\mu ^*_{w^*} \ne DA_{w^*}^W(R^*) = f_{w^*}^{R^*}\). Hence \(W \not \subset \{i\in N~|~\mu ^*_i = f_i^{R^*} \}\), a contradiction.

\({\varvec{\succeq ^{AM}.}}\) In order to derive a contradiction, suppose that \(DA^W \succ ^{AM} A\) for some mechanism A. By Lemmas 1 and 2, for all \(R_* \in \cup _{i\in W} {\mathcal {D}}_i\),

$$\begin{aligned} \{w \in N_{R_*}~|~w \text { can manipulate } DA^W \text { given preference } R_*\} = \emptyset . \end{aligned}$$
(A.5)

Thus, because \(DA^W \succ ^{AM} A\), for all \(R_* \in \cup _{i\in W} {\mathcal {D}}_i\),

$$\begin{aligned} \{w \in N_{R_*}~|~w \text { can manipulate } A \text { given preference } R_*\} = \emptyset . \end{aligned}$$
(A.6)

But by Lemma 2, (A.5) and (A.6) imply that for all \(R \in {\mathcal {D}}\) and for all \(w\in W\),

$$\begin{aligned} A_w(R) = DA^W_w(R) = f_w^R. \end{aligned}$$
(A.7)

Hence, \(A = DA^W\) by the definition of a matching, contradicting the assumption that \(DA^W \succ ^{AM} A\). \(\square \)

Proof of Proposition 5

Sketch of the proof.(i). For any h, any individual \(i\in N\), and any preference \(R_i \in {\mathcal {D}}_i\), it is possible to construct \(R^{R_i}_{-i}\) such that the profile \((R_i,R_{-i}^{R_i})\) mimics the Latin Square pattern of profile (3). Any of these Latin Square profiles admits h stable matchings. For any Latin Square profile, out of the h stable matchings, minimally PS-manipulable mechanisms must select either the men optimal or the women optimal matching. The upper bound \(\left( \frac{2}{h}\right) ^{h!}\) is obtained by considering the proportion of stable mechanisms that select one of these two stable matching in every Latin Square profile.

(ii). If a mechanism A is minimally AM-manipulable, then we cannot have \(A \succ ^{AM} DA\). By Lemmas 1 and 2, this implies that there exists an acceptor \(a\in N\), a proposer \(p \in N\), and a pair of preferences \((R_a, R_p) \in \bar{{\mathcal {D}}}_a \times \bar{{\mathcal {D}}}_p\) such that A always matches a and p with their most preferred achievable mate when they report \(R_a\) or \(R_p\), respectively.Footnote 21 In particular, a and p must match with their most preferred achievable mate given the Latin Square profiles \((R_a,R_{-a}^{R_a})\) and \((R_p,R_{-p}^{R_p})\). Because this only needs to be true for a single acceptor-proposer pair and for a single pair of profiles, this fact alone is not sufficient to prove that the proportion of mechanisms B such that \(B\succ ^{AM} DA\) is large. For every \(i\in N\) and every \(R_i \in {\mathcal {D}}_i\), it is however possible to construct sufficiently many variants of the Latin Square profiles \((R_i,R_{-i}^{R_i})\) to show that this proportion is in fact bounded below by \(1- \big (\frac{h}{(h-1)!} + \frac{1}{h((h-1)!)^2}\big )\). As a consequence, the proportion of minimally AM-manipulable mechanisms is at most \(\big (\frac{h}{(h-1)!} + \frac{1}{h((h-1)!)^2}\big )\). \(\square \)

Proof

Consider any \(i\in N\) and any \(R_i \in \bar{{\mathcal {D}}}_i\). Let \(i=w_1\) without loss of generality. Without loss of generality again, let the individuals in M be labeled in such a way that

$$\begin{aligned} {\begin{array}{*{10}ccc} R_{w_1}:&m_{h}&m_{(h-1)}&\dots&m_{2}&m_{{1} } \end{array}} \end{aligned}$$
(A.8)

A Latin Square profile \((R_i,R_{-i}^{R_i})\) generalizing profile (3) can then be constructed in the following way:

(A.9)

The preferences of the men in \((R_i,R_{-i}^{R_i})\) are constructed symmetrically to (A.9) with woman \(w_1\) appearing on the downward diagonal as in (3).

(i). For each \(R_i \in {\mathcal {D}}_i\), profile \((R_i,R_{-i}^{R_i})\) has h stable matchings, only two of which (the women and men optimal matchings) can be selected by a minimally PS-manipulable mechanism. Consider the construction of a stable mechanism A. Because there are h! preferences in \(\bar{{\mathcal {D}}}_i\), there are h! profiles \((R_i,R_{-i}^{R_i})\), one for each \(R_i \in \bar{{\mathcal {D}}}_i\). Among the \(h^{h!}\) possible choices of stable matchings for these h! profiles, only the \(2^{h!}\) that select the women or the men optimal matchings for each \((R_i,R_{-i}^{R_i})\) make it possible for A to be minimally PS-manipulable. Hence, the proportion of minimally manipulable mechanisms among the class of stable mechanisms is at most \(\left( \frac{2}{h}\right) ^{h!}\).

(ii). By Lemmas 1 and 2, for any stable mechanism A, if there exists no \(R_* \in \cup _{i\in N} \bar{{\mathcal {D}}}_i\) and no acceptor \(a\in N_{R_*}\) such that \(A_a(R_*,R_{-a}) = f_a^{(R_*,R_{-a})}\) for all \(R_{-a} \in \bar{{\mathcal {D}}}_{-a}\), then either \(A = DA\) or \(A\succ ^{AM} DA\). We are interested in the proportion of these mechanisms relative to the set of stable mechanisms.

Let \({\mathbb {P}}(X)\) denote the proportion of stable mechanisms A for which X is true. For every \(i\in N\), the preferences in \(\bar{{\mathcal {D}}}_i\) are labeled following some arbitrary order \(R_i^1,\dots ,R_i^{h!}\). The proportion we want to compute is equal to

$$\begin{aligned} \begin{aligned} 1- {\mathbb {P}} \big (&\cup _{\{i\in N|i \text { is an acceptor}\}} \cup _{k\in \{1,\dots ,h!\}} \\&[ A_i(R_i^k,R_{-i}) = f_i^{(R_i^k,R_{-i})} ~\text {for all}~R_{-i} \in \bar{{\mathcal {D}}}_{-i} ] \big ). \end{aligned} \end{aligned}$$
(A.10)

The expression in (A.10) is at least

$$\begin{aligned} 1- \sum _{\{i\in N|i \text { is an acceptor}\}~} \sum _{k\in \{1,\dots ,h!\}}{\mathbb {P}} \big (A_i(R_i^k,R_{-i}) = f_i^{(R_i^k,R_{-i})} ~\text {for all}~R_{-i} \in \bar{{\mathcal {D}}}_{-i} \big ). \end{aligned}$$
(A.11)

We can obtain a bound on (A.11) by bounding the term inside the double summation. For any profile \(R\in \bar{{\mathcal {D}}}\), let \(\sigma ^R(X)\) denote the proportion of stable matchings \(\mu \) for which X is true. Observe that

$$\begin{aligned} {\mathbb {P}} \big (A_i(R_i^k,R_{-i})&= f_i^{(R_i^k,R_{-i})} ~\text {for all}~R_{-i} \in \bar{{\mathcal {D}}}_{-i} \big ) \\&= \prod _{R_{-i} \in \bar{{\mathcal {D}}}_{-i}} \sigma ^{(R^k_i,R_{-i})}\big ( \mu _i=f_i^{(R^k_i,R_{-i})}\big ). \end{aligned}$$

For example, for the Latin Square profile \((R^k_i,R^{R^k_i}_{-i})\), we have \(\sigma ^{(R^k_i,R^{R^k_i}_{-i})}( \mu _i=f_i^{(R^k_i,R^{R^k_i}_{-i})}) = \frac{1}{h}\), which implies that

$$\begin{aligned} {\mathbb {P}} \big (A_i(R_i^k,R_{-i}) = f_i^{(R_i^k,R_{-i})} ~\text {for all}~R_{-i} \in \bar{{\mathcal {D}}}_{-i} \big ) \le \frac{1}{h}. \end{aligned}$$
(A.12)

A tighter bound for (A.11) can be obtained by tightening the bound in (A.12). This can be done by considering profiles different from the Latin Square profile (A.9). Specifically, I consider variations of (A.9) for which (a) the number of stable matchings and (b) the proportion of stable matchings that match i with her or his most preferred achievable mate are easy to compute.

In what follows, I use the relabeling introduced at the beginning of the proof of Proposition 5, with \(R_i^k = R_{w_1}\). The first variation of the Latin Square profile that is considered has \((h-1)\) stable matchings and is denoted by \((R_{w_1},R_{-w_1}^{R_{w_1}}(h-1,1))\). In \((R_{w_1},R_{-w_1}^{R_{w_1}}(h-1,1))\), the preferences of the women and of man \(m_h\) are as follows:

(A.13)

In (A.13), every woman ranks \(m_h\) first. Among the first \(h-1\) women, the sub-profile excluding \(m_h\) has a Latin Square structure of dimension \(h-1\) similar to (A.9). For \(w_h\) and \(m_h\), only the most preferred mate is specified.

In \((R_{w_1},R_{-w_1}^{R_{w_1}}(h-1,1))\), the preferences of men other than \(m_h\) are constructed symmetrically to the preferences of the women other than \(w_h\) in (A.13) with \(w_h\) ranked last and woman \(w_1\) appearing on the downward diagonal as in (3).

Observe that \(m_h\) and \(w_h\) match together in every stable matching given \((R_{w_1},R_{-w_1}^{R_{w_1}}(h-1,1))\), and \(m_h\) and \(w_h\) are therefore not achievable for any other man or woman. By analogy with (A.9), there are \((h-1)\) stable matchings among the remaining individuals \(\{m_1,\dots ,m_{h-1},w_1,\dots ,w_{h-1}\}\) due to the Latin Square structure of the profile once \(m_h\) and \(w_h\) are removed. There are therefore \((h-1)\) stable matchings given \((R_{w_1},R_{-w_1}^{R_{w_1}}(h-1,1))\), only one of which matches \(w_1\) with her most preferred achievable mate.

A natural variant of (A.13), denoted \((R_{w_1},R_{-w_1}^{R_{w_1}}(h-1,2))\), also has \((h-1)\) stable matchings. In \((R_{w_1},R_{-w_1}^{R_{w_1}}(h-1,2))\), the preferences of the women and of man \(m_1\) are as follows:

(A.14)

In (A.14), the first \(h-1\) women rank \(m_1\) last. Among the first \(h-1\) women, the sub-profile excluding \(m_1\) has a Latin Square structure of dimension \(h-1\) similar to (A.9). For \(w_h\) and \(m_1\), only the most preferred mate is specified.

In \((R_{w_1},R_{-w_1}^{R_{w_1}}(h-1,2))\), the preferences of men other than \(m_1\) are constructed symmetrically to the preferences of the women other than \(w_h\) in (A.13) with \(w_h\) ranked last and woman \(w_1\) appearing on the downward diagonal as in (3).

Similarly to \((R_{w_1},R_{-w_1}^{R_{w_1}}(h-1,1))\), there are \((h-1)\) stable matchings given \((R_{w_1},R_{-w_1}^{R_{w_1}}(h-1,2))\) only one of which matches \(w_1\) with her most preferred achievable mate.

It is easy to see how, for all \(k \in \{2,\dots , h-1\}\), the above constructions extend to profiles \((R_{w_1},R_{-w_1}^{R_{w_1}}(k,1))\) and \((R_{w_1},R_{-w_1}^{R_{w_1}}(k,2))\) that each admit k stable matchings only one of which matches \(w_1\) with her most preferred achievable mate. In \((R_{w_1},R_{-w_1}^{R_{w_1}}(h-2,1))\) for example, the first \(h-2\) women rank \(m_h\)and\(m_{h-1}\) first and, among the first \(h-2\) women, the sub-profile excluding \(m_h\)and\(m_{h-1}\) has a Latin Square structure of dimension \(h-2\). Also, \(w_h\) ranks \(m_h\) first and\(w_{h-1}\) ranks \(m_{h-1}\) first.

Together with the original Latin Square profile, we have therefore identified \(1+ 2(h-1)\) profiles with a partial Latin square structure and in which i’s preference is \(R^k_i\). In other words, we have identified a set of sub-profiles \(\{R_{-i}^1,\dots ,R_{-i}^{1+2(h-1)}\}\) such that the set of profiles \((R^k_i, R_{-i}^t)\) for \(t\in \{1,\dots , 1+ 2(h-1)\}\) consists of (a) the full Latin Square profile and (b) the \(2(h-1)\) partial Latin Square profiles described above.

There are \(h((h-1)!)^2\) ways to select stable matchings for these \(1+2(h-1)\) profiles. For example, a stable mechanism could select the first of the h stable matchings in the full Latin Square profile, the first of the \((h-1)\) stable matchings in \((R_{w_1},R_{-w_1}^{R_{w_1}}(h-1,1))\), the first of the \((h-2)\) stable matchings in \((R_{w_1},R_{-w_1}^{R_{w_1}}(h-2,1))\), and so on. A stable mechanism could also select the second of the h stable matchings in the full Latin Square profile, the first of the \((h-1)\) stable matchings in \((R_{w_1},R_{-w_1}^{R_{w_1}}(h-1,1))\), the first of the \((h-2)\) stable matchings in \((R_{w_1},R_{-w_1}^{R_{w_1}}(h-2,1))\), and so on. Of the \(h((h-1)!)^2\) possible selections for these \(1+2(h-1)\) profiles, only one always matches i with her or his most preferred achievable mate. Hence,

figureh

Using (A.15) in (A.11) shows that (A.10) is at least

$$\begin{aligned} 1-\sum _{\{i\in N|i \text { is acceptor}\}} \sum _{k\in \{1,\dots ,h!\}}\frac{1}{h((h-1)!)^2}. \end{aligned}$$
(A.16)

Because the fraction in (A.16) is independent of the indices used in the summations, (A.16) is equal to \(1- \frac{h (h!)}{h((h-1)!)^2} = 1- \frac{h}{(h-1)!}\).

Finally, we must account for the fact that DA itself might be one of the at most \(1- \frac{h}{(h-1)!}\) mechanisms A for which there exists no \(R_* \in \cup _{i\in N} \bar{{\mathcal {D}}}_i\) and no acceptor \(a\in N_{R_*}\) such that \(A_a(R_*,R_{-a}) = f_a^{(R_*,R_{-a})}\) for all \(R_{-a} \in \bar{{\mathcal {D}}}_{-a}\). Because we cannot have \(DA \succ ^{AM} DA\), we must not include DA itself when computing the upper bound.

Clearly, DA itself represents a very small proportion of the stable mechanisms. For example, only \(\frac{1}{h((h-1)!)^2}\) of the mechanisms select a combination of stable matchings for the full Latin Square profile and the \(2(h-1)\) variants described above that is compatible with the mechanism being DA. Hence, overall, the proportion of mechanisms B such that \(B \succ ^{AM} DA\) is at least \(1- \big (\frac{h}{(h-1)!} + \frac{1}{h((h-1)!)^2}\big )\). As a consequence, the proportion of minimally AM-manipulable mechanisms is at most \(\big (\frac{h}{(h-1)!} + \frac{1}{h((h-1)!)^2}\big )\).

\(\square \)

Proof of Proposition 6

Sketch of the proof.(i). See the sketch of the proof of Proposition 5.

(ii).\(A \succ ^{PS} DA\) does not hold for any mechanism A that, for at least one profile \(R^*\), selects a stable matching for which some acceptor a matches with \(f_a^R\). Such mechanisms abound. For example, a third of the stable mechanisms select matching \(\mu ^1\) for the Latin Square profile (3). Hence, only \((1-\frac{1}{3})\) of the stable mechanisms select a matching for profile (3) that allows \(A \succ ^{PS} DA\) . Considering variants of (3) yields the bound in the proposition. \(\square \)

Proof

(i). As shown in the proof of Proposition 5.(ii), the proportion of mechanisms B such that \(B \succ DA\) is at least \(1- \big (\frac{h}{(h-1)!} + \frac{1}{h((h-1)!)^2}\big )\).

(ii). If \(A \succ ^{PS} DA\), then A can never select the optimal stable matching of the accepting side whenever any acceptor has more than one achievable mate. This implies that for any acceptor a and any of the h! preferences \(R_a \in \bar{{\mathcal {D}}}_a\), we have \(A_a(R_a,R^{R_a}_{-a}) \ne f_a^{(R_a,R^{R_a}_{-a})}\), where the construction of \(R^{R_a}_{-a}\) is described in (A.9). Each \((R_a,R^{R_a}_{-a})\) has h stable matchings only one of which matches a with \(f_a^{(R_a,R^{R_a}_{-a})}\). Hence, for each \((R_a,R^{R_a}_{-a})\), there are \({h-1}\) ways to select a stable matching \(A(R_a,R^{R_a}_{-a})\) with \(A_a(R_a,R^{R_a}_{-a}) \ne f_a^{(R_a,R^{R_a}_{-a})}\). This implies that, of all the \(h^{h!}\) possible ways in which a stable mechanism A can select a stable matching for the h! profiles \((R_a,R^{R_a}_{-a})\), only \((h-1)^{h!}\) make it possible to have \(A \succ ^{PS} DA\). Therefore, at most \(\left( \frac{h-1}{h}\right) ^{h!}\) of the stable mechanisms A are such that \(A \succ DA\). \(\square \)

Proof of Theorem 2

(i)Consider any Latin Square profile \(R^{LS}\) as defined in (A.9) (if \(\#W \ne \#M\), see the argument in footnote 12). For any miniworst mechanism A, \(A_i(R^{LS}) \ne f_i^{R^{LS}}\) for all \(i\in N\).Footnote 22 The mechanism B constructed from A by changing the stable matching selected for \(R^{LS}\) to the men optimal or women optimal stable matching (and selecting the same matching as A otherwise) is such that \(A \succ ^{PS} B\). Hence A is not minimally PS-manipulable.

(ii). For any \(i\in N\) and any \(R_i \in {\mathcal {D}}^3_i\), it is possible to construct a Latin Square profile similar to (A.9) among the mates that are acceptable according to \(R_i\). Slightly abusing the notation, this profile is also denoted \((R_i,R^{R_i}_{-i})\). For example, if three mates are acceptable according to \(R_i\), let \(N^6 \subseteq N\) consist of (a) i, (b) i’s three acceptable mates, and (c) two more individuals on i’s side of the market. Profile \((R_i,R^{R_i}_{-i})\) then has the same structure as (A.9) among the individuals in \(N^6\). (For any \(i\in N^6\), any individual on the other side of the market that does not belong to \(N^6\) is unacceptable).

If mechanism A is miniworst, then for any \(i\in N\) and any \(R_i \in {\mathcal {D}}^3_i\), \(A_i(R_i,R^{R_i}_{-i}) \ne f_i^{(R_i,R^{R_i}_{-i})}\). Hence, for any \(R_* \in \cup _{i\in N} {\mathcal {D}}^3_i\),

$$\begin{aligned} \{i\in N_{R_*} ~|~ i \text { can manipulate } A \text { given } R_*\} = \{i\}. \end{aligned}$$
(A.17)

and A is clearly not minimally AM-manipulable. \(\square \)

Proof of Proposition 7

(i). Sufficiency. By Proposition 4, it is sufficient to show that for any miniworst mechanism, any profile R, and any stable matching \(\mu \), (15) does not hold. Because A is miniworst, (17) is false. That is, either

$$\begin{aligned} \{i\in N~|~A_i(R) = l_i^R \} = \{i\in N~|~\mu _i = l_i^R\}, \end{aligned}$$
(A.18)

or there exists \(i^*\in N\) such that

$$\begin{aligned} \mu _{i^*} = l_{i^*}^R \text { and } A_{i^*}(R) \ne l_{i^*}^R . \end{aligned}$$
(A.19)

By Lemma 1, individuals match with their least preferred achievable mates if and only if they are their mates’ most preferred achievable mate. Thus, if (A.18) holds,

$$\begin{aligned} \{i\in N~|~A_i(R) = f_i^R \} = \{i\in N~|~\mu _i = f_i^R\}. \end{aligned}$$
(A.20)

On the other hand, if (A.19) holds, we have

$$\begin{aligned} A_{ l_{i^*}^R}(R) \ne f_{ l_{i^*}^R}^R = i^* ~~\text {and}~~ \mu _{ l_{i^*}^R} = f_{ l_{i^*}^R}^R = {i^*}. \end{aligned}$$
(A.21)

If (A.20) holds, then the set of individuals who match with their most preferred achievable mate is the same in A(R) and \(\mu \). On the other hand, if (A.21) holds, then there is an individual \(l_{i^*}^R\) who matches with \(f_{ l_{i^*}^R}^R\) in \(\mu \), but not in A(R). In both cases, the set of individuals who match with their most preferred achievable mates in A(R) is not a superset of the set of individuals who match with their most preferred achievable mates in \(\mu \). That is,

$$\begin{aligned} \{i\in N~|~A_i(R) = f_i^R \} \not \supset \{i\in N~|~\mu _i = f_i^R \}. \end{aligned}$$
(A.22)

and (15) does not hold.

Necessity. I prove the contrapositive. Suppose that A is maximally PS-manipulable but A is not miniworst, i.e., there exists a profile \(R^*\) and a matching \(\mu ^*\) such that (17) holds. By Lemma 1, (17) implies

$$\begin{aligned} \{i\in N~|~A_i(R^*) = f_i^{R^*} \} \supset \{i\in N~|~\mu ^*_i = f_i^{R^*}\}. \end{aligned}$$
(A.23)

Now, construct mechanism B from A by setting \(B(R) = A(R)\) for all \(R \in {\mathcal {D}}\) with \(R \ne R^*\), and \(B(R^*) = \mu ^*\). By Lemma 2, because \(B(R) = A(R)\) for all \(R \ne R^*\), we have that for all \(R \ne R^*\),

$$\begin{aligned} \begin{aligned}&\{i\in N~|~i \text { can manipulate } A \text { given } R \}\\&\quad = \{i\in N~|~i \text { can manipulate } B \text { given } R \}. \end{aligned} \end{aligned}$$
(A.24)

Also, by (A.23) and Lemma 2,

$$\begin{aligned} \begin{aligned}&\{i\in N~|~i \text { can manipulate } A \text { given } R^* \} \\&\quad \subset \{i\in N~|~i \text { can manipulate } B \text { given } R^* \}. \end{aligned} \end{aligned}$$
(A.25)

Together, (A.24) and (A.25) imply that \(B \succ ^{PS} A\) and therefore A is not maximally PS-manipulable.

(ii). By (A.17), any miniworst mechanism is clearly maximally AM-manipulable.

(iii). For any \(i\in N\) and any \(R_i \in {\mathcal {D}}^3_i\), it is possible to construct a Latin Square profile similar to (A.9) among the mates that are acceptable according to \(R_i\). As explained in the text, it is in fact possible to construct two such Latin Square profiles that are distinct but share the properties of having at least as many (completely disctint) stable matchings as there are acceptable mates in \(R_i\). Slightly abusing the notation and terminology, these profiles are denoted \((R_i,R^{R_i}_{-i}(1))\) and \((R_i,R^{R_i}_{-i}(2))\) and called Latin Square profiles. Consider any mechanism A such that

  1. (a)

    for any of the first Latin Square profiles, A selects a stable matching that matches no individual with her or his most preferred achievable mate (i.e., for any \(i\in N\) and any \(R_i \in {\mathcal {D}}^3_i\), \(A_i(R_i,R^{R_i}_{-i}(1)) \ne f_i^{(R_i,R^{R_i}_{-i}(1))}\)), but

  2. (b)

    for any of the second Latin Square profiles, A selects the same matching as \(DA^M\) (i.e., for any \(i\in M\) and any \(R_i \in {\mathcal {D}}^3_i\), \(A_i(R_i,R^{R_i}_{-i}(2)) = f_i^{(R_i,R^{R_i}_{-i}(2))}\)).

By (a) and Lemma 2, for any \(R_* \in \cup _{i\in N} {\mathcal {D}}^3_i\),

$$\begin{aligned} \{i\in N_{R_*} ~|~ i \text { can manipulate } A \text { given } R_*\} = \{i\}, \end{aligned}$$

and A clearly is maximally AM-manipulable. However, by (b), for any second Latin Square profile of the form \((R_i,R^{R_i}_{-i}(2))\), A could have selected a “compromise” stable matching in which no individual matches with their least-preferred achievable mate. Hence, A is maximally AM-manipulable but not miniworst. \(\square \)

Proof of Proposition 8

As shown in (A.17), if a mechanism A is miniworst, then for any \(R_* \in \cup _{i\in N} {\mathcal {D}}^3_i\),

$$\begin{aligned} \{i\in N_{R_*} ~|~i \text { can manipulate } A \text { given } R_*\} = \{i\}. \end{aligned}$$

Hence, \(A \succ ^{AM} DA\).. \(\square \)

Proof of Theorem 3

(i). The proof is for \(\# W = \#M\). For the case \(\# W \ne \#M\), see the argument in footnote 12 as well as the caveat in footnote 22. Consider any Latin Square profile \(R^{LS}\) as defined in (A.9). Given \(R^{LS}\), median-stable mechanisms select a stable matching in which no individual matches with her or his most preferred achievable mate. The mechanism A constructed from median-stable mechanisms by changing the stable matching selected for \(R^{LS}\) to the men optimal or women optimal stable matching (and selecting the same matching as median-stable mechanisms otherwise) is such that \(MS \succ ^{PS} A\). Hence, median stable mechanisms are not minimally PS-manipulable.

(ii). For any \(i\in N\) and any \(R_i \in {\mathcal {D}}^3_i\), it is possible to construct a Latin Square profile similar to (A.9) among the mates that are acceptable according to \(R_i\). Slightly abusing the notation and terminology, this profile is also denoted \((R_i,R^{R_i}_{-i})\) and called a Latin Square profile. In \((R_i,R^{R_i}_{-i})\), median-stable mechanisms select a stable mechanism in which i does not match with her or his most preferred achievable mate. That is, for any \(i\in N\) any \(R_i \in {\mathcal {D}}^3_i\), and any median-stable mechanism MS, \(MS_i(R_i,R^{R_i}_{-i}) \ne f_i^{(R_i,R^{R_i}_{-i})}\). Thus, by Lemma 2, for any \(R_* \in \cup _{i\in N} {\mathcal {D}}^3_i\),

$$\begin{aligned} \{i\in N_{R_*}~|~ i \text { can manipulate } MS \text { given preference } R_*\} = \{i\}, \end{aligned}$$
(A.26)

and median-stable mechanisms are clearly not minimally AM-manipulable. \(\square \)

Proof of Proposition 9

(i). By (A.26) in the proof of Theorem 3.(ii) and Lemma 1, for any \(R_* \in \cup _{i\in N} {\mathcal {D}}^3_i\) and any median-stable mechanism MS,

$$\begin{aligned}&\{i\in N_{R_*}~|~ i \text { can manipulate } MS \text { given preference } R_*\} = \{i\}\\&\quad \supseteq \{i\in N_{R_*}~|~ i \text { can manipulate } DA \text { given preference } R_*\}, \end{aligned}$$

where inclusion is strict when i is a proposer in DA.

(ii). By Proposition 7, it is sufficient to show that median-stable mechanisms are not miniworst on the set of stable matchings. Let \(N^8 :=\{w_1,\dots , w_8, m_1, \dots , m_8\}\) and consider any profile R including the following sub-profile for individuals in \(N^8\):

$$\begin{aligned}&{\begin{array}{*{10}ccc} R_{w_1}:&{} m_3 &{} m_8 &{} m_7 &{} m_6 &{} m_5 &{} m_4 &{} m_2 &{} m_1 &{} \\ R_{w_2}:&{} m_2 &{} m_8 &{} m_7 &{} m_6 &{} m_5 &{} m_4 &{} m_1 &{} m_3 &{} \\ R_{w_3}:&{} m_1 &{} m_8 &{} m_7 &{} m_6 &{} m_5 &{} m_4 &{} m_3 &{} m_2 &{} \\ R_{w_4}:&{} m_8 &{} m_7 &{} m_6 &{} m_5 &{} m_4 &{} w_4 \\ R_{w_5}:&{} m_7 &{} m_6 &{} m_5 &{} m_4 &{} m_8 &{} w_5 \\ R_{w_6}:&{} m_6 &{} m_5 &{} m_4 &{} m_8 &{} m_7 &{} w_6 \\ R_{w_7}:&{} m_5 &{} m_4 &{} m_8 &{} m_7 &{} m_6 &{} w_7 \\ R_{w_8}:&{} m_4 &{} m_8 &{} m_7 &{} m_6 &{} m_5 &{} w_8 \end{array}} \\&{\begin{array}{*{10}ccc} R_{m_1}:&{} w_1 &{} w_2 &{} w_3 &{} m_1 \\ R_{m_2}:&{} w_3 &{} w_1 &{} w_2 &{} m_2\\ R_{m_3}:&{} w_2 &{} w_3 &{} w_1 &{} m_3 \\ R_{m_4}:&{} w_4 &{} w_5 &{} w_1 &{} w_2 &{} w_3 &{} w_6 &{} w_7 &{} w_8 \\ R_{m_5}:&{} w_8 &{} w_4 &{} w_1 &{} w_2 &{} w_3 &{} w_5 &{} w_6 &{} w_7 \\ R_{m_6}:&{} w_7 &{} w_8 &{} w_1 &{} w_2 &{} w_3 &{} w_4 &{} w_5 &{} w_6 \\ R_{m_7}:&{} w_6 &{} w_7 &{} w_1 &{} w_2 &{} w_3 &{} w_8 &{} w_4 &{} w_5 \\ R_{m_8}:&{} w_5 &{} w_6 &{} w_1 &{} w_2 &{} w_3 &{} w_7 &{} w_8 &{} w_4 \\ \end{array}} \end{aligned}$$

Observe that because no two women have the same most preferred man, matching every woman with her favorite man yields a stable matching. Thus, because the set of individuals who are married is the same in every stable matching (Roth and Sotomayor 1990), every individual in \(N^8\) is married in every stable matching. Therefore, because no man in \(\{m_1, m_2, m_3\}\) is acceptable to any woman in \(\{w_4, w_5, w_6, w_7, w_8\}\), but every man in \(\{m_1, m_2, m_3\}\) is acceptable to every woman in \(\{w_1,w_2,w_3\}\), all men in \(\{m_1, m_2, m_3\}\) must match with women in \(\{w_1,w_2,w_3\}\) in any stable matching (by definition, stable matchings are individually rational). As a consequence, all men in \(\{m_4, m_5, m_6, m_7, m_8\}\) also match with women in \(\{w_4, w_5, w_6, w_7, w_8\}\) in every stable matching.

Among \(\{m_1, m_2, m_3\} \cup \{w_1,w_2,w_3\}\), the stable sub-matchings are

figurei

Among \(\{m_4, m_5, m_6, m_7, m_8\}\cup \{w_4, w_5, w_6, w_7, w_8\}\), the stable sub-matchings are

figurej

Observe that in any stable matching that includes \(\mu ^{1}_{123}\) or \(\mu ^{2}_{123}\), the sub-matching among \(\{m_4, m_5, m_6, m_7, m_8\}\cup \{w_4, w_5, w_6, w_7, w_8\}\) must be either \(\mu _{45678}^1\) or \(\mu _{45678}^2\). Indeed, in any other combination that includes \(\mu ^{1}_{123}\) or \(\mu ^{2}_{123}\) (e.g., \((\mu ^{1}_{123},\mu _{45678}^4)\)), every man in \(\{m_4, m_5, m_6, m_7, m_8\}\) forms a blocking pair with every woman in \(\{w_1, w_2, w_3\}\). On the other hand, in stable matchings that includes \(\mu ^{3}_{123}\), the sub-matching among \(\{m_4, m_5, m_6, m_7, m_8\}\cup \{w_4, w_5, w_6, w_7, w_8\}\) can be any of the stable sub-matchings \((\mu _{45678}^1, \dots , \mu _{45678}^5)\). Overall, the stable matchings among individuals in \(N^8\) are

figurek

For every woman \(w\in \{w_1,w_2,w_3\}\), the stable matchings are ranked in the same way with

$$\begin{aligned} \mu ^9~R_{w}~\mu ^8~R_{w}~\mu ^7~R_{w}~\mu ^6~R_{w}~\mu ^5~R_{w}~\mu ^4~R_{w}~\mu ^3~R_{w}~\mu ^2~R_{w}~\mu ^1. \end{aligned}$$

Hence, given profile R, median-stable mechanisms match every woman in \(\{w_1,w_2,w_3\}\) with her match in \(\mu ^5\).

For every woman \(w\in \{w_4, w_5, w_6, w_7, w_8\}\), the stable matchings are also ranked in the same way with

$$\begin{aligned} \mu ^9~R_{w}~\mu ^8~R_{w}~\mu ^7~R_{w}~\mu ^6~R_{w}~\mu ^4~R_{w}~\mu ^2~R_{w}~\mu ^5~R_{w}~\mu ^3~R_{w}~\mu ^1. \end{aligned}$$

Hence, given profile R, median-stable mechanisms match every woman in \(\{w_4, w_5, w_6, w_7, w_8\}\) with her match under \(\mu ^4\).

Thus, given profile R, median-stable mechanisms match individuals in \(N^8\) according to matching \(\mu ^6\). In \(\mu ^6\), the set of \(i \in N^8\) who match with \(l_i^R\) is \(\{m_1,m_2,m_3\}\). But note that in \(\mu ^4\) the set of \(i \in N^8\) who match with \(l_i^R\) is empty. Hence, median-stable mechanisms are not miniworst on the set of stable matchings. \(\square \)

Proof of Proposition 10.(i)

See the proof of Proposition 9 (i). \(\square \)

B Independence of minimal PS- and minimal AM-manipulability

B.1 Minimal PS-manipulability does not imply minimal AM-manipulability

Consider profiles (3) and (5) for the case \(\#W = \#M = 3\). From these profiles, construct profiles \(\tilde{R}\) and \(\hat{R}\) by making \(w_1, w_2, w_3, m_1, m_2\) and \(m_3\) rank themselves in fourth position and make other individuals rank being self-matched first.

Consider any mechanism A such that (a) \(A(\tilde{R}) = DA^W(\tilde{R})\), (b) \(A(\hat{R}) = DA^M(\hat{R})\), and (c) \(A(R) = DA^W(R)\) for all \(R \in {\mathcal {D}}\backslash \{\tilde{R}, \hat{R} \}\).

By Proposition 4 and Lemma 1, any mechanism that always selects the outcome of either \(DA^M\) or \(DA^W\) is minimally PS-manipulable, and therefore, so is mechanism A. Also, no man m ever has a dominant strategy in A unless his preference \(R_m\) is such that m always has a single achievable mate in ever profile of the form \((R_m, R_{-m})\) (e.g., if m ranks being self-matched first). Indeed, the preferences of men \(m_1\) to \(m_3\) in profiles \(\tilde{R}\) and \(\hat{R}\) can be included in two Latin Square profiles. At most one of these Latin Square profiles is \(\hat{R}\), which implies that in the other, say \(\dot{R}\), the outcome is \(DA^W(\dot{R})\) and these men have an opportunity to manipulate A given their preference. For any other man and any other preference \(R_m\), if there exists a profile \((R_m, R_{-m})\) such that m has two achievable mates, then \(A_m(R_m, R_{-m}) \ne DA_m^M(R_m, R_{-m})\) and m can therefore again manipulate A given \(R_m\).

Thus, the collection of pairs \((i,R_i)\) such that i cannot manipulate A given preference \(R_i\) contains only women and pairs \((m,R_m)\) such that m has a single achievable mate for every profile of the form \((R_m, R_{-m})\). Clearly, because of (b), this collection does not include \((w_1, R^*_{w_1})\) because \(w_1\) can manipulate A given preference \(R^*_{w_1}\) by (b). Thus, A is not minimally manipulable by Proposition 3 (specifically, \(A \succ ^{AM} DA^W\)).

B.2 Minimal AM-manipulability does not imply minimal PS-manipulability

Consider any pair of preferences \(\tilde{R}_{w_4}\) and \(\tilde{R}_{m_4}\) such that

$$\begin{aligned} {\begin{array}{*{10}ccc} &{}\tilde{R}_{w_4}: &{} m_4 &{} m_2 &{} m_3 &{} m_1 &{} w_4 &{} \dots \\ &{}\tilde{R}_{m_4}: &{} w_4 &{} w_3 &{} w_1 &{} w_2 &{} m_4 &{} \dots \end{array}} \end{aligned}$$
(A.27)

Then consider the class of mechanisms \({\mathcal {A}}\) such that for any \(A \in {\mathcal {A}}\) and any \(R \in {\mathcal {D}}\),

$$\begin{aligned} \begin{aligned}&R_{w_4} = \tilde{R}_{w_4} \qquad \text {implies} \qquad A_{w_4}(R) = f_{w_4}^R, \text { and }\\&R_{m_4} = \tilde{R}_{m_4} \qquad \text {implies} \qquad A_{m_4}(R) = f_{m_4}^R. \end{aligned} \end{aligned}$$
(A.28)

Class \({\mathcal {A}}\) is non-empty because \(w_4\) and \(m_4\) rank each other first in \(\tilde{R}_{w_4}\) and \(\tilde{R}_{m_4}\). Therefore, if both \(R_{w_4} = \tilde{R}_{w_4}\) and \(R_{m_4} = \tilde{R}_{m_4}\), then \(f_{w_4}^R = m_4\) and \(f_{m_4}^R = w_4\). As a consequence, there exists mechanisms satisfying both constraints in (A.28).

There might also be mechanisms \(A^* \in {\mathcal {A}}\) for which there exists other preferences \(\hat{R}_i \ne R_{w_4}, R_{m_4}\) (with possibly \(i \ne w_4, m_4\)) such that for all \(R \in {\mathcal {D}}\),

$$\begin{aligned} \begin{aligned}&R_{i} = \hat{R}_{i} \qquad \text {implies} \qquad A^*_{i}(R) = f_{i}^R. \end{aligned} \end{aligned}$$
(A.29)

By Proposition 3, iteratively eliminating from \({\mathcal {A}}\) any mechanism for which the set of preferences satisfying (A.29) can be enlarged (with respect to inclusion) leads to a set \({\mathcal {A}}^*\) of minimally AM-manipulable mechanisms (the iteration must complete because the set of mechanisms and profiles is finite).

Now, consider any mechanism \({A}^* \in {\mathcal {A}}^*\). Because \({A}^*\) satisfies (A.28), \(A_i^*(R^*) = DA_i^W(R^*)\) for any \(i \in \{w_1, \dots , w_4, m_1,\dots , m_4\} \) and any profile \(R^*\) like the following one (the profile is such that \(R^{*}_{w_4} = \tilde{R}_{w_4}\) and is a Latin Square profile for \(w_1, \dots , w_4, m_1, \dots m_4\)):

figurel

For the same reason, \(A_i^*(R^{**}) = DA_i^M(R^{**})\) for any \(i \in \{w_1, \dots , w_4, m_1,\dots , m_4\} \) and any profile \(R^{**}\) like the following one (the profile is such that \(R^{**}_{m_4} = \tilde{R}_{m_4}\) and is a Latin Square profile for \(w_1, \dots , w_4, m_1, \dots m_4\)):

figurem

Now consider any third profile \(\dot{R}\) like the following one, where the preferences of \(w_1, \dots , w_4\), \(m_1, \dots m_4\) are a particular combination of the men’s and women’s preferences in the last two profiles (in particular, \(\dot{R}_{w_4} = \tilde{R}_{w_4} \) and \(\dot{R}_{m_4} = \tilde{R}_{m_4}\)).

figuren

Recall that \(A_i^*(R^*) = DA_i^W(R^*)\) and \(A_i^*(R^{**}) = DA_i^M(R^{**})\) for any \(A^* \in {\mathcal {A}}^*\). Therefore, every individual \(i \in \{w_1, w_2, w_3, m_1, m_2, m_3\}\) fails to have a dominant strategy in any \(A^*\in {\mathcal {A}}^*\) when that individual’s preference is \(\dot{R}_i\). This implies that the choice of a stable matching for profile \(\dot{R}\) is in some sense “unconstrained” by the requirement that \(A^*\) be minimally AM-manipulable, at least when it comes to the matching of individuals \(w_1, w_2, w_3, m_1, m_2\), and \(m_3\) (in particular, the matching of these individuals is not constrained by (A.28)).Footnote 23 That is, for every sub-matching \(\mu (\dot{R})\) among \(w_1, w_2, w_3, m_1, m_2,\) and \(m_3\) that is stable given \(\dot{R}\), there exists a mechanism \(A^{\mu (\dot{R})} \in {\mathcal {A}}^*\) such that \(A_i^{\mu (\dot{R})}(\dot{R}) = \mu _i(\dot{R})\) for all \(i \in \{w_1, w_2, w_3, m_1, m_2, m_3\}\).

Observe also that, given \(\dot{R}\), \(m_4\) and \(w_4\) match with one another in all stable matchings. Once these two individuals are removed, the remaining profile among individuals \(w_1, w_2, w_3, m_1, m_2\), and \(m_3\) is

figureo

which is a Latin Square profile equivalent to (5) (clearly, \(w_1, w_2, w_3, m_1, m_2\), and \(m_3\) can only match with one another in any stable matching given \(\dot{R}\)).

Thus, there exists a matching \(\mu ^*(\dot{R})\) that is stable given \(\dot{R}\) and such that none of \(w_1, w_2, w_3, m_1, m_2\), and \(m_3\) match with their most preferred achievable mate. By the above argument, there exists a mechanism \(A^{\mu ^*(\dot{R})} \in {\mathcal {A}}^*\) such that \(A_i^{\mu ^*(\dot{R})}(\dot{R}) = \mu _i^*(\dot{R})\) for all \(i \in \{w_1, w_2, w_3, m_1, m_2, m_3\}\). Therefore, \(A^{\mu ^*(\dot{R})}\) is minimally AM-manipulable but by Proposition 3, \(A^{\mu ^*(\dot{R})}\) fails to be minimally PS-manipulable (as it would be possible to match either the men or the women in \(\{w_1, w_2, w_3, m_1, m_2, m_3\}\) with their most preferred achievable mate given \(\dot{R}\) without affecting the matching of any other individual).

C Extensions to many-to-one matching

The results in this paper are for one-to-one matching. In this Section, I discuss possible extensions to many-to-one matching.

In the many-to-one problem (also known as “college admission” problem) each student \(s \in S\) matches with at most one college \(c \in C\), but colleges can admit up to \(q_c \ge 1\) students. In this model, students have preferences on the set of colleges (and themselves), whereas each college c has a preference on the set of subsets of students containing no more than \(q_c\) students (including the empty set). Famously, (pariwise-) stable matchings may not exist in this environment without further assumptions on the preferences of colleges over sets of students (Roth and Sotomayor 1990, Example 2.7.). One of the simplest restriction that restores the existence of stable matching consist in imposing that colleges’ preferences be responsive (see Roth 1985).

Under responsive preferences, Lemma 2 generalizes to the many-to-one environment, but for students only. In particular, the student-proposing DA is minimally manipulable in this more general environment as well. For colleges on the other hand, matching with their most preferred set of achievable students is only a necessary condition for non-manipulability. As Roth (1985) showed, colleges can sometimes manipulate stable mechanisms in spite of matching with their most preferred set of achievable students (even when, as in the example Roth (1985) provides, the profile admits a single stable matching). College-proposing DA might therefore not be minimally manipulable as instances where colleges can manipulate could be taken advantage of to reduce the manipulability for students.Footnote 24 Whether this is true is left as an open question. Similarly, we leave as an open question to compare the two versions of DA with fair stable mechanisms in the many-to-one context.Footnote 25

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Van der Linden, M. Deferred acceptance is minimally manipulable. Int J Game Theory 48, 609–645 (2019). https://doi.org/10.1007/s00182-018-0649-3

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Keywords

  • Matching
  • Deferred acceptance
  • Manipulability
  • Stability
  • Fairness

JEL

  • C78
  • D47
  • D82