Strategy-Proofness and Non-Obvious Manipulability of

Top-Trading-Cycles with Strategic Invitations

Shinnosuke Hamasaki, Taiki Todo and Makoto Yokoo

Graduate School of Information Science and Electrical Engineering,

Kyushu University, Fukuoka, 819-0395, Japan

hamasaki@agent.inf.kyushu-u.ac.jp, {todo, yokoo}@inf.kyushu-u.ac.jp

Keywords:

Two-Sided Matching, Top-Trading-Cycles Algorithm, Non-Obvious Manipulability.

Abstract:

Diffusion mechanism design is one of the recent trends in the literature of mechanism design. Its purpose

is to incentivize agents to diffuse the information about the mechanism to as many followers as possible, as

well as reporting their preferences. This paper is the ﬁrst attempt to consider diffusion mechanism design for

two-sided matching from the perspective of non-obvious manipulability. We focus on the top-trading-cycles

(TTC) mechanism for the many-to-one two-sided matching problem. We clarify the necessary and sufﬁcient

condition for the mechanism to satisfy strategy-proofness and non-obvious manipulability, respectively. We

also propose a new TTC-based matching mechanism that violates strategy-proofness but satisﬁes non-obvious

manipulability, which illustrates how we can handle strategic information diffusion in two-sided matching.

1 INTRODUCTION

As one of the active ﬁelds in the area of artiﬁcial in-

telligence, multi-agent systems have been attracting

considerable attention of researchers and practition-

ers. In a multi-agent system, multiple agents inter-

act with each other and the society containing these

agents makes a joint decision. Such a process is called

a multi-agent decision-making.

For the research of multi-agent decision-making,

game theory has played an important role. More

speciﬁcally, mechanism design is considered as a

mathematical foundation of multi-agent decision-

making, especially when agents are self-interested

and not cooperative with each other. The main pur-

pose of mechanism design is to develop decision-

making rules, also known as mechanisms, which in-

centivize selﬁsh agents to take desirable actions.

Strategy-proofness is a well-known incentive

property in the literature. It requires that for each

agent, reporting a true private information, which is

in many cases referred to a true type of an agent,

to a mechanism is a dominant strategy. While it is

quite appealing for achieving a socially-acceptable

outcome, there are a lot of negative results regarding

strategy-proofness, because requiring the existence of

dominant strategy equilibria is too demanding. Weak-

ening strategy-proofness and/or choosing other incen-

tive properties is then a natural direction.

In this paper we consider diffusion mechanism

design (Li et al., 2017), in which participation to a

decision-making is invitation-based. An agent can

participate in a decision-making only when other par-

ticipating agents invite her. In diffusion mechanism

design, strategy-proofness is rather demanding, since

it requires that telling a true preference and inviting

as many colleagues as possible is a dominant strat-

egy for every agent. Cho et al. (2022) showed several

impossibility theorems on strategy-proofness in diffu-

sion mechanism design for two-sided matching.

Given above impossibility theorems, in this paper

we consider non-obvious manipulability as a weaker

notion of incentive property, instead of strategy-

proofness. Non-obvious manipulability intuitively re-

quires that, for each agent, truth-telling is weakly bet-

ter than any manipulation in both the best- and worst-

cases. While several existing works have investigated

non-obvious manipulability in the literature of mech-

anism design for two-sided matching (please refer to

the next section for a survey), analysis of non-obvious

manipulability in diffusion mechanism design does

not exist, as far as the authors know.

To sum up, this paper is the ﬁrst attempt to

consider non-obvious manipulability in the diffusion

mechanism design for two-sided matching. We ﬁrst

show that, even if the incentive property is weak-

ened to non-obvious manipulability, known impos-

sibility results by Cho et al. (2022) come over. We

616

Hamasaki, S., Todo, T. and Yokoo, M.

Strategy-Proofness and Non-Obvious Manipulability of Top-Trading-Cycles with Strategic Invitations.

DOI: 10.5220/0013319100003890

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 17th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2025) - Volume 1, pages 616-623

ISBN: 978-989-758-737-5; ISSN: 2184-433X

then focus on the Top-Trading-Cycles (TTC) mecha-

nism (Abdulkadiro

glu and S

onmez, 2003), which is

known not to be manipulable in the standard settings

without strategic invitation, and provide two if-and-

only-if conditions for it to satisfy strategy-proofness

and non-obvious manipulability, respectively, in two-

sided matching with strategic invitation. We further

propose a new mechanism that violates SP but satis-

ﬁes NOM.

1.1 Related Works

Gale and Shapley (1962) initiated the research of two-

sided matching and proposed the seminal Deferred-

Acceptance mechanism. Crawford (1991) studied the

effect of having more students/colleges in the two-

sided matching. In their model the set of students

varies as a result of exogenous events, while in our

model it is due to the strategic actions of students.

Various extensions of two-sided matching have also

been studied, including school choice with diversity

constraints (Kurata et al., 2017), matchings with bud-

get constraints (Aziz et al., 2020), and uncertain pref-

erences (Rastegari et al., 2013; Todo et al., 2021).

Several works investigated strategy-proof mech-

anisms with monetary compensations from the per-

spective of diffusion mechanism design (Kawasaki

et al., 2020; Li et al., 2024). On the other hand,

there is limited research on diffusion mechanism de-

sign without money. Recently, Kawasaki et al. (2021)

and You et al. (2022) considered house allocation

over social networks, without monetary compensa-

tions. Another recent work by Ando et al. (2025)

studied strategy-proof social choice over social net-

works. Cho et al. (2022) dealt with two-sided match-

ing with strategic information diffusion. However, all

these works focused on strategy-proofness, and never

considered non-obvious manipulability.

Given the difﬁculties of designing strategy-proof

mechanisms, analysis based on the non-obvious ma-

nipulability (Troyan and Morrill, 2020) is one of the

recent trends. Ortega and Klein (2023) proposed a

two-sided matching mechanism that violates strategy-

proofness but satisﬁes the non-obvious manipulabil-

ity, although their model is quite standard and not

dealing with strategic invitation.

2 MODEL

Our model of two-sided matching over social net-

works is basically identical with Cho et al. (2022),

while we assume that every student is acceptable by

any college and the underlying social network among

students are restricted to be a tree. We will further de-

ﬁne some additional notations to formalize the prop-

erty of non-obvious manipulability.

In our model, there are two sets of agents, students

and colleges. Let C = {c

, c

, . . . , c

|C|

} be the set of

colleges, and let S = {s

, s

, . . . , s

|S|

} be the set of stu-

dents. We usually use c ∈ C and s ∈ S to represent

a college and a student without specifying their iden-

tiﬁers. The symbol ∅ denotes an “unmatched” sta-

tus for students. Furthermore, special agent o, called

moderator, corresponds to a trusted third party.

Each college c has a priority ≻

, given as a strict

ordering over S, specifying its one-to-one comparison

over students. As Cho et al. (2022) assumed, we do

not care about how colleges compare two subsets of

students. Let ≻

= (≻

)

c∈C

represent a proﬁle of the

priorities of colleges C. Each college c has its max-

imum quota q

∈ Z

, indicating the number of stu-

dents that college c can accept. Let q

:= (q

)

c∈C

Each student s has a preference ≻

, given as a

strict ordering over C ∪ {∅}. A notation c ≻

′

indi-

cates that s strictly prefers being assigned to c instead

of c

′

. Analogously, c ≻

∅ indicates that s strictly

prefers being assigned to c to being unmatched. Sym-

bol ≿

indicates the “weak preference” associated

with ≻

; since we focus on strict preferences, c ≿

′

indicates either c ≻

′

or c = c

′

. Let ≻

= (≻

)

s∈S

represent a preference proﬁle of students S.

Students are distributed over a social network. Let

⊆ S be the set of the neighbors of o, which are

also called the direct children of o. For each s, let

⊆ S \ {s} denote s’s neighbors. The neighborhood

relation is asymmetric, i.e., s

′

∈ r

does not imply

s ∈ r

′

. Given r

:= (r

)

s∈S

and r

, all the neighbor-

hood relations are deﬁned, specifying social network

G(r

, r

) among students and the moderator. In this

paper we assume that social networks are tree-shaped,

where the moderator is located at the root.

Matching m speciﬁes to which college each stu-

dent is assigned. Given matching m, let m(s) ∈

C ∪ {∅} denote the college (if any) to which student

s is assigned, and m(c) ⊆ S denote the set of stu-

dents (if any) with which college c is matched. We

abuse ≻

and write m ≻

′

(or m ≿

′

) instead of

m(s) ≻

′

(s) (or m(s) ≿

′

(s)).

Let θ

= (≻

, r

) denote the true type of student

s, and let θ = (θ

)

s∈S

denote a proﬁle of the stu-

dents’ true types. Let θ

−s

denote a proﬁle of the types

owned by the students except s. Analogously, given

subset (also called as a coalition) T ⊆ S, let θ

de-

note a proﬁle of the types owned by T , and let θ

−T

denote a proﬁle of the types owned by the students

except T . Let R(θ

) = {θ

′

= (≻

′

, r

′

) | r

′

⊆ r

} de-

note the set of reportable types by s with true type

Strategy-Proofness and Non-Obvious Manipulability of Top-Trading-Cycles with Strategic Invitations

617

, assuming that each s cannot pretend to be con-

nected to any student to whom s is not really con-

nected. When s reports r

′

as her neighbors, we say s

diffuses the information toward r

′

, or s invites r

′

. Let

′

= (θ

′

)

s∈S

∈ ×

s∈S

R(θ

) = R(θ) denote a reportable

type proﬁle. Analogously, given subset T ⊆ S, let θ

′

denote a proﬁle of the types reported by T , and θ

′

−T

proﬁle of the types reported by the students except T .

Given type proﬁle θ

′

, let

S(θ

′

) ⊆ S denote the set

of connected students to whom a path exists from o in

G(r

′

, r

). Given θ

′

, let M(θ

′

) denote a set of feasible

matchings m satisfying the followings:

1. consistency; for any s ∈

S(θ

′

) and any c ∈ C,

m(s) = c ⇔ s ∈ m(c)

2. max. quota constraint; for any c ∈ C, |m(c)| ≤ q

3. connectivity; for any s ∈ S, s ̸∈

S(θ

′

) ⇒ m(s) = ∅

Given true type proﬁle θ (which is not observable),

mechanism µ maps a reported proﬁle θ

′

∈ R(θ) into

feasible matching m ∈ M(θ

′

), while µ can use ≻

, q

and r

as parameters.

We further deﬁne some terms related to social

networks. Given θ

′

and s ∈

S(θ

′

), let d

(θ

′

) ∈ Z

be the distance (the number of edges in the shortest

path) from o to s in G(r

′

, r

). For any s ̸∈

S(θ

′

), let

(θ

′

) = ∞. Also, for any s ∈ S ∪ {o}, let δ

(θ

′

) be the

set of descendants of s in G(r

′

, r

Deﬁnition 1 (Strategy-Proofness (SP)). Given mech-

anism µ, arbitrarily chosen student s ∈ S, proﬁle θ

′

−s

∈

R(θ

−s

), true type θ

, and misreport θ

′

∈ R(θ

), let

m := µ(θ

, θ

′

−s

) and m

′

:= µ(θ

′

, θ

′

−s

). Then, µ satis-

ﬁes strategy-proofness (SP) if m(s) ≿

′

(s) holds.

Strategy-proofness requires that, for each student

s, telling a true type θ

is a dominant strategy, i.e., the

outcome under the truth-telling is weakly better than

the outcome under reporting a fake type θ

′

Deﬁnition 2 (Non-Obvious Manipulability (NOM)).

Given mechanism µ, arbitrarily chosen student s ∈ S,

a true proﬁle θ

′

−s

∈ R(θ

−s

), given preference ≻

, and

a type θ

′

of student s,

• B

≻

(θ

′

) := c ∈ C ∪ {∅} s.t. ∃θ

′

−s

∈ R(θ

−s

– µ

(θ

′

, θ

′

−s

) = c, and

– ∀θ

′′

−s

∈ R(θ

−s

), c ≿

(θ

, θ

′′

−s

• W

≻

(θ

) := c ∈ C ∪ {∅} s.t. ∃θ

′

−s

∈ R(θ

−s

– µ

(θ

, θ

′

−s

) = c, and

– ∀θ

′′

−s

∈ R(θ

−s

), µ

(θ

, θ

′′

−s

) ≿

Then, µ satisﬁes non-obvious manipulability if both

≻

(θ

) ≿

≻

(θ

′

) and W

≻

(θ

) ≿

≻

(θ

′

)

holds for any θ

:= (≻

, r

) and any θ

′

∈ R(θ

The condition on B

≻

(·) denotes the best-case in-

centive constraint, which requires that telling a true

type θ

is better than telling any other type θ

′

, under

the true preference ≻

, in the best-case. Here the best-

case is calculated by changing the reports of other stu-

dents S \ {s}. Analogously, the condition on W

≻

(·)

denotes the worst-case incentive constraint.

We also deﬁne stability, efﬁciency, and fairness

properties, which are used in Cho et al. (2022), for

obtaining our impossibility theorems.

The mutually-best property intuitively require

that, if there is a pair of a student and a college who

ranks them at the top with each other, such a pair is

matched. Some weaker variants of the mutually-best

properties are also deﬁned.

Deﬁnition 3 (Mutually-Best (MB)). Given θ

′

, we say

a pair of student s and college c is a mutually-best

pair (MB-pair) if c ≻

′

for any c

′

̸= c and s ≻

′

for any s

′

∈

S(θ

′

). Matching m is mutually-best if

any MB-pair (s, c) is matched as long as s ∈

S(θ

′

). A

mechanism is said to satisfy MB if it always returns a

mutually-best matching. Matching m is mutually-best

for direct children if any MB-pair (s, c) is matched as

long as s ∈

S(θ

′

) and s ∈ r

. A mechanism is said to

satisfy MB-D if it always returns a matching that is

mutually-best for direct children.

Non-wastefulness is a well-known notion of ef-

ﬁciency, which intuitively requires that, if a student

cannot enter a college that she strictly prefer to her

current assignment, it must be the case that the col-

lege is full. One of its weaker notions, introduced

here as weak non-wastefulness, only requires that, the

existence of such a student implies that either she is

already assigned to some college or the college she

prefers already have some student assigned.

Deﬁnition 4 (Weak Non-Wastefulness (WNW)). A

matching m is weakly non-wasteful (WNW) for given

′

if, for any s ∈

S(θ

′

) and any c ∈ C, c ≻

m(s) im-

plies either m(s) ̸= ∅ or |m(c)| > 0.

Fairness is a condition related to students’ envies

toward other students. A pair of a college c and a stu-

dent s forms a blocking coalition for a given match-

ing m if both c ≻

m(s) and s ≻

′

for some s

′

∈ m(c)

hold. In words, the blocking pair like them with each

other. Intuitively, a matching is fair if such a blocking

pair does not exist. In the deﬁnition below, a weaker

notion of fairness is also deﬁned.

Deﬁnition 5 (Fairness). For student s and college c,

assume c ≻

m(s). Then, s has (i) justiﬁed envy with

respect to priority toward student s

′

∈ m(c) if s ≻

′

and (ii) justiﬁed envy with respect to network toward

′

∈ m(c) if both s ≻

′

and s

′

̸∈ π

(θ

′

). Matching m

is fair (FR) for given θ

′

if, there is no student with jus-

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

618

Figure 1: An example showing the incompatibility between

NOM and MB, and the incompatibility among NOM, FR,

and MB-D. Each circle indicates students (or the modera-

tor), and each arrow indicates invitation.

Figure 2: An example showing the incompatibility among

NOM, FRN, and WNW.

tiﬁed envy with respect to priority. Matching m is fair

with respect to network (FRN) for given θ

′

if there is

no student with justiﬁed envy with respect to network.

Cho et al. (2022) showed the following theorems,

all of which requires SP as an incentive property.

Theorem 1 (Cho et al. (2022)). There exists no mech-

anism that simultaneously satisfy SP and MB.

Theorem 2 (Cho et al. (2022)). There exists no mech-

anism that simultaneously satisfy SP, FR, and MB-D.

Theorem 3 (Cho et al. (2022)). There exists no mech-

anism that simultaneously satisfy SP, FRN, and WNW.

Finally, we deﬁne the Top-Trading-Cycles (TTC)

mechanism for the classical two-sided matching prob-

lem (Abdulkadiro

glu and S

onmez, 2003).

Deﬁnition 6. In Round t = 1, 2, . . ., each student s ∈ S

points to her most preferred college, if any, among

those who still have a non-zero capacity. Each col-

lege c ∈ C points to the student who has the highest

priority at that college. For each cycle, each belong-

ing student is assigned to the college that she is point-

ing to. Remove that student, and reduce the quota of

the college by one. The algorithm terminates when all

the students are assigned or all the colleges have zero

capacity; otherwise, it proceeds to Round t + 1.

3 IMPOSSIBILITIES

We ﬁrst show three impossibility theorems, each of

which strengthens an existing result (Theorems 1, 2,

and 3) by replacing SP with NOM. The examples in

their proofs are almost identical with the original ones

in Cho et al. (2022), while our proofs are a bit more

complicated due to the weakened incentive property.

Theorem 4. There exists no mechanism that simulta-

neously satisfy NOM and MB.

Proof. Consider the social network shown in Fig. 1,

with two students, s

and s

, both of whom have pref-

erence c ≻ ∅. Note that θ

is deﬁned here as tuple

= (c

≻

∅, {s

}). There is one college c

with

priority s

≻

and quota q

= 1.

When s

sincerely reports θ

, implying that stu-

dent s

is invited, there is at least a case where s

re-

ports a preference c

≻

∅. In such a case, any mech-

anism satisfying MB must match s

to c

and leave s

unmatched. This is clearly the worst possible case for

. Thus, W

≻

(θ

) = ∅ holds.

On the other hand, when s

reports θ

′

= (c

≻

∅,

0), i.e., decides not to invite student s

, MB implies

that s

is matched to c

. Therefore, W

≻

(θ

′

) = c

holds. Since her true preference assumes c

≻

∅,

≻

(θ

′

) ≻

≻

(θ

) holds, violating NOM.

Theorem 5. There exists no mechanism that simulta-

neously satisfy NOM, FR, and MB-D.

Proof. Consider the social network shown in Fig. 1,

with two students, s

and s

, both of whom have pref-

erence c ≻ ∅. There is one college c

with priority

≻

and quota q

= 1.

When student s

does not invite student s

, the

MB-D condition requires that s

is matched to c

Thus, the worst-case outcome when s

does not in-

vite s

is c

. To guarantee NOM, it must be the case

that the worst-case outcome when s

sincerely reports

her true type, i.e., invites s

, must be also c

. How-

ever, if s

is assigned to college c

, student s

has a

justiﬁed envy with respect to priority toward s

, since

we have s

≻

. This is a violation to FR.

Theorem 6. There exists no mechanism that simulta-

neously satisfy NOM, FRN, and WNW.

Proof. Consider three students s

, s

and two col-

leges c

and c

. The social network among students

are given as Fig. 2, and their preferences are given as

≻ c

≻ ∅. Colleges have capacity q

= q

= 1,

and their priorities are given as s

≻ s

. Assume

that all the three students report their types truthfully.

To guarantee WNW, two students must be assigned to

two colleges, one-by-one. In other words, exactly one

student is unmatched.

If student s

is unmatched, she has a justiﬁed envy

with respect to network toward s

, since both colleges

prefer s

to s

. Thus it violates FRN. If student s

is unmatched, s

has a justiﬁed envy with respect to

network toward s

, which violates FRN.

Finally, if student s

is unmatched, consider the

worst-case outcomes. When s

invites s

, s

is un-

matched in the worst-case outcome. On the other

hand, if s

does not invite s

, s

is matched to c

the worst-case outcome, to guarantee FRN. Thus, the

worst-case condition of NOM is violated for s

These impossibility results have a quite negative

implication. Given original impossibility theorems

presented by Cho et al. (2022), it is quite natural to

Strategy-Proofness and Non-Obvious Manipulability of Top-Trading-Cycles with Strategic Invitations

619

consider designing new two-sided matching mecha-

nisms satisfying all the requirements in the theorems

except for SP, and guaranteeing NOM instead. Our

impossibility results therefore show that such a natu-

ral direction never provides any positive ﬁnding.

4 ACHIEVING

STRATEGY-PROOFNESS

We now consider restricting the target instances.

More precisely, we will give a necessary and sufﬁ-

cient conditions on parameters (e.g., colleges’ prior-

ities, quotas, and the underlying social network) for

TTC to satisfy SP and/or NOM. This section focuses

on SP, and the next section focuses on NOM.

Theorem 7. TTC satisﬁes SP if and only if

∀s ∈ S, ∀c s.t. q

≤ #ν

c,s

, ∀s

′

∈ δ

(θ

′

), s ≻

′

holds, where ν

c,s

:= {s

′′

∈ S | s

′′

̸= s, s

′′

≻

s}.

The notation ν

c,s

denotes the set of students s

′′

that

is more prioritized than student s at college c. The

condition intuitively requires that any student s has

a higher priority than any of her descendants at any

college c. Only the exception is where college c has

an enough capacity so that c can still accept student s

after accepting some of s’s descendants.

Proof. We ﬁrst show the sufﬁciency. When the con-

dition holds, we can guarantee that s is matched to

such a college c before any of her descendants s

′

∈ δ

Thus, s’s invitation strategy r

′

never affects the as-

signment of s. Also, if c has enough capacity, s can

be assigned to c under truth-telling. Thus, strategy-

proofness is guaranteed.

We then show the necessity. Assume that ∃s ∈ S,

∃c s.t. q

≤ #ν

c,s

, and ∃s

′

∈ δ

(θ

′

), it holds that s

′

≻

s. Now consider the case where student s

′

is a de-

scendant of student s, both s and s

′

have preference

c ≻ ∅ ≻ ···, arbitrarily chosen q

−1 students among

c,s

\ {s

′

} have preference c ≻ ···, and all the other

students have preference ∅ ≻ ···. In such a case, stu-

dent s is assigned to college c when she choose not to

invite s

′

(more precisely, the unique child of s who is

an ancestor of s

′

), but left unmatched when she invites

′

. This is a violation to SP.

5 ACHIEVING NON-OBVIOUS

MANIPULABILITY

We then consider achieving NOM by the TTC mech-

anism. The following theorem provides the necessary

and sufﬁcient condition for TTC to satisfy NOM.

Theorem 8. TTC satisﬁes NOM if and only if either

∀s ∈ S, ∀c s.t. q

≤ #ν

c,s

, ∀s

′

∈ δ

(θ

′

), s ≻

′

∀s ∈ S, ∀β

⊆ α

:= {c ∈ C | c ≻

∅},

∑

c∈β

≤ #

[

c,s

holds, where ν

c,s

:= {s

′′

∈ S | s

′′

̸= s, s

′′

≻

s} and

c,s

:= {s

′′

∈ S \ δ

(θ

′

) | s

′′

̸= s, s

′′

≻

s}.

The ﬁrst condition is exactly identical with The-

orem 7, since SP implies NOM. On the other hand,

the second condition requires that, for any subset β

of colleges that student s is willing to go, the sum

of their quotas must not exceed the number of stu-

dents that are more prioritized than s in at lease one

of these colleges. This condition resembles the well-

known Hall’s marriage theorem (Hall, 1935). Indeed,

our proof strategy for sufﬁciency essentially searches

a perfect matching between students and colleges.

Proof Sketch. First of all, it is obvious that the best-

case condition of NOM holds for every student s ∈ S;

consider the case where all the students except s re-

ports ∅ ≻ c for every c, and invites all their neigh-

bors. Then, every student except s points to herself,

and thus s is matched to the college that she ranks the

best, by deﬁnition of TTC. In other words, the LHS

of the best-case condition of NOM is the most pre-

ferred college of s. Therefore, the best-case condition

holds regardless of what the RHS is. Furthermore,

from Proposition 1 and the fact that TTC is resilient

to preference misreport, we can restrict our attention

to showing NOM only for the diffusion strategy.

Therefore, it sufﬁces to show that the worst-case

condition of NOM, restricted only for the diffusion

strategy, is satisﬁed if and only if the given conditions

are satisﬁed. From now on we will show both the

sufﬁciency and the necessity.

About the sufﬁciency, it is obvious that TTC sat-

isﬁes NOM, and even SP, if the ﬁrst condition is sat-

isﬁed, as we have already shown in Theorem 7. Here

we show that, the second condition is also sufﬁcient

to guarantee the worst-case condition of NOM, re-

stricted only for the diffusion strategy. As lemma 9

shows, not inviting any student achieves the best-

possible worst case. However, if the second condition

holds, we can ﬁnd a proﬁle of reports θ

′

−s

under which

all the remaining sheets in α

are ﬁlled. Thus, the

RHS of the worst-case condition of NOM is ∅. Since

TTC never assigns students to any college that they

are not willing to go, the RHS of the worst-case con-

dition of NOM is weakly better than ∅, which com-

plete the proof for the sufﬁciency.

We then show the necessity. Due to its complexity,

we will show an example and explain the intuition.

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

620

See Fig. 3, where there are four students s

, . . . , s

. We

assume that there are two colleges c

and c

, whose

quotas are set as q

= 2 and q

= 1. Now consider

student s

’s strategic invitation.

Assume that student s

’s preference is given as

≻ c

≻ ∅, and colleges’ priorities are given as

: s

≻ s

: s

≻ s

Here, s

≻

violates the ﬁrst condition of NOM,

and the second condition for NOM is also vio-

lated for s

and β

:= {c

, c

};

∑

c∈β

= 3 and

c∈β

c,s

= 2.

In this example, the worst-case outcome when s

sincerely invites s

is that both s

and s

are assigned

to college c

and s

is assigned to c

. On the other

hand, the worst-case outcome when s

does not invite

is that both s

and s

are assigned to c

and s

assigned to c

. In other words, the LHS of the worst-

case condition of NOM is c

, and the RHS is c

. Since

≻

holds, NOM is violated, which concludes

the sketch of the proof.

The following proposition and lemma are used in

the proof of Theorem 8.

Proposition 1. Assume a mechanism µ is resistant to

preference misreport for every student s, i.e., telling a

true preference dominates telling a false preference,

when s’s invitation strategy and all the other students’

reports are ﬁxed. Then, µ satisﬁes NOM if and only if

µ satisﬁes NOM only for the diffusion strategies.

Proof. The only-if direction is obvious, since we re-

strict possible manipulations by student s. We then

show the if direction. For the sake of contradiction,

assume that, under such a mechanism µ, there exists

student s, s’s true type θ

:= (≻

, r

), and s’s misreport

′

:= (≻

′

, r

′

) ∈ R(θ

) such that either

≻

(θ

′

) ≻

≻

(θ

) or W

≻

(θ

′

) ≻

≻

(θ

Note that µ is resistant to preference misreport.

Therefore, for a type θ

′′

:= (≻

, r

′

), both

≻

(θ

′′

) ≿

≻

(θ

′

) and W

≻

(θ

′′

) ≿

≻

(θ

′

)

holds; even for any ﬁxed θ

′

−s

, it holds that

(θ

′′

, θ

′

−s

) ≿

(θ

′

, θ

′

−s

). Thus, either

≻

(θ

′′

) ≿

≻

(θ

′

) ≻

≻

(θ

)

≻

(θ

′′

) ≿

≻

(θ

′

) ≻

≻

(θ

)

holds, which contradicts the assumption that µ satis-

ﬁes NOM for the diffusion strategies.

Figure 3: Example with Four Students.

Lemma 9. Given type θ

:= (≻

, r

) of student s ∈ S,

consider a type misreport θ

′

:= (≻

, r

′

) ∈ R(θ

) in

which student s is not misreporting her preference

and just consider changing her invitation strategy r

′

Then, under the TTC mechanism, the worst-case out-

come W

≻

(θ

′

) becomes the most preferred by student

s when s does not invite any student.

Proof Sketch. To consider the worst-case for student

s, we can arbitrarily change the preference of all the

other invited students. Therefore, from the deﬁni-

tion of TTC, when student s invites some students

′

, we can imitate the case where s does not invite

′

by just setting the preference ≻

′

such that ∅ ≻ ···.

In other words, there are weakly more possible out-

comes when s invites s

′

, implying that the worst-case

outcome is weakly worse when s invites s

′

The following example explains what the neces-

sary and sufﬁcient condition requires, and shows that

how TTC violates SP and satisﬁes NOM.

Example 1. Assume there are four students,

, . . . , s

, and two colleges, c

and c

. The social net-

work among students are given as Fig. 3, and student

has a preference c

≻ c

≻ ∅. Colleges have ca-

pacity q

= q

= 1, and their priorities are:

: s

≻ s

: s

≻ s

Note that s

≻

violates the condition in Theo-

rem 7, but satisﬁes the other condition in Theorem 8

for every s ∈ S. For example, if we choose β

, c

}, both

∑

c∈β

= 2 and #

c∈β

c,s

= 2

hold, which do not violate the other condition.

Here, a proﬁle of preferences of s

, . . . , s

exists,

under which s

has an incentive not to invite s

, say,

: c

≻ c

≻ ∅ s

: c

≻ ∅ ≻ c

: c

≻ c

≻ ∅

Student s

is unmatched under truth-telling, but would

be matched to c

if she does not invite s

However, student s

cannot get a better outcome in

both of the best- and worst-cases; s

is assigned to c

in the best-case under her sincere preference report,

and s

is left unmatched in the worst-case even if she

does not invite s

, e.g., consider the case where s

still willing to go to college c

Strategy-Proofness and Non-Obvious Manipulability of Top-Trading-Cycles with Strategic Invitations

621

Figure 4: An Example Showing that IBSI-TTC Violates SP.

6 NEW MECHANISM

SATISFYING NOM

As we have already shown in Section 3, just achiev-

ing NOM with other desirable properties is difﬁcult.

Therefore, in this section we ignore other properties

and focus on achieving NOM.

Deﬁnition 7 (Invitation-Based Stepwise-Improving

TTC (IBSI-TTC)). Let τ be an empty matching un-

der which no student is assigned to any college. Then,

in each Phase p = 1, 2, . . ., apply TTC for all the stu-

dents with distance less than or equal to p, except for

those who are at distance one and do not invite any

student. Let τ

be the outcome of Phase p. If all the

students who invite at least one student weakly prefers

to τ, then let τ := τ

; otherwise keep the current τ.

If all the students are at distance less than or equal to

p, the algorithm terminates and returns τ; otherwise

go to Phase p + 1.

The following example demonstrates how the new

algorithm works, and shows that it violates SP.

Example 2. There are six students, s

, . . . , s

, and

three colleges, c

, . . . , c

. The social network are

given as Fig. 4, and preferences are:

: c

≻ ∅ ≻ c

≻ c

: c

≻ c

≻ ∅ ≻ c

: c

≻ c

≻ ∅ ≻ c

: c

≻ c

≻ ∅ ≻ c

: c

≻ c

≻ ∅ ≻ c

: ···

Colleges have capacity q

= q

= 1, and their

priorities are given as follows:

: s

≻ s

: s

≻ s

: s

≻ s

Note that s

is a direct child of o and does not

invite any student. Such a student is ignored at all in

IBSI-TTC, and thus left unmatched in any case.

In Phase 1, IBSI-TTC applies TTC for s

. In the

tentative matching τ

, s

is matched to c

. Since this

is better than ∅ for s

, let τ := τ

In Phase 2, IBSI-TTC applies TTC for s

, s

, and

. Under the tentative outcome τ

, s

is matched to

, s

is matched to c

, and s

is matched to c

. Since

is weakly better than τ for both s

and s

, who in-

vited at least one student, let τ := τ

In Phase 3, IBSI-TTC applies TTC for all the stu-

dents. Under the tentative outcome τ

, s

is matched

to c

, s

is matched to c

and s

is matched to c

Since this outcome τ

is weakly better than (more pre-

cisely, identical to) the current τ for s

and s

, let

τ := τ

. We then get the ﬁnal outcome τ which assigns

to c

, s

to c

, and s

to c

Now consider the case where s

decides not to in-

vite s

. Then, Phase 3 differs from the above; under

the tentative outcome τ

in this case, s

is matched to

, s

is matched to c

and s

is matched to c

. This

is weakly better than the current τ for both s

and

. Thus, the ﬁnal outcome assigns s

to c

, which is

strictly better for s

than the above case. violating SP.

Intuitively, this mechanism keeps updating the

outcome by increasing the number of participating

students, based on the agreement of those who invited

at least one student. Thus, the candidate of ﬁnal out-

come, which is represented as τ in the description, is

weakly monotonically increasing for them. As a re-

sult, the following lemma holds. Due to the space

limitations, we omit the proof.

Lemma 10. For any direct child s ∈ r

who invites

at least one student, the ﬁnal outcome of IBSI-TTC is

weakly better than the ﬁrst tentative outcome τ

Indeed, the following theorem shows that IBSI-

TTC satisﬁes NOM, while it violates SP. As far as the

authors know, this is the ﬁrst example of mechanisms

that satisfy NOM but violate SP under the two-sided

matching model with information diffusion; the three

mechanisms proposed by Cho et al. (2022) satisﬁes

SP (or even stronger incentive properties), which im-

plies that they also satisfy NOM.

Theorem 11. IBSI-TTC satisﬁes NOM.

Proof. There are three categories of students; (i) stu-

dents who are not direct children of o (corresponding

to students s

, . . . , s

in Example 2), (ii) students who

are direct children of o and have at least one child (s

in Example 2), and (iii) students who are direct chil-

dren of o but do not have children (s

in Example 2).

First of all, for those students in the category (iii),

both the best- and worst-case conditions clearly hold,

since such a student is always unmatched, regardless

of what the other students report. In other words, both

LHS and RHS are ∅, in both of the conditions.

It is also obvious that the best-case condition of

NOM holds for every student s ∈ S in the categories

(i) and (ii); consider the case where all the students

except s reports ∅ ≻

c for every college c, and in-

vites all their neighbors. In this case, s is matched to

the college that she most prefers. In other words, the

LHS of the best-case condition of NOM is the most

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

622

preferred college of s. Therefore, the best-case condi-

tion holds regardless of what the RHS is.

We now show that the worst-case condition holds

for every s ∈ S in the categories (i) and (ii). For (i)

students s ̸∈ r

, when the parent of s does not invite s,

s is unmatched. Since IBSI-TTC never matches any

student to a college that she is not willing to go, this

is a worst-case, regardless of what s reports. Thus,

both LHS and RHS of the worst-case condition is ∅,

implying that the worst-case condition holds.

For (ii) students s ∈ r

who have at least one child,

let θ

:= (≻

, r

) be the true preference of s. Phase 1

of IBSI-TTC is resistant to preference misreport, from

the known property of TTC in the traditional setting.

Thus, when s arbitrarily misreports her preference, τ

gets weakly worse. Also, the tentative outcome at

Phase 1 is identical, regardless to s’s invitation strat-

egy r

′

. Let τ

′

be the tentative outcome at Phase 1 un-

der s’s arbitrary misreport θ

′

. We then have τ

≿

′

Furthermore, there is a possible proﬁle of reports

′

−s

∈ R(θ

−s

) by the other students so that students

in distance larger than one prefer to be unmatched.

Under such a proﬁle, the ﬁnal outcome coincides with

the Phase 1 outcome. That is, τ

is an upper bound of

the worst-case outcome, even from the viewpoint of

true preference ≻

of s, implying both

≿

≻

(θ

) and τ

′

≿

≻

(θ

′

Also, Lemma 10 implies W

≻

(θ

) ≿

. Thus,

≻

(θ

) ∼

≿

′

≿

≻

(θ

′

) holds, which guaran-

tees that the worst-case condition of NOM holds.

IBSI-TTC violates FR, FRN, WNW, MB, and

MB-D. This is mainly due to two facts. First, it is

based on TTC, which violates FRN (and thus FR).

Second, it ignores direct children of o if they do not

invite any students, which is totally wasteful and re-

sults in an unstable outcome. However, we believe

that the idea behind the IBSI-TTC mechanism, guar-

anteeing NOM only and not paying too much atten-

tion to achieve SP, will be a useful building block for

designing better NOM mechanisms in the future.

7 CONCLUDING REMARKS

Our model of two-sided matching with strategic invi-

tation is limited in the sense that the social network

among students are restricted as a tree-shaped. Han-

dling more general structures would be a promising

future direction. For each impossibility theorem, it

would also be required to show the independence of

the properties by providing mechanisms satisfying all

except one properties, though we strongly believe that

they are independent. Diffusion mechanism design

is still a new and developing model of mechanism

design. We believe there are further various exten-

sions to achieve relatively positive results, including

restricting preferences and allowing randomization.

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