Passively-Strictly Strong Nash Equilibrium in a Preference Revelation Game under the Student-Optimal Deferred Acceptance Algorithm

We revisit a college admission market and a related preference revelation game under the student-optimal deferred acceptance algorithm (SODA). Previous research has demonstrated the existence of a strictly strong Nash equilibrium (SSN) based on either an iterative deferred acceptance algorithm (DASSN) or the core of a corresponding house allocation problem (Core-SSN). We propose a new equilibrium concept called passively-strictly strong Nash equilibrium (P-SSN). It rules out a kind of deviation called passively weak deviation which includes students who were threatened to deviate. Then we show two preliminary existence results about P-SSN. (i) If the DA-SSN and the Core-SSN are not equivalent, then neither of them is a P-SSN. (ii) If the matching determined by the DA-SSN satisfies a property called irrelevance of low-tier agents, then the DA-SSN is also a P-SSN.


Introduction
A college admission market is usually formulated as a many-to-one matching problem.In this problem, each college can admit multiple students but each student can attend at most one college.This model adopts a primary solution concept called stable matching.We can use the student-optimal deferred acceptance algorithm (SODA) which is proposed by Gale and Shapley [1] to find a C. Y. Li et al. stable matching that all students weakly prefer it to any other stable matching 1 .
An important research topic on this problem is studying students' strategic behavior.Specifically, a college admission market can be formulated as a strategic game in which each college's true preferences over students are known publicly, but each student's preferences over colleges are revealed strategically.This game adopts a primary solution concept called strong Nash equilibrium (SN).It is a strategy profile under which none of the student groups can make all its members strictly better off by changing their joint strategies.Dubins and Freedman [3] demonstrated that the truth-telling strategy profile is a SN.However, Bando [4], Dubins and Freedman [3] and Sotomayor [5] all showed that it does not satisfy a stronger equilibrium concept, in the sense that none of the student groups can make all its members weakly better off and at least one member strictly better off by changing their joint strategies.
Bando [4] formally named this equilibrium concept as strictly strong Nash equilibrium (SSN).He then demonstrated the existence of two SSNs.The first one is obtained by iteratively applying the standard deferred acceptance algorithm (DA) so that we simply call it DA-SSN.This SSN has a very nice property that it always exists.Moreover, alternative algorithms can be found in Kesten [6] and Tang and Yu [7].The other one is obtained by constructing a corresponding house allocation problem which is defined in Shapley and Scarf [8] and then identifying the strict core.We simply call it Core-SSN.We note that its existence depends on whether the strict core exists.Under SODA, both of DA-SSN and Core-SSN yield a matching that weakly Pareto dominates the matching determined by students' true preferences.
In this research, we propose a new equilibrium concept called passivelystrictly strong Nash equilibrium (P-SSN).It is stronger than both DA-SSN and Core-SSN.It rules out a deviation called passively weak deviation.Such a deviating coalition includes members who become strictly worse off, which means such a student was actually unwilling to deviate.But if not doing so, she will receive an even worse outcome caused by the unilateral deviation of the rest members.In this sense, there exist students who "passively'' joined the deviating coalition.
Then we show two preliminary existence results about P-SSN.(i) If the DA-SSN and the Core-SSN are not equivalent, then neither of them is a P-SSN.
(ii) If the matching determined by the DA-SSN under SODA satisfies a property called irrelevance of low-tier agents, then it is also a P-SSN.In general, P-SSN is a quite restrictive equilibrium concept.But nevertheless, we will discuss some

College Admission Market
Let ( ) , , , ,  be the restriction of c  to singleton sets and the empty set.For each i I ∈ and c C ∈ , we say that i is ac- Let c q be the quota of college c.Let ( ) We assume that each college c's preferences satisfy responsiveness with quota c q which is defined in Roth [9] 2 .That is, c is strictly better off by replacing any student with a more preferred acceptable student in * c  , and if c has a vacant seat, then it is strictly better off by adding an acceptable student and worse off by adding an unacceptable student.Then we identify c  with * c  because only * c  is relevant to our analysis.

Given S I
⊆ and c C ∈ , each college c's choice set ( ) Ch S  is defined as the set which satisfies (i) A matching is defined as a function µ from I into C I  which satisfies (i) for each c C ∈ , ( ) ≤ and (ii) for each i I ∈ , ( ) We say that a matching µ is individually rational for students if ( ) for all i I ∈ , and is individually rational for colleges if ( ) ( ) ( ) ∈ .A matching is individually rational if it is individually rational for both students and colleges.We say that a matching µ is blocked by ( ) stable if it is individually rational and is not blocked.

Preference Revelation Game
be a strategic game defined on a college admission market ( ) , , , , . I is the set of players.Given a strategy profile I ′  , the outcome of this game is determined by SODA and is denoted be ( ) . Each student i I ∈ evaluates the outcome according to her true preferences i  .
For each i I ∈ , define  Dubins and Freedman [3] showed that the truth-telling strategy profile I  is always a SN, but it may not be a SSN.Hence the existence problem of SSN becomes one of the focuses in the later research.Bando [4] demonstrated the existence of the following two SSNs.

DA-SSN
Given a market ( ) , , , , be the profile of the restricted preferences.Then the iterative DA is defined as follows.
and proceed to the next step.
• Step ( )   and proceed to the next step.
This algorithm terminates in finite steps and yields a strategy profile ( ) , which is the DA-SSN.

Core-SSN
Given a market ( ) , , , , is a preference relationship for * I satisfying: (i) for any where i is acceptable for ( ) An allocation is defined as a bijection ( ) If x is a strict core allocation, then the strategy profile ( )  is the Core-SSN.Particularly, we note that this result depends on whether the strict core exists.
Under SODA, both of the DA-SSN and the Core-SSN yield a matching which is Pareto efficient, and weakly Pareto dominates the matching determined by students' true preferences.

Equilibrium Concept
In this section, we propose a new equilibrium concept which is stronger than SSN (and thus it is stronger than both DA-SSN and Core-SSN).First of all, we use a simple example to show how it is defined.
, , , , , , , C c c c c = and q q q q = = = = .The students' preferences and the colleges' preferences are given in the following table., , ,  i c c c The student-optimal stable matching µ is:

i i i i i
Consider the following deviating strategies However, if 4 i agrees to deviate and play the following strategy: , , , , , , yields the following matching ν ′ : We note that 4 i is strictly worse off in ν ′ than in * µ since ( ) . But nevertheless, she is strictly better off in ν ′ than in ν since ( ) . On the other hand, we note that ( ) ( ) In this sense, we can regard as 1 i 's and 2 i 's "threatening" strategy profile which forces 4 i to join the deviating coalition in order to avoid an even worse outcome caused by the unilateral deviation of them.
(Core-SSN) By constructing a house allocation problem as introduced in Section 2.2, we obtain the following strategy profile Core I  :

i i i i i
Consider the following deviating strategy , , , , , , However, if 1 i agrees to join 2 i and play the following deviating strategy: yields the following matching ν ′′′ :

i i i i i
Similarly, we observe that ( ) ( ) We can regard 2 i ′′  as 2 i 's threatening strategy which forces 1 i to deviate in order to avoid an even worse outcome caused by her unilateral deviation.
Example 1 presents a kind of deviating coalition in which some members were actually unwilling to deviate.They deviate not for achieving a better outcome, but for avoiding an even worse outcome which is caused by the unilateral deviation of the rest members.We name this deviation as passively weak deviation and give the formal definition as follows.( ) In a passively weak deviating coalition S, only a subgroup of members T are at least weakly better off.For each member in \ S T , although joining S will bring her an outcome which is worse than that in a SSN, she has to do so because the outcome caused by T's unilateral deviation is even worse for her.In this sense, each student in \ S T "passively" chooses to deviate.We say that a strategy profile * I  is a passively-strictly strong Nash equilibrium (P-SSN) if each coalition does not have any passively weak deviation.
We note that T S = holds when ( ) ( )  holds for all i S ∈ , and thus ( ) ( ) holds for all i S ∈ in this case.This means a weak deviation must also be a passively weak deviation, and thus a P-SSN must be a SSN.However, Example 1 shows that the converse may not be true.

Existence
In this section, we study the existence problem of P-SSN.Particularly, we examine under which conditions a DA-SSN or a Core-SSN would become a P-SSN.
We first introduce two strategies that will support our next results.One is the c-bottom strategy proposed by Bando [4].Given a market ( ) , , , , the set of colleges that i weakly prefers to c under i ′  .Then each i's -bottom Definition 2. (c-bottom strategy, Bando [4]) . We are interested in ( ) Here we note that the deviating coalition's joint strategies have a quite simple structure.Roth [10] and Bando [4] demonstrated that if a coalition S has a successful deviation (either in a strong sense or in a weak sense), then it equals to using a simplified strategy that each student in S reports the deviating outcome as her first choice.Therefore, we also consider this kind of simple deviation in this research.
Based on the previous results, we are ready to introduce our first finding.We show that if the DA-SSN and the Core-SSN yield different matchings under SODA, then neither of them is a P-SSN.In other words, we can always construct a passively weak deviation at either the DA-SSN or the Core-SSN.
Proposition 1.Let ( ) , , , , . Suppose that the Core-SSN exists, and let ( ) Proof.(i) We first show that we can construct a passively weak deviation at Core I  .By assumption, there exists i I ∈ such that ( ) ( ) as the set of students who weakly prefer c to their assignments.
Consider the strategy . We will show that ( ) ( ) Suppose that there exists i I ′ ∈ such that ( ) ( )  holds for all i I ∈ .This implies that ( ) ( ) By the stability of ν , we have that  .We note that ( ) ( ) ( ) ∈ .This implies that ν is also stable in the market ( ) , , , , . However, µ is the student-optimal matching in this market, which implies that ( ) ( ) Hence the only possibility is that ( ) ( )  for all i S + ∈ .The proof goes almost the same as that in (i), and thus we omit it here.□ However, the Core-SSN usually fails to exist since it depends on the existence of the strict core in the corresponding house allocation problem.Therefore, our second result focuses on the DA-SSN.We show that under SODA, if the DA-SSN yields a matching which satisfies a condition called irrelevance of low-tier agents, then it is also a P-SSN.DA q µ =   .Suppose that the iterative DA ends in K steps such that 1 K ≥ .Denote the set of last proposers in each step be k L such that 0 k K ≤ ≤ .We say that * µ satisfies irrelevance of low-tier agents if and only if pick any i I ∈ such that k i L ∈ , we have that ( ) ( ) This condition requires each student not to accept the assignment of any student who was identified as the last proposer in an earlier step than she was.
The next lemma is a direct application of Lemma 1.  Proof.Suppose that DA I  is not a P-SSN.Then there exists a coalition S and ( ) Otherwise, S has a weak deviation from DA I  and it cannot be a SSN.

Let
Suppose that there exists

Conclusion and Discussion
In this research, we revisit a college admission market and a related preference revelation game under SODA.We propose a new equilibrium concept called passively-strictly strong Nash equilibrium (P-SSN).It rules out a deviating coalition called passively weak deviation that includes members who become strictly worse off.In other words, they were actually unwilling to deviate.But if not doing so, they will receive an even worse outcome which is caused by the unilateral deviation of the rest members.In this sense, some students were threatened to join the deviating coalition.Then we show two preliminary existence results about P-SSN.
In general, P-SSN is a quite restrictive equilibrium concept that easily fails to exist.But nevertheless, we plan to examine the following two extensions of Definition 1. (i) ( ) ( ) ′  for all i T ∈ .This condition requires students who have threatened others to join the deviating coalition not become worse off even if they deviate unilaterally.That is, their threatening strategy will not hurt themselves even if the target students do not cooperate.(ii) ( ) ( ) j j j ν ν′  for all \ j S T ∈ . That is, each student who deviates passively should receive a strictly better outcome than that caused by the rest students' unilateral deviation.With those assumptions, we can check that the SSNs in Example 1 would become P-SSN.Such equilibrium concepts give inspiration to the refinement problem when multiple equilibria exist, which we will study in the future research.
extensions that possibly give inspiration to the refinement problem when multiple SSNs exist.The rest of this paper is organized as follows.Section 2 introduces a college admission market and a related preference revelation game under SODA.Section 3 defines our equilibrium concept with an example, and shows two preliminary results about its existence problem.Section 4 concludes and discusses 1 See Roth and Sotomayor [2] for a comprehensive introduction.C. Y. Li et al. 1246 some possible directions for the future research.
admission market.I is a finite set of students, and C is a finite set of colleges.Each student i has strict preferences i  over profile.Particularly, let * c


coalition does not have any weak deviation.When ( holds for all i S ∈ , we say that S has a strong deviation at * I  , and * I  is a SN if each coalition does not have any strong deviation. Next we look into the DA-SSN and the Core-SSN for this example.(DA-SSN)Applying the iterative DA which is introduced in Section 2.2, we obtain the following strategy profile DA I

Definition 1 .
(Passively weak deviation) Given a strategy profile S I ⊆ , we say that S has a passively weak deviation at *

iµ
-bottom strategy related to the true preference i  ( ( ) i µ -bottom strategy for short).Bando[4] further demonstrated the following results about ( ) i µ -bottom strategy.Lemma 1. (Bando [4]) Let ( ) be a market.Let µ be a stable matching in M .For each i I ∈ , let * i  be a ( ) i µ -bottom strategy re- lated to i  .Consider a coalition S and ( )

Definition 3 .
(Irrelevance of low-tier agents) Let µ satisfies irrelevance of lower-tier agents, then DA I  is a P-SSN.
and hence a contradiction occurs with the stability of * µ .On the other hand, since * µ satisfies irrelevance of low-tier agents,