We look at the well-known deferred-acceptance mechanism in a college admissions

Question

We look at the well-known deferred-acceptance mechanism in a college admissions problem. A college admissions problem is a five-tuple (I,S,q,P, ) such that I is a finite set of students; S is a finite set of schools; q = (q_s)_sEpsilonS is a capacity (or quota) vector. Each college s Epsilon S has q_s available seats. We assume that |I|lessthanorequalto Sigma_sEpsilonS q_s. If q_s is large enough, say q_s = |I|, then the college s may represent the option of remaining unassigned (or an outside option such as a private college) as it can be assigned to everybody. P = (p_i)_iEpsilonI is a (strict) preference profile. P_i is a strict preference relation over S. Let P be the set of all strict preferences on S. Given p_i Epsilon P, s P_i s' means that student i strictly prefers s to s'. Let R_i and I_i be the weak preference relation and the indifference relation associated with P_i. As P_i is a strict preference relation, we have s R_i s' equivalent s P_i s' or s I_i s'. Finally, =( _s)_sEpsilonS is a (strict) preference profile. Each college s has a strict preference relation _i over I. A matching is a correspondence Mu: I Union S rightarrow S Union I such that each student is assigned only one college and each college is assigned up to its capacity, i.e., for each i Epsilon I and each s Epsilon S, we have Mu(i) Subset S, Mu(s) Subset I, |Mu(i)| = 1, and |Mu(s)| lessorthenequalto q_s. We denote Mu(i) = s instead of Mu(i) = {s} as Mu (i) is a singleton and we have Mu(i) = s if i Epsilon Mu(s). Let Mode be the set of all matching's. Define Pareto efficiency for the students of a matching at P and . Define stability of a matching at P and .

Explanation / Answer

An economic state is called Pareto efficient, where there is a very efficient manner to allocate the resources. It is obtained when without making one party's situation worse, another party's situation cannot be improved. Pareto efficiency does not imply equality or unbiasedness. If an economic state is Pareto Efficient, each person is maximizing their utility. If we are given a limited amount of resources, then the decision of final allocation cannot be enhanced without causing harm to one of the participants.

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