Citation Context

...over P is NP-complete. 12 Proof. Since testing membership in P, the statements follows from the equivalence between optimizing over a poltope and implementing a separation oracle over over a polytope =-=[16]-=-. Although we have shown that testing whether a given assignment is ex post efficiency is NP-complete, we leave open the case when the assignment is uniform. We now show that in the random assignment ...

Citation Context

...ithm that can decompose any given random assignment (represented by a bistochastic matrix) into a convex combination of at most n2−n+ 1 deterministic assignments (represented by permutation matrices) =-=[18]-=-. 6 Birkhoff’s Algorithm. Birkhoff’s algorithm works as follows. We initialize i to 1. For a bistochastic matrix M , a permutation matrix Pi with respect to M is guaranteed to exist. M is set to M − λ...

Citation Context

...at depth in recent years [3, 9, 19, 4]. Similar analysis has been done for SD-efficient assignments where it has been shown that not only can an SD-efficient random assignment be computed efficiently =-=[13]-=-, a linear programming formulation can be used to check whether an assignment is SD-efficient or not [5]. However, to the best of our knowledge, 2 the complexity of testing ex post efficiency has not ...

Citation Context

...signment depends on the actual values. Keywords: Random assignment, efficiency JEL: C62, C63, and C78 Pareto optimality has been termed the “single most important tool of normative economic analysis” =-=[20]-=-. It appeals to the idea that there should not exist another possible outcome different from the social outcome which all the agents prefer. We consider Pareto optimality in the random assignment prob...

Citation Context

...o2, o3, o4} and the preferences % are as follows. 1 : o1, o2, o3, o4 2 : o1, o2, o3, o4 3 : o2, o1, o4, o3 4 : o2, o1, o4, o3 A random assignment p is a (n × n) matrix [p(i)(oj)] such that p(i)(oj) ∈ =-=[0, 1]-=- for all i ∈ N , and oj ∈ O, ; ∑ i∈N p(i)(oj) = 1 for all oj ∈ O; and∑ oj∈O p(i)(oj) = 1 for all i ∈ N . The value p(i)(oj) represents the probability of object oj being allocated to agent i. Each row...

Citation Context

... deterministic assignments is that they can allow for better ex ante fairness. It is well-known that any random assignment can be a result of a probability distribution over deterministic assignments =-=[12]-=-. A random assignment also a useful time-sharing interpretation whereby the probability of agent i getting object o is the fraction of time he will be matched to object o [see e.g., 21, 14]. In this p...

Citation Context

...which each agent weakly prefers and some agent strictly prefers with respect to the stochastic dominance relation. Ex post efficiency is a weaker requirement than stochastic dominance (SD) efficiency =-=[17]-=-. The main research problem in this paper is to understand the structure and complexity of efficient assignments in particular ex post efficiency assignments. We not only consider ex post efficiency a...

Citation Context

...ural computational problems related to efficiency of random assignments. Previously, computational aspects of Pareto optimal deterministic assignments have been studied in great depth in recent years =-=[3, 9, 19, 4]-=-. Similar analysis has been done for SD-efficient assignments where it has been shown that not only can an SD-efficient random assignment be computed efficiently [13], a linear programming formulation...

Citation Context

...per is that testing ex post efficiency is NP-complete. The result contrast with the followings facts (1) ex post stability in the two sided marriage setting can be tested in polynomial time via an LP =-=[24]-=-; (2) SD-efficiency can be tested in polynomial time; and (3) an ex post efficient assignment can be computed in polynomial time. One implication of the NP-completeness result is that the set of ex po...

Citation Context

...ural computational problems related to efficiency of random assignments. Previously, computational aspects of Pareto optimal deterministic assignments have been studied in great depth in recent years =-=[3, 9, 19, 4]-=-. Similar analysis has been done for SD-efficient assignments where it has been shown that not only can an SD-efficient random assignment be computed efficiently [13], a linear programming formulation...

Citation Context

... e.g., 2, 5, 13]. We show that robust efficiency can be checked in polynomial time if there are a constant number of agent types. We highlight that the well-known random serial dictatorship mechanism =-=[8]-=- is not robust ex post efficient. Our finding strengthens the observation of Bo1The uniform assignment also satisfies other properties such a probabilistic consistency [15] 3 gomolnaia and Moulin [13]...

Citation Context

...ural computational problems related to efficiency of random assignments. Previously, computational aspects of Pareto optimal deterministic assignments have been studied in great depth in recent years =-=[3, 9, 19, 4]-=-. Similar analysis has been done for SD-efficient assignments where it has been shown that not only can an SD-efficient random assignment be computed efficiently [13], a linear programming formulation...

Citation Context

...SDi q(i) for all i ∈ N and p(i) SDi q(i) for all i ∈ C. Hence q is not SD-efficient. SD-efficiency is a strictly stronger concept than robust ex post efficiency. Although Abdulkadiroğlu and Sönmez =-=[2]-=- did not explicitly define the concept robust ex post efficiency, they showed that a random assignment that has only one decomposition which is a randomization over Pareto optimal assignments does not...

Citation Context

...erial dictatorship mechanism [8] is not robust ex post efficient. Our finding strengthens the observation of Bo1The uniform assignment also satisfies other properties such a probabilistic consistency =-=[15]-=- 3 gomolnaia and Moulin [13] that random serial dictatorship is not SD-efficient. We show that whereas robust ex post efficiency is combinatorial, ex post efficiency is not. The finding that ex post e...

Citation Context

...te. Proof. The following problem is NP-complete: SerialDictatorshipFeasibility — check whether there exists a permutation of agents for which serial dictatorship gives a particular object to an agent =-=[22]-=-. We present a reduction from SerialDictatorshipFeasibility to the problem of checking whether there exists a Pareto optimal deterministic assignment consistent with the given random assignment Consid...

Citation Context

...ment such that q(i)(o) > 0 if and only if p(i)(o) > 0. Then in each Pareto optimal matrix that can be used to decompose 3The notion of an efficiency concept being combinatorial was first discussed in =-=[7]-=-. However, the setting was voting and not the random assignment problem. In voting, a lottery over alternatives is ex post efficient iff the support consists of Pareto optimal alternatives. Hence in v...

Citation Context

...ural computational problems related to efficiency of random assignments. Previously, computational aspects of Pareto optimal deterministic assignments have been studied in great depth in recent years =-=[3, 9, 19, 4]-=-. Similar analysis has been done for SD-efficient assignments where it has been shown that not only can an SD-efficient random assignment be computed efficiently [13], a linear programming formulation...

Citation Context

... it has been shown that not only can an SD-efficient random assignment be computed efficiently [13], a linear programming formulation can be used to check whether an assignment is SD-efficient or not =-=[5]-=-. However, to the best of our knowledge, 2 the complexity of testing ex post efficiency has not been settled. Testing ex post efficiency is also closely related to implemeting a random assignment with...