Abstract
In this paper, we study consistent enlargement of a solution. By computing it, one actually evaluates the extent to which the solution would have to be expanded in order to be well-defined and consistent. We show that the union of stable matchings and the matching recommended by a single-valued, well-defined, individually rational, and consistent solution is a minimal consistent enlargement of the core. Although individual rationality is sufficient it is not a necessity. Next, we show that for any fixed order on the set of agents in the society, the union of stable matchings and the serial dictatorship matching is a minimal consistent enlargement of the core.
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Notes
Note that, core extension is a particular extension of a core which picks exactly the core whenever it is non-empty.
The difference between consistent extension and consistent enlargement is that the latter must be a well-defined solution.
For any society \(A\), \(|A|\) denotes the cardinality of \(A.\)
The column \(\begin{array}{l} \underline{P_{i}} \\ j \\ k \\ l \end{array}\) represents \(j\,P_{i}\,k\,P_{i}\,l\).
For some problems \(p = (A,P)\in \mathbf {P},\,\varphi (p)\) can be emptyset.
As Gale and Shapley (1962) show \({\mathcal {S}}\) is not a well-defined solution.
The restricted preference profile \(P_{|A^{\prime }}\,\equiv (P_{i}^{\prime })_{i\in A^{\prime }}\) is a preference profile where for each agent \(i\in A^{\prime },\) a preference relation \(P_{i}^{\prime }\) over \(A^{\prime }\) is defined as for each \(i,j,k\in A^{\prime },\,j\,P_{i}^{\prime }\,k\) if and only if \(j\,P_{i}\,k\).
Note that \(\left( A^{\prime }\cup \mu \left( A^{\prime }\right) ,P_{|A^{\prime }\cup \mu \left( A^{\prime }\right) }\right) \ \)is simply the reduced problem of \(p\) with respect to \(A^{\prime }\cup \mu \left( A^{\prime }\right) .\)
Consistency implies weak consistency. But converse is not true. To see that consider a rule that is defined as follows:
$$\begin{aligned} \varphi (A,P)=\left\{ \begin{array}{l} PO(A,P)\quad \hbox {if}\,|A|>4 \\ \emptyset \quad \hbox {otherwise.} \end{array} \right. \end{aligned}$$Note that, \(\varphi \) satisfies weak consistency but it is not consistent.
Note that, minimal consistent core enlargement is not unique.
Note that, \(\tau (\widetilde{p})=\{(a_{1}),(a_{2}),(a_{3})\},\) for all other problems \(p\ne \widetilde{p},\) \(\tau (p)=(\mathcal {S}\vee \varphi _{\succ })(p)\).
References
Abraham, D. J., Biró, P., & Manlove, D.F., (2006). Almost Stable Matchings in the Roommate Problem. In T. Erlebach & G. Persiano (Eds.), Proceedings of WAOA 2005 (Vol. 3879, pp. 1–14)., Lecture Notes in Computer Science Berlin/Heidelberg: Springer.
Gale, D., & Shapley, L. (1962). College admissions and the stability of marriage. American Mathematical Monthly, 69, 9–15.
Inarra, E., Larrea, C., & Molis, E. (2013). Absorbing sets in the roommate problem. Games and Economic Behavior, 81, 165–178.
Inarra, E., Larrea, C., & Molis, E. (2008). Random paths to P-stability in the roommate problem. International Journal of Game Theory, 36, 461–471.
Klaus, B. (2013). Consistency and its converse for roommate problems. Cahier de recherches économiques du DEEP No. 13.12.
Klaus, B., Klijn, F., & Walzl, M. (2010a). Stochastic stability for roommate markets. Journal of Economic Theory, 145, 2218–2240.
Klaus, B., & Klijn, F. (2010b). Smith and Rawls share a room: Stability and medians. Social Choice and Welfare, 35, 647–667.
Nizamogullari, D., & Özkal-Sanver, İ. (2014). Characterization of the core in full domain marriage problems. Mathematical Social Sciences, 69, 34–42.
Özkal-Sanver, İ. (2010). Impossibilities for roommate problems. Mathematical Social Sciences, 59, 360–363.
Sasaki, H. (1995). Consistency and monotonicity in assignment problems. International Journal of Game Theory, 24, 373–397.
Sasaki, H., & Toda, M. (1992). Consistency and characterization of the core of two-sided matching problems. Journal of Economic Theory, 56, 218–237.
Sethuraman, J., & Teo, C. P. (2001). A polynomial-time algorithm for the bistable roommates problem. Journal of Computer and Systems Sciences, 63, 486–497.
Tan, J. J. M. (1990). A maximum stable matching for the roommate problems. BIT, 29, 631–640.
Thomson, W. (2013). Consistent Allocation Rules, University of Rochester, Unpublished monograph.
Thomson, W. (1990), The consistency principle. In Ichiishi, T., Neyman A., Tauman, Y., (Eds.), Game theory and applications. Proceedings of the 1987 international conference held at Ohio State University, Columbus, OH (pp.187–215.). New York: Academic Press.
Toda, M. (2006). Monotonicity and consistency in matching markets. International Journal of Game Theory, 34, 13–31.
Toda, M. (2005). Consistency and monotonicity in assignment problems. Games and Economic Behavior, 53, 248–261.
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Nizamogullari, D., Özkal-Sanver, İ. Consistent enlargements of the core in roommate problems. Theory Decis 79, 217–225 (2015). https://doi.org/10.1007/s11238-014-9470-z
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DOI: https://doi.org/10.1007/s11238-014-9470-z