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We consider both Nash and strong Nash implementation of various matching rules for college admissions problems. We show that all such rules are supersolutions of the stable rule. Among these rules the "lower bound" stable rule is implementable in both senses. The "upper bound" Pareto and individually rational rule is strong Nash implementable yet it is not Nash implementable. Two corollaries of interest are the stable rule is the minimal (Nash or strong Nash) implementable solution that is Pareto optimal and individually rational, and the stable rule is the minimal (Nash or strong Nash) implementable extension of any of its subsolutions. JEL Classification Numbers: C78, D78. IntroductionA class of public decision problems that has been extensively analyzed is the class of two-sided matching problems. (For an exposition of game theoretic modelling and analysis of two-sided matching problems see Roth and Sotomayor E20].) Most of the studies on these problems deal with one-to-one matching. Nevertheless in real life applications many-to-one matching is the most typical case, where one side consists of institutions and the other side consists of individuals: Colleges admit many students, first hire many workers, and hospitals employ many interns. On the other hand, students attend one college, workers work for one firm and interns work for one hospital.A college admissions problem 1 consists of two finite disjoint sets of agents (which are interpreted as sets of students and colleges), a vector of natural * We wish to thank Professor William Thomson for his efforts in supervision as well as his useful suggestions. We are grateful to the participants in his reading class, workshops at Bilkent University, University of Rochester, and in particular Jeffrey Banks, Stephen Ching, Bhaskar Dutta, Rangarajan Sundaram and an anonymous referee for their helpful comments.

h i g h l i g h t s • We provide an axiomatic characterization of Expected Scott-Suppes utility representation. • This can be used in applications that study intransitive indifference under uncertainty. • Our main result is the natural analog of vNM expected utility theorem for semiorders. • Our characterization provides an answer to the open problem noted by Fishburn (1968). • Our representation offers a decision-theoretical interpretation for epsilon equilibrium.

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We study the emergence of reference points in a bilateral, infinite horizon, alternating offers bargaining game. Players' preferences exhibit reference dependence, and their current offers have the potential to influence each other's future reference points. However, this influence is limited in that it expires in a finite number of periods. We first construct a subgame perfect equilibrium that involves an immediate agreement and study its properties. Later, we also show the existence of an equilibrium where agreement is reached with delay. We show that expiration lengths and initial reference points play a crucial role for the existence of this equilibrium. For instance, we show that equilibrium with a delayed agreement does not exist when the initial reference point is (0, 0). Finally, we provide comparative static analyses on model parameters, compare two variations of our model, and compare our findings with those of the closest paper to ours, Driesen et al. (Math Soc Sci 64:103-118, 2012).