# 2012 Working Papers: Operations Research and Decisions

**Pricing, replenishment, and timing of selling in a market with heterogeneous customers**, 33 pp.

S. Anily and R. Hassin

(Working Paper No. 24/2012)

We consider a deterministic pricing and replenishment model in which the retailer advertises a fixed price and the selling schedule, and customers can advance or delay their time of purchase incurring holding or shortage costs. We investigate the impact of heterogeneity in the customers’ reservation prices. We show that the resulting optimal solution may be very different from that obtained when customers are homogeneous. We identify nine types of possible optimal sales strategies, and compute their profits. In particular, the solution may contain sales at several discrete points of time between consecutive replenishment epochs with no sales between them.

**Matching of like rank and the size of the core in the marriage problem**, 12 pp.

R. Holzman and D. Samet

(Working Paper No. 25/2012)

When men and women are objectively ranked in a marriage problem, say by beauty, then pairing individuals of equal rank is the only stable matching. We generalize this observation by providing bounds on the size of the rank gap between mates in a stable matching in terms of the size of the ranking sets. Using a metric on the set of matchings, we provide bounds on the diameter of the core–the set of stable matchings–in terms of the size of the ranking sets and in terms of the size of the rank gap. We conclude that when the set of rankings is small, so are the core and the rank gap in stable matchings.

**Conditional belief types**, 26 pp.

A. Di Tillio, J.Y. Halpern and D. Samet

(Working Paper No. 26/2012)

We study type spaces where a player’s type at a state is a conditional probability on the space. We axiomatize these type spaces using conditional belief operators, and examine three additional axioms of increasing strength. First, introspection, which re- quires the agent to be unconditionally certain of her beliefs. Second, echo, according to which the unconditional beliefs implied by the condition must be held given the condition. Third, determination, which says that the conditional beliefs are the unconditional beliefs that are conditionally certain. The echo axiom implies that conditioning on an event is the same as conditioning on the event being certain, which formalizes the standard informal interpretation of conditioning in probability theory. The echo axiom also implies that the conditional probability given an event is a prior of the unconditional probability. The game-theoretic application of our model, which we treat in the context of an example, sheds light on a number of basic issues in the analysis of extensive form games. Type spaces are closely related to the sphere models of counterfactual conditionals and to models of hypothetical knowledge, and we discuss these relationships in detail.

**Belief consistency and trade consistency**, 21 pp.

E. Lehrer and D. Samet

(Working Paper No. 27/2012)

Interpersonal consistency can be described in epistemic terms as a property of beliefs, or in economic terms as the impossibility of certain trades. The existence of a common prior from which all agents' beliefs are derived is of the first kind. The non-existence of an agreeable bet, that is, a contingent zero-sum trade which is always favorable to all agents, is of the second kind. It is well established that these two notions of consistency are equivalent for finite type spaces but not for countable ones. We present three equivalences of epistemic consistency and economic consistency conditions for countable type spaces, defining in this way three levels of consistency of type spaces: weak consistency, consistency, and strong consistency. These three levels coincide in the finite case. We fully analyze the level of consistency of type spaces based on the knowledge structure of Rubinstein's email game. The new notion of belief consistency introduced here helps to justify the requirement of boundedness of payoff functions in countable type spaces by showing that in a large class of spaces there exists an agreeable unbounded bet even when a common prior exists.

**Regular games: Characterization and total balancedness of regular market games**, 27 pp.

S. Anily and M. Haviv

(Working Paper No. 22/2012)

The conventional definition of a cooperative game G(N, V ) with a set of players N = {1, . . . , n} and a characteristic function V, is quite rigid to alterations of the set of players N. Moreover, it necessitates a large input of size that is exponential in n. We show that the characteristic function of many games allows a simple, efficient and flexible presentation of the game. We call such games regular games. In such games each player is characterized by a vector of quantitative properties, and the characteristic function value of a coalition depends only on the vectors of properties of its members. We show that some regular games in which players can cooperate with respect to some of their resources and whose immediate formulation does not fit the framework of market games, can nevertheless be transformed into market games and hence they are totally balanced.