Search theory has been a major theme in Operations Research. This paper deals with a two-person zero-sum search game, called search allocation game (SAG), with a searcher and a target as players. In the SAG, the searcher distributes his searching resource in a search space to detect the target while the target moves to evade the searcher. Practical searching resource has a variety of properties on their effects and constraints. A flare fired by a drifting person in the water keeps its brightness within a limited range during some time. Namely, the flare has the property of long-distance effectiveness and temporal durability. There have been so far few researches focusing on the property of searching resource. From a general discussion about the property of searching resource, we notice that linear expressions of variables denoting the resource are crucial to formulate the SAG. In this paper, we propose two linear programming formulations to solve the SAG with linear expressions concerning the effect and the constraints of searching resource and to derive optimal strategies of the searcher and the target.

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This paper deals with a smuggling game with multiple stages, in which Customs and a smuggler participate. Customs and the smuggler are allowed to take an action of patrol and smuggling, respectively, within the limited number of chances. Customs obtains reward by the capture of the smuggler and the smuggler gets reward by the success of smuggling. The reward or the payoff of the game is brought at each stage and is assumed to be zero-sum. Almost all past researches modeled their games by the so-called complete information game and they assumed that each player knows the past strategies taken by his opponent or never knows them. Recently, we recognize that information is crucial to the results of the games. In this paper, we deal with a smuggling game with incomplete information, where information acquisition is asymmetric between players and is disadvantageous to Customs, and we evaluate the value of information by developing a computational methodology to derive Bayesian equilibrium.

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For functions defined on integer lattice points, discrete versions of the Hessian matrix have been considered in various contexts. In discrete convex analysis, for example certain combinatorial properties of the discrete Hessian matrices are known to characterize M^〓-convex and L^〓-convex functions, which can be extended to convex functions in real variables. The relationship between convex extensibility and discrete Hessian matrices is not fully understood in general, and unfortunately, some vague or imprecise statements have been made in the literature. This note points out that the positive semidefiniteness of the discrete Hessian matrix does not imply nor is implied by convex extensibility of discrete functions.

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A class of methods of apportionment called the relaxed divisor methods is introduced here. It is theoretically shown that their biases between small and large states decrease and eventually approach zero as the house size increases. Because the methods of Hill and Webster are examples of the class, they both are practically unbiased for the house size large enough. However, computer simulations performed here strongly suggest that Webster is the first method to practically attain a state of unbiasedness when the house size increases from 200 to 43,500.

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Since Briec developed a family of the least distance based inefficiency measures satisfying weak monotonicity over weakly efficient frontier, the existence of a least distance based efficiency measure satisfying strong monotonicity on the strongly efficient frontier is still an open problem. This paper gives a negative answer to the open problem and its relaxed open problem. Modifying Briec's inefficiency measures gives an alternative solution to the relaxed open problem, that can be used for theoretical and practical applications.

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We discuss the optimal allocation problem in combinatorial auctions, where the items are allocated to bidders so that the sum of the bidders' utilities is maximized. In this paper, we consider the case where utility functions are given by quadratic functions; the class of such utility functions has a succinct representation but is sufficiently general. The main aim of this paper is to show the computational complexity of the optimal allocation problem with quadratic utility functions. We consider the cases where utility functions are submodular and supermodular, and show NP-hardness and/or polynomial-time exact/approximation algorithms. These results are given by using the relationship with graph cut problems such as the min/max cut problem and the multiway cut problem.

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