Uniqueness of Minimax Strategy in View of Minimum Error Discrimination of Two Quantum States

This study considers the minimum error discrimination of two quantum states in terms of a two-party zero-sum game, whose optimal strategy is a minimax strategy. A minimax strategy is one in which a sender chooses a strategy for a receiver so that the receiver may obtain the minimum information about quantum states, but the receiver performs an optimal measurement to obtain guessing probability for the quantum ensemble prepared by the sender. Therefore, knowing whether the optimal strategy of the game is unique is essential. This is because there is no alternative if the optimal strategy is unique. This paper proposes the necessary and sufficient condition for an optimal strategy of the sender to be unique. Also, we investigate the quantum states that exhibit the minimum guessing probability when a sender’s minimax strategy is unique. Furthermore, we show that a sender’s minimax strategy and a receiver’s minimum error strategy cannot be unique if one can simultaneously diagonalize two quantum states, with the optimal measurement of the minimax strategy. This implies that a sender can confirm that the optimal strategy of only a single side (a sender or a receiver but not both of them) is unique by preparing specific quantum states.

Maximin and Minimax Strategies in Two-Players Game with Two Strategic Variables

We examine maximin and minimax strategies for players in a two-players game with two strategic variables, [Formula: see text] and [Formula: see text]. We consider two patterns of game; one is the [Formula: see text]-game in which the strategic variables of players are [Formula: see text]’s, and the other is the [Formula: see text]-game in which the strategic variables of players are [Formula: see text]’s. We call two players Players A and B, and will show that the maximin strategy and the minimax strategy in the [Formula: see text]-game, and the maximin strategy and the minimax strategy in the [Formula: see text]-game are all equivalent for each player. However, the maximin strategy for Player A and that for Player B are not necessarily equivalent, and they are not necessarily equivalent to their Nash equilibrium strategies in the [Formula: see text]-game nor the [Formula: see text]-game. But, in a special case, where the objective function of Player B is the opposite of the objective function of Player A, the maximin strategy for Player A and that for Player B are equivalent, and they constitute the Nash equilibrium both in the [Formula: see text]-game and the [Formula: see text]-game.

A game theory appraisal of the insurance hypothesis: Specific polymorphisms in the energy homeostasis network as imprints of a successful minimax strategy

The existence of specific polymorphisms in genes of key hormones of the energy homeostasis network that have been shown to predispose to obesity and the so-called metabolic syndrome provides further biological support for the proposed insurance hypothesis. In a broader sense, such polymorphisms can be understood as biological imprints of an evolutionarily successful minimax strategy employed by ancient Homo sapiens subpopulations in a one-player game against nature.

A robust modification of the K-means method for clustering under interval-valued data

Предложена робастная модификация метода K-средних для решения задачи кластеризации при условии, что элементы обучающей выборки являются интервальными. Существующие методы кластеризации в большинстве либо основаны на замене интервальных данных их точными аналогами, например, центрами интервалов, либо используют специальные метрики расстояния между гиперпрямоугольниками (многомерными интервалами) или между точкой и гиперпрямоугольником, например расстояние Хаусдорфа. В отличие от существующих методов, идеей, лежащей в основе предлагаемого алгоритма, является трансформация интервального характера неопределенности во множество распределений весов примеров и расширение обучающей выборки. При этом новые элементы обучающей выборки, являющиеся точками исходных интервалов, имеют неопределенные веса, назначенные таким образом, чтобы не нарушить исходную структуру обучающей выборки, не внося никакой дополнительной необоснованной информации. Другой идеей является использование минимаксной стратегии для обеспечения робастности. Показано, что новый алгоритм отличается от стандартного алгоритма K-средних этапом решения простой задачи линейного программирования. Также показано, что в простейшем случае, когда элементы исходной обучающей выборки имеют одинаковые веса, предлагаемый алгоритм сводится к тому, что выбираются точки гиперпрямоугольников, находящиеся от текущего центра тяжести на максимальном расстоянии. Полученные результаты можно рассматривать в рамках теории Демпстера-Шейфера. Предлагаемый алгоритм целесообразно применять в случае больших интервалов данных или при малом объеме обучающей выборки. A robust modification of the K-means method for solving a clustering problem under interval-valued training data is proposed. The existing methods of clustering are mainly based on the replacement of interval-valued data with their point-valued representations, for example, with centers of intervals, or they use some special distance metrics between hyper-rectangles (multi-dimensional intervals) or between points and hyper-rectangles, for example, the Hausdorff distance. In contrast to the existing methods, the first idea underlying the proposed algorithm is transferring of interval uncertainty to sets of example weights and to an extension of the training set. At that, new elements of the training set, being points approximating intervals, have imprecise weights assigned such that they do not change an initial structure of training data and do not introduce additional unjustified information. The second idea is to use the minimax strategy for providing the robustness. It is shown in the paper that the new algorithm differs from the standard K-means algorithm by a step of solving a simple linear programming problem. It is also shown in the paper that in the simplest case when all elements of the training set have identical weights, the proposed algorithm is reduced to the choice of a point inside hyper-rectangles, which are located on the largest distance from the center of a cluster. The obtained results can be considered also in the framework of Dempster-Shafer theory. The proposed algorithm is useful when the intervals of data are rather large and when the training set is small.

Optimal allocation of the limited oral cholera vaccine supply between endemic and epidemic settings

The World Health Organization (WHO) recently established a global stockpile of oral cholera vaccine (OCV) to be preferentially used in epidemic response (reactive campaigns) with any vaccine remaining after 1 year allocated to endemic settings. Hence, the number of cholera cases or deaths prevented in an endemic setting represents the minimum utility of these doses, and the optimal risk-averse response to any reactive vaccination request (i.e. the minimax strategy) is one that allocates the remaining doses between the requested epidemic response and endemic use in order to ensure that at least this minimum utility is achieved. Using mathematical models, we find that the best minimax strategy is to allocate the majority of doses to reactive campaigns, unless the request came late in the targeted epidemic. As vaccine supplies dwindle, the case for reactive use of the remaining doses grows stronger. Our analysis provides a lower bound for the amount of OCV to keep in reserve when responding to any request. These results provide a strategic context for the fulfilment of requests to the stockpile, and define allocation strategies that minimize the number of OCV doses that are allocated to suboptimal situations.