What Decision to Make in a Conflict Situation Under Interval Uncertainty: Efficient Algorithms for the Hurwicz Approach

In this paper, we show how to take interval uncertainty into account when solving conflict situations. Algorithms for conflict situations under interval uncertainty are known under the assumption that each side of the conflict maximizes its worst-case expected gain. However, it is known that a more general Hurwicz approach provides a more adequate description of decision making under uncertainty. In this approach, each side maximizes the convex combination of the worst-case and the best-case expected gains. In this paper, we describe how to resolve conflict situations under the general Hurwicz approach to interval uncertainty. 1 Conflict Situations Under Interval Uncertainty: Formulation of the Problem and What Is Known So Far How conflict situations are usually described. In many practical situations – e.g., in security – we have conflict situations in which the interests of the two sides are opposite. For example, a terrorist group wants to attack one of our assets, while we want to defend them. To fully describe such a situation, we need to describe: – for each possible strategy i of one side and – for each possible strategy j of the other side, what will be the resulting gain uij to the first side (negative if it is a loss), and the gain vij to the other side. A conflict situation is when we cannot improve v without worsening u. An example of a conflict situation in a zero-sum game, when the gain of one side is the loss of another side, i.e., when vij = −uij ; see, e.g., [9]. While zero-sum games are a useful approximation, they are not always a perfect description of the situation. For example, the main objective of the terrorists may be publicity. In this sense, a small attack in the country’s capital may not 2 B. J. Kubica, A. Pownuk, and V. Kreinovich cause much damage but it will bring them a lot of media attention, while a more serious attack in a remote location may be more damaging to the country, but not as media-attractive. To take this difference into account, we need, for each pair of strategies (i, j), to describe both: – the gain uij of the first side and – the gain vij of the second side. In this general case, we do not necessarily have vij = −uij [9]. How to describe this problem in precise terms. It is a well-known fact that in conflict situations, instead of following one of the deterministic strategies, it is beneficial to select a strategy at random, with some probability. For example, if we only have one security person available and two objects to protect, then we have two deterministic strategies: – post this person at the first objects and – post him/her at the second object. If we exactly follow one of these strategies, then the adversary will be able to easily attack the other – unprotected – object. It is thus more beneficial to every time flip a coin and assign the security person to one of the objects at random. This way, for each object of attack, there will be a 50% probability that this object will be defended. In general, each corresponding strategy of the first side can be described by the probabilities p1, . . . , pn of selecting each of the possible strategies, so that n ∑