Interval multi-objective linear programming (IMOLP) ímodels are one of the methods to tackle uncertainties. In this paper, we propose two methods to determine the efficient solutions in the IMOLP models through the expected value, variance and entropy operators which have good properties. One of the most important properties of these methods is to obtain different efficient solutions set according to decision makers’ preferences as available information. We first develop the concept of the expected value, variance and entropy operators on the set of intervals and study some properties of the expected value, variance and entropy operators. Then, we present an IMOLP model with uncertain parameters in the objective functions. In the first method, we use the expected value and variance operators in the IMOLP models and then we apply the weighted sum method to convert an IMOLP model into a multi-objective non-linear programming (MONLP) model. In the second method, the IMOLP model using the expected value, variance and entropy operators can be converted into a multi-objective linear programming (MOLP) model. The proposed methods are applicable for large scale models. Finally, to illustrate the efficiency of the proposed methods, numerical examples and two real-world models are solved.

Traditionally, game theory problems were considered for exact data, and the decisions were based on known payoffs. However, this assumption is rarely true in practice. Uncertainty in measurements and imprecise information must be taken into account. The interval-based approach for handling such uncertainties assumes that one has lower and upper bounds on payoffs. In this paper, interval bimatrix games are studied. Especially, we focus on three kinds of support set invariancy. Support of a mixed strategy consists of that pure strategies having positive probabilities. Given an interval-valued bimatrix game and supports for both players, the question states as follows: Does every bimatrix game instance have an equilibrium with the prescribed support? The other two kinds of invariancies are slight modifications: Has every bimatrix game instance an equilibrium being a subset/superset of the prescribed support? It is computationally difficult to answer these questions: the first case costs solving a large number of linear programs or mixed integer programs. For the remaining two cases a sufficient condition and a necessary condition are proposed, respectively.

This paper deals with solving interval system of linear equations. The problem is to find a nonnegative algebraic solution. Based on sign function approach and using interval center and radius arithmetic operations, we propose an algorithm for computation of an algebraic interval solution vector. We also discuss fundamental properties of this solution vector, such as existence and uniqueness. Further, the nonnegative solution algorithm has been extended to other sign-restricted approach. Numerical examples of interval system of linear equations show efficiency of the algorithms presented.

The leading approach to solving large imperfect information games is to pre-calculate an approximate solution using a simplified abstraction of the full game; that solution is then used to play the original, full-scale game. The abstraction step is necessitated by the size of the game tree. However, as the original game progresses, the remaining portion of the tree (the subgame) becomes smaller. An appealing idea is to use the simplified abstraction to play the early parts of the game and then, once the subgame becomes tractable, to calculate a solution using a finer-grained abstraction in real time, creating a combined final strategy. While this approach is straightforward for perfect information games, it is a much more complex problem for imperfect information games. If the subgame is solved locally, the opponent can alter his play in prior to this subgame to exploit our combined strategy. To prevent this, we introduce the notion of subgame margin, a simple value with appealing properties. If any best response reaches the subgame, the improvement of exploitability of the combined strategy is (at least) proportional to the subgame margin. This motivates subgame refinements resulting in large positive margins. Unfortunately, current techniques either neglect subgame margin (potentially leading to a large negative subgame margin and drastically more exploitable strategies), or guarantee only non-negative subgame margin (possibly producing the original, unrefined strategy, even if much stronger strategies are possible). Our technique remedies this problem by maximizing the subgame margin and is guaranteed to find the optimal solution. We evaluate our technique using one of the top participants of the AAAI-14 Computer Poker Competition, the leading playground for agents in imperfect information setting

It is a well known fact that in extensive form games with perfect information, there is a Nash equilibrium with support of size one. This doesn't hold for games with imperfect information, where the size of minimal support can be larger. We present a dependency between the level of uncertainty and the minimum support size. For many games, there is a big disproportion between the game uncertainty and the number of actions available. In Bayesian extensive games with perfect information, the only uncertainty is about the type of players. In card games, the uncertainty comes from dealing the deck. In these games, we can significantly reduce the support size. Our result applies to general-sum extensive form games with any finite number of players.

We study possibilistic nonlinear regression models with crisp and/or interval data. Herein, the task is to compute tight interval regression parameters such that all observed output data (either crisp or interval) are covered by the range of the nonlinear interval regression function. We propose a method for determination of interval regression parameters based on the tolerance approach developed by the authors for the linear case. We define two classes of nonlinear regression models for which efficient algorithms exist. For other models, we provide some extensions allowing to calculate lower and upper bounds on the widths of the optimal interval regression parameters. We also discuss other approaches to interval regression than the possibilistic one. We illustrate the theory by examples.