Selection-Based Approach to Cooperative Interval Games

Bok, Jan; Hladík, Milan

doi:10.1007/978-3-319-27680-9_3

Cited by 6 publications

(7 citation statements)

References 17 publications

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“…The minus sign is used for Moore's subtraction operator there. Also, in our previous paper [8,9] we used minus sign for Moore's subtraction.…”

Section: Interval Analysismentioning

confidence: 99%

“…Finally, we define selection based classes of interval games. The paper [8] discusses their properties and relations with the previous classes.…”

Section: W|(t ) ≤ |W|(s)mentioning

confidence: 99%

“…We continue in the line of research started in [8] (there is also an updated version on arXiv [9]). We focus on selections, that is on possible outcomes of interval games.…”

Section: Introductionmentioning

confidence: 99%

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On convexity and solution concepts in cooperative interval games

Bok¹

2018

Preprint

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Cooperative interval game is a cooperative game in which every coalition gets assigned some closed real interval. This models uncertainty about how much the members of a coalition get for cooperating together.In this paper we study convexity, core and the Shapley value of games with interval uncertainty. Our motivation to do so is twofold. First, we want to capture which properties are preserved when we generalize concepts from classical cooperative game theory to interval games. Second, since these generalizations can be done in different ways, mainly with regard to the resulting level of uncertainty, we try to compare them and show their relation to each other.

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“…The minus sign is used for Moore's subtraction operator there. Also, in our previous paper [8,9] we used minus sign for Moore's subtraction.…”

Section: Interval Analysismentioning

confidence: 99%

“…Finally, we define selection based classes of interval games. The paper [8] discusses their properties and relations with the previous classes.…”

Section: W|(t ) ≤ |W|(s)mentioning

confidence: 99%

See 1 more Smart Citation

On convexity and solution concepts in cooperative interval games

Bok¹

2018

Preprint

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“…The complete (sufficient and necessary) characterization of the second and third support set invariancy remains as an open problem. It seems to be a difficult task because of double appearance of the matrix A in (5) and B in (6). Generally, such dependences in interval linear systems are very hard to deal with.…”

Section: Resultsmentioning

confidence: 99%

Support Set Invariancy for Interval Bimatrix Games

Hladík

2019

Int. J. Unc. Fuzz. Knowl. Based Syst.

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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.

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“…This stemmed various approaches to uncertainty, resulting in models differing in complexity, applicability, and other qualitative criteria. To name a few such models, let us highlight fuzzy cooperative games [6,19,20], multi-choice games [6], cooperative interval games [1,2,4], fuzzy interval games [18], games under bubbly uncertainty [25], ellipsoidal games [36], and games based on grey numbers [24].…”

Section: Introductionmentioning

confidence: 99%

1-convex extensions of incomplete cooperative games and the average value

Bok¹,

Černý²

2021

Preprint

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Partially defined cooperative games are a generalisation of classical cooperative games in which the worth of some of the coalitions is not known. Therefore, they are one of the possible approaches to uncertainty in cooperative game theory. The main focus of this paper is the class of 1-convex cooperative games under this framework.For incomplete cooperative games with minimal information, we present a compact description of the set of 1-convex extensions employing its extreme points and its extreme rays. Then we investigate generalisations of three solution concepts for complete games, namely the τ -value, the Shapley value and the nucleolus. We consider two variants where we compute the centre of gravity of either extreme games or of a combination of extreme games and extreme rays. We show that all of the generalised values coincide for games with minimal information and we call this solution concept the average value. Further, we provide three different axiomatisations of the average value and outline a method to generalise several axiomatisations of the τ -value and the Shapley value into an axiomatisation of the average value.We also briefly mention a similar derivation for incomplete games with defined upper vector and indicate several open questions.

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Selection-Based Approach to Cooperative Interval Games

Cited by 6 publications

References 17 publications

On convexity and solution concepts in cooperative interval games

On convexity and solution concepts in cooperative interval games

Support Set Invariancy for Interval Bimatrix Games

1-convex extensions of incomplete cooperative games and the average value

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