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Register a new user1-convex extensions of partially defined cooperative games and the average value
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Added by
Jan Bok
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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 $tau$-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 emph{average value}. Further, we provide three different axiomatisations of the average value and outline a method to generalise several axiomatisations of the $tau$-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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Partially defined cooperative games are a generalisation of classical cooperative games in which payoffs for some of the coalitions are not known. In this paper we perform a systematic study of partially defined games, focusing on two important classes of cooperative games: convex games and positive games. In the first part, we focus on convexity and give a polynomially decidable condition for extendability and a full description of the set of symmetric convex extensions. The extreme games of this set, together with the lower game and the upper game, are also described. In the second part, we study positivity. We characterise the non-extendability to a positive game by existence of a certificate and provide a characterisation for the extreme games of the set of positive extensions. We use both characterisations to describe the positive extensions of several classes of incomplete games with special structures. Our results complement and extend the existing theory of partially defined cooperative games. We provide context to the problem of completing partial functions and, finally, we outline an entirely new perspective on a connection between partially defined cooperative games and cooperative interval games.
In some games, additional information hurts a player, e.g., in games with first-mover advantage, the second-mover is hurt by seeing the first-movers move. What properties of a game determine whether it has such negative value of information for a particular player? Can a game have negative value of information for all players? To answer such questions, we generalize the definition of marginal utility of a good to define the marginal utility of a parameter vector specifying a game. So rather than analyze the global structure of the relationship between a games parameter vector and player behavior, as in previous work, we focus on the local structure of that relationship. This allows us to prove that generically, every game can have negative marginal value of information, unless one imposes a priori constraints on allowed changes to the games parameter vector. We demonstrate these and related results numerically, and discuss their implications.
We investigate a class of weighted voting games for which weights are randomly distributed over the standard probability simplex. We provide close-formed formulae for the expectation and density of the distribution of weight of the $k$-th largest player under the uniform distribution. We analyze the average voting power of the $k$-th largest player and its dependence on the quota, obtaining analytical and numerical results for small values of $n$ and a general theorem about the functional form of the relation between the average Penrose--Banzhaf power index and the quota for the uniform measure on the simplex. We also analyze the power of a collectivity to act (Coleman efficiency index) of random weighted voting games, obtaining analytical upper bounds therefor.
An average-time game is played on the infinite graph of configurations of a finite timed automaton. The two players, Min and Max, construct an infinite run of the automaton by taking turns to perform a timed transition. Player Min wants to minimise the average time per transition and player Max wants to maximise it. A solution of average-time games is presented using a reduction to average-price game on a finite graph. A direct consequence is an elementary proof of determinacy for average-time games. This complements our results for reachability-time games and partially solves a problem posed by Bouyer et al., to design an algorithm for solving average-price games on priced timed automata. The paper also establishes the exact computational complexity of solving average-time games: the problem is EXPTIME-complete for timed automata with at least two clocks.
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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