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Register a new userConvexity and positivity in partially defined cooperative games
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Added by
Jan Bok
Publication date
2020
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 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.
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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.
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.
Recent superhuman results in games have largely been achieved in a variety of zero-sum settings, such as Go and Poker, in which agents need to compete against others. However, just like humans, real-world AI systems have to coordinate and communicate with other agents in cooperative partially observable environments as well. These settings commonly require participants to both interpret the actions of others and to act in a way that is informative when being interpreted. Those abilities are typically summarized as theory f mind and are seen as crucial for social interactions. In this paper we propose two different search techniques that can be applied to improve an arbitrary agreed-upon policy in a cooperative partially observable game. The first one, single-agent search, effectively converts the problem into a single agent setting by making all but one of the agents play according to the agreed-upon policy. In contrast, in multi-agent search all agents carry out the same common-knowledge search procedure whenever doing so is computationally feasible, and fall back to playing according to the agreed-upon policy otherwise. We prove that these search procedures are theoretically guaranteed to at least maintain the original performance of the agreed-upon policy (up to a bounded approximation error). In the benchmark challenge problem of Hanabi, our search technique greatly improves the performance of every agent we tested and when applied to a policy trained using RL achieves a new state-of-the-art score of 24.61 / 25 in the game, compared to a previous-best of 24.08 / 25.
Two intimately related new classes of games are introduced and studied: entropy games (EGs) and matrix multiplication games (MMGs). An EG is played on a finite arena by two-and-a-half players: Despot, Tribune and the non-deterministic People. Despot wants to make the set of possible Peoples behaviors as small as possible, while Tribune wants to make it as large as possible.An MMG is played by two players that alternately write matrices from some predefined finite sets. One wants to maximize the growth rate of the product, and the other to minimize it. We show that in general MMGs are undecidable in quite a strong sense.On the positive side, EGs correspond to a subclass of MMGs, and we prove that such MMGs and EGs are determined, and that the optimal strategies are simple. The complexity of solving such games is in NP&coNP.
We study two forms of a symmetric cooperative game played by three players, one classical and other quantum. In its classical form making a coalition gives advantage to players and they are motivated to do so. However in its quantum form the advantage is lost and players are left with no motivation to make a coalition.
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