Positivity and convexity in incomplete cooperative games

TL;DR

The paper develops a systematic theory for incomplete cooperative games, concentrating on positive and convex extensions. It provides duality-based certificates and a complete description of the set of positive extensions via dividends and extreme games, including explicit forms for several structured incomplete games. It then analyzes symmetric convex extensions, offering linear-time extendability tests, reduced-form representations, and a full extreme-point description in the symmetric setting. Finally, it highlights a novel connection between incomplete cooperative games and cooperative interval games, suggesting avenues for unification and further research.

Abstract

Incomplete cooperative games generalise the classical model of cooperative games by omitting the values of some of the coalitions. This allows to incorporate uncertainty into the model and study the underlying games as well as possible payoff distribution based only on the partial information. In this paper we perform a systematic study of incomplete games, focusing on two important classes of cooperative games: positive and convex games. Regarding positivity, we generalise previous results for a special class of minimal incomplete games to general setting. We characterise non-extendability to a positive game by the existence of a certificate and provide a description of the set of positive extensions using its extreme games. The results are then used to obtain explicit formulas for several classes of incomplete games with special structures. The second part deals with convexity. We begin with considering the case of non-negative minimal incomplete games. Then we survey existing results in the related theory of set functions, namely providing context to the problem of completing partial functions. We provide a characterisation of extendability and a full description of the set of symmetric convex extensions. The set serves as an approximation of the set of convex extensions. Finally, we outline an entirely new perspective on a connection between incomplete cooperative games and cooperative interval games.

Figures (4)

• Figure 1: Examples of line charts of symmetric convex games in their reduced forms. The figure on the left depicts a game $(N,s)$ where $s(1) > 0$, the graph on the right a situation where $s(1) < 0$. The slopes of the line segments are bounded by convexity of the function.
• Figure 2: The construction of a $C^n_\sigma$-extension of $(N,\mathcal{X},\sigma)$ where $\mathcal{X} = \{x_1,x_2,x_3,x_4\}$, using the line chart of $(N,\mathcal{X},\sigma)$. The value $s(k)$ lies on the line segment connecting $(x_3,\sigma(x_3))$ and $(x_4,\sigma(x_4))$.
• Figure 3: An example of a reduced game $(N,\mathcal{X},\sigma)$ with $\mathcal{X} = \left\{0,1,2,4,6\right\}$ where the condition $\frac{\underline{s}(3) + \underline{s}(5)}{2} \ngeq \underline{s}(4)$ from Theorem \ref{['thm:symconv']} is not satisfied. This implies that $(N,\underline{s})$ is not a $C^n_\sigma$-extension of $(N,\mathcal{X},\sigma)$.
• Figure 4: Examples of a violation of convexity of the line chart of both $(N,s_1)$ and $(N,s_2)$. The full lines depict the line chart of $(N,\underline{s})$ and the dotted lines depict the line charts of $(N,s_1)$ and $(N,s_2)$. On the left, the situation where $k < m$ is shown. We have values $s^k(i) = s_1(k)$ and $s^k(k)=s_1(k)$, yet $s_1(m)$ is too small. Similarly, on the right, the situation where $m < k$ is shown, with $s^k(i) = s_2(k)$, $s^k(k) = s_2(k)$. However, in this case, the value $s_2(m)$ is too big.

Theorems & Definitions (55)

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