A New Value for Cooperative Games on Intersection-Closed Systems

TL;DR

The paper proposes the uniform-dividend value (UD-value) for incomplete cooperative games and proves its unique determination when the known coalitions form an intersection-closed family. It shows that the UD-value equals the expected Shapley value over all positive extensions, connecting to the framework of complete extensions and Δ-values that recursively determine surpluses. A unified framework and axiomatic comparisons position UD relative to the R-value and IC-value, highlighting differences in fairness and monotonicity properties. Empirical results indicate UD and R are typically close, while IC often distributes value more uniformly, and UD remains well-defined for a wide range of systems as the number of players grows.

Abstract

We introduce a new allocation rule, the uniform-dividend value (UD-value), for cooperative games whose characteristic function is incomplete. The UD-value assigns payoffs by distributing the total surplus of each family of indistinguishable coalitions uniformly among them. Our primary focus is on set systems that are intersection-closed, for which we show the UD-value is uniquely determined and can be interpreted as the expected Shapley value over all positive (i.e., nonnegative-surplus) extensions of the incomplete game. We compare the UD-value to two existing allocation rules for intersection-closed games: the R-value, defined as the Shapley value of a game that sets surplus of absent coalition values to zero, and the IC-value, tailored specifically for intersection-closed systems. We provide axiomatic characterizations of the UD-value motivated by characterizations of the IC-value and discuss further properties such as fairness and balanced contributions. Further, our experiments suggest that the UD-value and the R-value typically lie closer to each other than either does to the IC-value. Beyond intersection-closed systems, we find that while the UD-value is not always unique, a surprisingly large fraction of non-intersection-closed set systems still yield a unique UD-value, making it a practical choice in broader scenarios of incomplete cooperative games.

Figures (6)

• Figure 1: Average $\ell_1$-norms of differences between the R-value, UD-value, and IC-value for all intersection-closed set systems with $n = 3$ players. Each set system is represented on the x-axis by its integer encoding, and the y-axis shows the average norm of the differences between the values, computed over 100 randomly generated games games with values selected uniformly from $[0,1]$.
• Figure 2: Frequencies of each $\ell_1$-norm difference being the smallest, second largest, or largest for all intersection-closed set systems with $n = 3, 4, 5, 6$ players. The y-axis shows the frequency of each rank, aggregated over sampled or exhaustively evaluated systems, highlighting consistent trends as $n$ increases.
• Figure 3: Histograms of the average differences between value pairs for $n = 3, 4, 5, 6$. Each histogram shows the frequency distribution of average differences divided into intervals of width 0.1 from 0 to 1.2.
• Figure 4: Average $\ell_1$-norms of distances between the R-value, UD-value, IC-value and the Equal Division Rule for 3 players. Each set system is represented on the x-axis by its integer encoding, while the y-axis shows the average distance between the values and ED, computed over 100 randomly generated games with values selected uniformly from $[0,1]$.
• Figure 5: Frequencies of each value’s distance from the ED rule being the closest, second closest, and largest for $n = 3, 4, 5, 6$. The y-axis shows the frequency of each rank, aggregated over sampled or exhaustively evaluated systems, higlighting consistent trends as $n$ increases.

Theorems & Definitions (24)

• Definition 1
• Lemma 1
• Definition 2
• Definition 3
• Proposition 2
• proof
• Definition 4
• Theorem 3
• Proposition 4
• proof