Abstract
In this paper we put forward \(\varepsilon \)-MC nets, a novel succinct rule-based representation scheme for large cooperative games. First, we provide a polynomial algorithm that reaches the proposed representation by exploiting the agents’ estimates over marginal contributions, along with their acceptable information loss, \(\varepsilon \), regarding these estimates. Then we introduce the notion of equivalence classes of agents, and exploit it to (i) obtain an even more compact representation; and (ii) derive new, previously unheld, beliefs over the value of unobserved agent collaboration patterns. Moreover, we present theoretical and empirical results on the information loss arising from this “representational compression”, and on the degree of succinctness achieved. Notably, we show that an arbitrary number of merges to reach the compressed representation, exhibits an information loss that does not exceed \(\varepsilon \). Finally, we provide theoretical guarantees for the coalitional relative error and the Shapley value in the \(\varepsilon \)-MC net with respect to the initial representation.
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Notes
- 1.
This work is an improved version of our earlier work presented in [13].
- 2.
Note that a pattern may consist of only one literal, representing singletons, thus we can assume that \(i\rightarrow val \equiv i\wedge \bot \rightarrow val\), where \(\bot \) is the empty clause.
- 3.
The threshold denotes the minimum similarity degree for two agents in order for them to be equivalent, and depends on the problem at hand. In our experimental evaluation we demonstrate how to employ specific correlation metrics to this purpose.
- 4.
To ensure the practicality of the algorithm for large n and m, we note that our implementation samples an agent in \(\varOmega _{\text {equiv}}\), and checks the conditions for mutual or equivalent agents. Thus the complexity of our implementation, is \(\mathcal {O}(n\cdot m^2)\).
- 5.
The cardinality of positive/negative sets in pattern c also considers the common literal.
- 6.
We consider equivalences only on positive literals.
- 7.
In case of ambiguities, we count the collaboration pattern once.
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Acknowledgements
E. Streviniotis has been supported by the Onassis Foundation - Scholarship ID: G ZR 012-1/2021-2022. A. Georgara has been supported by TAILOR (H2020-952215), 2019DI17, Crowd4SDG (H2020-872944), CI-SUSTAIN (PID2019-104156GB-I00) and Enzyme Advising Group.
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Streviniotis, E., Georgara, A., Chalkiadakis, G. (2022). \(\varepsilon -\)MC Nets: A Compact Representation Scheme for Large Cooperative Game Settings. In: Memmi, G., Yang, B., Kong, L., Zhang, T., Qiu, M. (eds) Knowledge Science, Engineering and Management. KSEM 2022. Lecture Notes in Computer Science(), vol 13370. Springer, Cham. https://doi.org/10.1007/978-3-031-10989-8_15
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