Noise Robust Core-stable Coalitions of Hedonic Games

Prashant Trivedi, Nandyala Hemachandra
Proceedings of The 14th Asian Conference on Machine Learning, PMLR 189:1038-1053, 2023.

Abstract

In this work, we consider the coalition formation games with an additional component, ‘noisy preferences’. Moreover, such noisy preferences are available only for a sample of coalitions. We propose a multiplicative noise model (equivalent to an additive noise model) and obtain the prediction probability, defined as the probability that the estimated PAC core-stable partition of the \emph{noisy} game is also PAC core-stable for the \emph{unknown noise-free} game. This prediction probability depends on the probability of a combinatorial construct called an ‘agreement event’. We explicitly obtain the agreement probability for $n$ agent noisy game with $l\geq 2$ support noise distribution. For a user-given satisfaction value on this probability, we identify the noise regimes for which an estimated partition is noise robust; that is, it is PAC core-stable in both the noisy and noise-free games. We obtain similar robustness results when the estimated partition is not PAC core-stable. These noise regimes correspond to the level sets of the agreement probability function and are non-convex sets. Moreover, an important fact is that the prediction probability can be high even if high noise values occur with a high probability. Further, for a class of top-responsive hedonic games, we obtain the bounds on the extra noisy samples required to get noise robustness with a user-given satisfaction value. We completely solve the noise robustness problem of a $2$ agent hedonic game. In particular, we obtain the prediction probability function for $l=2$ and $l=3$ noise support cases. For $l=2$, the prediction probability is convex in noise probability, but the noise robust regime is non-convex. Its minimum value, called the safety value, is 0.62; so, below 0.62, the noise robust regime is the entire probability simplex. However, for $l \geq 3$, the prediction probability is non-convex; so, the safety value is the global minima of a non-convex function and is computationally hard.

Cite this Paper

BibTeX
@InProceedings{pmlr-v189-trivedi23a, title = {Noise Robust Core-stable Coalitions of Hedonic Games}, author = {Trivedi, Prashant and Hemachandra, Nandyala}, booktitle = {Proceedings of The 14th Asian Conference on Machine Learning}, pages = {1038--1053}, year = {2023}, editor = {Khan, Emtiyaz and Gonen, Mehmet}, volume = {189}, series = {Proceedings of Machine Learning Research}, month = {12--14 Dec}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v189/trivedi23a/trivedi23a.pdf}, url = {https://proceedings.mlr.press/v189/trivedi23a.html}, abstract = {In this work, we consider the coalition formation games with an additional component, ‘noisy preferences’. Moreover, such noisy preferences are available only for a sample of coalitions. We propose a multiplicative noise model (equivalent to an additive noise model) and obtain the prediction probability, defined as the probability that the estimated PAC core-stable partition of the \emph{noisy} game is also PAC core-stable for the \emph{unknown noise-free} game. This prediction probability depends on the probability of a combinatorial construct called an ‘agreement event’. We explicitly obtain the agreement probability for $n$ agent noisy game with $l\geq 2$ support noise distribution. For a user-given satisfaction value on this probability, we identify the noise regimes for which an estimated partition is noise robust; that is, it is PAC core-stable in both the noisy and noise-free games. We obtain similar robustness results when the estimated partition is not PAC core-stable. These noise regimes correspond to the level sets of the agreement probability function and are non-convex sets. Moreover, an important fact is that the prediction probability can be high even if high noise values occur with a high probability. Further, for a class of top-responsive hedonic games, we obtain the bounds on the extra noisy samples required to get noise robustness with a user-given satisfaction value. We completely solve the noise robustness problem of a $2$ agent hedonic game. In particular, we obtain the prediction probability function for $l=2$ and $l=3$ noise support cases. For $l=2$, the prediction probability is convex in noise probability, but the noise robust regime is non-convex. Its minimum value, called the safety value, is 0.62; so, below 0.62, the noise robust regime is the entire probability simplex. However, for $l \geq 3$, the prediction probability is non-convex; so, the safety value is the global minima of a non-convex function and is computationally hard.} }
Endnote
%0 Conference Paper %T Noise Robust Core-stable Coalitions of Hedonic Games %A Prashant Trivedi %A Nandyala Hemachandra %B Proceedings of The 14th Asian Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Emtiyaz Khan %E Mehmet Gonen %F pmlr-v189-trivedi23a %I PMLR %P 1038--1053 %U https://proceedings.mlr.press/v189/trivedi23a.html %V 189 %X In this work, we consider the coalition formation games with an additional component, ‘noisy preferences’. Moreover, such noisy preferences are available only for a sample of coalitions. We propose a multiplicative noise model (equivalent to an additive noise model) and obtain the prediction probability, defined as the probability that the estimated PAC core-stable partition of the \emph{noisy} game is also PAC core-stable for the \emph{unknown noise-free} game. This prediction probability depends on the probability of a combinatorial construct called an ‘agreement event’. We explicitly obtain the agreement probability for $n$ agent noisy game with $l\geq 2$ support noise distribution. For a user-given satisfaction value on this probability, we identify the noise regimes for which an estimated partition is noise robust; that is, it is PAC core-stable in both the noisy and noise-free games. We obtain similar robustness results when the estimated partition is not PAC core-stable. These noise regimes correspond to the level sets of the agreement probability function and are non-convex sets. Moreover, an important fact is that the prediction probability can be high even if high noise values occur with a high probability. Further, for a class of top-responsive hedonic games, we obtain the bounds on the extra noisy samples required to get noise robustness with a user-given satisfaction value. We completely solve the noise robustness problem of a $2$ agent hedonic game. In particular, we obtain the prediction probability function for $l=2$ and $l=3$ noise support cases. For $l=2$, the prediction probability is convex in noise probability, but the noise robust regime is non-convex. Its minimum value, called the safety value, is 0.62; so, below 0.62, the noise robust regime is the entire probability simplex. However, for $l \geq 3$, the prediction probability is non-convex; so, the safety value is the global minima of a non-convex function and is computationally hard.
APA
Trivedi, P. & Hemachandra, N.. (2023). Noise Robust Core-stable Coalitions of Hedonic Games. Proceedings of The 14th Asian Conference on Machine Learning, in Proceedings of Machine Learning Research 189:1038-1053 Available from https://proceedings.mlr.press/v189/trivedi23a.html.

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