[Paper Review] Learning Cooperative Games

Maria-Florina Balcan, Ariel D. Procaccia+1 more|arXiv (Cornell University)|Apr 30, 2015

Machine Learning and Algorithms30 references21 citations

TL;DR

This paper introduces a PAC learning framework for cooperative games, showing that stable payoff divisions can be learned efficiently from polynomially many random samples. It establishes that games like network flow, threshold task, and induced subgraph games are efficiently PAC-learnable, and links learnability to core stability, enabling likely stability with minimal sampling and computational cost.

ABSTRACT

This paper explores a PAC (probably approximately correct) learning model in cooperative games. Specifically, we are given $m$ random samples of coalitions and their values, taken from some unknown cooperative game; can we predict the values of unseen coalitions? We study the PAC learnability of several well-known classes of cooperative games, such as network flow games, threshold task games, and induced subgraph games. We also establish a novel connection between PAC learnability and core stability: for games that are efficiently learnable, it is possible to find payoff divisions that are likely to be stable using a polynomial number of samples.

Motivation & Objective

• To address the computational and informational barrier in cooperative game theory, where full knowledge of coalition values is required for solution concepts like the core.
• To explore whether cooperative games can be efficiently learned from a polynomial number of random samples of coalitions and their values.
• To establish a connection between PAC learnability and core stability, showing that stable payoff divisions can be found with high probability using few samples.
• To identify which well-known classes of cooperative games are efficiently PAC-learnable under the PAC model.
• To extend learning-theoretic insights to cooperative solution concepts, particularly focusing on stability and feasibility of payoff allocations.

Proposed method

• Adopt the PAC learning model to learn an unknown characteristic function v: 2^N → R from m i.i.d. samples of coalitions and their values.
• Introduce a relaxed notion of stability called 'PAC stabilizability', where payoff divisions are likely to resist deviations by coalitions drawn from a distribution D.
• Design a polynomial-time algorithm that outputs a feasible payoff division (not exceeding the cost of stability) that is likely to be stable under future deviations.
• Reduce learning problems in specific game classes (e.g., network flow, threshold task, induced subgraph) to known learnable problems (e.g., linear functions, XOS, DNF formulas) via structural transformations.
• Use reductions from known hardness results (e.g., learning DNF, intersection of hyperplanes) to show intractability for classes like k-vector WVGs and MC-nets.
• Prove that coalitional skill games are efficiently PAC-learnable when player skills are known, even if task requirements are unknown.

Experimental results

Research questions

• RQ1 Can cooperative games be efficiently learned from a polynomial number of random samples of coalitions and their values?
• RQ2 Is there a connection between PAC learnability and the stability of payoff divisions in the core?
• RQ3 Which well-known classes of cooperative games are efficiently PAC-learnable under the PAC model?
• RQ4 Can we achieve stable, feasible payoff divisions using only a polynomial number of samples, even when the core is empty?
• RQ5 What are the computational limits of PAC learning in cooperative games, particularly for classes like MC-nets and k-vector weighted voting games?

Key findings

• Network flow games are efficiently PAC-learnable using path queries, but not in general without such queries.
• Threshold task games and induced subgraph games are efficiently PAC-learnable.
• k-vector weighted voting games are not efficiently PAC-learnable unless P = NP, due to the NP-hardness of learning the intersection of k hyperplanes.
• MC-nets are not efficiently PAC-learnable, as learning MC-nets reduces to learning DNF formulas, which is intractable.
• Coalitional skill games are efficiently PAC-learnable when player skills are known, even if task requirements are unknown.
• The paper establishes that any cooperative game is PAC stabilizable: a stable, feasible payoff division can be computed in polynomial time using a polynomial number of samples, with high probability.