Learning Cooperative Games Maria-Florina Balcan, Ariel D. Procaccia and Yair Zick (to appear in IJCAI 2015)

Cooperative Games Players divide into coalitions to perform tasks Coalition members can freely divide profits. How should profits be divided?

Cooperative Games

The Core

Learning Coalitional Values 6 I want the forest cleared of threats!

Learning Coalitional Values 7 I’ll pay my men fairly to do it.

Learning Coalitional Values 8 But, what can they do?

Learning Coalitional Values 9 I know nothing!

Learning Coalitional Values Let me observe what the scouting missions do

Learning Cooperative Games We want to find a stable outcome, but the valuation function is unknown. Can we, using a small number of samples, find a payoff division that is likely to be stable? 11

PAC Learning 12

PAC Learning 13

PAC Learning 14

PAC Stability 15

Stability via Learnability 16

Stability via Learnability 17

Simple Games 18

PAC Stability in Simple Games 19

Simple Games 20

Simple Games 21

Simple Games 22

Simple Games 23

PAC Stability in Simple Games Only Sam appeared in all observed winning coalitions: he is likely to be a veto player; pay him everything. 24

PAC Stability in Simple Games Theorem: simple games are PAC stabilizable (though they are not generally PAC learnable). What about other classes of games? We investigate both PAC learnability and PAC stability of some common classes of cooperative games. 25

Network Flow Games We are given a weighted, directed graph Players are edges; value of a coalition is the value of the max. flow it can pass from s to t. s t

Network Flow Games Theorem: network flow games are not efficiently PAC learnable unless RP = NP. Proof idea: we show that a similar class of games (min-sum games) is not efficiently learnable (the reduction from them to network flows is easy).

Network Flow Games

Network flow games are generally hard to learn. But, if we limit ourselves to path queries, they are easy to learn! Theorem: the class of network flow games is PAC learnable (and PAC stabilizable) when we are limited to path queries.

Network Flow Games s t

Network Flow Games s t

s t

s t

Threshold Task Games [Chalkiadakis et al., 2011]

Threshold Task Games

Additional Results Induced Subgraph Games [Deng & Papadimitriou, 1994]: PAC learnable, PAC stabilizable if edge weights are non-negative

Additional Results

Conclusions Handling uncertainty in cooperative games is important! -Gateway to their applicability. -Can we circumvent hardness of PAC learning and directly obtain PAC stable outcomes (like we did in simple games)? -What about distributional assumptions? Thank you! Questions?

Additional Slides

Shattering Dimension and Learning

Reverse Engineering a Game