CS6283 Topics in Computer Science IV: Computational Social Choice

1. CS6283 Topics in Computer Science IV: Computational Social Choice Instructor: YairZick 2017

2. Cooperative Games

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

4. Matching Games 12 1 2 5 1 4 7 3 5 6 3 3 5 1 2 4 We are given a weighted graph Players are nodes; value of a coalition is the value of the max. weighted matching on the subgraph. Applications: markets, collaboration networks.

5. 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. Applications: computer networks, traffic flow, transport networks.

6. Weighted Voting Games We are given a list of weights and a threshold. Each player has a weight ; value of a coalition is if its total weight is more than (winning), and 0 otherwise (losing). Applications: models parliaments, UN security council, EU council of members, linear classifiers.

8. Cost Sharing Splitting a taxi fare or a bill Sharing the cost of a public project

9. Cooperative Games Computing Optimal Coalition structures is a non-trivial problem in itself! • A set of players - • Characteristic function - • – value of a coalition . • – a partition of ; a coalition structure. • Imputation: a vector satisfying efficiency: for all in

10. Cooperative Games A game is called simpleif is monotoneif for any : is superadditiveif for disjoint : is convexif for & :

11. Dividing Payoffs in Cooperative Games The core and the Shapley value

13. The core is a polyhedron: a set of vectors in that satisfies linear constraints

14. For three players,

15. 1 2 3

16. The Core of a Cooperative Game Is never empty Assigns equal payoffs to equal contributors? Is poly-time computable? ? ? ? ? ? ? ? ? ? ?

17. Is the Core Empty? Core-Empty: given a game , is the core of empty? Note that we are “cheating” here: a naïve representation of is a list of vectors We are generally dealing with • Games with a compact representation • Oracle access to … and obtaining algorithms that are

18. Is the Core Empty? Simple Games: a game is called simpleif for all . Coalitions with value are winning; those with value are losing. A player is called a veto player if she is a member of every winning coalition (can’t win without her).

19. Theorem: let be a simple game; then iff has veto players.

20. Key Concept – Computational Complexity A Quick and Dirty Guide to NP-hardness

24. Theorem: Core-Empty is NP-hard 5 1 2 7 9 6 3 4 8 Proof: we will show this claim for induced-subgraph games Players are nodes, value of a coalition is the weight of its induced subgraph.

25. Lemma: the core of an induced subgraph game is not empty iff the graph has no negative cut. Proof: we will show first that if there is no negative cut, then the core is not empty. Consider the payoff division that assigns each node half the value of the edges connected to it Need to show that for all .

26. Since there are no negative cuts, the last expression is at least Note: we haven’t shown efficiency, i.e. , but that’s trivial from the above.

27. Now, suppose that there is some negative cut; i.e. there is some such that Take any imputation ; then Where again

28. Therefore: So, it is either the case that or ; hence cannot be in the core.

29. Lemma: deciding whether a graph has a negative cut is NP-complete Proof: we reduce from the Max-Cut problem. Given a weighted, undirected graph , where for all , and an integer , is there a cut of whose weight is more than ?

30. 5 1 2 7 9 6 3 4 8 10 0 We write . We define a graph with capacities as follows for all for all

31. 5 1 2 7 9 6 3 4 8 10 0 The lowest weight cut in this graph must separate and . It must also have exactly edges with capacity . Therefore, it is strictly negative iff the original graph has a cut with weight at least .

32. Discussion This proof is one of the first complexity results on the stability of cooperative games, and appears in Deng & Papadimitriou’s seminal paper “On the Complexity of Cooperative Solution Concepts” (1994). The payoff division is special: it is in fact the Shapley valuefor induced graph games.

33. Learning Cooperative Games Learning Theory Meets Cooperative Games

34. Unknown Coalitional Values Often, agent capabilities are unknown. What’s the optimal coalition structure? Can we find a stable outcome? Observe agents’ coalitional values compute solutions based on observations ?

35. Learning from Examples with probability (probably), output a function that has an error probability of over (approximately correct). Observe i.i.d samples from a distribution Probably approximately correct (PAC)

36. with probability (probably), output a payoff division that is -stable over (approximately stable). Observe i.i.d samples from a distribution We require that the amount of money allocated is minimal: if the core is not empty Probably approximately stable

37. Stability via Learnability Given samples Stability grounded in observed agent behavior Useful if we are interested in resistance to a specific type of deviation (e.g. only from very small sets). Theorem: any coalitional game is PAC stabilizable Proof: equivalent to learning linear separators.

38. What functions can be stabilized? Many common valuations are not PAC learnable (though they are all PAC stabilizable)

39. Network Flow Games We are given a weighted, directed graph 10 1 7 1 7 3 5 6 3 3 5 1 2 4 Players are edges; value of a coalition is the value of its max. flow.

40. Network Flow Games Proof idea: we show that a similar class of games (min-sum games) is not efficiently learnable. Theorem: network flow games are not efficiently PAC learnable unless .

41. Network Flow Games 1-min-sum games: linear functions. -min-sum games are defined by vectors

42. Network Flow Games Construct -clause CNF from Argue that PAC approximates Assume that PAC approximates -min sum games are not PAC learnable It is known that CNF formulas with clauses are hard to learn if . We reduce hardness for -clause-CNF formulas to hardness for -min-sum.

43. Network Flow Games Theorem: the class of network flow games is PAC learnable when we only observe path flows. Network flow games are generally hard to learn. But, if we only observe flows on paths, they are easy to learn.

44. Network Flow Games 10 1 7 s 1 7 3 5 6 t 3 3 5 1 2 4 Proof idea: Suppose we are given the input Define for every

45. Network Flow Games 2 2 2 s 2 t 2 2 Proof idea: Suppose we are given the input Define for every

46. Network Flow Games 5 2 5 5 2 s 2 5 t 5 2 2 Proof idea: Suppose we are given the input Define for every

47. Network Flow Games 1 5 5 1 s 1 5 t 1 1 5 2 1 2 Proof idea: Suppose we are given the input Define for every

48. Core Restrictions The Cost of Stability and Graph Restricted Games

49. Core Extensions What if the core is empty? Players cannot generate enough revenue to satisfy all demands. We can increase the total value with a subsidy

50. The Cost of Stability As an LP: If then the core is not empty. The value of in an optimal solution of the above LP is called the cost of stability of , and referred to as