1. Nash Equilibria in Competitive Societies Eyal Rozenberg Roy Fox

2. Overview Introducing an interesting model Presenting and proving some results: Characterizing Nash equilibria Bounding the price of anarchy Applying the results: Competitive facility location Competitive k-median problem Selfish routing

3. The Model Focus on games in which Each player has a set of available acts The action of a player is a subset of her available acts Some of the actions are feasible, though some may not be

4. Action Profiles A profile is a vector of actions, one for each player Profile operators

5. Utility Functions Each player has a private utility function There is a social utility function Assumptions All functions are measured in the same units The social utility function is submodular A player loses more from dropping out than the society does

6. Set Functions A function is called non-decreasing if A function is called submodular if The discrete directed derivative of at in direction is

7. Submodular Functions Equivalent definitions

8. Notation Set of available acts Action space - set of feasible actions Strategy space - set of distributions on,

9. Notation (2) Private utility function Social utility function For convenience, require For, define

10. Utility Systems Submodular social utility function Validity: It follows that

11. Ascent Lemma Denote Ascent Lemma (special case): Generally:

12. Basic Utility Systems Equality in 2 nd requirement 3 rd requirement follows For every submodular there exists a utility system - the basic one

13. Mixed Strategies Requirements hold is sumbodular Ascent Lemma holds

14. Example: The Oil Game There are nations, each having barrels of oil The utility of each nation is the square root of the number of barrels it exports The social utility is the sum of private utilities

15. For an optimal solution and any Nash equilibrium If is non-decreasing The Price of Anarchy

16. Improved Bound For a non-decreasing, submodular, define its discrete curvature

17. Pure Strategy Equilibrium In a basic utility system, there is a pure strategy Nash equilibrium The game graph is acyclic Nodes - pure strategies Edges - improving changes for some player If player improves from to

18. Facility Location Problem A bipartite graph Locations - cost of building a facility Markets - value of serving customers Edges - cost of serving from k-median problem - restricted action set For a choice of locations The actual cost of serving is The price charged from is

19. Utility Functions The player is maximizing No consumer surplus The player inadvertently maximizes the total surplus

20. Competitive Version (CFL) Cost for firm of building a facility in Value of serving Cost for firm of serving from For a choice of locations The cost for firm of serving is The winning firms are The actual cost of serving is The price charged from is, for some

21. Utility Functions in CFL Denote Firm is maximizing The consumer surplus is The total surplus is

22. CFL Fits the Model The total surplus is submodular Marginal costs are supermodular In the absence of fixed costs, the total surplus is non-decreasing The system is basic When a player joins, the increase in consumer surplus matches the decrease in the other players’ profits

23. Results for CFL In the absence of fixed costs There is a pure strategy Nash equilibrium These results are tight

24. Selfish Routing There are many copies of each path The amount of flow is the number of copies chosen by a player Player gains value from each unit of flow she routes The social utility is the sum of private utilities

25. Selfish Routing Fits the Model The social surplus is submodular Latencies is supermodular The 2 nd requirement holds The system is valid We can restrict the action sets to allow only the correct amount of flow

26. Results for Selfish Routing Choose to get the Roughgarden-Tardos double-rate result If is non-decreasing

27. Questions?