1. Raga Gopalakrishnan University of Colorado at Boulder Sean D. Nixon (University of Vermont) Jason R. Marden (University of Colorado at Boulder) Stable Utility Design for Distributed Resource Allocation

2. Resource Allocation Allocate agents to resources to optimize system-level objective Wireless Frequency Selection F1F1 F2F2 F3F3 F1F1 F2F2 F3F3 ? frequency

3. Wireless Access Point Assignment ? Resource Allocation Allocate agents to resources to optimize system-level objective

4. ? Sensor Coverage Resource Allocation Allocate agents to resources to optimize system-level objective

5. ? Sensor Coverage Allocate agents to resources to optimize system-level objective Distributed Resource Allocation

6. ? Sensor Coverage Design local control policies for agents that result in desirable global behavior (convergence to an allocation optimizing system-level objective) Distributed Resource Allocation

7. Design local control policies for agents that result in desirable global behavior (convergence to an allocation optimizing system-level objective) Distributed Resource Allocation Game Theoretic Control Model agents as players in a non-cooperative game Equilibria correspond to stable allocations Goal is to design the game such that equilibria exist (stability) are efficient are easy to converge to UTILITY DESIGN (static) LEARNING DESIGN (dynamic)

8. Formal Model

9. DESIGN

10. Formal Model

12. S1 S2 D1 D2 6 1 6 1 1 6 1 ?+? Example: Network formation

13. S1 S2 D1 D2 6 1 6 1 1 6 1 3+3 A Nash equilibrium Also optimal! 1+5 Unique Nash equilibrium Suboptimal Example: Network formation

14. Key feature: Distribution rules outcome ?+? S1 S2 D1 D2 6 1 6 1 1 6 1 Example: Network formation

15. Formal Model

16. Most prior work studies two distribution rules Marginal Contribution (MC) [ Wolpert and Tumer 1999 ] average marginal contribution over player orderings Shapley Value (SV) [ Shapley 1953 ] externality experienced by all other players Both guarantee PNE in all games! Question: Are there other such distribution rules? Prior Work: NO, for any given welfare function. [G., Marden, Wierman 2013]

17. Most prior work studies two distribution rules Marginal Contribution (MC) [ Wolpert and Tumer 1999 ] Shapley Value (SV) [ Shapley 1953 ] Both guarantee PNE in all games! Question: Are there other such distribution rules? Prior Work: NO, for any given welfare function. [G., Marden, Wierman 2013] Observation: Many practical problems involve “single-selection”: agents select a single resource. Question: Are there other such distribution rules if we only require equilibrium existence for all single-selection games ? Our Answer: No, not for all welfare functions.

18. Single-Selection Scenario [Marden and Wierman 2013] Our Results (characterizations): The only linear budget-balanced distribution rules that guarantee PNE in all single-selection games, for all welfare functions, are weighted Shapley values. Given any linear welfare function with no dummy players, the only budget-balanced distribution rules that guarantee PNE in all single-selection games are weighted Shapley values. Given any welfare function, the only budget-balanced distribution rules that guarantee PNE in all two-player single-selection games are weighted Shapley values. [G., Nixon, Marden 2013]

19. Concluding Remarks Consequences of the restriction to weighted Shapley values: Resulting game is a weighted potential game for which several learning dynamics converge to PNE. It is hard for agents to compute their utilities. Open Problems: Obtaining a tighter characterization of stable distribution rules for a given welfare function. Obtaining the characterization when budget- balance is relaxed. Optimizing the “weights” for efficiency.

20. Ragavendran Gopalakrishnan University of Colorado at Boulder Sean D. Nixon (University of Vermont) Jason R. Marden (University of Colorado at Boulder) Stable Utility Design for Distributed Resource Allocation