Characterizing distribution rules for cost sharing games Raga Gopalakrishnan Caltech Joint work with Jason R. Marden & Adam Wierman

Cost sharing games: Self-interested agents make decisions, and share the incurred cost among themselves. Lots of examples: Network formation games Facility location games Profit sharing games Key Question: How should the cost be shared?

Cost sharing games: Lots of examples: Network formation games Facility location games Profit sharing games S1 S2 D1 D2 Key Question: How should the cost be shared? Self-interested agents make decisions, and share the incurred cost among themselves.

Cost sharing games: Lots of examples: Network formation games Facility location games Profit sharing games Key Question: How should the cost be shared? Self-interested agents make decisions, and share the incurred cost among themselves.

Cost sharing games: Lots of examples: Network formation games Facility location games Profit sharing games Key Question: How should the cost be shared? Self-interested agents make decisions, and share the incurred cost among themselves.

Cost sharing games: Lots of examples: Network formation games [Jackson 2003][Anshelevich et al. 2004] Facility location games [Goemans et al. 2000] [Chekuri et al. 2006] Profit sharing games [Kalai et al. 1982] [Ju et al. 2003] Huge literature in Economics Growing literature in CS New application: Designing for distributed control [Gopalakrishnan et al. 2011][Ozdaglar et al. 2009][Alpcan et al. 2009] Key Question: How should the cost be shared? Self-interested agents make decisions, and share the incurred cost among themselves.

Cost sharing games (more formally): set of agents/players set of resources S1 S2 D1 D2 Example:

Cost sharing games (more formally): set of agents/players set of resources

Cost sharing games (more formally): set of agents/players set of resources common base welfare function

Cost sharing games (more formally): set of agents/players set of resources resource-specific coefficients welfare function

Cost sharing games (more formally): set of agents/players set of resources resource-specific coefficients welfare function

Cost sharing games (more formally): set of agents/players set of resources resource-specific coefficients distribution rule welfare function

Requirements on the distribution rule The distribution rule should be: (i) Budget-balanced (ii) “Stable” and/or “Fair” (iii) “Efficient”

Requirements on the distribution rule The distribution rule should be: (i) Budget-balanced (ii) “Stable” and/or “Fair” (iii) “Efficient”

Requirements on the distribution rule [Gillies 1959] [Devanur et al. 2003] [Chander et al. 2006] The distribution rule should be: (i) Budget-balanced (ii) “Stable” and/or “Fair” (iii) “Efficient” Lots of work on characterizing “stability” and “fairness” Nash equilibrium Core [von Neumann et al. 1944] [Nash 1951] [Moulin 1992] [Albers et al. 2006]

Requirements on the distribution rule [Gillies 1959] [Devanur et al. 2003] [Chander et al. 2006] The distribution rule should be: (i) Budget-balanced (ii) “Stable” and/or “Fair” (iii) “Efficient” Lots of work on characterizing “stability” and “fairness” Nash equilibrium Core [von Neumann et al. 1944] [Nash 1951] [Moulin 1992] [Albers et al. 2006]

Requirements on the distribution rule The distribution rule should be: (i) Budget-balanced (ii) “Stable” and/or “Fair” (iii) “Efficient” Has good Price of Anarchy and Price of Stability properties

The Shapley value [] The Shapley value [Shapley 1953] A player’s share of the welfare should depend on their “average” marginal contribution Note: There is also a weighted Shapley value

Properties of the Shapley value approximations are often tractable [Castro et al. 2009]

Research question: Are there distribution rules besides the (weighted) Shapley value that always guarantee a Nash equilibrium?

Research question: Our (surprising) answer: NO, for any submodular welfare function. “decreasing marginal returns” Are there distribution rules besides the (weighted) Shapley value that always guarantee a Nash equilibrium? natural way to model many real-world problems

The inspiration for our work Our result

The inspiration for our work Our result

Consequences

Proof Sketch “contributing coalition” “magnitude of contribution”

Proof Sketch (A single T-Welfare Function) Don’t allocate welfare to any player

Proof Sketch (General Welfare Functions) Don’t allocate welfare to any player Allocate welfare only to players in these formed coalitions, independent of others

Proof Sketch (General Welfare Functions) Don’t allocate welfare to any player Allocate welfare only to players in these formed coalitions, independent of others Weights of common players in any two coalitions must be linearly dependent

Research question: Are there distribution rules besides the (weighted) Shapley value that always guarantee a Nash equilibrium? Cost Sharing Games Our answer: NO, for any submodular welfare function. what about for other welfare functions? Understand what causes this fundamental restriction – perhaps some structure of action sets?

Characterizing distribution rules for cost sharing games Raga Gopalakrishnan Caltech Joint work with Jason R. Marden & Adam Wierman

References [von Neumann et al. 1944] [Nash 1951] [Shapley 1953] [Gillies 1959] [Shapley 1967] [Kalai et al. 1982] [Moulin 1992] [Goemans et al. 2000] [Ui 2000] [Devanur et al. 2003] [Jackson 2003] [Ju et al. 2003] [Anshelevich et al. 2004] [Conitzer et al. 2004] [Albers et al. 2006] [Chander et al. 2006] [Chekuri et al. 2006] [Alpcan et al. 2009] [Ozdaglar et al. 2009] [Chen et al. 2010] [Gopalakrishnan et al. 2011] [Marden et al. 2011]