characterizing distribution rules for cost sharing games raga gopalakrishnan caltech joint work with...

Characterizing distribution rules for cost sharing gamesRaga 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 gamesFacility location gamesProfit sharing games

Key Question: How should the cost be shared?

Cost sharing games:

Lots of examples:Network formation gamesFacility location gamesProfit 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 gamesFacility location gamesProfit 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 gamesFacility location gamesProfit 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):

𝐺=(𝑁 ,𝑅 , {𝒜𝑖 }𝑖∈𝑁 ,𝒲 , {𝒰𝑖}𝑖 ∈𝑁)

set of agents/players

set of resources

action set of agent

welfare functionutility function

of agent

S1

S2

D1

D2

Example:

Cost sharing games (more formally):

𝐺=(𝑁 ,𝑅 , {𝒜𝑖 }𝑖∈𝑁 ,𝒲 , {𝒰𝑖}𝑖 ∈𝑁)

set of agents/players

set of resources

action set of agent

welfare functionutility function

of agent

Assumption: is separable across resources

set of agents choosingresource in allocation

Cost sharing games (more formally):

𝐺=(𝑁 ,𝑅 , {𝒜𝑖 }𝑖∈𝑁 , {𝒲𝑟 }𝑟∈𝑅 , {𝒰𝑖}𝑖∈𝑁 )

set of agents/players

set of resources

action set of agent

welfare function at resource

utility function of agent

Assumption: is scalable

𝑣𝑟∈ℝ++¿ ¿common base

welfare function

Cost sharing games (more formally):

𝐺=(𝑁 ,𝑅 , {𝒜𝑖 }𝑖∈𝑁 , {𝑣𝑟 }𝑟∈𝑅 ,𝑊 , {𝒰𝑖}𝑖∈𝑁)

set of agents/players

set of resources

action set of agent

resource-specific coefficients

utility function of agent

welfare functio

n

Cost sharing games (more formally):

𝐺=(𝑁 ,𝑅 , {𝒜𝑖 }𝑖∈𝑁 , {𝑣𝑟 }𝑟∈𝑅 ,𝑊 , {𝒰𝑖}𝑖∈𝑁)

set of agents/players

set of resources

action set of agent

resource-specific coefficients

utility function of agent

Assumption: Utility functions are also separable/scalable

welfare functio

n

common base distribution rule(portion of welfare at to agent )

Cost sharing games (more formally):

𝐺=(𝑁 ,𝑅 , {𝒜𝑖 }𝑖∈𝑁 , {𝑣𝑟 }𝑟∈𝑅 ,𝑊 , 𝑓 )

set of agents/players

set of resources

action set of agent

resource-specific coefficients

distribution

rule

welfare functio

n

Goal: Design the distribution rule

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

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

∑𝑖∈𝑆

𝑓 (𝑖 ,𝑆 )=𝑊 (𝑆)

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”

Nashequilibrium

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”

Nashequilibrium

Core

[von Neumann et al. 1944] [Nash 1951]

[Moulin 1992][Albers et al. 2006]

Requirements on the distribution ruleThe 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 [Shapley 1953]A player’s share of the welfare should depend on their“average” marginal contribution

Example: If players are homogeneous,

Note: There is also a weighted Shapley value

𝑓 (𝑖 ,𝑆 )= ∑𝑇⊆𝑆 ¿ 𝑖}¿ ¿¿ ¿

¿

Players are assigned ‘weights’

Properties of the Shapley value

+ Guaranteed to be in the core for “balanced” games [Shapley 1967]

+ Results in a potential game [Ui 2000]

+ Guarantees the existence of a Nash equilibrium

- Often intractable to compute [Conitzer et al. 2004]

- Not “efficient” in terms of social welfare e.g. Price of Anarchy/Stability

[Marden et al. 2011]

approximations are often tractable

[Castro et al. 2009]

Research question:

If so: can designs be more efficient and/or more tractable?If not: we can optimize over to determine the best design!

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 workTheorem (Chen, Roughgarden, Valiant):There exists a welfare function , for which no

distribution rules other than the weighted Shapley value guarantee a Nash equilibrium in all games.

[Chen et al. 2010]

Our resultTheorem:For any submodular welfare function , no

distribution rules other than the weighted Shapley value guarantee a Nash equilibrium in all games.

A game is specified by

The inspiration for our workTheorem (Chen, Roughgarden, Valiant):Given all games posses a

Nash equilibrium if and only if is a weighted Shapley value.

[Chen et al. 2010]

Theorem:Given and any submodular all games posses a

Nash equilibrium if and only if is a weighted Shapley value.

Our result

Can obtain the best distribution rule by optimizing the player weights,

Can always work within a potential game• Small, well-defined class of games• Several learning algorithms for Nash

equilibrium Fundamental limits on tractability and efficiency

Theorem:For any submodular welfare function , no

distribution rules other than the weighted Shapley value guarantee a Nash equilibrium in all games.

Our result

Consequences

Proof SketchFirst step: Represent using a linear basis

– Define a -welfare function:

– Given any , there exists a set , and a sequence of weights indexed by , such that:

Proof technique: Establish a series of necessary conditions on

𝑊≔ ∑𝑇 ∈𝒯 𝑞𝑇𝑊

𝑇

“contributing coalition”

“magnitude of contribution”

Proof Sketch (A single T-Welfare Function)

is completely specified by

⇓ is a weighted Shapley value

𝜔 𝑖❑={ 𝑓 (𝑖 ,𝑇 )

𝑞𝑇

𝑖∈𝑇

arbitrary 𝑖∉𝑇

What is requiredof

Proof technique: Establish a series necessary conditions on

⇓

is not formed in

is formed in

Don’t allocate welfare to any player

Allocate welfare only to players in ,

independent of others

is the basis weighted Shapley value

corresponding to , with weights

Key challenge: Each basis might use different !

Proof Sketch (General Welfare Functions)

What is requiredof

Proof technique: Establish a series necessary conditions on

no coalition from is formed in

Don’t allocate welfare to any player

Allocate welfare only to players in these formed coalitions,

independent of others

a coalition from is formed in

𝑓 ≔ ∑𝑇∈𝒯 𝑞𝑇 𝑓

𝑇

⇓

𝑓 ≔ ∑𝑇∈𝒯 𝑞𝑇 𝑓

𝑇

Proof Sketch (General Welfare Functions)

What is requiredof

Proof technique: Establish a series necessary conditions on

no coalition from is formed in

Don’t allocate welfare to any player

Allocate welfare only to players in these formed coalitions,

independent of others

⇓ is a weightedShapley value

Weights of common players in any two coalitions must be linearly dependent

is submodular

⇓a coalition from is formed in

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 gamesRaga 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]