Embed Size (px)
DESCRIPTION
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. Key Question: How should the cost be shared?. - PowerPoint PPT Presentation
Citation preview
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]
