Raga Gopalakrishnan Caltech CU-Boulder Adam Wierman (Caltech) Jason Marden (CU-Boulder)
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Raga GopalakrishnanCaltech CU-Boulder
Adam Wierman (Caltech)Jason Marden (CU-Boulder)
Potential games are necessary to ensure pure Nash equilibria in cost sharing games
• Model for distributed resource allocation problems.
• Self-interested agents make decisions and share the resulting cost.
cost sharing games
Network formation[ Anshelevich et al. 2004 ]
S1
S2
D1
D2
61 6
1
1
6
1?+?
cost sharing games
Network formation[ Anshelevich et al. 2004 ]
S1
S2
D1
D2
61 6
1
1
6
13+3
A Nash equilibrium
Also optimal!
1+5
Unique Nash
equilibriumSuboptimal
cost sharing games
Key feature:Distribution rules
outcome!
Network formation[ Anshelevich et al. 2004 ]
?+?
S1
S2
D1
D2
61 6
1
1
6
1
Can we understand this
better?
cost sharing games
• Network formation games
• Facility location games
• Congestion games
• Routing games
• Multicast games
• Coverage games
• …
[ Anshelevich et al. 2004 ] [ Corbo and Parkes 2005 ] [ Fiat et al. 2006 ] [ Albers 2009 ][ Chen and Roughgarden 2009 ] [ Epstein, Feldman, and Mansour 2009 ] [ … ]
[ Rosenthal 1973 ] [ Milchtaich 1996 ] [ Christodoulou and Koutsoupias 2005 ][ Suri, Tóth, and Zhou 2007 ] [Bhawalkar, Gairing, and Roughgarden 2010 ] [ … ]
[ Roughgarden and Tardos 2002 ] [ Kontogiannis and Spirakis 2005 ][ Awerbuch, Azar, and Epstein 2005 ] [ Chen, Chen, and Hu 2010 ] [ … ]
[ Marden and Wierman 2008 ][ Panagopoulou and Spirakis 2008 ] [ … ]
[ Vetta 2002 ] [ Hoefer 2006 ] [ Dürr and Thang 2006 ] [ Chekuri et al. 2007 ] [ Hansen and Telelis 2008 ] [ … ]
[ Chekuri et al. 2007 ] [ Cardinal and Hoefer 2010 ][ Bilò et al. 2010 ] [ Buchbinder et al. 2010 ] [ … ]
[ Johari and Tsitsiklis 2004 ][ Panagopoulou and Spirakis 2008 ][ Marden and Effros 2009 ][ Harks and Miller 2011 ][ von Falkenhausen and Harks 2013 ] [ … ]
Key feature:Distribution rules
outcome!
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
Extensions: weighted and generalized weighted versions
parameterized by “weight system”
Most prior work studies two distribution rules
Marginal Contributions (MC+)[ Wolpert and Tumer 1999 ]
average marginal contribution over player
orderings
Shapley Values (SV+)[ Shapley 1953 ]
externality experienced by all other players
Extensions: weighted and generalized weighted versions
parameterized by “weight system”
Both guarantee PNE in all games!
Question: Are there other such distribution rules?Short answer: NO!
for any fixed cost functions
don‘t guarantee PNE in
all games
guarantee PNE in
all games
SV+
don‘t guarantee PNE in
all games
guarantee PNE in
all games
MC+
all distribution rules
SV+
MC+
guarantee potential
game
≡
1
2
don‘t guarantee PNE in
all games
guarantee PNE in
all games?
𝑮=(𝑵 ,𝑹 , {𝓐𝒊 }𝒊∈𝑵 , {𝑾 𝒓 }𝒓∈𝑹 , { 𝒇𝑾 }𝑾 ∈ {𝑾 𝒓 })
set of players
set of resources
action set of player
local welfare function distribution rule
S1
S2
D1
D2
Example:
Formal model
𝟏
𝟐𝟒
𝟕
𝟑
𝟓
𝟔𝑾 𝟒 ( {𝟏 ,𝟐} )
𝓤𝒊 (𝒂)=∑𝒓∈𝒂 𝒊
𝒇𝑾 𝒓 (𝒊 , {𝒂 }𝒓 )𝓤𝟏 (𝒂)𝒇𝑾 𝟒 (𝟏 , {𝟏 ,𝟐} )
“welfare” = revenue / negative cost
player 1’s share of
𝑮=(𝑵 ,𝑹 , {𝓐𝒊 }𝒊∈𝑵 , {𝑾 𝒓 }𝒓∈𝑹 , { 𝒇𝑾 }𝑾 ∈ {𝑾 𝒓 })𝑮=¿
Shapley Values (SV+)(parameterized by weight system )
𝑮=¿
Marginal Contributions (MC+)(parameterized by weight system )
Distribution rules:
0+0
S1
S2
D1
D2
61 6
1
1
6
13+3
𝒇𝑺𝑽+¿𝑾 [𝝎 ] ( 𝒊 ,𝑺)=∑
𝝅ℙ𝝎 (𝝅 ) ⋅ (𝑾 (𝑷 𝒊
𝝅∪ {𝒊 })−𝑾 (𝑷 𝒊𝝅 )) ¿
𝒇 𝑺𝑽+¿𝑾 [(𝟏 ,𝟏) ] (𝟏, {𝟏 ,𝟐 })=𝟑¿
𝒇 𝑺𝑽+¿𝑾 [(𝟓 ,𝟏) ] (𝟏, {𝟏 ,𝟐 })=𝟏¿1+5
𝒇 𝑴𝑪+¿𝑾 [(𝟏 ,𝟏) ] (𝟏 ,{𝟏 ,𝟐 })=𝟎 ¿
𝒇 𝑴𝑪+¿𝑾 [(𝟓 ,𝟏) ] (𝟏 ,{𝟏 ,𝟐 })=𝟎 ¿
𝒇𝑴𝑪+¿𝑾 [𝝎 ] (𝒊 ,𝑺 )=𝝎 𝒊 (𝑾 (𝑺 )−𝑾 (𝑺− {𝒊 }) )¿
𝑮=(𝑵 ,𝑹 , {𝓐𝒊 }𝒊∈𝑵 , {𝑾 𝒓 }𝒓∈𝑹 , { 𝒇𝑾 }𝑾 ∈ {𝑾 𝒓 })
“all games”
𝑾 𝒓∈𝕎 𝒇𝑾∈ 𝒇 𝕎
𝓖 (𝑵 ,𝕎 , 𝒇𝕎 )
S1
S2
D1
D2
61 6
1
1
6
1
𝓖 ( {𝟏 ,𝟐 }, {𝑾 𝟏 ,𝑾 𝟐 } , {𝒇 𝑾 𝟏 , 𝒇𝑾 𝟐 })
6 1
D1
D2
11
1
6 6
6 6
61
1
don‘t guarantee PNE in
all games
guarantee PNE in
all games
𝒇𝑺𝑽+¿𝕎
′
[𝝎 ] ¿
The inspiration for our work[ Chen, Roughgarden, and Valiant 2010 ]
budget-balancedguarantee potential
game
There exists :
don‘t guarantee PNE in
all games
guarantee PNE in
all games
𝒇 𝑺𝑽 +¿𝕎[𝝎 ] ¿
Our characterizationFor any :
?
𝕎 ′=𝕎
actual welfare
distributed
?
don‘t guarantee PNE in
all games
guarantee PNE in
all games
𝒇𝑴𝑪+¿𝕎
′ ′
[𝝎 ]¿
don‘t guarantee PNE in
all games
guarantee PNE in
all games
𝒇𝑺𝑽+¿𝕎
′
[𝝎 ] ¿
The inspiration for our work[ Chen, Roughgarden, and Valiant 2010 ]
budget-balancedguarantee potential
game
There exists :
don‘t guarantee PNE in
all games
guarantee PNE in
all games
𝒇 𝑺𝑽 +¿𝕎[𝝎 ] ¿
Our characterizationsFor any :
𝒉(⋅)
Tractability
Consequences
Efficiency
Incentive compatibili
ty
• If budget-balance is required, set . Just optimize over .
• Theorem: If budget-balance is not required, weights () don’t matter! Just optimize over .
Budget-balance
Four other important properties:
don‘t guarantee PNE in
all games
guarantee PNE in
all games
𝒇𝑺𝑽+¿𝕎
′
[𝝎 ] ¿
Tractability
Consequences
Efficiency
Incentive compatibili
ty
Budget-balance
Four other important properties:
Easier to control budget-balance
More tractable
“preprocessing”
don‘t guarantee PNE in
all games
guarantee PNE in
all games
𝒇𝑴𝑪+¿𝕎
′ ′
[𝝎 ]¿
don‘t guarantee PNE in
all games
guarantee PNE in
all games
𝒇𝑺𝑽+¿𝕎
′
[𝝎 ] ¿
exponential time!
𝒉(⋅)
Tractability
Consequences
Efficiency
Incentive compatibili
ty
Budget-balance
Four other important properties:
𝓤𝒊 (𝒂)=𝒗 𝒊 (𝒂)+ ∑𝒓 ∈𝒂𝒊
𝒇𝑾 𝒓 (𝒊 , {𝒂 }𝒓 )
private values
Future work: Design incentive compatible cost sharing mechanisms for a
noncooperative setting
𝓤𝒊 (𝒂)=∑𝒓∈𝒂 𝒊
𝒇𝑾 𝒓 (𝒊 , {𝒂 }𝒓 )
Basis representation: Any can be written as
“coalition”“contributio
n of the coalition”
Proof sketchRestrict to the case where where is arbitrary
Restrict to the case of characterizing only budget-balanced
𝑾= ∑𝑻⊆𝑵
𝒒𝑻𝑾𝑾 𝑻
𝑾𝑻 (𝑺 )={𝟏 , 𝑻 ⊆𝑺𝟎 , 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞
Technique: Establish a series of necessary conditions on
[ Shapley 1953 ]
Inclusion functions: Special welfare functions :
is said to be a “contributing” coalition in if
Proof sketch
How much of should get?
Does contain a contributing
coalition of that contains ?
no
yes
𝒇𝑾 (𝒊 ,𝑺 )=𝟎
should not depend on the “noncontributi
ng” players
𝒇𝑾≔ ∑𝑻⊆𝑵
𝒒𝑻𝑾 𝒇𝑾𝑻
DECOMPOSITON
Proof: Establish a “fairness” condition on
1
Proof sketch
𝒇𝑾≔ ∑𝑻⊆𝑵
𝒒𝑻𝑾 𝒇 𝑻
DECOMPOSITON
Proof: Establish a “fairness” condition
1
𝒇𝑾≔ ∑𝑻⊆𝑵
𝒒𝑻𝑾 𝒇 𝑺𝑽 +¿𝑻 [𝝎𝑻 ]¿
𝒇𝑾≔ ∑𝑻⊆𝑵
𝒒𝑻𝑾 𝒇 𝑺𝑽 +¿𝑻 [𝝎∗] ¿
𝒇𝑾≔ 𝒇 𝑺𝑽+¿𝑾 [𝝎∗] ¿
≝
CONSISTENCY
Proof: Construct a “universal” equivalent
to all
2
Ragavendran GopalakrishnanCaltech CU-Boulder
Adam Wierman (Caltech)Jason Marden (CU-Boulder)
Potential games are necessary to ensure pure Nash equilibria in cost sharing games
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