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