Characterizing distribution rules for cost sharing games

Characterizing distribution rules for cost sharing games

Thesis defenseMay 28,

2013byRaga Gopalakrishnan

Computing and Mathematical Sciences, Caltech

Distributed resource allocation problems• Allocate scarce resources to distributed

agents, such that some objective function is optimized.

• Several examples throughout this talk…

Example: Network formation

S1

S2

D1

D2

61

6

1

1

6

1

Problems:• NP-hard, not scalable,

not reliable, …• Agents are people; not

everyone may be happy with the outcome.

Centralized optimization:• Build the optimal

network (cost 10).• Recover this cost

from S1 and S2.

[ Jackson 2003 ]

Example: Network formation

S1

S2

D1

D2

61

6

1

1

6

1

Distributed solution:• Let sources play a

noncooperative game by choosing the edges they want and pay for them.

• Q: How to share the cost of the common edge?

Outcome depends on

how this cost is shared!

[ Jackson 2003 ]

Example: Network formation game

S1

S2

D1

D2

61

3+3

1

1

6

1

S1

S2

D1

D2

61

5+1

1

1

6

1

optimal network is a Nash

equilibrium

sub-optimal Nash

equilibrium

Q: How to share the cost of the common edge?

Option 1: S1 pays 5 S2 pays 1

Option 2: S1 pays 3 S2 pays 3

[ Anshelevich et al. 2004 ]

Cost sharing games

Network formation

Facility location

Multi-project management

S1

S2

D1

D2

? ??

????

??

?

Common Feature:The distribution rules used

to share the global cost/revenue determine the

players’ local utility functions,


[ Vetta 2002 ]

Cost sharing games

Network formation

Facility location


S1

S2

D1

D2

? ??

????

??

?

Common Feature:The distribution rules used

to share the global “welfare” determine the

players’ local utility functions,

and hence the outcome!


[ Vetta 2002 ]

Cost sharing games

Network formation

Facility location


S1

S2

D1

D2

? ??

????

??

?

Goal:Design distribution

rules that result in a “desirable” outcome.


[ Vetta 2002 ]

Goal of the thesis:

Network formation

Facility locationMulti-projectmanagement

S1

S2

D1

D2

? ??

????

???

Characterize the space of all distribution rules

that result in a “desirable” outcomefor a broad class of cost sharing games.

[ G., Marden, Wierman. NetGCooP 2011. ][ G., Marden, Wierman. EC 2013. ] [ full version under submission to MOR. ]

Goal of the thesis:

Network formation


S1

S2

D1

D2

? ??

????

???


that result in a stable outcomefor a broad class of cost sharing games.


“equilibrium”+efficiency,

+tractability, …

Scenario: Heterogeneous multi-server system

lFCFS

scheduling

m1

m2

mm

?

Variations – distributed resource allocationScheduling games in service systems

1Focus: Designing dispatch policies[ G., Doroudi, Wierman. MAMA 2011. ]

Staffing games in service systems

2Focus: Investigating square-root staffing policy for large systems[ Working paper, in preparation for submission to OR. ]Joint work with Ward and Wierman.

staffing

?

Variations – distributed resource allocation

? ? ?

? ? ?

? ? ?

Scenario:Content distribution

network

Content caching and replicationFocus: Investigating for and computing pure Nash equilibria[ G., Kanoulas, Karuturi, Rangan, Rajaraman, Sundaram. LATIN 2012. ]


Scenario: Distributed control

Sensor coverage

?

Wireless frequency selection

F1 F2 F3 F1 F2 F3

?

frequency

frequency

Wireless access point assignment

?

• No people involved! Objective function is not cost/revenue.

• pretend that agents are people, model them as players in a “cost” sharing game.

Game theoretic controlFocus: Designing the entire game[ G., Marden, Wierman. HotMetrics 2010. ]


VM

VM

VM

VM

VM

VM

VM

VM?

??

?

?

Scenario:Bandwidth allocation in

datacenter networks

Bandwidth allocation in multi-tenant datacentersFocus: Designing a robust bandwidth allocation scheme[ working paper ]Joint work with Attar, Jeyakumar, Narayana, Prabhakar.

Goal of the thesis:

Network formation


S1

S2

D1

D2

? ??

????

???


that result in a stable outcomefor a broad class of cost sharing games.


Model

players (researchers)

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

𝟒𝟑

𝟏 𝟐

𝟓 𝟔

𝟒𝟑

𝟏 𝟐

𝟓 𝟔

Model

players (researchers) resources (projects)

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

𝒓𝟏

𝒓𝟑

𝒓𝟐

𝒓𝟒

𝟒𝟑

𝟏 𝟐

𝟓 𝟔

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

𝒓𝟏

𝒓𝟑

𝒓𝟐

𝒓𝟒

𝓐𝟒= {{𝒓 𝟏 } , {𝒓𝟑 ,𝒓𝟒 }}

𝒂𝒊∈𝓐𝒊⊆𝟐𝑹

𝒂𝟒=¿

𝓐𝟔

𝓐𝟓

𝓐𝟒

𝓐𝟑

𝓐𝟐

𝓐𝟏

𝟒𝟑

𝟏 𝟐

𝟓 𝟔


𝒓𝟏

𝒓𝟑

𝒓𝟐

𝒓𝟒

𝑾 𝒓 :𝟐𝑵→ℝ

𝑾 𝒓𝟑( {𝟒 ,𝟔} )

Denote the set of all (distinct) local welfare functions:𝕎𝑮=¿𝒓∈𝑹 {𝑾 𝒓 }

Global welfare:

𝓦 (𝒂 )=∑𝒓∈𝑹

𝑾 𝒓 ({𝒂}𝒓 )

𝟒𝟑

𝟏 𝟐

𝟓 𝟔


𝒓𝟏

𝒓𝟑

𝒓𝟐

𝒓𝟒

𝑾 𝒓𝟑( {𝟒 ,𝟔} )

𝒇 𝒓 :𝑵 ×𝟐𝑵→ℝ

𝒇𝒓𝟑 (𝟒 , {𝟒 ,𝟔 })

𝒇 𝒓 𝟑(𝟔 , {𝟒 ,𝟔} )

Scalability assumption:𝑾 𝒓 𝒊=𝑾 𝒓 𝒋

⟹ 𝒇 𝒓 𝒊= 𝒇 𝒓 𝒋

Denote by𝒇 𝒓 𝒇𝑾 𝒓

𝟒𝟑

𝟏 𝟐

𝟓 𝟔

Model 𝑮=(𝑵 ,𝑹 , {𝓐𝒊 }𝒊∈𝑵 , {𝑾 𝒓 }𝒓∈𝑹 , { 𝒇𝑾 }𝑾 ∈𝕎𝑮

)

𝒓𝟏

𝒓𝟑

𝒓𝟐

𝒓𝟒

𝑾 𝒓𝟑( {𝟒 ,𝟔} )

𝒇𝑾 :𝑵 ×𝟐𝑵→ℝ

𝒇𝒓𝟑 (𝟒 , {𝟒 ,𝟔 })

𝒇 𝒓 𝟑(𝟔 , {𝟒 ,𝟔} )




𝟒𝟑

𝟏 𝟐

𝟓 𝟔

Model 𝑮=(𝑵 ,𝑹 , {𝓐𝒊 }𝒊∈𝑵 , {𝑾 𝒓 }𝒓∈𝑹 , 𝒇𝕎𝑮)

𝒓𝟏

𝒓𝟑

𝒓𝟐

𝒓𝟒

𝑾 𝒓𝟑( {𝟒 ,𝟔} )

𝒇𝑾 :𝑵 ×𝟐𝑵→ℝ

𝒇𝒓𝟑 (𝟒 , {𝟒 ,𝟔 })

𝒇 𝒓 𝟑(𝟔 , {𝟒 ,𝟔} )




𝟒𝟑

𝟏 𝟐

𝟓 𝟔

Model𝒓𝟏

𝒓𝟑

𝒓𝟐

𝒓𝟒 𝟏

𝟒

𝟐𝟒

𝟔

𝟐

𝟑

𝟑

𝟔

𝟓An allocation𝒂= (𝒂𝟏 ,𝒂𝟐 ,⋯ ,𝒂𝒏 )

Players’ overall utility is the sum of their shares across all the resources

they choose𝓤𝒊 (𝒂)=∑𝒓∈𝒂 𝒊

𝒇 𝒓 (𝒊 , {𝒂}𝒓 )

Example: 𝒂𝟒={𝒓𝟑 ,𝒓 𝟒 }𝓤𝟒 (𝒂)= 𝒇 𝒓𝟑(𝟒 , {𝟒 ,𝟔 } )+ 𝒇 𝒓𝟒

(𝟒 , {𝟏 ,𝟐 ,𝟒 } )𝟒

𝓐𝟔

𝓐𝟓

𝓐𝟒

𝓐𝟑

𝓐𝟐

𝓐𝟏

𝑮=(𝑵 ,𝑹 , {𝓐𝒊 }𝒊∈𝑵 , {𝑾 𝒓 }𝒓∈𝑹 , 𝒇𝕎𝑮)

Model𝒓𝟏

𝒓𝟑

𝒓𝟐

𝒓𝟒 𝟏

𝟒

𝟐𝟒

𝟔

𝟐

𝟑

𝟑

𝟔

𝟓An allocation𝒂= (𝒂𝟏 ,𝒂𝟐 ,⋯ ,𝒂𝒏 )

Players’ overall utility is the sum of their shares across all the resources

they choose𝓤𝒊 (𝒂)=∑𝒓∈𝒂 𝒊

𝒇 𝒓 (𝒊 , {𝒂}𝒓 )

Example: 𝒂𝟒={𝒓𝟑 ,𝒓 𝟒 }𝓤𝟒 (𝒂)= 𝒇 𝒓𝟑(𝟒 , {𝟒 ,𝟔 } )+ 𝒇 𝒓𝟒

(𝟒 , {𝟏 ,𝟐 ,𝟒 } )𝟒

Design!

𝑮=(𝑵 ,𝑹 , {𝓐𝒊 }𝒊∈𝑵 , {𝑾 𝒓 }𝒓∈𝑹 , 𝒇𝕎𝑮)

Model

a broad model that includes:

• Multi-project management

• Network formation games

• Facility location games• Multicast games• Congestion games• Routing games• Coverage games• …

𝑮=(𝑵 ,𝑹 , {𝓐𝒊 }𝒊∈𝑵 , {𝑾 𝒓 }𝒓∈𝑹 , 𝒇𝕎𝑮)


[ Vetta 2002 ]

[ Chekuri et al. 2007 ]

[ Rosenthal 1973 ]

[ Roughgarden and Tardos 2002 ]

[ Marden and Wierman 2008,2013 ]

𝟒𝟑

𝟏 𝟐

𝟓 𝟔 𝓐𝟔

𝓐𝟓

𝓐𝟒

𝓐𝟑

𝓐𝟐

𝓐𝟏 𝒓𝟏

𝒓𝟑

𝒓𝟐

𝒓𝟒

Goal:

𝟒𝟑

𝟏 𝟐

𝟓 𝟔

Given the set of players

𝓐𝟔

𝓐𝟓

𝓐𝟒

𝓐𝟑

𝓐𝟐

𝓐𝟏 𝒓𝟏

𝒓𝟑

𝒓𝟐

𝒓𝟒

Goal:

𝟒𝟑

𝟏 𝟐

𝟓 𝟔

Given the set of players and the local welfare functions,

𝑾 𝒓𝟑( ∙ ) 𝑾 𝒓 𝟒

( ∙ )

𝑾 𝒓𝟏( ∙ ) 𝑾 𝒓𝟐

( ∙ )

𝓐𝟔

𝓐𝟓

𝓐𝟒

𝓐𝟑

𝓐𝟐

𝓐𝟏

Goal:

𝟒𝟑

𝟏 𝟐

𝟓 𝟔

Given the set of players and the local welfare functions, design local distribution rules

𝒇𝑾 𝒓 𝟏 ( ∙ ,∙ )

𝑾 𝒓𝟑( ∙ ) 𝑾 𝒓 𝟒

( ∙ )

𝒇𝑾 𝒓 𝟐 ( ∙ ,∙ )

𝒇𝑾 𝒓 𝟑 ( ∙ ,∙ ) 𝒇𝑾 𝒓 𝟒 ( ∙ , ∙ )

𝑾 𝒓𝟏( ∙ ) 𝑾 𝒓𝟐

( ∙ )

𝓐𝟔

𝓐𝟓

𝓐𝟒

𝓐𝟑

𝓐𝟐

𝓐𝟏

Goal:

𝟒𝟑

𝟏 𝟐

𝟓 𝟔


that result in a “desirable” outcome

𝓐𝟔

𝓐𝟓

𝓐𝟒

𝓐𝟑

𝓐𝟐

𝓐𝟏

Goal:

𝒇𝑾 𝒓 𝟏 ( ∙ ,∙ )

𝑾 𝒓𝟑( ∙ ) 𝑾 𝒓 𝟒

( ∙ )

𝒇𝑾 𝒓 𝟐 ( ∙ ,∙ )

𝒇𝑾 𝒓 𝟑 ( ∙ ,∙ ) 𝒇𝑾 𝒓 𝟒 ( ∙ , ∙ )

𝑾 𝒓𝟏( ∙ ) 𝑾 𝒓𝟐

( ∙ )

𝟒𝟑

𝟏 𝟐

𝟓 𝟔


that result in a “desirable” outcomeregardless of the set of resources and players’

action sets.

𝓐𝟔

𝓐𝟓

𝓐𝟒

𝓐𝟑

𝓐𝟐

𝓐𝟏

Goal:

𝒇𝑾 𝒓 𝟏 ( ∙ ,∙ )

𝑾 𝒓𝟑( ∙ ) 𝑾 𝒓 𝟒

( ∙ )

𝒇𝑾 𝒓 𝟐 ( ∙ ,∙ )

𝒇𝑾 𝒓 𝟑 ( ∙ ,∙ ) 𝒇𝑾 𝒓 𝟒 ( ∙ , ∙ )

𝑾 𝒓𝟏( ∙ ) 𝑾 𝒓𝟐

( ∙ )

𝓖 (𝑵 ,𝕎 , 𝒇𝕎 )Class of all games

with set of players ,set of local welfare functions ,

corresponding local distribution rules

𝑮=(𝑵 ,𝑹 , {𝓐𝒊 }𝒊∈𝑵 , {𝑾 𝒓 }𝒓∈𝑹 , 𝒇𝕎𝑮)


that result in a “desirable” outcomeregardless of the set of resources and players’

action sets.

Goal:

Given any and ,design

such that all games in are “desirable”.

𝓖 (𝑵 ,𝕎 , 𝒇𝕎 )Class of all games

with set of players ,set of local welfare functions ,

corresponding local distribution rules

Goal:

𝑮=(𝑵 ,𝑹 , {𝓐𝒊 }𝒊∈𝑵 , {𝑾 𝒓 }𝒓∈𝑹 , 𝒇𝕎𝑮)

“desirable” properties for a gameStatic:

• Stability• Equilibrium concepts

• Efficiency• Price of anarchy• Price of stability

• Budget-balance

• Tractability• Poly-time

computable

• Locality• Separable

• …

Dynamic:• Existence of

distributed learning rules that converge to one of the stable outcomes

• Existence of distributed learning rules that converge to the most efficient stable outcome

• Good convergence rates

• Simple, intuitive learning dynamics perform well, e.g., “best-response”

• …




• Budget-balance


computable


• …






• …




• Stability• pure Nash

equilibrium

• dominant strategy equilibrium

• mixed Nash equilibrium

• correlated equilibrium

• coarse-correlated equilibrium

• …

∀ 𝒊 𝒂𝒊𝑵𝑬∈𝐚𝐫𝐠𝐦𝐚𝐱

𝒂 𝒊∈𝓐𝒊

𝓤𝒊 (𝒂𝒊 ,𝒂−𝒊𝑵𝑬 )

An allocation/outcome that satisfies:



equilibrium

• Efficiency





Price of Anarchy (PoA)Ratio of the optimum

welfare to the welfare of the worst Nash

equilibrium.

Price of Stability (PoS)Ratio of the optimum

welfare to the welfare of the best Nash equilibrium.

Goal: Given any and , design such that all games in are “desirable”.



equilibrium

• Efficiency





Price of Anarchy (PoA)Ratio of the optimum

welfare to the welfare of the worst Nash

equilibrium.

Price of Stability (PoS)Ratio of the optimum

welfare to the welfare of the best Nash equilibrium.

Goal: Given any and , design such that all games in possess a pure Nash equilibrium.

Existing distribution rulesThe Shapley value [ Shapley 1953 ]

A player’s share of the welfare should depend on their

“average” marginal contribution

Intuition:• Imagine the players in arriving one at a time to the

resource, according to some order.• When player arrives, depending on when he

arrived, he sees some subset of players already present.

• Player can be thought of as contributing .• The Shapley value is his expected marginal

contribution over all possible orders, assuming each order is equally likely.

• Example: when all players are ‘identical’, .

𝒇 𝑺𝑽𝑾 (𝒊 ,𝑺 )= ∑

𝑻⊆𝑺¿𝒊 }¿ ¿¿ ¿¿

Properties of the Shapley valueStatic:


equilibrium


• Budget-balance


computable


• …






• …

Static:• Stability

• pure Nash equilibrium


• Budget-balance


computable


• …






• …

Properties of the Shapley value

Static:• Stability

• pure Nash equilibrium


• Budget-balance


computable


• …






• …




equilibrium






Potential gameThere exists a

player-independent

function , called the potential

function, which encodes utility

differences due to unilateral

deviations by any player.

∀ 𝒊 ∀𝒂−𝒊 ∀𝒂𝒊 ,𝒂𝒊′ 𝓤𝒊 (𝒂𝒊 ,𝒂−𝒊 )−𝓤𝒊 (𝒂𝒊

′ ,𝒂−𝒊 )=𝚽 (𝒂𝒊 ,𝒂−𝒊 )−𝚽 (𝒂𝒊′ ,𝒂− 𝒊 )

• Potential game• guarantees stability

(pure Nash equilibrium), good dynamic properties


• Budget-balance

• Tractability• Poly-time computable


• …





• Budget-balance



• …


+ Optimal in many specific settings, e.g., network coding, network formation. [ Marden and Effros 2009 ] [ Roughgarden 2009]

‒ Lower bound of 2 for submodular welfare functions. [ Marden and Wierman 2013 ]




• Budget-balance



• …





• Budget-balance



• …


‒ Requires computing exponentially many marginal contributions

‒ Intractable in general [ Matsui and Matsui 2000 ]

Existing distribution rulesExtensions of the Shapley value

• Weighted Shapley value : :• Parameterized by a vector of strictly positive player

weights • Expected marginal contribution according to a

probability distribution (that depends on ) with full support on all orders.

• Generalized weighted Shapley value : :• Parameterized by a weight system , where is a vector

of strictly positive player weights, and is an ordered partition of the set of players.

• Expected marginal contribution according to a probability distribution (that depends on ) with support only on those orders that respect : players in arrive before players in when .

• They lead to weighted/generalized weighted potential games. Their properties are similar to those of the Shapley value.

Existing distribution rulesThe marginal contribution

[ Wolpert and Tumer 1999 ]

A player’s share of the welfare is theirmarginal contribution to the welfare

• Similar extensions: weighted marginal contribution ( ) and generalized weighted marginal contribution ( ) can be defined.

𝒇 𝑴𝑪𝑾 (𝒊 ,𝑺 )=𝑾 (𝑺)−𝑾 (𝑺 ¿{𝒊¿})




• Budget-balance



• …

Properties of the marginal contribution




• Budget-balance



• …


+ Potential function is the same as the welfare function!



• Efficiency• Price of anarchy• Price of stability = 1

• Budget-balance



• …


+ Potential function is the same as the welfare function!

‒ Price of anarchy can be arbitrarily bad. [ Marden and Wierman 2013 ]




• Budget-balance



• …


‒ Not budget-balanced in general.




• Budget-balance



• …


+ Requires computing just one marginal contribution!

Research question:

If so: how do they perform?If not: we can optimize over for efficiency!

Are there distribution rules besides the Shapley value and marginal contribution families that always guarantee a pure Nash equilibrium for all games in ?

Cost sharing games

Research question:

Our (surprising) answer:

NO, for any given local welfare functions .

Cost sharing games

Are there distribution rules besides the Shapley value and marginal contribution families that always guarantee a pure Nash equilibrium for all games in ?

This is a much bigger “NO” than last year’s!

The inspiration for our workTheorem (CRV,MW):There exists a for which any budget-balanced distribution rules that guarantee a pure Nash equilibrium for all games must be equivalent to generalized weighted Shapley values on .[Chen et al. 2010, Marden and Wierman 2013]

Our resultTheorem:For any , any distribution rules that guarantee a pure Nash equilibrium for all games must be equivalent to generalized weighted Shapley values on some “ground” welfare functions , corresponding to the actual welfares distributed.

Cost sharing games

𝑮=(𝑵 ,𝑹 , {𝓐𝒊 }𝒊∈𝑵 , {𝑾 𝒓 }𝒓∈𝑹 , 𝒇𝕎𝑮)

If budget-balance is desired, then .

The inspiration for our workTheorem (CRV,MW):There exists a local welfare function for which all games in possess a pure Nash equilibrium for a budget-balanced if and only if there exists a weight system for which .[Chen et al. 2010, Marden and Wierman 2013]

Our resultTheorem:For any , all games in possess a pure Nash equilibrium if and only if there exists a weight system , and a mapping that maps each local welfare function to a corresponding “ground” welfare function such that . Moreover,

Cost sharing games

first

actual welfaredistributed

The inspiration for our workTheorem (CRV,MW):There exists a local welfare function for which all games in possess a pure Nash equilibrium for a budget-balanced if and only if there exists a weight system for which .[Chen et al. 2010, Marden and Wierman 2013]

Our second resultTheorem:For any , any distribution rules that guarantee a pure Nash equilibrium for all games must be equivalent to generalized weighted marginal contributions on some “ground” welfare functions .

Cost sharing games

PropositionTheorem:For any two welfare functions and , and any weight system ,

Any welfare function can be represented as a linear combination of inclusion functions as follows:

where, for each , the inclusion function is defined as:

denotes the set of contributing coalitions of . denotes the magnitude of their contributions in .

Shapley value = marginal contribution

PropositionTheorem:For any two welfare functions and , and any weight system ,

Our second resultTheorem:For any , all games in possess a pure Nash equilibrium if and only if there exists a weight system , and a mapping that maps each local welfare function to a corresponding “ground” welfare function such that . Moreover,

,where denotes the bijection that maps to .

Cost sharing games

call this map as

What’s different from last time?Then Now

The space of desirable

distribution rules:Weighted Shapley

values on , .

• Single ‘base’ welfare function scaled at each resource

• Scalability:

• Any submodular • Technical

assumption to exclude zero weights

• Budget-balance

[ NetGCooP 2011 ]




values on , .


• Scalability:



• Budget-balanceThe space of

desirable distribution rules:Weighted Shapley

values on , .



values on , .

[ NetGCooP 2011 ]




values on , .


• Scalability:



• Budget-balance

• Arbitrary welfare functions from at each resource

• Weak Scalability:

• Any



values on , .



values on , .

[ NetGCooP 2011 ]




values on , .

The space of desirable distribution rules:

Generalized weighted Shapley values on , .


• Scalability:



• Budget-balance



• Any • No technical

assumption

[ NetGCooP 2011 ]




values on , .


Generalized weighted Shapley values on , .


• Scalability:



• Budget-balance




assumption• Not necessarily

budget-balanced[ NetGCooP 2011 ] [ EC 2013 ]




values on , .


Generalized weighted Shapley values on ,

+ equivalent characterization


• Scalability:



• Budget-balance




assumption• Not necessarily

budget-balanced[ EC 2013 ][ NetGCooP 2011 ]

Theorem (marginal contributions):For any , all games in possess a pure Nash equilibrium if and only if there exists a weight system , and a mapping that maps each local welfare function to a corresponding “ground” welfare function such that . Moreover,

,where denotes a special bijection that maps to .

Our characterizationsTheorem (Shapley values):For any , all games in possess a pure Nash equilibrium if and only if there exists a weight system , and a mapping that maps each local welfare function to a corresponding “ground” welfare function such that . Moreover,

Static:


equilibrium


• Budget-balance


Consequences of our characterizationsDynamic:

• Existence of distributed learning rules that converge to one of the stable outcomes







• Budget-balance


Consequences of our characterizations

• Potential games are necessary to ensure pure Nash equilibria in our model of cost sharing games!

• Both and , where , result in the same (weighted) potential game, with the same potential function .

• This provides a (previously unknown) closed form expression for the potential function for the weighted Shapley value.

(Weighted)

(Weighted)• Potential game

• guarantees stability (pure Nash equilibrium), good dynamic properties


• Budget-balance


Consequences of our characterizations

• New structure on distribution rules for optimizing for PoA/PoS.

• Optimize over the weights and ground welfare functions .

• If budget-balance is required• Set = . Just optimize over .

• If budget-balance is not required• Further reduction in the

space of parameters is possible.

• Theorem: Weights don’t matter!• Just optimize over .

• Future work: Can we obtain a precise characterization of the budget-balance vs. efficiency tradeoff?

(Weighted)• Potential game

• guarantees stability (pure Nash equilibrium), good dynamic properties


• Budget-balance


Consequences of our characterizations• A design through Shapley

values provides control over budget-balance, since is the actual welfare distributed.• But, Shapley values are

hard to compute.• A design through marginal

contributions is tractable, however, is not the actual welfare distributed.• To control budget-balance,

we need to start with desired to be distributed, and then use the exponential time transformation to get .

• Can be thought of as a “preprocessing” step.

Recall: Basis representation of as a linear combination of inclusion functions:

where

“contributing coalition”

“magnitude of

contribution”

Proof sketch of main resultRestrict to the case (single welfare function)

𝑾= ∑𝑻∈𝓣 𝒒𝑻𝑾

𝑻𝑾𝑻 (𝑺 )={𝟏 , 𝑻 ⊆𝑺

𝟎 , 𝐨𝐭𝐡𝐞𝐫𝐰𝐢𝐬𝐞

Reduction to characterizing budget-balanced distribution rules:• Notice that does not directly affect strategic

behavior (it only does so through ).• and are the same for any two • Pick so that is a budget-balanced rule for .

Proof technique: Establish a series of necessary conditions on

Proof sketch when Proof technique: Establish a series of necessary conditions on

is completely specified by

𝒇 𝑻≔ 𝒇 𝑮𝑾𝑺𝑽𝑻 [𝝎𝑻 ]

where and

What is requiredof

is not formed in

is formed in

Don’t allocate welfare to any

player

Allocate welfare only to players in ,

independent of others

𝝀𝒊𝑻={𝒇 𝑻 (𝒊 ,𝑻 ) , 𝒊∈𝑺𝟏

𝑻

arbitrary , 𝒊∈𝑺𝟐𝑻

Proof sketch (arbitrary )Proof technique: Establish a series of necessary conditions on

What is requiredof

no coalition from is formed in

Don’t allocate welfare to any

playerAllocate welfare only to players in

these formed coalitions,

independent of others

a coalition from is formed in

𝒇𝑾≔ ∑𝑻∈𝓣 𝒒𝑻 𝒇

𝑻

𝒇𝑾≔ ∑𝑻∈𝓣 𝒒𝑻 𝒇 𝑮𝑾𝑺𝑽

𝑻 [𝝎𝑻 ]

𝒇𝑾≔ ∑𝑻∈𝓣 𝒒𝑻 𝒇 𝑮𝑾𝑺𝑽

𝑻 [𝝎∗ ]

𝒇𝑾≔ 𝒇 𝑮𝑾𝑺𝑽𝑾 [𝝎∗]

Universal weight system ( equivalent to

all the )

Key proof techniquesBasis framework:• Crucial to the modular structure of the proof.• Separates the following two steps:

– Focus on basis distribution rules• Independent of welfare function

– Focus on consistency of these distribution rules• Depends on the welfare function• Separate into cases (different sign combinations of the basis

coefficients), and provide a counterexample template for each case.

– Allows easily extending the proof to the case of multiple welfare functions.

• Exposes fundamental equivalence of Shapley value and marginal contribution rules.

Inclusion-exclusion framework:• Crucial to show that distribution rules must be

representable in the same basis framework.• Helps build inductive counterexamples by singling out

any desired inclusion function (unanimity game) from an arbitrary welfare function.

Limitations of our characterizations• Our goal was to characterize “universal” cost

sharing designs – that guarantee pure Nash equilibrium in all games with arbitrary resources and action sets.

• However, if the class of games of interest have more structure on the action sets, our characterizations claim nothing!

• There may be distribution rules other than generalized weighted Shapley values that guarantee pure Nash equilibria.

• E.g., “Single-selection” coverage games. [ Marden and Wierman 2008 ]

• Future work: What structure on action sets causes the restriction to Shapley value designs?

Extensions• Recall that players’ utility functions are given by:

• In addition, suppose players also have (possibly heterogeneous) private valuations over the set of outcomes, so that:

• Suppose . Then, our model and results extend to this situation if the distribution rules cannot depend on these values.

• But, more efficient designs could be possible if the distribution rules can depend on these private values.• Additional desirable property: “incentive

compatibility”

• (Challenging!) Future work: Can we design efficient, incentive compatible cost sharing mechanisms for such noncooperative settings?

Characterizing distribution rules for cost sharing games

Thesis defenseMay 28,

2013byRaga Gopalakrishnan

Computing and Mathematical Sciences, Caltech

Documents

Characterizing distribution rules for cost sharing games