Resource allocation in local public good networks: An ...svandana/test_g000006.pdf · Resource...

Introduction Local public goods problem Decentralized mechanism Future directions

Resource allocation in local public good networks:An implementation theory perspective

by

Shruti SharmaYahoo! Labs, Bangalore, India

and

Demos TeneketzisEECS, University of Michigan, Ann Arbor

Brazilian Workshop of the Game Theory Society

University of Sao Paulo, Brazil

July 29 – August 4, 2010

Shruti Sharma BWGT 2010, Sao Paulo 1 / 31


Outline

1 IntroductionMotivationLiterature surveyContribution of our work

2 Resource allocation in local public goods networksThe model (M)The resource allocation problem (PD)

3 A decentralized resource allocation mechanismIdeas for construction of game formResults

4 Future directions


Introduction Local public goods problem Decentralized mechanism Future directionsMotivation Literature Servey Contribution

Motivation

In networks individuals’ actions inuence the performance of directly connectedneighbors.

Influence of individuals’ actions can propagate in the entire network and affect itsperformance

Local public goods networks

Pollution abatement program by a jurisdiction

New library built by a university

Spread of information and innovation in social/research networks

Users’ attention received by an advertisement is influenced by the presence ofother advertisers on a webpage

· · ·



Motivation

Local public good vs. public good

A local public good is accessible to and inuences the utilities of individuals in aparticular locality (neighborhood) within a big network.

A public good is accessible to and inuences the utilities of all individuals.

Challenges for local public goods networks

Information about the network is often localized.

Individuals have private information about the network or their own characteristics.

Individuals may be selfish who care only about their own benefit in the network.



Literature survey

Tiebout’56 and successive works:

Study network formation problems in which individuals choose where to locate based onthe local public goods expenditure patterns of various municipalities.

Bramoulle, Kranton’07 and Yuan:

Study influence of selfish users’ behavior on local public goods provision in networkswith fixed links.

Investigate relation between the structure of networks and the existence and nature ofNash equilibria (NE) of users’ effort levels in those networks.

None of the NE of analyzed games result in a local public goods provision that achievesoptimum social welfare.

Our work:

We design a mechanism that can implement the optimum social welfare in NE.

Follows the philosophy of Groves, Ledyard’77, Hurwicz’79, Walker’81, Chen’02 in thecontext of local public goods provision.



Contribution of our work

Set Ri

Set Cj

ij

• The formulation of a problem of local public goods provision in the framework ofimplementation theory.

• Specification of a game form (decentralized mechanism) that(i) implements in Nash equilibria the optimal solution of the corresponding

centralized local public goods provision problem.(ii) is individually rational.(iii) results in budget balance at all Nash equilibria and off equilibrium.


Introduction Local public goods problem Decentralized mechanism Future directionsModel Resource allocation problem

Outline




4 Future directions



The model (M)

Set Ri

Set Cj

ij

Set of N users N := {1, 2, . . . ,N}; one network operator.

User i ∈ N has to take an action ai ∈ Ai .Assumption: For each i ∈ N ,

Ai is a convex and compact set in R that includes 0.Ai is user i ’s private information.



The model (M)

Set Ri

Set Cj

ij

Set of N users N := {1, 2, . . . ,N}; one network operator.

User i ∈ N has to take an action ai ∈ Ai .Assumption: For each i ∈ N ,

Ai is a convex and compact set in R that includes 0.Ai is user i ’s private information.

Each user obtains a utility from the actions of a subset of network users.



The model (M)

Set Ri

Set Cj

ij

Neighbor setsSet of users that affect user i : Ri := {k ∈ N | k → i}



The model (M)

Set Ri

Set Cj

ij


Set of users affected by user j : Cj := {k ∈ N | j → k}



The model (M)

Set Ri

Set Cj

ij


Set of users affected by user j : Cj := {k ∈ N | j → k}

Assumption:

i affects i ∀ i ∈ N .Ri and Ci are independent of users’ action profile aN := (ak )k∈N .Every user i ∈ N knows its neighbor sets Ri and Ci .The network operator knows Ri and Ci for all i ∈ N .



The model (M)

Set Ri

Set Cj

ij

Users’ utilitiesUtility of user i resulting from action profile aRi := (ak )k∈Ri is ui (aRi ).

Assumption:

For all i ∈ N , ui : R|Ri | → R is concave in aRi .ui (aRi ) = 0 for ai /∈ Ai .ui is private information of user i .

Each user i ∈ N is non-cooperative/selfish and is self utility maximizer.



The model (M)

Set Ri

Set Cj

ij

TaxEach user i ∈ N pays a tax ti (>,<,=) 0.

Assumption: Network operator redistributes the tax:∑

i∈N ti = 0.

Aggregate utility of user i : uAi (aRi , ti ) : R

|Ri |+1 → R ∪ {−∞}.

uAi (aRi , ti ) :=

−ti + ui (aRi ), if ai ∈ Ai

−∞, otherwise.(1)



The model (M)

Set Ri

Set Cj

ij

Static network:The set N of users, the neighbor sets Ri and Ci , i ∈ N , the action spacesAi , i ∈ N , and the utility functions ui , i ∈ N , remain fixed.



The resource allocation problem (PD)

Problem (PC )

max(aN ,tN )

∑i∈N

uAi (aRi , ti )

s.t.∑i∈N

ti = 0(2)

≡ max(aN ,tN )∈D

∑i∈N

ui (aRi )

where, D :={(aN , tN ) ∈ R2N | ai ∈ Ai ∀ i ∈ N ;

∑i∈N

ti = 0} (3)

Objective

To develop a game form that

• implements in Nash equilibria the optimal solution of Problem (PC ).

• is individually rational.

• results in budget balance.



Why Nash equilibrium as a solution concept?

Possible solution concepts under incomplete information

Bayesian Nash – Users in Model (M) do not possess prior beliefs about theutility functions and action sets of other users.

Dominant strategy – Impossibility results for the existence of non-parametricefficient dominant strategy mechanisms in classical public good environments.

Vickrey-Clarke-Groves (VCG) mechanisms –

Do not guarantee budget balance.Require infinite message space to communicate continuous utility functions.

Nash equilibrium – Existence results for non-parametric, individually rational,budget-balanced Nash implementation mechanisms for classical private andpublic goods environments.



Why Nash equilibrium as a solution concept?

Interpretation of Nash equilibria for Model(M)

Groves and Ledyard, “Incentive compatibility since 1972.” –

“We do not suggest that each user knows all of system environmentwhen it computes its message.

We do suggest, however, that the complete information Nash game-theoretic equilibrium messages may be possible stationary messagesof some unspecified dynamic message exchange process.”


Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results

Outline




4 Future directions



Ideas for construction of game form

Construction of message space

Message exchange should contain informationfor determining optimal allocation (a∗N , t

∗N ).

A user should contribute in determining theactions of its neighbors that affect its utility.

A user should pay for its neighbors’ actionsthat contribute to its utility.

Set Ri

Set Cj

ij

Possible message space

Each user should communicate a message consisting of two components:

• Proposal for actions the user wants its neighbors to take.

• Proposal for price the user wants to pay for the actions of each of its neighbors.




Construction of outcome function

Nash implementation⇒

Price taking behavior must be induced.

A user should not control its NE price.

Given NE price, a user should be able tochoose actions that maximize its utility.

Set Ri

Set Cj

ij

Difficulty

• A user may propose arbitrary prices.

Resolving difficulty

Penalty is added to the tax of each user.

Budget balance⇒

Additional tax to balance the flow of money.

Should not depend on a user’s own message.




Construction of outcome function

Individual rationality is achieved if

Price taking behavior is induced.

Given the prices, each user can controlthe actions that affect its utility.

⇒For any given NE prices, user i can guarantee atleast its initial utility by forcing all its neighbors’actions to be 0.

Set Ri

Set Cj

ij



The game form

Message space

User i ∈ N sends to the network operator amessage mi ∈Mi := R|Ri | × R|Ri |

+

mi := ( ai Ri, πi Ri

);

ai Ri∈ R|Ri |, πi Ri

∈ R|Ri |+ , i ∈ N ,

(4)

where,

ai Ri:= ( ai k )k∈Ri

and πi Ri:= ( πi k )k∈Ri , i ∈ N .

(5)

Set Ri

Set Cj

ij

ai k is user i ’s action proposal for user k ∈ Ri .

πi k is the price that user i proposes to pay for the action of user k ∈ Ri .



The game form

Outcome function

Action profile

Network operator determines user i ’s action ai

from its neighbors’ (set Ci ) messages;

aj (mCj ) =1|Cj |

∑k∈Cj

akj , j ∈ N , (6)

where,mCj := (mk )k∈Cj . (7)

Set Ri

Set Cj

ij



The game form

Outcome function

Tax profile

Set Cj

ij

h

k

l

p

1

2

3

4

5

0

Ilj = 3Cj(3) = l

Ijj = 4Cj(4) = j

Iij = 5Cj(5) = i

Ikj = 2Cj(2) = k

Ihj = 1 = Iij + 1Cj(1) = h

Ipj = 0

Indices 1, 2, . . . , |Cj | assigned in cyclic order to users in each set Cj , j ∈ N .

Iij : Index of user i ∈ N associated with set Cj , j ∈ N .

Cyclic order indexing⇒ if Iij = |Cj |, then Iij + 1 = 1, Iij + 2 = 2, . . .Indices Iij remain fixed throughout the time period of interest.

Cj(n): User with index n ∈ {1, 2, . . . , |Cj |} in set Cj .



The game form

Outcome function

Tax profile

Set Cj

ij

h

k

l

p

1

2

3

4

5

0

Ilj = 3Cj(3) = l

Ijj = 4Cj(4) = j

Iij = 5Cj(5) = i

Ikj = 2Cj(2) = k

Ihj = 1 = Iij + 1Cj(1) = h

Ipj = 0

ti ((mCj )j∈Ri ) =∑j∈Ri

lij (mCj ) aj (mCj ) +∑j∈Ri

πi j

(ai j − a

Cj(Iij+1)

j

)2−∑j∈Ri

πCj(Iij+1)

j

(a

Cj(Iij+1)

j − aCj(Iij+2)

j

)2

Term 1 (price taking) Term 2 (penalty) Term 3 (budget balancing)(8)

where, lij (mCj ) = πCj(Iij+1)

j− πCj(Iij+2)

j , j ∈ Ri , i ∈ N . (9)



Results

Theorem 1

Let m∗N be a Nash equilibrium of the game induced by the proposed game formand users’ utility functions.

Let (a∗N , t∗N ) := (aN (m∗N ), tN (m∗N )) :=

((ai (m∗Ci

))i∈N , (ti ((m∗Cj)j∈Ri ))i∈N

)be the action and tax profiles at m∗N determined by the game form.

Then,

(a) Each user i ∈ N weakly prefers (a∗Ri, t∗i ) to its initial allocation (0, 0).

Mathematically,

uAi

(a∗Ri

, t∗i)≥ uA

i

(0, 0

), ∀ i ∈ N .

(b) (a∗N , t∗N ) is an optimal solution of Problem (PC ).



Results

Theorem 2Let a∗N be the optimum action profile of Problem (PC ). Then,

(a) There exist a set of personalized prices l∗ij , j ∈ Ri , i ∈ N , such that

a∗Ri= arg max

ai∈Aiaj∈R, j∈Ri\{i}

−∑j∈Ri

l∗ij aj + ui (aRi ), ∀ i ∈ N .

(b) There exists at least one Nash equilibrium (NE) m∗N of the game induced bythe proposed game form and users’ utility functions such that aN (m∗N ) = a∗N .

If t∗i :=∑

j∈Ril∗ij a∗j , i ∈ N , the set of all NE m∗N = (m∗i )i∈N = ( ai ∗Ri

, πi ∗Ri)

that result in (a∗N , t∗N ) is characterized by the solution of:

1|Ci |

∑k∈Ci

ak ∗i = a∗i , i ∈ N ,

Cj(Iij+1)π∗j −Cj(Iij+2)π∗j = l∗ij , j ∈ Ri , i ∈ N ,

πi ∗j

(ai ∗j −

Cj(Iij+1)a∗j)2

= 0, j ∈ Ri , i ∈ N ,

πi ∗j ≥ 0, j ∈ Ri , i ∈ N .



Outline




4 Future directions



Future directions

Development of effcient mechanisms that can compute NE.

Admission and topology control for local public good networks.

Implementation mechanisms for local public good networks underdynamic situations.

Networks with multiple network operators.



Thank you!


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