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
Introduction Local public goods problem Decentralized mechanism Future directions
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
Shruti Sharma BWGT 2010, Sao Paulo 2 / 31
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
· · ·
Shruti Sharma BWGT 2010, Sao Paulo 3 / 31
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
· · ·
Shruti Sharma BWGT 2010, Sao Paulo 3 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsMotivation Literature Servey Contribution
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.
Shruti Sharma BWGT 2010, Sao Paulo 4 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsMotivation Literature Servey Contribution
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.
Shruti Sharma BWGT 2010, Sao Paulo 4 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsMotivation Literature Servey Contribution
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.
Shruti Sharma BWGT 2010, Sao Paulo 5 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsMotivation Literature Servey Contribution
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.
Shruti Sharma BWGT 2010, Sao Paulo 6 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsModel Resource allocation problem
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
Shruti Sharma BWGT 2010, Sao Paulo 7 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsModel Resource allocation problem
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.
Shruti Sharma BWGT 2010, Sao Paulo 8 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsModel Resource allocation problem
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.
Shruti Sharma BWGT 2010, Sao Paulo 8 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsModel Resource allocation problem
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.
Shruti Sharma BWGT 2010, Sao Paulo 9 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsModel Resource allocation problem
The model (M)
Set Ri
Set Cj
ij
Neighbor setsSet of users that affect user i : Ri := {k ∈ N | k → i}
Shruti Sharma BWGT 2010, Sao Paulo 10 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsModel Resource allocation problem
The model (M)
Set Ri
Set Cj
ij
Neighbor setsSet of users that affect user i : Ri := {k ∈ N | k → i}
Set of users affected by user j : Cj := {k ∈ N | j → k}
Shruti Sharma BWGT 2010, Sao Paulo 11 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsModel Resource allocation problem
The model (M)
Set Ri
Set Cj
ij
Neighbor setsSet of users that affect user i : Ri := {k ∈ N | k → i}
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 .
Shruti Sharma BWGT 2010, Sao Paulo 12 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsModel Resource allocation problem
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.
Shruti Sharma BWGT 2010, Sao Paulo 13 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsModel Resource allocation problem
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)
Shruti Sharma BWGT 2010, Sao Paulo 14 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsModel Resource allocation problem
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.
Shruti Sharma BWGT 2010, Sao Paulo 15 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsModel Resource allocation problem
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.
Shruti Sharma BWGT 2010, Sao Paulo 16 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsModel Resource allocation problem
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.
Shruti Sharma BWGT 2010, Sao Paulo 16 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsModel Resource allocation problem
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.
Shruti Sharma BWGT 2010, Sao Paulo 16 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsModel Resource allocation problem
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.
Shruti Sharma BWGT 2010, Sao Paulo 17 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsModel Resource allocation problem
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.”
Shruti Sharma BWGT 2010, Sao Paulo 18 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results
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
Shruti Sharma BWGT 2010, Sao Paulo 19 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results
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.
Shruti Sharma BWGT 2010, Sao Paulo 20 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results
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.
Shruti Sharma BWGT 2010, Sao Paulo 20 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results
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.
Shruti Sharma BWGT 2010, Sao Paulo 20 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results
Ideas for construction of game form
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.
Shruti Sharma BWGT 2010, Sao Paulo 21 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results
Ideas for construction of game form
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.
Shruti Sharma BWGT 2010, Sao Paulo 21 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results
Ideas for construction of game form
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.
Shruti Sharma BWGT 2010, Sao Paulo 21 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results
Ideas for construction of game form
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.
Shruti Sharma BWGT 2010, Sao Paulo 21 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results
Ideas for construction of game form
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.
Shruti Sharma BWGT 2010, Sao Paulo 21 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results
Ideas for construction of game form
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
Shruti Sharma BWGT 2010, Sao Paulo 22 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results
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 .
Shruti Sharma BWGT 2010, Sao Paulo 23 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results
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 .
Shruti Sharma BWGT 2010, Sao Paulo 23 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results
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
Shruti Sharma BWGT 2010, Sao Paulo 24 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results
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 .
Shruti Sharma BWGT 2010, Sao Paulo 25 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results
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)
Shruti Sharma BWGT 2010, Sao Paulo 26 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results
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 ).
Shruti Sharma BWGT 2010, Sao Paulo 27 / 31
Introduction Local public goods problem Decentralized mechanism Future directionsGame form Results
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 .
Shruti Sharma BWGT 2010, Sao Paulo 28 / 31
Introduction Local public goods problem Decentralized mechanism Future directions
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
Shruti Sharma BWGT 2010, Sao Paulo 29 / 31
Introduction Local public goods problem Decentralized mechanism 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.
Shruti Sharma BWGT 2010, Sao Paulo 30 / 31
Introduction Local public goods problem Decentralized mechanism Future directions
Thank you!
Shruti Sharma BWGT 2010, Sao Paulo 31 / 31