Post on 04-Aug-2020
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NECTAR : Nash Equilibrium CompuTationAlgorithms and Resources
Sirshendu AroshArjun SureshAravind S. R.
Deepak Vishwakarma
Mentors : Rohith D. Vallam, Premmraj H.Faculty Advisor: Prof Y. Narahari
Department of Computer Science & AutomationIndian Institute of Science, Bangalore
Bengaluru, India.
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Organization of the Presentation
NECTAR: OverviewOur Contribution
Co-operative GameCoreShapley ValueNucleolus
Mechanism DesignEx-Post EfficiencyDSIC, BICDictatorshipIndividual Rationality
Bayesian Games
Future work
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NECTAR : Nash Equilibrium CompuTation Algorithmsand Resources
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NECTAR : Nash Equilibrium CompuTation Algorithmsand Resources
NECTAR at a glance
Figure: Architectural View
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NECTAR : Nash Equilibrium CompuTation Algorithmsand Resources
Figure: Previous View Figure: Present View
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NECTAR
NECTAR: Detailed View
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COOPERATIVE GAMES: CORE, SHAPLEY VALUE,NUCLEOLUS
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CO-OPERATIVE GAME
Definition
Co-operative Games
Co-operative games emphasize participation, challenge,and fun rather than defeating someone.
A co-operative game may involve switching teams so that everyone ends up on the winning team.
We consider Co-operative games in coalition form.
The coalitional form of an n-person game is given by the pair (N, v),where N is the set of players and v is areal-valued function, called the characteristic function of the game, defined on the power set of N satisfying
v(φ) = 0 and
if S and T are disjoint coalitions (S ∩ T ) = φ , then v(S) + v(T ) ≤ v(S ∪ T )
Example:
N = {A, B,C}v(A) : 1v(B) : 2v(C) : 3v(AB) : 4v(BC) : 7v(AC) : 5v(ABC) : 15
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Core
Definition
The set of all payoff allocations of a Transferable Utility (TU)game that are individually rational, coalitionally rational, andcollectively rational.Then, Core(N, v) =
(x1, ..., xn) ∈ Rn :n∑
i=1
xi = v(N);∑i∈c
(x)i > v(C),∀C ⊆ N
Example:
N = {A,B,C}v(A) : 1 −→ x(A) > 1v(B) : 2 −→ x(B) > 2v(C) : 3 −→ x(C) > 3v(AB) : 4 −→ x(A) + x(B) > 4v(BC) : 7 −→ x(B) + x(C) > 7v(AC) : 5 −→ x(A) + x(C) > 5v(ABC) : 15 −→ x(A) + x(B) + x(C) = 15
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Shapley Value
Axioms of Shapley Value
Efficiency: Σi∈Nφi(v) = v(N)
Symmetry: If i and j are such that v(S ∪ {i}) = v(S ∪ {j})for every coalition S not containing i and j, thenφi(v) = φj(v)
Dummy Axiom: If i is such that v(S) = v(S ∪ {i}) forevery coalition S not containing i, thenφi(v) = 0Additivity: If u and v are characteristic functions, thenφ(u + v) = φ(u) + φ(v)
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Shapley Value
Definition
Shapely Value is the exactly one mapping φ : R2n−1 −→ Rn that satisfies all the Shapely Value axioms.
This function satisfies ∀i ∈ N, ∀v ∈ R2n−1 ,
φi (v) = ΣC⊆N−i|C|!(n − |C| − 1)!
n!v(c ∪ i)− v(c)
Example:In the previous example,
φA(v) = ((v(AB)− v(B)) \ 3!) + ((v(AC)− v(C)) \ 3!) + (2 ∗ (v(A)− v(φ)) \ 3!) + (2 ∗ (v(ABC)− v(BC)) \ 3!)= (4− 2) \ 6 + (5− 3) \ 6 + 2 ∗ (1− 0) \ 6 + 2 ∗ (15− 7) \ 6= 11 \ 3
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Shapley Value
Alternate Shapley Value Calculation
Alternatively, Shapley value can be expressed in terms of the possible orders of players in N.Let O : {1, .....N} −→ {1, ....N} be a permutation that assigns to each position k the player O(k) Given apermutation O, let Prei (O) denote the set of predecessors of the player i in the order O(i.e. Prei (O) = {O(1), ......,O(K − 1)}, if i = O(k)).The Shapley value can now be given as
shi (v) = ΣO∈π(N)
1
n!v(Prei (O) ∪ i)− v(Prei (O))
Example:In the previous example, The set of all permutations are {ABC, ACB, BAC, CAB, BCA, CBA}
φA(v) = (v(A) + v(A) + v(BA)− v(B) + V (CA)− v(C) + v(BCA)− v(BC) + v(CBA)− v(CB)) \ 3!= (1 + 1 + 4− 2 + 5− 3 + 15− 7 + 15− 7) \ 6= 11 \ 3
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Polynomial Calculation of the Shapley Value based onsampling
Reference
Polynomial Calculation of the Shapley Value based on sampling by Javier Castro, Daniel Gomez,Juan Tejadapublished on Computers and Operations Research, date:15 April 2008.
Estimation of Shapley Value
The population of sampling process P will be set of all possible orders i,e P = π(N)
The vector parameter under study will be sh = sh1, ....., shn
The characteristics observed in each sampling unit O ∈ π(N) are marginal contributions of the players inthe order O i.e;X(O) = (X(O)1, ....., X(O)n) where X(O)i = v(Prei (O) ∪ i)− v(Prei (O))
The estimate of parameter sh , sh will be the mean of the marginal distributions over the sample Mi.e.sh = (sh1, ...., shm) where shi = (1/m)ΣO∈M X(O)i
Finally the selection process used to determine the sample M will take any order with probability (1/n!)
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Nucleolus
DefinitionExcess is a measure of the inequity of an imputation x for acoalition S is defined ase(x ,S) = v(S)− Σj∈Sxj
DefinitionDefine O(x) as the vector of excesses arranged in decreasing(nonincreasing) order
Let X = {x : Σnj=1xj = v(N)} be the set of efficient allocations
We say that a vector v ∈ X is a Nucleolus if for every x ∈ X wehave,O(v) ≤L O(x)
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Nucleolus
Properties of NucleolusThe nucleolus of a game in coalitional form exists and isuniqueThe nucleolus is group rational, individually rational, andsatisfies the symmetry axiom and the dummy axiomIf the core is not empty, the nucleolus is in the core
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Snap View Shapley Value
Output Output
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Snap View of Core and Nucleolus
Output Output
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MECHANISM DESIGN
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Social choice function
DefinitionEx-Post Efficiency:
SCF f(θ) is Pareto optimal .The outcome f(θ1, ....., θn) is Pareto optimal if there doesnot exist any x ∈ X such that
ui(x , θi) > ui(f (θ), θi)∀i ∈ N and ui(x , θi) > ui(f (θ), θi) forall i ∈ N
DefinitionDictatorship:A SCF is said to be Dictatorial if for any agent d , the followingproperty satisfies
∀θ ∈ Θ, f (θ) is such that ud (f (θ), θd ) > ud (x , θd )∀x ∈ X
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Incentive compatibility
DefinitionDominant Strategy Incentive Compatibility(DSIC):A SCF is DSIC, if the direct revelation mechanismD = ((Θi)i∈N , f (.)) has a weakly dominant strategy equilibrium
sd (.) = (sd1 (.), ......, sd
n (.)) in which sdi (θi) = θi∀θi ∈ Θ, i ∈ N
ui(f (θi , θ−i), θi) > ui(f (θ)i , θi)∀i ∈ N, ∀θi ∈ Θi ,∀θ−i ∈ Θ−i ,∀θi ∈ Θi
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Incentive compatibility
DefinitionBayesian Incentive Compatibility(BIC):A SCF is BIC if the direct revelation mechanismD = ((Θi)i∈N , f (.)) has a Bayesian Nash equilibrium
s∗(.) = (s∗1(.), ......, s∗n(.)) in which s∗i (θi) = θi∀θi ∈ Θ, i ∈ N
Eθ−i [ui(f (θi , θ−i), θi)|θi ] > Eθ−i [ui(f (θ)i , θi)|θi ],∀i ∈ N, ∀θi ∈Θi ,∀θ−i ∈ Θ−i ,∀θi ∈ Θi
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Individual rationality
Definition
Ex-Post Individual Rationality Constraints:
when any agent i is given a choice to withdraw from the mechanism at the ex-post stage
to ensure the agent i’s participation, we must satisfy the following ex-post participation constraints
ui (f (θi , θ−i )θi ) > ui (θi )
Definition
Interim Individual Rationality Constraints:
let the agent i be allowed to withdraw from mechanism only at an interim stage
interim participation(or individual rationality) constraints for agent i require that
Ui (θi \ f ) = Eθ−i[ui (f (θi , θ−i ), θi \ θi ] > ui (θi )θi ∈ Θi
Definition
Ex-Ante Individual Rationality constraints:
Let agent i be refuse to participate in the mechanism only at ex-ante stage
Ex-Ante participation constraints for agent i require that
Ui (f ) = Eθ [ui (f (θi , θ−i ), θi )] > Eθi[ui (θi )]
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Input format
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Snap View of EPE and DSIC
EPE DSIC
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Snap View of BIC and Ex-Post IR
BIC Ex-Post IR
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Snap View of IIR and Ex-Ante IR
IIR Ex-Ante IR
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BAYESIAN GAMES
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Bayesian Games: Definition, Notation
DefinitionA Bayesian game is a game with incomplete information and itis given by tuple Γ =< N, (Θi), (Si), (pi), (ui) >
N =1,2,...,N is a set of playersΘi is the set of types of the player i where i=1,2,...,n.Si is the set of actions or pure strategies of player i wherei=1,2,...,n.
Consistency of Beliefs
We say beliefs (pi)i∈N in a Bayesian game are consistent iffthere is some common prior distribution over the set of typeprofiles Θ. For each player’s beliefs,it can be given by
pi(θ−i |θi) =p(θi , θ−i)
Σt−i∈θ−ip(θi , t−i)∀θi ∈ Θi ;∀θ−i ∈ Θ−i ;∀i ∈ N
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Type Agent Representation and the Selten Game
Reinhard Selten proposed a representation of Bayesiangames.The idea is to represent every possible type of every playeras an agent or player in the new game.Given a Bayesian game
Γ =< N, (Θi), (Si), (pi), (ui) >
The Selten Game is an equivalent strategic form gameΓs =< Ns,Sj
s,Uj >
Each player in the original Bayesian game is now replacedwith a number of type agents.
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Bayesion Game to Selten Game
InputCommon Prior distributionPlayer Utilities for every Prior
ComputationSelten Players Ns = ∪i∈NΘiSelten Players Strategies Ss
θi= Si for every θi ∈ Θi
Player Belief probabilities
pi (θ−i |θi ) =p(θi , θ−i )
Σt−i∈θ−ip(θi , t−i )∀θi ∈ Θi ;∀θ−i ∈ Θ−i ;∀i ∈ N
Payoff computation in Selten Gameuθi (sθi , s−θi ) = Σt−i∈Θ−i pi (t−i|θi )ui (θi ,t−i , sθi , st−i ))
OutputSelten Game in NFG formUse any implemented algorithm to calculate equilibria
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Snap View of input and output
Input Output
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Future Work
Implementation of Additional Solution ConceptsUse of more Approximation AlgorithmsImprovement of GUIProvision of a web-based InterfaceReplacement of cplex with an open-source LP Solver
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Thank You !!!
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