nectar : nash equilibrium computation algorithms and … · nectar co-operative game deﬁnition...

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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.

Dept of CSA Game Theory Mini Project Presentation

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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 at a glance

Figure: Architectural View


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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 !!!


nectar : nash equilibrium computation algorithms and … · nectar co-operative game deﬁnition...

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