csa-logo-color.p Nectar NECTAR : Nash Equilibrium CompuTation Algorithms and Resources Sirshendu Arosh Arjun Suresh Aravind S. R. Deepak Vishwakarma Mentors : Rohith D. Vallam, Premmraj H. Faculty Advisor: Prof Y. Narahari Department of Computer Science & Automation Indian Institute of Science, Bangalore Bengaluru, India. Dept of CSA Game Theory Mini Project Presentation
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NECTAR : Nash Equilibrium CompuTation Algorithms and … · Nectar CO-OPERATIVE GAME Definition Co-operative Games Co-operative games emphasize participation, challenge,and fun rather
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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
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 )
The set of all payoff allocations of a Transferable Utility (TU)game that are individually rational, coalitionally rational, andcollectively rational.Then, Core(N, v) =
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.
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}
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
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
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