Computation of the Nucleolus

1. Computation of the Nucleolus Presented by Jeffrey Smith Aaron Hardin

2. Contents Core What is the core? Definitions What is the Nucleolus? Some properties of the Nucleolus Uniqueness of the Nucleolus Conclusion References

3. Core By superadditivity, grand coalition is most beneficial, but what are the individual payoffs? Introduced to Game Theory by Gillies in 1959 (though the concept first appeared in a paper on economic theory by Edgeworth in 1881) Core is the set of payoff vectors for which no other coalition can offer better payoffs

4. What is the core • Imputation = distribution of payoff to members of a coalition • Dominated if there exists a coalition C that can have a greater or equal payoff to each player in C • Core is simply the set of non-dominated imputations • Could also be the empty set

5. Some definitions • Excess is the gain that players in a coalition S can obtain if they withdraw from the grand coalition N under payoff x and instead take the payoff v(S). • Lexicographic ordering is basically an alphabetic ordering. • An example is a most significant digit radix sort. • A pre-imputation is an efficient payoff vector. • The strong epsilon (ε) core: Cε(v)={xϵℝN:∑iϵNxi=v(N); ∑iϵSxi ≥v(S)-ε,∀ S⊆N} • The least-core is the intersection of all non-empty strong ε-cores. • A characteristic function game is a pair [N, v] where N is the set of players and v is a real-valued function defined over subsets of N (the coalitions)(N. Megiddo 1974)

6. What is the nucleolus Mathematically (Schmeidler 1969) : Let [N,v] be a game where v:2N→ℝ let xϵℝN be a payoff vector. The excess of x for a coalition S⊆N is the quantity v(S)-∑iϵSxi

7. What is the nucleolus Now let θ(x) ϵℝ2N be a vector of excesses of x, arranged in non-increasing order. In other words, θi(x)≥θj(x),∀i<j. Notice that x is in the core of v if and only if it is a pre-imputation and θ1(x)≤0.

8. What is the nucleolus To define the nucleolus, we consider the lexicographic ordering of vectors in ℝ2N For two payoff vectors x,y we say θ(x) is lexicographically smaller than θ(y) if for some index k, we have θi(x)=θi(y),∀i<k and θk(x)<θk(y).

9. What is the nucleolus The nucleolus of v is the lexicographically minimal imputation based on this ordering.

10. What is the nucleolus In (nearly) English (Maschler, Peleg & Shapley 1979): Start with the least-core. Record the coalitions for which the right-hand side of the inequality in the definition of Cε(v) cannot be further reduced without making the set empty. Continue decreasing the right-hand side for the remaining coalitions, until it cannot be reduced without making the set empty.

11. What is the nucleolus Record the new set of coalitions for which the inequalities hold at equality Continue decreasing the right-hand side of remaining coalitions and repeat this process as many times as necessary until all coalitions have been recorded. The resulting payoff vector is the nucleolus.

12. Some properties • Nucleolus is always unique. (Driessen 1988) • If the core is non-empty, the nucleolus is in the core. • The nucleolus is always in the kernel. • The kernel is contained in the bargaining set. • So the nucleolus is always in the bargaining set. (Driessen 1988)

13. Some properties Nucleolus of a compound game is the combination of the nucleoluses of the components. (N. Megiddo 1974) As a point of the kernel of the game the nucleolus reflects strength relations between players and symmetry properties of the characteristic function. (N. Megiddo 1974)

14. Uniqueness of the nucleolus • The nucleolus of a convex set consists of one outcome at most. (Schmeidler 1969) • It is enough to prove θ(x)=θ(y) and x≠y => θ(1/2(x+y))≺θ(x) • Let x,y ϵRr and η:Rr→Rr be the ordering function: ηi(z)=max{min{zj|jϵA}|A⊂{1,2,…,r} and Ā=i} • η(x+y)≾ η(x)+ η(y) • Let ηt(x)=xit, ηt(y)=yjt, ηt(x+y)=xkt+ykt, t=1…r xk1≤xi1 and yk1≤yj1 => xk1+yk1≤xi1+yj1 • Case 1: η(x+y) = η(x)+ η(y) • By induction on r, the proof is complete • Case 2: η(x+y) < η(x)+ η(y), see next slide

15. Uniqueness of the nucleolus • This brings us to 2θ(1/2(x+y)) ≾ θ(x)+θ(y) • Case 1: 2θ(1/2(x+y)) < θ(x)+θ(y) • The proposition is proved • Case 2: 2θ(1/2(x+y)) = θ(x)+θ(y) • Using the assumption θ(x)=θ(y) • We have v(Skt)-x(Skt)=v(Skt)-y(Skt) for each t • Equivalently, x(S)=y(S) for each S • This implies x=y, a contradiction

16. Conclusion The core is the non-dominated imputations. The nucleolus is in the kernel which is in the bargaining set. When the core is non-empty the nucleolus is inside it. “One of the most appealing properties of the nucleolus, as a solution concept, is its uniqueness.” (Schmeidler 1969)

17. references Nimrod Megiddo NUCLEOLUSES OF COMPOUND SIMPLE GAMES 1974 David Schmeidler THE NUCLEOLUS OF A CHARACTERISTIC FUNCTION GAME 1969 Theo DriessenCooperative Games, Solutions and Applications 1988 M. Maschler, B. Peleg, L. Shapley Geometric properties of the kernel, nucleolus, and related solution concepts 1979

18. References Definition of Nucleolus adapted from wikipedia, the article credits (Schmeidler 1969) and (Maschler, Peleg & Shapley 1979), so we have done the same, no other sources were found with the same text.