The lexicographic kernel of a game lexicographically maximizes the surplusses $s_{ij}$ (rather than the excesses as would the nucleolus). We show that an element in the lexicographic kernel can be computed efficiently, provided we can efficiently compute the surplusses $s_{ij}(x)$ corresponding to a given allocation $x$. This approach improves previously obtained results and allows us to determine a kernel element without appealing to Maschler transfers in the execution of the algorithm. \u

Faigle, U.

Kern, W.

Kuipers, J.

The lexicographic kernel of a game lexicographically maximizes the surplusses $s_{ij}$ (rather than the excesses as would the nucleolus). We show that an element in the lexicographic kernel can be computed efficiently, provided we can efficiently compute the surplusses $s_{ij}(x)$ corresponding to a given allocation $x$. This approach improves previously obtained results and allows us to determine a kernel element without appealing to Maschler transfers in the execution of the algorithm

Kern, Walter

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Computing an element in the lexicographic kernel of a game

Kuipers, Jeroen

Faigle, Ulrich

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Faculty of Mathematical SciencestUniversity of TwenteThe NetherlandsP.O. Box 2177500 AE EnschedeThe NetherlandsPhone: +31-53-4893400Fax: +31-53-4893114Email: memo@math.utwente.nlwww.math.utwente.nl/publicationsMemorandum No. 1663Computing an element in thelexicographic kernel of a gameU. Faigle1, W. Kern and J. Kuipers2November, 2002ISSN 0169-26901Zentrum für Angewandte Informatik Köln, Universität zu Köln, Weyertal 80, D-50931 Köln2Department of Mathematics, Maastricht University, PO Box 616, NL-6200 MD MaastrichtComputing an element in the lexicographickernel of a gameUlrich FaigleZentrum f̈ur Angewandte InformatikUniversiẗat zu Köln, Weyertal 80, D-50931 K̈olnWalter KernFaculty of Electrical Engineering,Mathematics and Computer ScienceUniversity of Twente, P.O.Box 217, NL-7500 AE EnschedeJeroen KuipersDepartment of MathematicsMaastricht UniversityP.O. Box 616, NL-6200 MD MaastrichtAbstractThe lexicographic kernel of a game lexicographically maximizes the surplussessij (rather than the excesses as would the nucleolus). We show that an element inthe lexicographic kernel can be computed efficiently, provided we can efficientlycompute the surplussessij(x) corresponding to a given allocationx. This ap-proach improves the results in [2] and allows us to determine a kernel elementwithout appealing to Maschler transfers in the execution of the algorithm.Keywords:Kernel, lexicographic, computational complexity.AMS Classification:90C27, 90D12.1 IntroductionThe lexicographic kernelis a solution concept for cooperative games, introduced byKalai (cf. [9]). The basic idea is to lexicographically maximize the vector of surplussessij (rather than the vector of excesses as in the definition of the nucleolus). So thissolution concept tries to combine characteristic features of both the nucleolus and thekernel.The lexicographic kernel is a polytope which (similar to the nucleolus) is containedin the intersection of the least core with the kernel. In contrast to the nucleolus, it is a“geometrical locus” ([9]) in the sense that it is completely determined by the (least) coreas a subset of the euclidian space. M. Yarom ([9]) investigates continuity aspects of thissolution concept and presents a bargaining procedure converging to the lexicographickernel.The purpose of our work is to present an algorithm that computes an element in thelexicographic kernel (in a finite number of steps). Our approach is closely related1to the one we used in [2] for finding an element in the kernel. Under weak assump-tions, concerning the efficient computability of the surplusses, the algorithm is efficient(i.e., has polynomial running time). The algorithm presented here offers a substantialimprovement with respect to the algorithm (and its analysis) derived in [2]: As thelexicographic kernel is a subset of the kernel, we are now able to completely eliminateMaschler’stransfer steps(cf. [8]) for the computation of a kernel element.2 Basic definitionsWe consider(cooperative) games(N, c), whereN = {1, . . . , n} is the set of playersandc : 2N → R a cost function, assigning a costc(S) to everycoalition S ⊆ N .We assume throughout thatc(∅) = 0. An allocation is a vectorx ∈ Rn satisfyingx(N) = c(N), where we use the standard shorthand notationx(S) =∑i∈Sxi .We letX ⊆ Rn denote the set of allocations. Relative to a givenx ∈ X, theexcessofa coalitionS ⊆ N is defined ase(S, x) = c(S)− x(S) .The minimum (non-trivial) excess is then given byemin(x) = minS 6=∅,Ne(S, x) .A related notion is thesurplussij(x) of playeri against playerj, wheresij(x) = min{e(S, x) | S ⊆ N, i ∈ S, j 6∈ S}.With these notions, we are ready to introduce the followingsolution concepts. Forε ∈ R, theε-core is defined asε-core(c) = {x ∈ X | emin(x) ≥ ε}.Thusε = 0 yields the well-knowncore. If ε is the unique maximum number for whichtheε-core is nonempty, we obtain the so-calledast core.Thepre-kernelK(c) is defined asK(c) = {x ∈ X | sij(x) = sji(x) ∀i 6= j}.The pre-nucleolusη(c) is the unique allocationx ∈ X that lexicographically maxi-mizes the(2n − 2)-dimensional vector obtained by arranging the non-trivial excessese(S, x), ∅ 6= S 6= N , in nondecreasing order.Replacing the excesses by the surplusses in the last definition, we obtain the so-calledlexicographic pre-kernelKlex(c). ThusKlex(c) is the set of allocationsx ∈ X thatlexicographically maximize then(n − 1)-dimensional vector obtained by arrangingthe surplussesij(x) in nondecreasing order.2Remark: An allocationx ∈ X is calledindividually rational if x(i) ≤ c({i}) for alli ∈ N . Restricting oneself to the setX∗ = {x ∈ X | x(i) ≤ c({i}) ∀i ∈ N}of individually rational allocations, one arrives at slightly modified solution concepts.For example, thenucleolusresp. thelexicographic kernelis obtained by replacingXwith X∗ in the definitions above. From our point of view, however, there is no reason(other than tradition) to restrict ourselves toX∗, i.e., to distinguish between singletoncoalitions and others in advance. We therefore work with the above “pre-solution con-cepts”. It is straightforward to modify the algorithm we present in Section 4 so that itcomputes elements in the lexicographic kernel.It is well-known that the (pre-)nucleolus is contained in the intersection of the (pre-)kernel with the least core. A similar relation holds for the lexicographic (pre-)kernel.Proposition 2.1 Klex(c) ⊆ K(c) ∩ least core (c).Proof: The inclusionKlex(c) ⊆ least core(c), follows directly from the definitions. ToproveKlex(c) ⊆ K(c), assumex ∈ Klex(c) and order the surplusses non-decreasingly:si1j1(x) ≤ . . . ≤ simjm(x).If x 6∈ K(c), there exists a smallest indexk such that the pair(i, j) = (ik, jk) satisfiessij(x) < sji(x), say,sij(x) = sji(x) − 2α for someα > 0. In this case, we couldexecute atransferof sizeα and pass fromx to the allocationx = x + αej − αei(with ei and ej being thei-th resp. j − th unit vector inRn). We claim that theα-transfer yields a lexicographically larger vector of surplusses, contradicting our as-sumption thatx ∈ Klex(c).Indeed, the transfer yieldssij(x) = sij(x) + α > sij(x). Suppose that neverthelessthe claim is false and there exist players`,m, however, such thats`m(x) ≤ sij(x) and s`m(x) < s`m(x) .LettingS (with ` ∈ S andm /∈ S) be such thats`m(x) = c(S)− x(S), we then musthavej ∈ S andi /∈ S and, therefore,sji(x) ≤ c(S)− x(S) < s`m(x) + α ≤ sij(x) + α ,which contradicts the equalitysji(x) = sij(x) + 2α. 2We end this section by mentioning that the lexicographic (pre-)kernel does not neces-sarily contain the (pre-)nucleolus,cf. [5],[9].3 Our computational modelIn principle, a game(N, c) can be described by a complete list of all2n cost valuesc(S), S ⊆ N . Relative to this notion ofinput sizemost computational game-theoreticproblems are trivially easy (efficiently solvable). However, measuring the input size3this way is often not adequate. For example, in the case of a minimum spanning treegame we arenot given such a list of2n cost values, but rather a weighted graph onn + 1 vertices, from which we can easily infer the costc(S) for any given coalitionS ⊆ N .For this reason, a more adequate (and more interesting) model is used (cf. [2] for moredetails and additional motivation). We consider a fixed classC of games. Each game(N, c) ∈ C has acompact descriptionin terms of• The finite setN = {1, . . . , n} of players• An upper bound〈c〉 on the maximum size of a cost value,i.e., 〈c〉 ≥ maxS⊆N〈c(S)〉.• An algorithm (“oracle”) which, on inputS ⊆ N , computes the correspondingcostc(S).(Here, we assume that all costsc(S) are rational numbers. The size〈r〉 of a rationalnumberr = p/q is the number of bits necessary to representp andq in binary.)We consideralgorithms for the classC. The input for such an algorithmA is a game(N, c) ∈ C, presentedvia the player setN , the upper bound〈c〉 and access to the ora-cle for computing the cost values. There may also be additional input such as,e.g., anallocationx of size (encoding length)〈x〉. Therunning timeof A is measured in termsof the number of elementary (bit) operations plus calls to the oracle for computing cer-tain c-values. Correspondingly, we say thatA is efficient, if the number of elementaryoperations and oracle calls is polynomially bounded inn, 〈c〉 and〈x〉. (See [2] for aconcrete example.)For certain classes of games (e.g., minimum spanning tree games,cf. [1]), computingemin(x) or sij(x) for a given allocationx is NP-hard. For such classes of games wecan hardly expect to compute the nucleolus or elements in the (lexicographic) kernelefficiently. We therefore assume that our classC of games satisfies(CCM) There exists an efficient algorithmA which, on input(N, c) ∈ C andallocationx ∈ X, efficiently computes the numbermin(x).As shown in [2], this assumption is tantamount to the efficient computability of thesurplusses ij(x). Furthermore, not only the surplussessij(x) can be computed effi-ciently, but we can also identify in polynomial time a coalitionS ⊆ N containingi,but notj, with c(S)− x(S) = sij(x).Computingemin(x) can be done efficiently, for example, whenc (and hencec − x)is submodular (cf. [8]). Hence (CCM) holds, for example, for any class of convexgames. A concrete example is provided,.g., by Mediggo’s [6] tree games. There are,however, also non-convex games that satisfy (CCM). An interesting case is,e.g., theclass of (non-bipartite) matching games (cf. [4]).4 The lexicographic pre-kernelWe consider a fixed classC of games satisfying (CCM). Assume(N, c) ∈ C and letIdenote the set of pairs(i, j) of playersi 6= j. We then consider the problemε1 := max εsij(x) ≥ ε, (i, j) ∈ Ix ∈ X.(1)4Observe that a constraintsij(x) ≥ ε actually corresponds to2n−2 linear constraints ofthe formc(S)− x(S) ≥ ε .So (1) is a linear progeam and the its set of feasible solutions(x, ε) forms a polyhedronP ⊆ Rn+1.Given a vector(x, ε) ∈ Rn+1, (CCM) allows us to check efficiently whether(x, ε) ∈P . Moreover, in case(x, ε) 6∈ P , we can efficiently determine a correspondingviolatedinequality, i.e., a linear inequality from the constraints in (1) that is violated by(x, ε).Indeed, assume that, say,sij(x) < ε holds for some(i, j) ∈ I. As pointed out at theend of Section 3, (CCM) also allows us to compute efficiently a corresponding coalitionS ⊆ N with i ∈ S, j 6∈ S ande(S, x) = sij(x). Thenc(S)− x(S) ≥ εis one of the constraints in (1) that is violated by(x, ε).This observation, together with standard results on the ellipsoid method (cf. also [2]),yields an efficient algorithm for solving (1). Note that (1) is feasible and bounded, sooptimal solutions exist. (The corresponding optimalε1 defines the least core.)Our next step is to identify the setI1 ⊆ I of pairs (i, j) for which the constraintsij(x) ≥ ε1 is necessarily tight whenever(x, ε1) is an optimal solution of (1). This isstraightforward: For each(i1, j1) ∈ I, we solveεi1j1 := max εsi1j1(x) ≥ εsij(x) ≥ ε1 (i, j) ∈ I\{(i1, j1)}x ∈ X.(2)and include(i1, j1) into I1 if and only if εi1,j1 = ε1. (Note thatεi1,j1 ≥ ε1 holds ingeneral.)By definition, each(i1, j1) ∈ I thus admits a correspondingx = x(i1, j1) such thatsij(x) ≥ ε1, for all (i, j) ands(i1, j1)(x) = ε1 if and only if (i1, j1) ∈ I1. Taking theaveragex +1|I|∑(i,j)∈Ix(i, j),we obtain an allocationx ∈ X. Due to the concavity of thesij , this vectorx solves(1) with sij(x) ≥ ε1 being tight exactly when(i, j) ∈ I1. So we conclude that indeedI1 is the set of pairs for which the constraintsij(x) ≥ ε in (1) is necessarily tight at anoptimum solution(x, ε1) of (1).Having computedI1 ⊆ I, we proceed to solveε2 := max εsij(x) ≥ ε1 (i, j) ∈ I1sij(x) ≥ ε (i, j) ∈ I\I1(3)with optimumε2 > ε1 and determine a corresponding setI2 in a similar way. Afterat mostr = O(n2) iterations, we end with a complete description of the lexicographicpre-kernelKlex = {x ∈ X | sij(x) ≥ εk, (i, j) ∈ Ik, k = 1, . . . , r}5and somex ∈ Klex (obtained while computingεr andIr).There is one problem left. To prove efficiency of our algorithm, we have to analyze thesize of the numbersε1, . . . , εr that we compute iteratively. But this is easy by usingthe followinga posterioriargument. Relative to the partitionI = I1 ∪ . . .∪ Ir that wehave constructed, the valuesε1, . . . , εr are uniquely determined by the solution of thefollowing lexicographic maximization problemlex−max (ε1, . . . , εr)s.t. sij(x) ≥ εk, (i, j) ∈ Ik, k = 1, . . . , rx ∈ Kwith n + r variablesx1, . . . , xn, ε1, . . . , εr. The optimum is attained at a vertex of thefeasible setP . Such a vertex has components polynomially bounded in the dimensionn + r = O(n2) and the maximum size of a coefficient in the system of inequalitiesdescribingP . Hence, in particular, the optimum valuesε1, . . . , εr are polynomiallybounded inn and〈c〉, as required.Summarizing, we have provedTheorem 4.1 If C satisfies (CCM), the problem of computing an element in the lexico-graphic pre-kernel is efficiently solvable (for games inC). 2References[1] U. Faigle, S.P. Fekete, W. Hochstättler and W. Kern:On the complexity of testingcore membership for min cost spanning tree games, Int. Journal of Game Theory26, 361–366 (1997).[2] U. Faigle, W. Kern, and J. Kuipers:On the computation of the nucleolus of acooperative game, Int. Journal of Game Theory 30(1), 79–98, (2001).[3] M. Justman: Iterative processes with “nucleolar” restrictions, Int. Journal ofGame Theory 6, 189–212, (1977).[4] W. Kern, D. Paulusma:Matching games: The least core and the nucleolus, Math.of Operations Research, to appear.[5] M. Maschler, B. Peleg:Stable sets and stable points of set valued dynamic sys-tems with applications to game theory, SIAM Journal of Control and Optimiza-tion 14, 985–995 (1976).[6] N. Megiddo: Computational complexity of the game theory approach to cost al-location for a tree, Mathematics of Operations Research 3, 189-196 (1987).[7] A. Schrijver: A combinatorial algorithm for minimizing submodular functions instrongly polynomial time, Journal of Combinatorial Theory B80, 346–355 (2000).[8] R.E. Stearns:Convergent transfer schemes forn-person games, Trans. Amer.Math. Soc. 134, 449–459 (1968).[9] M. Yarom: The lexicographic kernel of a cooperative game, Math. of OperationsResearch 6(1), 88–100 (1981).6

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