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The nucleolus of a matrix game and other nucleoli

Potters, J.A.M.; Tijs, S.H.

Publication date:1993

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Citation for published version (APA):Potters, J. A. M., & Tijs, S. H. (1993). The nucleolus of a matrix game and other nucleoli. (Reprint series /CentER for Economic Research; Vol. 103). Unknown Publisher.

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for~mic Research ii~ iuii ~ ~I iiiu iiiii uui iuii iui mi

The Nucleolus of a Matrix Gameand Other Nucleoli

byJos A.M. Potters

andStef H. Tijs

Reprinted from Mathematics of OperationsResearch, Vol. 17, No. 1, 1992

, Q~~ Reprint SeriesJ~~~ ~

~,~OJQ~ ~ no. 103

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ISSN 0924-7874

1992

forEconomic Research

The Nucleolus of a Matrix Gameand Other Nucleoli

byJos A.M. Potters

andStef H. Tijs

Reprinted from Mathematics of OperationsResearch, Vol. 17, No. 1, 1992

Reprint Seriesno. 103

MATHEMATICS OF OPEHATIONS RLSE~RCHvol. 17, No. 1, February IY42PnnisJ ~n U.U

THE NUCLEOLUS OF A MATRIX GAMEAND OTHER NUCLEOLI~`

JOS A. M. POTTERS ANI~ STGF H. TIJS

We define the nucleolus of a continuous convex map F: fl -~ RT on a convex compact set

Il e R". A.t special caxs we obtain known notions as nucleulus, prenucleolus and weightrdnucleulus of a TU-game with (or without) coalitiun structure. Alsu the nucleolus of a matrixgame turns out lo be an inleresting special case. II appcars that the nuclcolus of a matrix

game coincides with tha set of Dresher optimal strategy pairs of tbe game. This implies, inparticular, that the nucleolus cnnsisls of lhe proper equilibria of Ihe matrix game. To each

(zero-normalized) TU-game one can construct a matrix game-the excess gamc-such thatthe nucleolus of Ihe TU-game cuincides with the uniyue proper optimal strategy of playcr 11

in the excess game. Also for other nucleoli oL TU-games a suitahlc matrix game can beconstructed where the nucleolus under consideration is related to the nucleolus of the matrix

game in an analogous way. A balancedness condition is given characterizing nucleoluselements of a matrix game. ll is shown that this balancedness result implíes thc knownbalancedness characterizations of Kohlberg, Sobolev, Owen and Wallmeier.

0. Introduction. In Dresher U961) an interesting multi-step procedure is de-scribed to select from the optimal strategy set of a player in a matru game a subset ofoptimal strategies which exploit mistakes of the opponent optimally. This Dresherprocedure which refines the set of equilibrium points of a matrix game turns out to beequal to the refinement of Myerson (1978) as it is shown in van Damme (1983). So theset of Dresher saddle points of a matrix game form the set of proper equilibria ofthe game. Later on we will call this set-and for good reasons-the nucleolus of thematrix game. The elements in the nucleolus of a player in a matrix game can becharacterízed by an interesting balancedness property. This characterization is ob-tained by using an old result of Bohnenblust et aL (1950) (cf. Gale and Sherman 1950)which states that the set of equalizers of a player coincides with the set of all purestrategies used in at least one optimal strategy.

The nucleoli of coalitional games (Schmeidler 1969, Grotte 1970, Sobolcv 1975,Owen 1977) are closely related to the nucleolus of well-chosen matrix games.More precisely, one can construct to a 0-normalized n-person coalitional game a(2" - 2) x n matrix game-the excess game-in such a way that, e.g., the nucleolusof Schmeidler (1969) is the unique Dresher optimal strategy of player II in the excessgame. The strategy set of player II in the excess game is just the set of imputations ofthe coalitional game. The columns of the excess game have the excesses of theextreme points of the imputation set as coordinates. Also for the prenucleolus andweighted nucleolus of a coalitional game with or without coalition structure one candefine suitable excess games. From this correspondence between the nucleoli ofcoalítional games and the nucleolus of suitably chosen matrix game it is easy tointerpret the balancedness results for matrix games in the case of coalitional games

'Reccived May 12, 19R9; revised July 16, 1990.,1MS 1980 subjecr classificarion. Primary: 90D12.LIOR 1973 subjcct cfassificatian. Main: Games.OR~MS Inder 1978 subjecr cfassèjcarion. Primary: 234 Games~cooperative.Kry H~ords. Malrix games, excess game, nucleolus, optimal stratcgy.

164

O164-7h5X~92~170I ~OIM~f01.25Copyri~M O 1992, The Insrirme or Mana{emcnl Sciences~Opcnr~om Rcxarch Sacmiy of Amcnu

NUCLEOLUS OF A MATRIX GAME AND OTHER NUCLEOLI ]ÁS

and to prove the results of Kohlberg (1971), Sobolev (1975), Owen (1977) andWallmeicr (1980).

The paper is organized as follows. First we define the most abstract concept-thenucleolus of a convex continuous map of a compact convex subset of R" intoR'"-and describe a procedure for finding the nucleolus. Using this concept it is easyto introduce in a unified way the nucleolus of a matrix game and the nucleoli ofcoalitional game. The "nuclcolus hunting" procedure for convex maps coincides withthe Dresher procedure in case of a matrix game. Nezt we derive the balancednesscondition for matrix gamcs and the relation with [he set of proper equilibria. Finally,we construct to the diKerent nucleoli of coalitional games their excess games and usethe results of the nucleolus of a matrix game to obtain results in cooperative gametheory.

1. The nucleolus N(fI, F). In this section we define the central concept of thispaper-the nucleolus.

L.et II be a nonempty compact convex subset of R" and F: 11 -. Rm a continuousconvex map. The following two properties are easy to prove.

. lf ~ c Nm then the function g(x) - max(Ft(x)Ij e~} is continuousand convex.

. If .~ c Nm and w E R such that F.(x) ~ w for all x e II and for each(1.2) j E~ there is a point x~ E I] where F~(z~) C w, then F~(x) G w for all

j E~ for some point x E I] (e.g., the barycenter of the points (z~)~ E 9).

For any m E N we dcfine the map B: Rm --~ Rm which orders thc coordinates of apoint of Rm in a weakly decreasing order, i.e., B(x) - y iff there is a permutation T ofNm -(], .. , m) such that y; - x,l;l for all i e Nm and yl ~ yZ ~ ~~~ ~ ym.

Further, we define the lexicographic order {I~x by X~~x y iff [here is an indexk~ m such that x; - y; for all i G k and xk c yk or x- y.

The following properties are well known:

(1.3) . The lexicographic order ~~x is transitive, complete and asymmetric.

(1.4) -Ifx,yERm,zER'then9(x) ~I~xB(y) -~B(x,z) ~I~,B(y,z).

We define the nucleolus N(CI, F) by

N((1,F):-{xE I1IBo F(x) ~IrR 9oF(y) forall yE ll~.

TIiEORCM 1. The nucleolus N(fl, F) is a nonempty compact conuex subset of 17and the nrap F is constant on N(11, F).

PROOF. In this proof we make the following construction to be used in the sequelmany times. Let l(a - Il and ~o - fd. If ~-1 ~ Nm we define inductively g; :-max{F~Ij ~~,-I), a continuous and convex function (by (1.1)), w; :~ min{g;(z)Ix eIl,-I), well-defined by the continuity of g;, CI; :- {x E 17;-Ilg;(x) - w;), the set ofpoints where g,lfli-1 attains its minimal value; the set Il; is compact (by thecontinuity of g;) and convex (by the convezity of g;), ~; :- { j E NmIF.(x) ~ w; for atlX E (I~}.

We can make the following observations:- For all j~~-1 there is a point x~ E I7;-I such that F~(x~) c wi-1 and F(xk) ic

w,-.I fur all k~.~-1, k~ j. Then F(X) G w,-I for all j~.~i-1 if x is the

166 JOS A. M. POTTGRS Rc STEF H. TI1S

barycentcr of thc point x~, j E.~-, (by ( 1.2)). This mcans that N~, ~ R.,-, and~, ~ ~-,.

- Suppose that to cvcry j~~,-, therc is a point x~ E Tl, such that F~(x~) ~ w,.Then F~(x) c w, for all j~~,-, i( x is the barycenter of thc points x~, j~.~-,(by (1.2)) and hcncc g,(x) c w; and z E Il, in contradiction with thc dcfinition of w,.So, there is an indcx j E .~, `,~,-,.

- Hcnce, eventually we have .~, - N", for some s E N.Wc shall provc that fl, - N(Il, F).- We prove that for i - 1, ..., s, if x E TI, and B a F( y) ~,~x B o F( z), thcn y E fl,.

First we notice that for all j e.~ `,~-, and x E Il„ we have F~(x) ~ w, and by thedefinition of fl; we have F.(x) - w;. Suppose that y e Ilw-, with k ~ i. ThenF~(x) - F~(y) for all j E~3k-, and gF.(y) ~ gk(x) - wk (by (1.4)). Sincc wk -min{gk(x)Ix e(Ik-1) we have equality and y E Ilk. Since we can repcat this argu-ment while k~ i we find finally y e Il;.

- For i - s we find that, if x E Il, and 9 o F( y) ~,t1 B o F(x), then y E fl, andF~(y) - FJ(x) - wk( j) for all j E Nm where k( j) is determined by j E~;,.;~, `~k,~,-,.This means that F(x) - F( y) and that there is no point y E Il such that 8 0F(Y) ~,~x 6 o F(x): Ilr c N(Il, F).

- If conversely y E N(CI, F) and x E Il~ then B o F( y) ~,e, B o F(x) gives y E Il,.Conclusion: I7, - N(Il, F) is nonempty convex and compact and if x E N(Il, F)

then N(11, F) -(y E IIIF(y) - F(x)}. QEDFor use later on we recall that, for i - 1, ..., s,

(1.S) I7; - {x E IÏIF~(x) - wk~1~ forall j E~~

where k(j) - min{k e NI1 E.~k) is.For an axiomatic characterization of the nucleolus correspondence we refer to

Maschler, Potters and Tijs (1991).The following examples'play a crucial role in the remaining sections.ExAMPr.E 1. The nucleo[us of a twoperson zero-sum game. L,et A be an m x n-

matrix and M(N) the set of rows (columns) of A. L.et A1 and A~~ be set ofprobability measures on M and N.

If we take II - Ar and F z(F: p--~ -pAe~, j e N} then Nr(A) .- N(iI, F).If we take II- Arr and F-{F;: q-~ e;Aq, i e M} then NJr(A) :- N(II, F).Finally we define N(A) -- NJ(A) x N~~(A)-the nucleolus of the matrix game A.In the next section we shall prove that N(A) is the set of proper equilibria of the

matrix game A.EXAMPLE 2. The nucleoli of a cooperatiue game with side payments. (a) L.et (N, u)

be a game in characteristic function form. If we take TI - I(u), the set of imputationsof the game (N,v) and F-{Fs}s~N.srN.e where Fs is the excess function ofcoalition S defined by Fs(x) -- v(S) - E;Es x; for all points x E I(u), then we findthe nucleolus of the game N(u) as introduced in Schmeidler (1969).

(b) Let (N, u) be a cooperative game and T- {T„ ..., Tk} be a partition of N (i.e.,a coalition structure). We take

[I - I(v,,T)

.-{xeR"~x(T,)-u(T,)fori- 1,...,kandx;3u(i)foralliEN}.

The functions F- (Fs}s~N,s.N.eare the excess functions as before. The nucleoluswhich we find in this situation is denoted by N(u, .~T ) ( cf. Owen 1977 and Wallmcier1980).

NUCLEOLUS OF A MATRIX GAME AND OTHER NUCLEOLI li)Í

(c) Let (N,u) be a cooperative game and !'(u) the set of pre-imputations, i.e.,{x E R"~E, E Nx; - v( N)}. L.et F- {Fs}s ~ N, s. N.o be the excess functions as before.In this situation we cannot apply the theory of the preceding section immediatelysince 1'(u) is not compact. But after one step in the construction of Thcorem 1 wefind a compact (ll for

[]1 c(x E I'(u)~u(i) - x; ~ wl for all i e N}.

The nucleolus of this pair (!'(u), F) is called the pre-nucleolus N'(u) of the game (cf.Davis and Maschler 1965, Sobolev 1975).

(d) Let (N, u) be a cooperative game and w: 2N -~ Rt, a weighting map. If wetake (1 - I(u) and

F - {FS: x E I(U) -'~ ((U(S) -X(S)~W(S))}ScN,S~N.O~

then we find the weighted nucleolus Nw(u). ]n the literature the weights w(S) - ISI-1(cf. Grotte 1970) and w(S) - f(~S~)-1 with f: {0, 1,2,...,~N~} -~ R,t increasing (cf.Wallmeier 1980) are mentioned.

Of course Examples 2(a) through (d) can be combined to, for example, Nw(u, .5r),the weighted pre-nucleolus of a cooperative game (N, u) with coalition structure .T

REMARK. From Theorem 1 of ~1 we infer immediately that all these nucleoli arenonempry and convex. Since, furthermore, the function F is constant on the nucleo-lus N(11, F) we find in Examples 2(a)-(d) one-point sets (consider the functions Ft;~,i E N ).

RI~MARK. In ~4 we shall introduce excess games which reduce Examples 2(a)-(d)to special cases of Example 1.

2. The nucleolus N(A) of a matrix game A. In this section we discuss thenucleolus of a matrix game. It turns out that the procedure described in the proof ofTheorem 1 is just the Dresher procedure (cf. Dresher 1961). From this result and atheorem of van Damme (1983) we conclude that the nucleolus of a matrix game is theset of (weakly) proper equilibria of the game.

First we recall shortly the main well-known facts about matrix games. The value ofa matrix game A is given by

val(A) :- max minpA~~ - min maxe;Aq.pEA~ JEN QEG~~ ÍEM

The set of optimal strategies of player I is defined by

Or(A) :- {p E ArI pAe~ ~ val(A) for all j E N}.

Similarly the set of optimal strategies of player I1 is given by

Ofr( A) :- {q E Gril e;Aq ~ val( A) for all i E M}.

Further we denote the carrier of a strategy p e A~ by C~(p) .- {i e M~p; ~ 0), thecarrier of the set of optimal strategies of player I by Cr(A) -- (JoEo~,,rCr(p) andrh~ equalizer set

{i e M~e;Aq - val(A) forall q e O~~(A)}

by E~(A). The carricrs C~~(y), C~~(A) and the equalizer E~~(A) are dcfined similarly.

](g JOS A. M. POTr[RS ác STEF H. TIJS

A theorem o( Bohncnblust, Karlin and Shaplcy (1950) (cf. also Galc and Shcrman1950) states the equality of C~(A) and E;(A) and of C~;(A) and E;;(A).

The procedurc described in Dresher (19fi]) constructs from a matrix gamc A asequence of matrix games Ak in the following way:

We start with the matrix game A' - A. In the next game AZ player I has the purcstrategies M`C;(A), player II has the strategy space O;;(A) and the payofís arc

e;Aq, e; ~ Ct(A) and q E O~~(A). It Ak-' has been defined and C;(A"-') ~ 0~-'then [he pure strategies of player [ in the game Ak are the pure strategies of playcr I

not used in the optimal strategies of the preceding games and the set of strategies ofplayer II is the set OI;( Ak-' ). After finitely many, say k, steps the set M is exhaustedand the set of Dresher-0ptimal strategies oj player I1 is the sct Oi;( Ak ). L.et ~k bcthe set of all pure strategies of player ] used in the first k stcps of the Dreshcrprocedure. Then

val(Ak) - min max e;Aq - min max F;(q).qEOi~íAk-~)Í~l.~-~ 9E0~~(A~-~)iE:~-~

If we compare this process with the construction of ~1 we see immediately thatIIk - OtJ(Ak), wk - val(Ak) and ~k -~k. This means that N~~(A) equals the setDtt(A) of Dresher-optimal points o( player II. Van Damme ( 1983) has proved thatthe set D~(A) x Dtt(A) coincides with the set of (weakly) proper equilibria of thematríx game and therefore we find

THEOREM 2. The nucleolus oj a matrix game is the set oj proper equilibria oj thegame.

CoROt.u.RV. The set oj proper equilibria oj a matrix game is conuex ared exchartge-able.

REMARK. The set of perfect equilibria dcesn't have this property (cf. Borm et al.1988).

3. A balancedness criterion for the nucleolus of a matrix game. In this sectionwe prove a balancedness result for the nucleolus of a matrix game.

Let A be an m x n-matrix game. The following lemma gives an alternativedescription of the data 17k, ~k and wk occurring in the Dresher procedure of ~2.

For k- 1, ..., s we define an m x n-matrix A~k) with ith row

- (A; ~- (wI - wka))eN if i EA~k) - (I

A; f (w~ - wk)eN if i~

~k-I,

.~k - I .

This sequence of games {A~k)) has the advantage that the strategy sets of the playersare the same for all games A~k).

By a simple induction argument one can prove

LEMMA 3. For k~ ], ..., s we haue(a) Ilk ~ Drt(Atk)).(b) val(Atk)) - val(AtIJ) ~ wI.(C) ~k ~ Et(Atk)) a Ci(AtkJ).

To formulate the main result of this section we need the tollowing definition.We call a subset R c M balanced with respect to A on a set C c N if there are a

vector p E R~, a vector r E R; and a real number a E R such that pA - r~ aeN,Ct(p) ~ R and Ctt(r) n C ~ (ó.

NUCLEOLUS OF A MATRIX GAME AND OTHER NUCLEOLI 169

For q E 11~~ we define for all t E R

~~(q~A) -{i E M~Fr(9) - e;Aa 3 t}.

Thcn wc have the following characterization of elements of N~~(A) (and of N~(A)).

THEOREM 4. A strategy q E 0~~ is an element ojthe nucleolus N~t(A) ijand only ijjor all t E R the set .~,(qI A) is balanced with respect to A on the set Crt(q). A similarresult holds jor the nucleolus N~(A).

PROOF. ~ Suppose that q E Ntr(A). Then q E Ifk - Orr(Atk~) for all k-1,...,s (by L.emma 3). Hence, there is a strategy p e Gt such that Ct(p) - Ct(A~kr)-~k (by Lemma 3) and r e RN such that

pAtk~ - r- val(Atkt)eN with Ctr(r) fl Crr(9) -~-

Thcn

pA - r - (val(Atkt) - ~ P;(wi - wkt~~)~eN.111 ; E C~,

T'his means that ~9k is balanced with respect to A on Crr(q).We have to prove that for every t E R there ís an index k such that ~,(qI A) -~k.L.et to be the largest value of t such that ~,(qI A) ~ ~o, .~~, ...,.~, and suppose

that wk, ~ C to 6 wk. There is an index jo E~, (qIA) such that F;a(q) - to (for,otherwise, t~ is not maximal!). Since F~o(q) E(w„ ... , w,), we find to - wk and

~~„(qI A) ~ ~k ~ M.Suppose that j e~, (q~A) and j~~k. This means that F(q) ~ to - wk and,

therefore, gk t~(q) ~ wk, i.e., q~ Ilk r~ in contradiction with q e Ntr(A).c Suppose that q E OtJ satisfies the balancedness conditions and suppose that

q E [Ik-~ `ITk. Then there is an index jo ~~k-~ Wlth Wk G F.a(q) -- t ~ WR-~.From the balancedness of ~,(qI A) we infer that there is a strategy p E t1 f wi[h

Cr(p) -~,(qIA), a vector r E R;' with Crt(r) fl Ctt(q) ~ 0 and a real numbera E R such that pA - r- aeN. Then

pAtk~ - r-( a t ~ p;(w~ - wkt;t)~eN -- ueN.1 tEc,(v) J

- This means that w~ - val(A~k~) ~ á and pA~k~q - á.. Foreach index j E~,(q~A)we have e;Aq ~ t and e~Atk~q ~ t t(w~ - wk) 1 w„

if moreover, j~~k-,. For i E .~k-~ we have e;Atktq - w~ because q E Ilk-~. So,pA~"~y ~ w~ sincc C~(p) -~,(qIA) and, in particular, p~~ 1 0.

The contradiction between these two inequalities shows that the assumptionq E Clk-~ `flk is false. QED

4. The Kohlberg criterion for nucleoli of cooperative games. In this section weshow that the nucleolus eoncepts for ecwperative games can be found as the set ofDreshcr strategics (for player II) N~f(A) for well-chosen matrix games A. Thisenabl~s us to 'translate' the theorcm of the foregoing section into balancednessthcorcros for thc nuclcolus, prenuclcolus and nucleolus of games with coalitionstructurc.

170 JOS A. M. POT7'ERS 6c STEF H. TI1S

We start with a rather general situation. L.ct fl c R" be a polytope (i.e., the convexhull of finitely many points) and let F„ i E M be a~ne functions. Supposc that E~,j- l, ..., l are thc extremc points of Il and Iet ~ bc the standard simplcx in R~. L.ctrb: 0--~ TI be thc affine map ~(q) - E;-Iq;E;. Thcn wc can dcfine thc matrix (gamc)A by A;i - F,.( E~) for all i e M and j- 1, ...,1. Each point x E Il can bc writtcn as

a convcx expression in the extreme points E~, j- 1, ...,! and we say that E; is in thccarrier (C(x)) of a point x E fl if it is possihle to use the extrcmc point in a convcxezpression of x. Note that for evcry point x e fl there is a point q E t] such thatrd(q) - x and C(x) - C(q).

It is an immcdiate consequence of the following Iemma that N~r(A) - Q~-IN(Il, F).

LEMMA S. If A: II' ~ fl is an aJfine map onto TI and F is a fnite set oj continuousconuex functions on II , then N( I7', F o A)-~-1 N(11, F).

The proof of this lemma is straightforward and is thercfore omittcd.From this result and Theorem 4 we find the following statements: x e N(Il, F) a

rb-1(s) c N~~(A) a the set ~,(qIA) is balanced on C(q) with respect to the matrixA for all q e~-1(x) and all real numbers t. Note that ~,(qIA) -{i e MIF,(z) -e;Aq ~ r) -~,(xIR) if q E rb-1(x). Hence we find the following provisional result

Let x be a point of [7. Then x E N(CI, F) if and only if for all real numbers t E Rthere are positive real numbers {p;);E9,~,In1 and a real number a such that

(4.1) ~ D;Fi(E,) ~ a; E ~,t,~in t

with equality if E; e C(x).L.et (N, u) be a cooperative game. We assume, for simplicity, that the game is

zero-normalized i.e. v(i) - 0 for all i E N. L.et w: 2N -~ Rtt be a weight function.L.et Il be a convex polytope in RN with extreme points {Ei}~-1 t. L,et {Fs)s.O.N bethe set of (weighted) excess functions of the game (N,u). In this case we call thematrix game A defined in the same way as before the excess game of the game (N, u)with respect to the set fI. In this situation equivalence (4.1) becomes:

If x e II, then the following statements are equivalent: .(1) x e N(Il, F) (in fact x ~ N(II, F)).(2) for all real numbers t e R there are positive numbers {ps)sE.9,c~1n1 and a real

number a such that

(4.2) ~~ Psw(S)es, Ei~ ~~ Psw(S)u(S) - a-: á for all j - 1,...,(`S E Q, S E ~I,

where (. ,-) denotes the inner product in RN, Sd, is a shorthand notation for~,(x117) and Ei, j- 1, ... ,1 are the extreme points of fl as bcfore. Furthermore,there is equality if Ei E C(x). If we substitute ys :- psw(S) we find

(4.3) ~~yses, Ei 1 ~ á`s

with equality for Ei e C(x).In the remaining part of this section we investigate the meaning o( (4.3) for special

choices of il.( 1) (weighted) nucleolus: If TI - J(u), the map rd is very simple q--~ u(N)q and

(4.3) gives (Esyses)i ~ u(N)-lá with equality if xi ~ 0. Since Esyses has at least

NUCLEOLUS OF A MATHIX GAME AND OTttER NUCLEOLI 171

one positive coordinate the number u(N)-lir ~ 0. After normalization we find thewell-known results o( Kohlberg ( 1971) and Wallmeier ( 1980).

(2) prenucleolus: Let R be a real number such that R~ max u(S) - ISI~INIu(N)and II -{x E.~`(u)Ix, t R y 0 for all i E N). Thcn the nuclcolus N(Il, F) is noton the buundary of Il for a point on the boundary has at least one coordinatcx, --R and thcrcfore maximal exccss ~ R but the point z -- u(N)~INIeN hasmaximal cxcess ~ R. If wc substitute E~ - -R(1, 1,..., 1) t(u(N) t INIR)e~ andá-(c(N) t INIR)-I(á t REslSlys) we find, since C(x) -{E~Ij E N},

(4.4) ~ Yses - áeN.SE~,

After normalization (á ~ 0) we find the balancedness result of Sobolev (1975).(3) nucleolus for games with coalitional structure: For II -.1(u, J-) the situation

is a little more complicated since fI is no longer a simplex but a product of simplices.The extreme points of II are the points E~ -- u(TI)e;~l~ t.~. tu(Tk)e;~k~ whereJ-{i(1), ..., i(k )) and i(s) - J tl T, for s- 1, ..., k. The cartier C(x) of a pointx E íl consists of these extreme points E~ with J - {i(1),...,i(k)) and x;~,~ ~ 0 fors- I, ..., k. Using equation (4.3) an easy calculation shows that there are realnumbers nl, az, . .. , ak such that

(4.5) ~ Yses ~ aler, t ... t~ker.sE~

with equality for the coordinates j with x~ ~ 0. This characterization ot the nucleolusfor games with coalitional structure can be found in Wallmeier (1980) (cf. also Owen1979).

If we call ~ Tbalanced on C(x) if (4.5) holds then we can summarize our resultsby

THEOREM 5. (a) (Kohlberg 1971) An imputation x E .J(u) is the (weighted)nucleolus of the gome (N, u) if and only if the collections ~,(x~u,(w)) are balaneed onC(x) jor al! t e R with ~,(xlu,(w)) ~ 0.

(b) (Soboleu 1975) A pre-impuradon x e,~'(u) is the prenucleolus ofa game (N, u)ij and only ij rhe collections ~,(xl u) are balanced jor a!! t e R where the collection isnonempty.

(c) (Owen 1977, Wallmeier 1980) An element x e.p(u, .`T ) is the nucleolus oj thegame ( N,u) with coalition structure .`T if and only if the co!lections ~,(xlu) are.Tbalanced jor al! releuant rea! numbers t E R.

ReMARK. The nucleolus of a matrix game is a really more general concept thanthe nucleolus of a cooperative game. Some properties remain true (e.g., the balanced-ness results, cf. Theorem 4) but other properties become false (no longer a one-pointconcept, ( semi-kontinuiry). Therefore there is little hope that one may succeed toprove our Theorem 4 írom Kohlberg's result.

5. Some properties of the nucleoli of cooperative games. In this section we giveanother proof of some properties of the nucleoli of cooperative games.

(1) It is wcll known in the literature (cf. Maschler et al. 1979) that the prenucleolusand the nuclculus of a cooperative game are the same if the game is 0-monotonic, i.e.,if thc 0-normalization of the game is monotonic. We shall give an alternative proof.

172 JOS A. M. POTrr:Rti tc STI..r H. TI1S

Pkoor. Let n bc a (1-monotonic gamc and x E I'(r) `I(I ). Thcn x, ~ r(i) for atleast one index i E N. L.et ( E R bc thc largcst numhcr such that ~9,(xll )~ P1. Wcprove that every clcment S E a~l,(xlu) contains playcr i. This fullows from the factthat

v(S U (i~) -x(SV (i~) ~ r~(S) -x(.S) f u(i) -x, ~ ~~(.S) -x(S)

for all S c N`i (the inequality is a consequence of the 0-monotonicity of the game).So, the collection ~,(xlu) cannot he balanccd and x ~ N'(u). Thercfore, N`(r)

E I(n) and N`(n) - N(u). QED(2) The prenucleolus of a game satisfics the reduced game property (cf. Sobolcv

1975 and Davis and Maschler 1965). Wc shall rcprovc this propcrty using the resultsof the preceding sections.

L.et (N, u) be a cooperative game, x E 1'(u) and T c N a nonempty coalition. Thereduced game u,.T is defined by

u~.T(S) :- max u(SVR) -x(R)RnT-0

ifScT,S~Tandu~T(T):-u(N)-x(N`T).

We shall reprove the following property of the prenucleolus (reduced game property).If x is the prenucleolus of a game (N,e) and T is a nonempty coalition, then

N'(u~.T) - x~T (the restriction of x to T).

PROOF. Note that the excess e(S, z~Tlu, T) - maxRnT-e e(S V R, xlu). Thismeans that, for all t E R, ~,(xiT~us T) - ~,(xlu) n T. From Theorem 5 we inferthat ~,(x~u) is a balanced collection. Then ~,(xiT~u, T) is balanced on T for allt E R. Using Theorem 5 again we find that xiT is the prenucleolus of the reducedgame u~,T. QED

(3) In the literature there are several places where it is proved that the nucleolus orprenucleolus with or without weights and coalition structures are continuous func-tions of the data. On the other hand it is known that the set of proper equilibriumpoints of a matrix game is not upper semicontinuous. In this section we shallinvestigate where the difference comes from and reprove the continuity of thenucleoli.

We start with an example showing the discontinuity of the map A~ N~~(A). Take,for n E N,

~,q„ .- (0 -1 ~A ~- 10 -1)'

Note that in the matrices A„ the first strategy of player 1 dominates the secondstrategy ( weakly) and therefore the equilibrium point (el,el) is the unique perfect(proper) equilibrium ( cf. van Damme 1983). So, we have N~~(A„) - {el}. In thematrix A the second strategy of player II dominates the first one and therefore(el, eZ) is the unique proper equilibrium: N11(A) - ( ez}. Hence, there is no continuityin the point A.

Finally we reprove the continuity of the different type of nucleoli as function ot thegame and the weights.

L.et {u~)~EN be a sequence of cooperative games converging to the game u and{w„}~E N a sequence of weight functions converging to the weight function w. Further,

NUCL[(ILUS OP A MATRIX GAME AND OTHfR NUCLr:O1.I 173

Ict x„ bc thc wcightcd nuclcolus N,,. (n„) for n- 1, 2, .... Supposc that thc scqucncc(x„1,,,~ N convcrgcs to x E RN. Wc will prove that x- N,,.(r~).

Lct {A„}„ E N bc thc scqucncc of exccss gamcs of {t~, , w„)„ F N and A thc cxccssgamc of r, w. Then wc havc A„ -~ A and q„ -~ y if q„ corresponds to x„ and qcorresponds with x undcr thc aftinc map ~. In ordcr to provc that q - Nrr(A) wc tryto show that .49,(ylA) is balanccd on thc carricr Crr(q) for all t E R.

Lct t E R and choosc e~ 0 such that ,y9,(q~A) - .~,-ze(ylA).Choose N(e) E N such that, for all pure strategies S of player I in the excess

gamcs and all n~ N(e), Iers~A„q„ - ers'Aql c e and C(q„) ~ C(q). From this weinfcr

S E ~r(q~A) ~ S E ~,-~(q~l A„1 ~ S E ~r-2~(q~A) -~i(q~A).

Then ~r(q~A) -.~,-,(q„~A„) is balanced on C(q„)with respect to A„ (for n ~ N(e)).So, certainly ~,(qI A) is balanced on C(q) with respect to A,,. In the case ofcooperatiue games the words "with respect to A„" can be omitted and we find thatq -~(x) is in the nucleolus Nrt(A) by Theorem 5. QED

For the prenucleolus and in games with coalitional structure the proof of thecontinuity goes along the same lines.

Note that the proof we gave above can be followed also for the nucleolus of matrixgames up [o the point

".~,(q~A) is balanced on C(q) with respect to A„"

but now the addition "with respect to A„" is very important. Let us reconsider ourcountcrexamplc. There we have ~,(eIl A) - (et, ez} for t 5 0 and we can balance therows of A„ on C(et) -{et) by

nfl(0'n)}ntl(0,-1)-(0,0)

but we cannot do the same for the limit matrix A because

Pt(0,0) f Pz(0, -1) - (0, -Pz)

and the second coordinate is smaller than the first one if p E AZ.

Acknowledgements. We thank Peter Borm for his careful reading of thismanuscript. By his cooperation the readability of this manuscript was greatly im-proved.

ReferencesAumann, R. 1. and Drèze, J. H. U974). Cooperative Games with Coalition Structuro. Intemat. 1. Game

Theory 3 217-237.Bohnenblust, H. F., Karlin, S. and Shapley, L. S. (1950). Solutions of Discrete Two-Person Games. .1nn. oj

Marh. Srudies 2451-72.Borm, P. E. M., lansen, M. J. M., Potters, J. A. M. and Tijs, S. H. (1988). On the Structure of the Sct of

Undominated Equilibria in Bímatrix Games. Internal Report of NICI, University of Nijmcgen, 88N1C1 09.

van Damme, E. E. C. (1983). Rejinemrnts oJ the Nash Equilibrrum Concept. Springer Verlag, Berlin.Davis, M. and Maschler, M. (1965). The Kernel of a Cooperative Game. Naual Res. Lagisr. Quart. 12

223-259.Dresher, M. (1961). Gomes ojStrategy. Prentice-Hall Inc., Englewood CIiRs, N), 71-73.Gale, D. and Sherman, S. (1950). Solutions of Finite Two-Person Games. .Inn, of Math. Srudies 24 37-49.

174 JOS A. M. POTTERS á STEF H. TIJS

Grotte, J. H. (197U). Compwatiun of and Observations on the Nucleolus, lhe Normalizrd Nucleolus andthe Central Games. M.S. Thesis, Cornell Univrrsity, Ithaca, NY.

Kohlberg, E. (197U. On the Nuclrolusof a Charactrristic Functiun Game. SIAMJ. Appl. Muth. 2062-65.Maschler, M., Pclrg, B. and Shaplry, L. S. (1979). The Kcrnel and the Bargaining Srl for Convex Games.

bttrrnar. J. Gume 7beory 1 73-93.

, Poucrs, 1. A. M. and Tijs, S. H. (1991). The Abstracl Nuclrolus and the Reduced Game Property.Diuussion paper, Depanment of Mathematics, Univcrsity of Nijmngen.

Mycrson, R. B. (1978). Refincments of Ihe Nash Equilibrium Concept. lntrrnut. J. Gmnc 77irory 7 73-NU.

Owen, G. (1977). A Generalization o( the Kohltxrg Criterion. lnrernur. J. Gume Thror~ 6 249-L55.Schmeidler, D. (1969). The Nucleolus of a Characteristic Funclron Gama. SIAM J. Appl. Murh. 17

1163-117U.Sobolev, A. L(1975). A Characterization of Optimality Principles in Cooperative Games by Functional

Equations. ( Russian) Murh. Merhods Socia! Sci. 6 94-I51.Wallmeier, E. (19liU). Der f-Nukleolus als L.Bsungskonzept (ur n-Personenspiele in Funktíonsform. Diplo-

marbcit, Universit~t Munster.

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No. 73 E. Bomhoff, Currency convertibility: when and how? A contribution to theBulgarian debate, Kredit und Kapital, vol. 24, no. 3, 1991, pp. 412 - 431.

No.74 H. Keuzenkamp and F. van der Ploeg, Perceived constraints for Dutchunemployment policy, in C. de Neubourg ( ed.), The Arr oj FuU Employment -Unemployment Policy in Open Economies, Contributions to Economie Analysis203, Amsterdam: Elsevier Science Publishers B.V. (North-Holland), 1991, pp. 7- 37.

No. 75 H. Peters and E. van Damme, Characterizing the Nash and Raitfa bargainingsolutions by disagreement point axions, Mathematiet ojOpemtions Research, vol.16, no. 3, 1991, pp. 447 - 461.

No.76 P.J. Deschamps, On the estimated variances of regression ecefficients inmisspecified error components modeLs, Econometric Theory, vol. 7, no. 3, 1991,pp. 369 - 384.

No. 77 A. de Zeeuw, Note on 'Nash and Stackelberg solutions in a differential gamemodel of capitalism', Joutnal ojEconomic Dynamicr and Corurol, vol. 16, no. 1,1992, pp. 139 - 145.

No. 78 J.R. Magnus, On the fundamental bordered matrix of linear estimation, in F. vander Ploeg (ed.), Advanced Lectures in Quantitative Economics, London-Orlando:AcadCmic Press Ltd., 1990, pp. 583 - 604.

No. 79 F. van der Ploeg and A. de Zeeuw, A differential game of international pollutioneontrol, Systents and Control Letters, vol. 17, no. 6, 1991, pp. 409 - 414.

No. 80 Th. Nijman and M. Verbeek, The optimal choice of controls and pre-experimen-tal observations, Joumal oj Econometrics, vol. 51, no. 1~2, 1992, pp. 183 - 189.

No. 81 M. Verbeek and Th. Nijman, Can cohort data be treated as genuine panel data?,Empincal Economics, vol. 17, no. i, 1992, pp. 9- 23.

No. 82 E. van Damme and W. Guth, Equilibrium selection in the Spence signaling game,in R. Selten (ed.), Game Equilibnum ModeLr II - Methods, Momis, mut Markeu,

Berlin: Springer-Verlag, 1991, pp. 2G3 - 288.

No. 83 R.P. Gilles and P.H.M. Ruys, Characterization of economic agents in arbitrarycommunication structures, Nieuw Archiej voor Wukundt, vol. 8, no. 3, 1990, pp.325 - 345.

No. 84 A. de Zeeuw and F. van der Ploeg, 15ifference games and policy evaluation: aconceptual framework, O.cjord Economic Papers, vol. 43, no. 4, 1991, pp. 612 -636.

No. 85 E. van Damme, Fair division under asymmetric information, in R. Selten (ed.),Rational Internction - Essays in Honor ojJohn C Harsanyi, Berlin~Heidelberg:

Springer-Verlag, 1992, pp. 121 - 144.

No. 86 F. de long, A. Kemna and T. Kloek, A contribution to event study methodologywith an application to the Dutch stock market, Jouma! oj Banlcing and Fitwrue,voL 16, no. 1, 1992, pp. 11 - 36.

No. 87 A.P. Barten, The estimation of mixed demand systems, in R. Bewley and T. VanHoa (eds.), Contributionr to Consumer Demand and Ecunometrics, Fssays inHonow ojHe~ui Thei1, Basingstoke: The Macmillan Press Ltd., 1992, pp. 31 - 57.

No. 88 T. Wansbeek and A. Kapteyn, Simple estimators for dynamic panel data modelswith errors in variables, in R. Bewley and T. Van Hoa (eds.), Contributions toConsumer Demand and Econometrics, Essvys in Nonow oj Henri Theil,Basingstoke: The Macmillan Press Ltd., 1992, pp. 238 - 251.

No. 89 S. Chib, J. Osiewalski and M. Steel, Posterior inference on the degrees of

freedom parameter in multivariate-t regression models, Economics Letters, vol.37, no. 4, 1991, pp. 391 - 397.

No. 90 H. Peters and P. Wakker, Independence of irrelevant alternatives and revealedgroup preferences, Econometrica, vol. 59, no. 6, 1991, pp. 1787 - 1801.

No. 91 G. AlogoskouCts and F. van der Plceg, On budgetary policies, growth, and

external deficits in an interdependent world, Journal oj the lapanGCe andIruemationol Economies, vol. 5, no. 4, 1991, pp. 305 - 324.

No. 92 R.P. Gilles, G. Owen and R. van den Brink, Games with permission structures:The conjunctive approach, lntemationa! Journal oj Garru Theory, vol. 20, no. 3,1992, pp. 277 - 293.

No. 93 JA.M. Potters, I.J. Curiel and S.H. Tijs, Traveling salesman games, MathematicolProgtnmming, vol. 53, no. 2, 1992, pp. 199 - 211.

No. 94 A.P. lurg, M.J.M. Jansen, JA.M. Pottcrs and S.H. Tijs, A symmetrization forfinite two-person games, Zeiucluift f'w t~perotiotu Researclt - Methodr and ModeLsoj Operatiorts Research, vol. 36, no. 2, 1992, pp. 111 - 123.

No. 95 A. van den Nouweland, P. Borm and S. Tijs, Allocation rules for hypergraphcommunication siwations, lntemationu! Jounta! oj Game Theory, vol. 20, no. 3,1992, PP. 255 - 2G8.

No. 96 E.J. Bomho(f, Monetary reform in Eastern Europe, European Economic Review,vol. 36, no. 2~3, 1992, pp. 454 - 458.

No. 97 F. van der Plceg and A. de Zeeuw, International aspects of pollution control,Environmemal and Resourre Ecortomict, vol. 2, no. 2, 1992, pp. I 17 - 139.

No. 98 P.E.M. Borm and S.H. Tijs, Strategic claim games corresponding to an NTU-game, Games and Economic Behavior, vol. 4, no. 1, 1992, pp. 58 - 71.

No. 99 A. van Soest and P. Kooreman, Coherenry of the indirect translog demandsystem with binding nonnegativity constraints, Jouma! oj Econometrics, vol. 44,no. 3, 1990, pp. 391 - 400.

No. ]00 Th. ten Raa and E.N. Wolff, Secondary products and the measurement ofproductivity growth, Regional Science and Urban Economics, vol. 21, no. 4, 1991,pp. 581 - 615.

No. 101 P. Kooreman and A. Kapteyn, On the empirical implementation of some gametheoretic models o[ household labor supply, TheJournal ojHuntan Resources, vol.25, no. 4, 1990, pp. 584 - 598.

No. 102 H. Bester, Bertrand eyuilibrium in a di(ferentiated duopoly, IntenwtionalEconomic Review, vol. 33, no. 2, 1992, pp. 433 - 448.

No. 103 1.A.M. Potters and S.H. Tijs, The nucleolus of a matrix game and other nucleoli,Alathematics of Opemtions Research, vol. 17, no. 1, 1992, pp. 1G4 - 174.

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