The First Steps with Alexia, the Average Lexicographic Value Tijs, S.H.

Transcription

1 Tlburg Unversty The Frst Steps wth Alexa, the Average Lexcographc Value Tjs, S.H. Publcaton date: 2005 Lnk to publcaton Ctaton for publshed verson (APA): Tjs, S. H. (2005). The Frst Steps wth Alexa, the Average Lexcographc Value. (CentER Dscusson Paper; Vol ). Tlburg: Operatons research. General rghts Copyrght and moral rghts for the publcatons made accessble n the publc portal are retaned by the authors and/or other copyrght owners and t s a condton of accessng publcatons that users recognse and abde by the legal requrements assocated wth these rghts. - Users may download and prnt one copy of any publcaton from the publc portal for the purpose of prvate study or research - You may not further dstrbute the materal or use t for any proft-makng actvty or commercal gan - You may freely dstrbute the URL dentfyng the publcaton n the publc portal Take down polcy If you beleve that ths document breaches copyrght, please contact us provdng detals, and we wll remove access to the work mmedately and nvestgate your clam. Download date: 05. jul. 2018

2 No THE FIRST STEPS WITH ALEXIA, THE AVERAGE LEXICOGRAPHIC VALUE By Stef Tjs November 2005 ISSN

3 The frst steps wth Alexa, the average lexcographc value Stef Tjs CentER and Department of Econometrcs and Operatons Research, Tlburg Unversty; Department of Mathematcs, Unversty of Genoa Abstract The new value AL for balanced games s dscussed, whch s based on averagng of lexcographc maxma of the core. Exactfcatons of games play a specal role to fnd nterestng relatons of AL wth other soluton concepts for varous classes of games as convex games, bg boss games, smplex games etc. Also exactfcatons are helpful to assocate fully defned games to partally defned games and to develop soluton concepts there. Keywords: cooperatve game, average lexcographc value, exact game, partally defned game. JEL code: C71

4 2 1 Introducton Two soluton concepts are domnant n game theory: the Nash equlbrum set (NEset) and the core. The mportance of the frst concept was recognzed mmedately, the hstory of the core s more complex. It turns out that there are remarkable smlartes fonelooksattheroleofthene-setnnon-cooperatvegametheoryandtheroleofthe core n cooperatve game theory. Non-cooperatve games wthout Nash equlbra as well as cooperatve games wth an empty core are not very attractve for a game theorst and also not n practce. In such cases one can stll hope for the exstence of approxmate Nash equlbra or approxmate core elements. In case of a determned non-cooperatve game an agreement of the players on some NE gves a certan stablty because n a play none of the players can proft n unlaterally devatng from the agreed equlbrum. Smlarly, f n a game a core element s proposed no subgroup of players can perform better n splttng off. In case the NE-set s large or the core s large there s room for a selecton theory or a refnement theory. To fnsh wth the smlartes, for both soluton concepts, there are axomatzatons usng consstency and converse consstency. In ths paper we concentrate on balanced cooperatve games, whch are games wth a non-empty core [4], and ntroduce for such games a new core selecton, the AL-value. Just as n the defnton of the well-known Shapley value [8] an averagng of n! vectors takes place whch correspond to the n! possble orders of the players n an n-person game. For the Shapley value the vectors are the margnal vectors of the game, for the AL-value the vectors are the lexcographcal optmal ponts n the core. The outlne of the paper s as follows. In secton 2 I ntroduce the AL-value, dscuss some nterestng propertes and treat some examples. It turns out that on the cone of convex games [9] the AL-value and the Shapley value concde. Secton 3 deals wth exact games and the exactfcaton operator on games. The AL-value s an addtve functon on the cone of exact games. Characterstc for the AL-value s the INVEXproperty (the nvarance w.r.t. exactfcaton). Further, for smplex games [1, 11] and also for dual smplex games or 1-convex games [1, 3] t s shown that the AL-value of such a game concdes wth the Shapley value of the exactfcaton of the game. In secton 4 the AL-value s studed for the cone of bg boss games [5] and t concdes there wth the τ-value [10] and the nucleolus [6]. In secton 5 the exactfcaton operator s adapted to treat a famly of partally defned games, whch gves a possblty to defne for such games also the AL-value and other values. Secton 6 ndcates topcs for further research.

5 2 The average lexographc value AL 3 Gven a balanced n-person < N,v > and gven an orderng σ =(σ(1),σ(2),...,σ(n)) of the players n N, the lexcographc maxmum of the core C(v) wth respect to σ s denoted by S σ (v). It s the unque pont of the core C(v) wth the propertes: (S σ (v)) σ(1) =max{x σ(1) x C(v)}, (S σ (v)) σ(2) =max{x σ(2) x C(v) wth x σ(1) = (S σ (v)) σ(1) },...,(S σ (v)) σ(n) =max{x σ(n) x C(v) wth (x σ(1),x σ(2),...,x σ(n 1) )= ((S σ (v)) σ(1), (S σ (v)) σ(2),...,(s σ (v) σ(n 1) ))}. Note that S σ (v) s an extreme pont of the core for each σ. The average lexcographc value of <N,v>s the average over all S σ (v).e. AL(v) = S σ (v), where Π(N) denotes the set of n! orderngs of N. 1 n! Example 2.1. Let <N,v>bea2-person balanced game wth N = {1, 2}. Then v(1, 2) v(1) + v(2) and C(v) =conv({f 1,f 2 }) wth f 1 =(v(n) v(2),v(2)), f 2 = (v(1),v(n) v(1)). Further Π(N) ={(1, 2), (2, 1)}, S (1,2) (v) =f 1,S (2,1) (v) =f 2, So, AL(v) = 1 (f 1 + f 2 )=(v(1) + 1 (v(1, 2) v(1) v(2)), v(2) + 1 (v(1, 2) v(1) v(2)), the standard soluton for the 2-person game <N,v>. Example 2.2. Let <N,v>be the 3-person convex game wth N = {1, 2, 3}, v() =0 for each N, v(s) =10f S =2and v(n) =30. Then S (1,2,3) (v) =(20, 10, 0) = m (3,2,1) (v), S (1,3,2) (v) =(20, 0, 10) = m (2,3,1) (v),...,s (3,2,1) (v) =(0, 10, 20) = m (1,2,3) (v). Here m σ (v) s the margnal vector w.r.t. σ wth m σ σ(k)(v) =v(σ(1),...,σ(k)) v(σ(1),σ(2),...,σ(k 1)) for each k N. So, AL(v) =(10, 10, 10) = 1 S σ (v) = 1 m σ (v) =φ(v) where 3! 3! σ =(σ(3),σ(2),σ(1)), the reverse order of σ, and φ(v) s the Shapley value of <N,v>. Theorem 2.3. For each convex game <N,v>: AL(v) =φ(v). Proof. Note that for each σ Π(N) : S σ (v) = m σ (v), where σ = (σ(n),σ(n 1),...σ(2),σ(1)). Theorem 2.4. Let <N,v>be a balanced smplex game [1,11].e. a game where C(v) s equal to the non-empty mputaton set I(v) ={x R n n x = v(n), x v({}) for =1 each N}. Then AL(v) =CIS(v), the center of the mputaton set. n Proof. Note that I(v) =conv{f 1 (v),f 2 (v),...,f n (v)} and CIS(v) = 1 f k (v) where n k=1 (f k (v)) = v() for N\{k} and (f k (v)) k = v(n) v(). Because S σ (v) = N\{k}

6 4 f σ(1) (v) for each σ Π(N) we obtan AL(v) = 1 n! f σ(1) (v) = 1 n n f k (v) =CIS(v). For dual smplex games (also called 1-convex games) (see [3],[11]) we gve wthout proof the followng results. Theorem 2.5. Let <N,v>beabalanced n-person game wth C(v) = I (v) = conv({g 1 (v),g 2 (v),...,g n (v)}), where (g k (v)) = v (k) =v(n) v(n\{k}) for k and (g k (v)) k = v(n) v (). N\{k} Then AL(v) = ENSR(v), where ENSR s the rule whch splts equally the nonseparable rewards. AL(v) s also equal to the nucleolus [6] and the τ-value [10] of (N,v). It wll be clear that AL satsfes the followng propertes: IR (Indvdual ratonalty), EFF (effcency), S-equvalence, CS (core selecton) and SYM (symmetry). Also DUM (the dummy property) holds for AL because for each balanced game <N,v>the AL-value AL(v) s an element of the core and for each x C(v) : v() x = n x k x k v(n) v(n\{}) k=1 k N\{} So, f s a dummy player, then x = v() for each core element and, especally AL (v) = v() for a dummy player. In the next secton we consder two other propertes of AL : INVEX, ADD E. k=1 3 Exact games Exact games are ntroduced by Schmedler [7] and they play an nterestng role n ths secton. Recall that a game s an exact game f for each coalton S 2 N \{φ} there s an element x S C(v) such that x S = v(s). Let us denote by EX N the set of exact S games wth player set N. In fact EX N s a cone of games and one easly sees that AL : EX N R n s addtve. We call ths nterestng property ADD E :AL(v +w) =AL(v)+ AL(w) for each v, w EX N.

7 5 Note that for each balanced game <N,v>theresaunqueexactgame<N,v E > wth the same core as the orgnal game. Ths exactfcaton <N,v E > of <N,v>s defned by v E (φ) =0and v E (S) =mn{ x x C(v)} for each S 2 N \{φ} S So, C(v E ) = C(v) for each balanced game < N,v > and v E = v ff < N,v > s exact. Note that an nterestng property for AL s: f for <N,v>,<N,w>we have C(v) = C(w) φ, then AL(v) = AL(w). Ths property s equvalent wth the property INVEX : AL(v) =AL(v E ) for each balanced game <N,v>,where INVEX stands for nvarant w.r.t. exactfcaton. In vew of theorem 2.3 ths INVEX-property of AL gves the possblty to prove that for some games <N,v>theAL-valueof<N,v>concdes wth the Shapley value φ(v E ) of the exactfcaton <N,v E > of <N,v>.Thssthecaseforthosegame<N,v> for whch the exactfcaton s convex. Ths holds e.g. for smplex games, dual smplex games and also for 2- and 3-person balanced game. So we obtan Theorem 3.1. () If <N,v>s a balanced 2-person game or a 3-person game, then AL(v) =φ(v E ). () For each smplex game <N,v>we have AL(v) =φ(v E ). () For each dual smplex game <N,v>we have AL(v) =φ(v E ). Proof of () only. Let <N,v>be a smplex game. Then C(v) = I(v) =conv{f 1 (v),f 2 (v),...,f n (v)}. So v E (N) =v(n) and for each S 2 N \{φ, N} : v E (S) =mn{ x x C(v)} =mn{ f k k {1, 2,...,n}} =mn{ v(), v(n) S S S v()} = v(). Ths mples that v E s a sum of convex games namely v E = N\S S n v()u {} +(v(n) n v(k))u N (where u S denotes the unanmty game wth u S (T )=1 =1 k=1 f S T and u S (T )=0otherwse). So, <N,v E > s a convex game and AL(v) = AL(v E )=φ(v E ). Now we gve a 4-person exact game <N,v>, where φ(v) =φ(v E ) AL(v). Ths game s a slght varant of an example n [2] on p. 91.

8 6 Example 3.2. Let ε (0, 1] and let <N,v>be the game wth N = {1, 2, 3, 4}, v(s) =7, 12, 22 f S =2, 3, 4 respectvely and v(1) = ε, v(2) = v(3) = v(4) = 0. Note that <N,v>s not convex because v(1, 2, 3) v(1, 2) = 5 <v(1, 3) v(1) = 7 ε. Note further that Ext(C(v)) has the maxmum number of 24 extreme ponts: () 12 extreme ponts whch are permutatons of (10,5,5,2), () 9 extreme ponts whch are permutatons of (7,7,8,0) but wth frst coordnate unequal to 0, () (ε, 7 ε, 7 ε, 8+ε), (ε, 7 ε, 8 ε, 7 ε) and (ε, 8 ε, 7 ε, 7 ε). From ths follows that <N,v>s an exact game, and that each lexcographc maxmum S σ (v) s equal to a permutaton of the vector (10,5,5,2), where each such permutaton corresponds to two orders. So, AL(v) =(5 1, 5 1, 5 1, 5 1 ) and s unequal to φ(v E )=φ(v) =( ε, ε, ε, ε). 4 Bg boss games and the average lexcographc value Bg boss games are ntroduced n [5] and further dscussed n [1] and [11]. Recall that an n-person game <N,v>s a bg boss game wth n as bg boss f the followng three condtons hold: 1. Bg boss property: v(s) =0for all S wth n/ S. 2. Monotoncty property: v(s) v(t ) for all S, T 2 N wth S T. 3. Unon property: v(n) v(s) M (v) for each S 2 N wth n S. N\S

9 7 It s well-known that the core of a bg boss game wth n as bg boss s gven by C(v) ={x R n 0 x M (v) for each N\{n}, n x = v(n)} and the τ-value by τ(v) =( 1M 2 1(v), 1M 2 2(v),..., 1M 2 n 1(v), v(n) 1 M 2 (v)). =1 The extreme ponts of a bg boss game <N,v>wth n as bg boss are of the form P T where T N\{n} and P T = M (v) f T, P T =0f N\T {n} and P T = v(n) n M (v). For each σ Π(N) the lexocographc maxmum S σ (v) equals T P T (σ), where T (σ) ={ N\{n} σ() <σ(n)}. Theorem 4.1. Let < N,v > be a bg boss game wth n as bg boss. Then AL(v) =τ(v). Proof. For each N\{n} : AL (v) = 1 n! 1 M n! (v) {σ Π(N) σ() <σ(n)} = 1M 2 (v) =τ (v) By EFF of τ andalthenalsoal n (v) =τ n (v). =1 (S σ (v)) = 1 n! n 1 (P T (σ) ) = Letuslookattheexactfcaton<N,v E > of the bg boss game <N,v>wth n as a bg boss. () For S N\{n} we have v E (S) = mn T N\{n} S P T = S P φ =0 () For S wth n S we have v E (S) = mn T N\{n} S = P N\{n} S P T = mn (v(n) M (v)) T N\{n} T \S =(v(n) n 1 M (v)) + M (v). =1 S Ths mples that v E s a non-negatve combnaton of convex unanmty games: v E =(v(n) n 1 =1 M (v))u {n} + N\{n} M (v)u {,n} So, v E s a convex game (and also a bg boss game) and the extreme ponts of C(v) and of C(v E ) concde. So we obtan τ(v) =AL(v) =AL(v E )=φ(v E ). Theorem 4.2. The AL-value of a bg boss game equals the Shapley value of the exactfcaton of the bg boss game.

10 5 An approach to handle partally defned games 8 Cases where a player set N s confronted wth the problem of dvdng v(n), where not for each subcoalton of N the worth s gven, are dscussed extensvely n the lterature. I wll consder specal balanced partally defned games. These are games <N,v,F >, where N s the player set, F s a subset of 2 N, contanng N and φ and v : F Rhas the propertes v(φ) =0and C F (v) :={x R n n x = v(n), x v(s) for all F 2 N } =1 S s a non-empty and bounded set. For such a balanced F-game v one can study the exact extenson v :2 N R where v(s) =mn{ S x x C F (v)} where we have a real extenson f <N,v,F > has the exactness property: v(s) = v(s) for S F. Gven a soluton Ψ for games <N,v>one can defne a soluton Ψ for balanced partally defned game by Ψ(N,v,F) =Ψ(N, v) It s nterestng to study AL n such stuatons. 6 Concludng remarks Further research on the average lexcographc value wll nclude () monotoncty propertes of AL, () contnuty propertes of AL, () consstency propertes of AL, (v) axomatzatons, (v) numercal aspects, (v) cones wth a perfect kernel system and AL, (v) relatons wth other core selectons, (v) extensons of the AL-value for non-balanced games, (x) more relatons wth other soluton concepts.

11 9 References [1] Branze, R., Tjs, S. (2001). Addtvty regons for solutons n cooperatve game theory, Lbertas Mathematca 21: [2] Derks, J., Kupers,J. (2002). On the number of extreme ponts of the core of a transferable utlty game, n: Chapters n Game Theory (Eds. P. Borm and H. Peters), Kluwer Academc Publshers, pp [3] Dressen, T., Tjs, S. (1983). The τ-value, the nucleolus and the core for a subclass of games, Methods of Operatons Research 46: [4] Glles, D.B. (1953). Some theorems on n-person games. Dssertaton, Department of Mathematcs, Prnceton Unversty. [5] Muto, S., Nakayama, M., Potters, J., Tjs, S. (1988). On bg boss games, The Economc Studes Quarterly 39: [6] Schmedler, D. (1969). The nucleolus of a characterstc functon game, SIAM Journal of Appled Mathematcs 17: [7] Schmedler, D. (1972). Cores of exact games, Journal of Mathematcal Analyss and Applcatons 40, [8] Shapley, L. (1953). A value for n-person games, n: Contrbutons to the Theory of Games II (Eds. H.W. Kuhn and A.W. Tucker), Annals of Mathematcs Studes No. 28, Prnceton Unversty Press, pp [9] Shapley, L. (1971). Cores of convex games, Internatonal Journal of Game Theory 1: [10] Tjs, S. (1981). Bounds of the core of a game and the τ-value, n: Game Theory and Mathematcal Economcs (Eds. O. Moeschln and D. Pallaschke), North-Holland Publshng Company, pp [11] Tjs, S., Branze, R. (2002). Addtve stable solutons on perfect cones of cooperatve games, Internatonal Journal of Game Theory 31: