European Journal of Operational Research

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1 European Journal of Operatonal Research 198 (2009) Contents lsts avalable at ScenceDrect European Journal of Operatonal Research journal homepage Stochastcs and Statstcs Convex mult-choce games Characterzatons and monotonc allocaton schemes R. Branze a, *,1, S. Tjs b,1, J. Zarzuelo c a Faculty of Computer Scence, Alexandru Ioan Cuza Unversty, Iasß, Romana b CentER and Department of Econometrcs and Operatons Research, Tlburg Unversty, The Netherlands c Department of Appled Economcs IV, Faculty of Economcs and Busness Admnstraton Basque Country Unversty, Blbao, Span artcle nfo abstract Artcle hstory Receved 3 Aprl 2008 Accepted 24 September 2008 Avalable onlne 1 October 2008 Keywords Mult-choce games Convex games Margnal games Weber set Monotonc allocaton schemes Ths paper focuses on new characterzatons of convex mult-choce games usng the notons of exactness and superaddtvty. Furthermore, level-ncrease monotonc allocaton schemes (lmas) on the class of convex mult-choce games are ntroduced and studed. It turns out that each element of the Weber set of such a game s extendable to a lmas, and the (total) Shapley value for mult-choce games generates a lmas for each convex mult-choce game. Ó 2008 Publshed by Elsever B.V. 1. Introducton Mult-choce games were ntroduced by Hsao and Raghavan (1993a,b) to allow players n a cooperatve envronment to exert any of a fnte number of sutable actvty levels for the stuaton at stake. An extenson of ths model of cooperatve games was ntroduced by Nouweland et al. (1995) to cope wth stuatons where dfferent players mght have dfferent sets of actvty levels to partcpate wth when cooperatng wth other players. Results on mult-choce games can be also found n Nouweland (1993), Kljn et al. (1999), Calvo et al. (2000), Calvo and Santos (2000), Peters and Zank (2005), Grabsch and Lange (2007), Grabsch and Xe (2007). Addtonally, the reader can look at the survey on multchoce cooperatve games n Branze et al. (2005, 2008). Multchoce cooperatve games have been a useful tool for modelng nteracton of players n economc and operatons research stuatons n whch they may have dfferent optons for cooperaton, varyng from non-cooperaton (a partcpaton level 0) to a maxmal partcpaton level. In partcular, mult-choce games can be seen as an approprate analytcal tool for modelng cost allocaton stuatons n whch commodtes are ndvsble goods that are only avalable at a certan fnte number of levels. Our am s to extend * Correspondng author. Tel E-mal addresses branzer@nfoas.ro, branzer@nfo.uac.ro (R. Branze). 1 Fnancal support from the Basque Country Unversty for the research vst n May 2006 s gratefully acknowledged. Ths research has been partally supported by the Unversty of the Basque Country (Project 9/UPV /2003) and DGES Mnstero de Educacíon y Cenca (Project SEJ ). some results concernng tradtonal convex games for convex mult-choce games. Recall that a tradtonal cooperatve game s a par hn; v, where N s a set of players and v s a characterstc functon v 2 N! R wth vð;þ ¼ 0. A game hn; v s called convex f vðs [ TÞþvðS \ TÞ P vðsþþvðtþ for all S; T # N. Convex games are balanced games,.e. the core (cf. Glles (1953)) of such a game s nonempty, where CðvÞ ¼ x 2 R P N 2N x ¼ vðnþ; P 2S x P vðsþ for each S 2 2 N g. Trvally, subgames of convex games are also convex. Recall that for T N, the subgame of v based on S, ðs; v S Þ, s obtaned from hn; v by restrctng attenton to S,.e. v S ðtþ ¼vðTÞ for all T 2 2 S. It s well known that convex games are exact games and total exact games (.e. games whose all subgames are also exact) are convex (cf. Bswas et al. (1999)). Recall that a game hn; v s called exact f for each S 2 2 N n f;g there s an x 2 CðvÞ wth P 2S x ¼ vðsþ. For T N, the margnal game of v based on T s defned by v T ðsþ ¼ vðs [ TÞ vðtþ for each S # N n T. It s well known that a convex game s superaddtve and games whose margnal games are all superaddtve are convex (cf. Branze et al. (2004a), and Martnez-Legaz (1997, 2006)). Recall that a game hn; v s called superaddtve f vðs [ TÞ P vðsþþvðtþ for all S; T # N wth S \ T ¼;. Our work on convex mult-choce games n ths paper s based on defntons and results from Nouweland et al. (1995) and Branze et al. (2005, 2008), that we brefly recall n Secton 2. Then, n Secton 3, we gve new characterzatons of convex mult-choce games usng the notons of exactness and superaddtvty. Inspred by Sprumont (1990), we ntroduce the noton of a levelncrease monotonc allocaton scheme (lmas) for convex multchoce games n Secton 4, and prove that each element of the Weber set of a convex mult-choce game s extendable to a lmas /$ - see front matter Ó 2008 Publshed by Elsever B.V. do /j.ejor

2 572 R. Branze et al. / European Journal of Operatonal Research 198 (2009) We also show there that the (total) Shapley value of a convex mult-choce game (cf. Nouweland et al. (1995)) generates a lmas of the game. We conclude n Secton Prelmnares on mult-choce games Let N be a set of players, usually of the form {1, 2,..., n}, that consder cooperaton n a mult-choce envronment,.e. each player 2 N has a fnte number of feasble partcpaton levels whose set we denote by M ¼f0; 1;...; m g, where m 2 N wth N ¼f1; 2;...g. We consder the product M N ¼ Q 2N M. Each element s ¼ðs 1 ; s 2 ;...; s n Þ2M N specfes a partcpaton profle for players and s referred to as a mult-choce coalton. So, a multchoce coalton ndcates the partcpaton level of each player. Then, m ¼ðm 1 ; m 2 ;...; m n Þ s the players maxmal partcpaton profle that plays the role of the grand coalton, whereas 0 ¼ð0; 0;...; 0Þ plays the role of the empty coalton. We also use the notaton M þ ¼ M nf0g and M N þ ¼ MN nf0g. A cooperatve mult-choce game s a trple hn; m; v, where v M N! R s the characterstc functon wth vð0þ ¼0 that specfes the players potental worth, vðsþ, when they jon ther efforts at any actvty level profle s ¼ðs 1 ;...; s n Þ. For s 2 M N we denote by ðs ; kþ the partcpaton profle where all players except player play at levels defned by s whle player plays at level k 2 M. A useful partcular case s ð0 ; kþ, when only player s actve. We defne the carrer of s by carðsþ ¼f2 Njs > 0g. For s; t 2 M N we use the notaton s 6 t ff s 6 t for each 2 N and defne s ^ t ¼ðmnðs 1 ; t 1 Þ;...; mnðs n ; t n ÞÞ and s _ t ¼ðmaxðs 1 ; t 1 Þ;...; max ðs n ; t n ÞÞ. We denote the set of all mult-choce games wth player set N and maxmal partcpaton profle m by MC N;m. Often, we dentfy a mult-choce game hn; m; v wth ts characterstc functon v. A game v 2 MC N;m s called superaddtve f vðs _ tþ P vðsþþvðtþ for all s; t 2 M N wth s ^ t ¼ 0. A game v 2 MC N;m s called convex f vðs ^ tþþvðs _ tþ P vðsþþvðtþ for all s; t 2 M N ð21þ Relaton (2.1) s equvalent wth vðs þ tþ vðsþ P vðs þ tþ vðsþ; ð22þ for all s;s; t 2 M N satsfyng s 6 s;s ¼ s for all 2 carðtþ and s þ t 2 M N. Relaton (2.2) s obtaned from relaton (2.1) by puttng s and s þ t n the roles of s and t, respectvely. In fact, every game satsfyng relaton (2.2) s convex. In ths paper, relaton (2.2), whch can be seen as the mult-choce extenson of the property of ncreasng margnal contrbutons for coaltons n tradtonal convex games, wll play a key role. Clearly, a convex mult-choce game s superaddtve. In the sequel, we denote the class of convex multchoce games wth player set N and maxmal partcpaton profle m by CMC N;m. Let v 2 MC N;m. We defne M ¼ fð; jþj 2 N; j 2 M g and M þ ¼fð; jþj 2 N; j 2 M þ g. A (level) payoff vector for the game v s a functon x M! R, where for 2 N and j 2 M þ ; x j denotes the payoff to player correspondng to a change of actvty level of ths player from j 1toj, and x 0 ¼ 0 for all 2 N. One can represent a payoff vector for a game v as a P 2N m -dmensonal vector whose coordnates are numbered by the correspondng elements of M þ, where the frst m 1 coordnates represent payoffs for successve levels of player 1, the next m 2 coordnates are payoffs for successve levels of player 2, and so on. For each s 2 M N, the payoff of s accordng to x s XðsÞ ¼ P 2NP s j¼1 x j. Clearly, XðmÞ ¼ P 2N P m j¼1 x j. Let x and y be two payoff vectors for the game v. We say that x s weakly smaller than y f for each s 2 M N XðsÞ 6 YðsÞ A level payoff vector x M! R s called effcent f XðmÞ ¼vðmÞ, and s called level-ncrease ratonal f, for all 2 N and j 2 M þ ; x j s at least the ncrease n worth that player can obtan on hs/her own (.e. workng alone) when he/she changes hs/her actvty level from j 1 to j, that s x j P vðje Þ vððj 1Þe Þ, or, equvalently, x j P vð0 ; jþ vð0 ; j 1Þ. Here, e s the -th standard vector n R N. A level payoff vector whch s both effcent and level-ncrease ratonal s called an mputaton. We denote by IðvÞ the set of mputatons of v 2 MC N;m. The core CðvÞ of a game v 2 MC N;m conssts of all x 2 IðvÞ that satsfy XðsÞ P vðsþ for all s 2 M N,.e. CðvÞ ¼fx 2 IðvÞjXðsÞ P vðsþ for each s 2 M N g A game whose core s nonempty s called a balanced game. We call a balanced mult-choce game hn; m; v an exact game f for each s 2 M N there s an x 2 CðvÞ such that XðsÞ ¼vðsÞ. Let v 2 MC N;m and let u 2 M N. We denote by M N u the subset of MN consstng of mult-choce coaltons s such that s 6 u. The subgame of v wth respect to u; hn; u; v u, s defned by v u ðsþ ¼ vðsþ for each s 2 M N u. We defne the margnal game of v based on u (or the u- margnal game of v), hn; m u; v u,byv u ðsþ ¼ vðs þ uþ vðuþ for each s 2 M N m u. Margnal games of a mult-choce game quantfy the mpact of ncreases of partcpaton levels of players for each gven partcpaton profle n the mult-choce game at stake. The set C mn ðvþ of mnmal core elements of v s defned as fx 2 CðvÞj 9= y 2 CðvÞ st y x and y s weakly smaller than xg Two mportant soluton concepts for mult-choce games, namely the Shapley value and the Weber set (cf. Nouweland et al. (1995)), are based on margnal payoff vectors whch are defned by usng admssble orderngs. Let v 2 MC N;m.Anadmssble orderng (for v) s a bjecton r M þ!f1;...; P 2N m g satsfyng rðð; jþþ < r ðð; j þ 1ÞÞ for all 2 N and j 2f1;...; m 1g. The number of admssble orderngs for v s P 2N m Q! 2N ðm!þ; we denote the set of all admssble orderngs for a game v by NðvÞ. Now, let r 2 NðvÞ and k 2f1;...; P 2N m g. Denote by s r;k the coalton defned by s r;k ¼ max ðfj 2 M jrðð; jþþ 6 kg[f0gþ; for all 2 N. The coalton s r;k s the partcpaton profle reached after k steps accordng to the orderng r. The margnal vector w r;v M! R of v correspondng to r s defned by ¼ vs r;rðð;jþþ vs r;rðð;jþþ 1 ; w r;v j for all 2 N and j 2 M þ. In general, the margnal vectors w r;v ; r 2 NðvÞ, of a mult-choce game v are not necessarly mputatons. For multchoce games, several dfferent Shapley-lke values are known, where the contrbuton on games wth precedence constrants (cf. Fagle and Kern (1992)) played an mportant role. In ths paper we use the Shapley value (cf. Nouweland et al. (1995)) UðvÞ of v 2 MC N;m whch s defned as the average of all margnal vectors of w r;v,.e. Q UðvÞ ¼ U j ðvþ ; U j ðvþ ¼ P 2Nðm!Þ X w r;v! j 2N;j2M þ 2N m r2nðvþ The Weber set, WðvÞ, of a mult-choce game v s the convex hull of the margnal vectors of v,.e. WðvÞ ¼cofw r;v jr 2 NðvÞg. Basc results for convex mult-choce games that are used n ths paper are v 2 CMC N;m ff WðvÞ ¼coðC mn ðvþþ (Theorems and n Branze et al. (2005)); f v 2 CMC N;m then WðvÞ CðvÞ and WðvÞ CðvÞ s possble (see Theorem 11.9 and Example n Branze et al., 2005). Grabsch and Xe (2007) proposed notons related to the core and the Weber set of a mult-choce game, and showed that n case of convexty there s stll equalty between that core and that Weber set of the game. 3. New characterzatons of convex mult-choce games In ths secton, we extend two characterzatons for classcal convex games to the mult-choce settng. Lemma 3.1. Let v 2 CMC N;m and let u 2 M N þ. Then, v u 2 CMC N;m u.

3 R. Branze et al. / European Journal of Operatonal Research 198 (2009) Proof. Note that for s; t 2 M N m u we have v u ðs _ tþþv u ðs ^ tþ ¼vððs _ tþþuþþvððs ^ tþþuþ 2vðuÞ ¼ vððs þ uþ_ðt þ uþþ þ vððs þ uþ^ðt þ uþþ 2vðuÞ P vðs þ uþþvðt þ uþ 2vðuÞ ¼ v u ðsþþv u ðtþ; where the nequalty follows from the convexty of hn; m; v. Snce each convex game s also superaddtve, we conclude from Lemma 3.1 that f v 2 CMC N;m then all ts margnal games are superaddtve. Next, we prove that the converse also holds true. Ths result has been ndependently obtaned for tradtonal cooperatve games hn; v by Branze et al. (2004a) and Martnez-Legaz (1997, 2006). Theorem 3.1. Let v 2 MC N;m. Then the followng assertons are equvalent () For each u 2 M N þ, the u-margnal game v u of v s superaddtve; () v s a convex game. Proof. We stll need to prove that ())(). Suppose that v u s superaddtve. Then (2.1) holds true for all s; t 2 M N wth s ^ t ¼ 0 because v ¼ v 0 s superaddtve. For s ^ t ¼ f 0, take p ¼ s f and q ¼ t f. Snce hn; m f ; v f s superaddtve, we obtan 0 6 v f ðp _ qþ v f ðpþ v f ðqþ ¼ vðp _ q þ f Þ vðpþfþ vðqþfþþvðfþ ¼ vðs _ tþ vðsþ vðtþþvðs^tþ;.e. v s convex. h For a tradtonal cooperatve game hn; v, Bswas et al. (1999) (see also Azrel and Lehrer, 2007) proved that the game s convex f and only f each subgame hs; v, wth S N, s an exact game. In the sequel, we prove that a smlar characterzaton holds true n the mult-choce settng. Proposton 3.1. Each convex mult-choce game v s an exact game. Proof. Accordng to Theorem n Branze et al. (2005), for v 2 CMC N;m ; WðvÞ ¼coðC mn ðvþþ, mplyng that all margnal vectors w r;v are core elements. Take r such that s s one of the ntermedate coaltons between 0 and m. Then, XðsÞ ¼w r;v ðsþ ¼vðsÞ. h Theorem 3.2. Let v 2 MC N;m. Then the followng assertons are equvalent () hn; m; v s convex; () hn; u; v u s exact for each u 2 M N þ. Proof. () ) () follows from Proposton 3.1 because each subgame of a convex game s convex, and hence exact. () ) () Take s; t 2 M N. Snce the subgame v s_t s exact, there s x 2 Cðv s_t Þ such that Xðs ^ tþ ¼v s_t ðs ^ tþ ¼vðs ^ tþ. Now, usng Xðs _ tþ ¼v s_t ðs _ tþ ¼vðs _ tþ, we obtan vðs _ tþþvðs^tþ ¼Xðs _ tþþxðs^tþ ¼XðsÞþXðtÞ P vðsþþvðtþ We note that the two characterzatons of convex mult-choce games provded by Theorems 3.1 and 3.2 do not represent a beneft from the computatonal pont of vew from the orgnal defnton of a convex mult-choce game, but they are nterestng from the theoretcal pont of vew. h 4. Level-ncrease monotonc allocaton schemes Inspred by Sprumont (1990) (see also Hokar, 2000; Thomson, 1983, 1995) who ntroduced and studed the nterestng noton of populaton monotonc allocaton scheme (pmas) for tradtonal cooperatve games, we ntroduce here for mult-choce games the noton of level-ncrease monotonc allocaton scheme (lmas). Recall that a pmas for a (tradtonal) cooperatve game hn; v s an allocaton scheme ½a S; Š S22 N nf;g;2s such that () ða S; Þ 2S 2 Cðv S Þ for each S 2 2 N n f;g, where v S s the subgame correspondng to S; () a S; 6 a T; for all S; T 2 2 N n f;g wth S T and 2 S. Let v 2 MC N;m and let t 2 M N þ. For 2 N, denote the set f1; 2;...; t g by M t. A scheme a ¼½at j Št2MN þ s called a level-ncrease 2N;j2M t monotonc allocaton scheme (lmas) f () a t 2 Cðv t Þ for all t 2 M N þ (stablty condton); () a s j 6 at j for all s; t 2 MN þ wth s 6 t, for all 2 carðsþ and for all j 2 M s (level-ncrease monotoncty condton). The level-ncrease monotoncty condton mples that, f the scheme s used as regulator for the (level) payoff dstrbutons, n the mult-choce subgames players are pad for each one-unt level-ncrease (weakly) more n larger coaltons than n smaller coaltons. We notce that a necessary condton for the exstence of a lmas for a mult-choce game v s the exstence of core elements for v t for each t 2 M N. But ths s not suffcent, as n the case of tradtonal cooperatve games whch can be seen as mult-choce games where each player has exactly two partcpaton levels. A suffcent condton for the exstence of a lmas s the convexty of the game as we see n Theorem 4.1. Let v 2 MC N;m and x 2 WðvÞ. Then we call x lmas extendable f there exsts a lmas ½a t j Št2MN þ such that a m 2N;j2M t j ¼ x j for each 2 N and j 2 M þ. In the next theorem, we show that convex mult-choce games have a lmas. Specfcally, we prove that each Weber set element of a convex mult-choce game s lmas extendable. In the proof, restrctons of r 2 NðvÞ to subgames v t of v wll play a role. It wll be useful to look at such r as beng a sequence of flags f ; 2 N, sgnalng the players turns to one-unt level ncrease accordng to ther sets of partcpaton levels. Then, for each t 2 M N þ, the restrcton of r to t, denoted here by r t, can be obtaned from the sequence of flags of r by removng (notaton * ) for each player 2 N exactly m t flags f startng from the back of that sequence. We llustrate ths procedure n Example 4.1. Example 4.1. Consder a convex mult-choce game hn; m; v wth N ¼f1; 2; 3g; m ¼ð2; 1; 2Þ and r 1 2 NðvÞ expressed n terms of flags as r 1 ¼ðf 3 ; f 1 ; f 3 ; f 2 ; f 1 Þ. Note that ths orderng generates the followng maxmal chan of mult-choce coaltons n M N ð0; 0; 0Þ! f 3 ð0; 0; 1Þ! f 1 ð1; 0; 1Þ! f 3 ð1; 0; 2Þ! f 2 ð1; 1; 2Þ! f 1 ð2; 1; 2Þ Now, consder the mult-choce coalton t ¼ð1; 1; 1Þ and the correspondng subgame hn; t; v t. Then, the restrcton of r 1 to t s the orderng r 1 t whch can be expressed n terms of flags as ðf 3 ; f 1 ; ; f 2 ; Þ; t generates the followng maxmal chan of multchoce coaltons n M N t ð0; 0; 0Þ! f 3 ð0; 0; 1Þ! f 1 ð1; 0; 1Þ! f 2 ð1; 1; 1Þ Theorem 4.1. Let v 2 CMC N;m and let x 2 WðvÞ. Then x s lmas extendable.

4 574 R. Branze et al. / European Journal of Operatonal Research 198 (2009) Proof. Snce x s n the convex hull of the margnal vectors w r;v of v, t suffces to prove that each margnal vector w r;v s lmas extendable, because then the rght convex combnaton of these lmas extensons gves a lmas extenson of x. Take r 2 NðvÞ and defne ½a t j Št2MN þ by a t 2N;j2M t j ¼ wrt;vt j for each t 2 M N þ, 2 N and j 2 Mt, where r t s the restrcton of r to t (obtaned va the procedure descrbed above and llustrated n Example 4.1). We clam that ths scheme s a lmas extenson of w r;v. Clearly, a m j ¼ w r;v for each 2 N and j 2 M þ j snce v m ¼ v. Further, each mult-choce subgame v t, t 2 M N þ, s a convex game, and snce w rt;vt 2 Wðv t Þ and Wðv t ÞCðv t Þ (cf. Theorem 11.9 n Branze et al. (2005)), t follows that ða t j Þ 2N;j2M t 2 Cðv t Þ. Hence, the scheme satsfes the stablty condton. To prove the level-ncrease monotoncty condton, take s; t 2 M N þ wth s 6 t; 2 carðsþ, and j 2 Ms M t. We have to show that a s j 6 at j. Now, as j ¼ ¼ vðu wrs;vs j ; jþ vðu ; j 1Þ, where ðu ; jþ s the ntermedary mult-choce coalton n the maxmal chan generated by the restrcton of r to s, when player ncreased hs/her partcpaton level from j 1 to j. Smlarly, a t j ¼ ¼ vðu wrt;vt j ; jþ vðu ; j 1Þ. Note that, snce s 6 t, n the maxmal chan generated by r s the turn of to ncrease hs/her partcpaton level from j 1toj wll come not later than the same turn n the maxmal chan generated by r t, mplyng that ðu ; jþ 6 ðu ; jþ. Furthermore, ðu ; jþ 6 m. Then a s j ¼ vðu ; jþ vðu ; j 1Þ 6 vðu ; jþ vðu ; j 1Þ ¼a t j ; where the nequalty follows from the convexty of v. Specfcally, we used relaton (2.2) wth ðu ; j 1Þ n the role of s; ðu ; j 1Þ n the role of s, and ð0 ; 1Þ n the role of t. Hence, ½a t j Št2MN þ s a lmas 2N;j2M t extenson of w r;v. Further, the total Shapley value (cf. Nouweland et al. (1995)) of a convex mult-choce game, whch s the scheme ½U j ðv t ÞŠ t2mn þ 2N;j2M t wth the Shapley value of each mult-choce subgame v t, s a lmas. One can represent a lmas as a defectve jm N þ jjmþ j-matrx, whose rows correspond to mult-choce coaltons and whose columns correspond to elements of M þ arranged accordng to the ncreasng orderng for players and for each player wth respect to hs/her partcpaton levels. In each row t there s a core element of the mult-choce subgame v t, wth * for all components x j, wth 2 N and j 2ft þ 1;...; m g. h Example 4.2. Consder the convex mult-choce game hn; m; v wth N ¼f1; 2g; m ¼ð2; 1Þ; vðð0; 0ÞÞ ¼ 0; vðð1; 0ÞÞ ¼ 5; vðð2; 0ÞÞ ¼ 6; vðð0; 1ÞÞ ¼ 3; vðð1; 1ÞÞ ¼ 9; vðð2; 1ÞÞ ¼ 13. There are three orderngs on M þ ¼fð1; 1Þ; ð1; 2Þ; ð2; 1Þg r 1 ¼ðf 1 ; f 1 ; f 2 Þ; r 2 ¼ðf 1 ; f 2 ; f 1 Þ and r 3 ¼ðf 2 ; f 1 ; f 1 Þ. The correspondng margnal vectors w r1 ;v ; w r2 ;v r3 ;v ; w are extendable to the followng level-ncrease monotonc schemes Then, the total Shapley value UðvÞ generates the lmas 5. Concludng remarks Our man contrbuton on convex mult-choce games n ths paper s the new noton of lmas and the related results. The ntated study on lmas could be completed by takng nto account the exstng lterature on pmas, because a pmas can be seen as a specal nstance of lmas. Frst, we note that for a classcal convex game not only the Shapley value (cf. Shapley (1953)), but also the constraned egaltaran soluton (cf. Dutta and Ray (1989)) generates a pmas of that game (see also Hokar, 2000). However, a smlar result does not hold for all convex mult-choce games because the constraned egaltaran soluton of such a game s not necessarly a core element (see Example 4.2 n Branze et al., 2007). Second, snce convexty of a mult-choce game s a suffcent condton for the exstence of a lmas, an nterestng topc for further research s to characterze the class of those mult-choce games that possess a lmas. We recall that Sprumont (1990) characterzed the class of classcal cooperatve games that possess a pmas. Now, we notce that the way n whch level-ncrease monotonc schemes are ntroduced and studed n ths paper s qute smlar to the way to extend populaton monotonc allocaton schemes for classcal convex cooperatve games to partcpaton monotonc allocaton schemes (pamas) for convex games wth fuzzy coaltons (cf. Branze et al. (2004b)). Recall that a fuzzy coalton s a vector s 2½0; 1Š N, where s 2½0; 1Š represents the partcpaton level of player, 2 N, and the characterstc functon v assgns to every fuzzy coalton a real number, wth vð0; 0;...; 0Þ ¼0. Then, fuzzy games can be seen as mult-choce games where all players have a contnuum of possble partcpaton levels. However, there s nether a drect relaton between the concepts of lmas and pamas nor a straghtforward way to go from pamas to lmas or the way around. There are three man reasons for ths the dfference between the noton of convex fuzzy game (see Defnton 7.1 and Theorem 7.9 n Branze et al., 2005) and that of convex mult-choce game; the dfference between the (Aubn) core of a fuzzy game and that of the core of a mult-choce game; buldng blocks for a pamas for convex fuzzy games are players total payoffs n fuzzy coaltons nstead of level payoff vectors as n the case of a lmas for convex mult-choce games. Acknowledgements We thank two anonymous referees for ther useful comments and the Edtor for suggestng us a new sgnfcant reference. References Azrel, Y., Lehrer, E., Extendable cooperatve games. Journal of Publc Economc Theory 9, Bswas, A.K., Parthasarathy, T., Potters, J.A.M., Voorneveld, M., Large cores and exactness. Games and Economc Behavor 28, Branze, R., Dmtrov, D., Tjs, S. 2004a. 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