- Email: [email protected]
Egoist’s dilemma with interval data
Applied Mathematics and Computation 183 (2006) 94–105 www.elsevier.com/locate/amc
Egoist’s dilemma with interval data G.R. Jahanshahloo, F. Hosseinzadeh Lotfi, S. Sohraiee
*
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
Abstract In this paper, we investigate the problems of consensus-making among individuals or organizations with multiple criteria for evaluating performance when the players are supposed to be egoistic and the score for each criterion for a player is supposed to be an interval. Egoistic means that each player sticks to his superiority regarding the criteria. The concept that is developed in optimization leads the problem to a dilemma called ‘egoist’s dilemma’. Some parts of game theory are considered to propose a solution. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Optimization; Game theory; Cooperative game
1. Introduction Consider n players each have m criteria for evaluating their competency or ability, which is represented by interval. For example, consider a usual classroom examination, the higher score for a criterion is, the better player is judged to perform that criterion [4,6,7]. For example let the players be three company A, B, C with three criteria, budget, number of employee and production of companies each of them are supposed to be variable in an interval (In special case when data are ratio the prove is considered in [5]). Now the problem is allocated a certain amount of fellowship in accordance to their score at these three criteria. All players are supposed to be selfish or egoistic in the sense that they insist on their own advantage on the scores [8–10]. However, they must reach a consensus in order to get the fellowship. This paper uses the deterministic situation [1–3,5] in order to propose a new scheme for allocating or imputing the given benefit under the framework of optimization. The sections of this paper are as follows. In Section 2, the basic models are considered in which our assumption on optimization problem are introduced. Also the meaning of ‘‘Egoist’s dilemma’’ is represented in Section 2 and some properties of problem are proved as Theorem. Section 3, investigates a special case of ‘‘Egoist’s dilemma’’ in which both superiority and inferiority of players are considered as criteria to evaluate the contribution of their fellowship. In Section 4, a numerical presentation of real data from a commercial bank of Iran is considered. At last, some concluding remarks follow in Section 5. *
Corresponding author. E-mail address: [email protected] (S. Sohraiee).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.058
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
95
2. Basic models of the game The basic models and structure of the game are introduced in this section. 2.1. Selfish behavior and the egoist’s dilemma Let ½xlij ; xuij be the score of player j in the criterion i for i = 1, . . . , m and j = 1, . . . , n and xlij > 0; xuij > 0. It is assumed that the higher the interval score for a criterion is, the better the player is judged to perform as regard to that criterion. Each player k has a right to choose two sets of nonnegative weights wk ¼ ðwk1 ; . . . ; wkm Þ to the criteria that are most preferable to the player. Using the weight wk, the relative scores of player k to the total score are defined as follows: Pm k l Pm k u i¼1 wi xik i¼1 wi xik P Pm k P P ; : ð1Þ n m n u k u l l w ð ðx þ x ÞÞ w ð ij i¼1 i j¼1 ij i¼1 i j¼1 ðxij þ xij ÞÞ The denominators represent the total score of all players as measured by players k weight selection. While the numerators indicates player k’s self-evaluation as a lower and upper bound and lower bound. Therefore, the statement (1) represents player’s k’s lower and upper relative importance in accordance to the xlik ; xuik . We assume that the weighted score are transferable. Player k wishes to maximize the ratios by selecting the most preferable weights wk for each of xlik ; xuik as follows: Pm k l i¼1 wi xik Max Pm k P ð2Þ n u l w i¼1 i j¼1 ðxij þ xij Þ s:t: Max s:t:
wki P 0; i ¼ 1; . . . ; m: Pm k u i¼1 wi xik Pm k P n u l i¼1 wi j¼1 ðxij þ xij Þ wki P 0;
ð3Þ
i ¼ 1; . . . ; m:
The motivation behind this problem is that the player k’s wishes to maximize his lower relative efficiency by (2), and upper relative efficiency by (3). As we can see in (2) the lower weighted sum of its record to the total weighted sum of all player records is maximized and in (3) the upper weighted sum of its record is maximized. Without loss of generality the following construct is added to the problem, in order to make the problem linear: n X ðxlij þ xuij Þ ¼ 1; i ¼ 1; . . . ; m: ð4Þ j¼1
We can do this with divide the xlij ; xuij to the will have the following linear problems: m X wki xlik cl ðkÞ ¼ Max
Pn
l j¼1 ðxij
þ xuij Þ. The problems (2) and (3) are changed by (4) and we
i¼1 m X
s:t:
wki ¼ 1;
ð5Þ
i¼1
wki
P 0;
cu ðkÞ ¼ Max s:t:
i ¼ 1; . . . ; m:
m X
wki xuik
i¼1 m X
wki ¼ 1;
i¼1
wki
P 0;
i ¼ 1; . . . ; m:
ð6Þ
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G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
After solving problems (5) and (6), if the optimal value of the problem (5) and (6) are cl(k) and cu(k) respectively, then cl(k) + cu(k) may be considered Pmas optimal value of the problem. Now, the problem is to maximize the objectives (5) and (6) on the simplex i¼1 wki ¼ 1. Apparently, the optimal solutions are given by assigning 0 1 to wkiðkÞ and wkiðk0 Þ for the criterion i(k) and i(k ), such that xliðkÞ ¼ Maxfxlik ji ¼ 1; . . . ; mg and xuiðk0 Þ ¼ Maxfxuik ji ¼ 1; . . . ; mg, respectively. Therefore, the optimal values will be as follows: cl ðkÞ ¼ xliðkÞ
and
cu ðkÞ ¼ xuiðk0 Þ ;
k ¼ 1; . . . ; n:
ð7Þ
The deterministic value cl(k) + cu(k) indicates the highest relative score of player k which is obtained by the optimal weight selecting behavior. The optimal weight wkiðkÞ and wkiðk0 Þ may differ from one player to another. Theorem 1 n X ðcl ðkÞ þ cu ðkÞÞ P 1:
ð8Þ
k¼1
Proof. Let the optimal weight for the player k be wlk ¼ ðwl1k ; . . . ; wlmk Þ and wuk ¼ ðwu1k ; . . . ; wumk Þ i.e. wliðkÞk ¼ 1 and wlik ¼ 0; i 6¼ iðkÞ and wuiðk0 Þk ¼ 1 and wuik ¼ 0; i 6¼ iðk 0 Þ. Then we have n n n n X m n X m n n X X X X X X X ðcl ðkÞ þ cu ðkÞÞ ¼ cl ðkÞ þ cu ðkÞ ¼ wlik xlik þ wuik xuik ¼ xliðkÞk þ xuiðk0 Þk k¼1
k¼1
P
n X
k¼1
xl1k þ
k¼1
n X
k¼1
i¼1
k¼1
i¼1
k¼1
k¼1
xulk ¼ 1:
k¼1
xliðkÞk
P xl1k and xuiðk0 Þk P xu1k and the last equality follows from the row-wise The inequality above follows from normalization. This proportion asserts that, if each player sticks to his egoistic sense of value and insists on getting the portion of the benefit as designated by cl(k) and cu(k), the sum of shares usually exceeds 1 and hence cl(k) + cu(k)cannot fulfill the role of division of the benefit. If eventually the sum of cl(k) and cu(k) turns out to be 1, all players will agree to accept the division cl(k) + cu(k), since this is obtained by most preferable weight selection. The latter case will occur when all players have the same and common optimal weight selection. More correctly we have the following theorem. h Pn l u Theorem 2. The equality k¼1 ðc ðkÞ þ c ðkÞÞ ¼ 1 holds if and only if our data satisfies the condition l l l u u x1k ¼ x2k ¼ ¼ xmk and x1k ¼ x2k ¼ ¼ xumk ; k ¼ 1; . . . ; n. That is, each player has the same score with respect to the m criteria. Proof. The ‘if’ part can be seen as follows. Since cl ðkÞ ¼ xl1k and cu ðkÞ ¼ xu1k for all k, we have n n n n n X X X X X ðcl ðkÞ þ cu ðkÞÞ ¼ cl ðkÞ þ cu ðkÞ ¼ xl1k þ xu1k ¼ 1: k¼1
k¼1
k¼1
k¼1
k¼1
xl11
> xl21 and xu11 > xu21 then there must be two columns The ‘only if’ part can be proved as follows. Suppose 0 l l u u h 5 1 and h 5 1such that x1h < x2h and x1h0 < x2h0 otherwise the second row sum can not attain 1. Thus we have cl ð1Þ P xl11
and
cl ðhÞ P xl2h > xl1h
cu ð1Þ P xu11 ; and
cu ðh0 Þ P xu2h0 > xu2h0 ;
cl ðjÞ P xl1j ; j 6¼ 1; h and
cu ðjÞ P xu2j ; j 6¼ 1; h0 :
Hence it holds that n n n n n n n X X X X X X X ðcl ðkÞ þ cu ðkÞÞ ¼ cl ðkÞ þ cu ðkÞ P xl1j þ xl2h þ xu2j þ xu2h0 > xl1j þ xu1j ¼ 1: k¼1
k¼1
k¼1
j¼1 j6¼h
j¼1 j6¼h0
j¼1
j¼1
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
97
This leads to a contradiction. Therefore, player 1 must have the same score in all criteria. The same relation must hold for the other players. In the above case, only one criterion is needed for describing the game and the division proportional to this score is a fair division. However, such situations might occur only in rare instances. In the majority of cases, we have n X ðcl ðkÞ þ cu ðkÞÞ > 1: k¼1
We may call this the ‘‘Egoist dilemma’’.
h
2.2. Assumption on the game and fair division The following statements are assumed among players in order to attain a fair division but we know that each player sticks to his superiority. Now the laws of game are as follows: (1) All players agree not to break off the game. (2) All players are willing to negotiate with each other to attain a reasonable and fair division z = (z1, . . . , zn). 2.3. Coalition with additive property Let the coalition S be a subset of players set N = (1, . . . , n). The record for coalition S is defined by X X xli ðSÞ ¼ xlij ; xui ðSÞ ¼ xuij ; i ¼ 1; . . . ; m: j2S
ð9Þ
j2S
These coalitions aim to maximize the outcomes cl(S), cu(S) as follows: m X wi xli ðSÞ; cl ðSÞ ¼ Max i¼1
s:t:
m X
wi ¼ 1;
i¼1
wi P 0; i ¼ 1; . . . ; m; m X cu ðSÞ ¼ Max wi xui ðSÞ;
ð10Þ
i¼1
s:t:
m X
wi ¼ 1;
i¼1
wi P 0;
i ¼ 1; . . . ; m:
ð11Þ
The cl(S) and cu(S), with cl(/) = 0 and cu(/) = 0, defines a characteristic function of the coalition S. Thus a game is represented by (N, c). Definition 1. A function f is called sub-additive, if for any S N and T N with S \ T = / the following statement holds: f ðS [ T Þ 6 f ðSÞ þ f ðT Þ: Definition 2. A function f is called super-additive, if for any S N and T N with S \ T = / the following statement holds: f ðS [ T Þ P f ðSÞ þ f ðT Þ:
98
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
Theorem 3. The characteristic function c is sub-additive, i.e. for any S N and T N with S \ T = / we have cl ðS [ T Þ þ cu ðS [ T Þ 6 cl ðSÞ þ cl ðT Þ þ cu ðSÞ þ cu ðT Þ:
ð12Þ
Proof. By renumbering the indexes, we have S ¼ f1; . . . ; hg;
T ¼ fh þ 1; . . . ; kg and
S [ T ¼ f1; . . . ; kg:
For these sets, considering the optimal solution as defined in (7), we have the following statement: cl ðS [ T Þ þ cu ðS [ T Þ ¼ max i
k X
xlij þ max i
j¼1
6 max i
h X
i
l
xuij
j¼1
xlij þ max
j¼1
l
k X
k X
xlij þ max i
j¼hþ1 u
h X
xuij þ max i
j¼1
u
¼ c ðSÞ þ c ðT Þ þ c ðSÞ þ c ðT Þ:
k X
xuij
j¼hþ1
Theorem 4 cl ðN Þ þ cu ðN Þ ¼ 1: Proof l
u
c ðN Þ þ c ðN Þ ¼
m X i¼1
wi
n X
xlij
þ
j¼1
m X
wi
i¼1
n X
xuij
m X
¼
j¼1
n X
wi
i¼1
! ðxlij
þ
xuij Þ
¼
j¼1
m X
wi ¼ 1:
i¼1
2.4. Another expression of the game Let us define another game (N, v) by X X vl ðSÞ ¼ cl ðjÞ cl ðSÞ and vu ðSÞ ¼ cl ðjÞ cu ðSÞ: j2S
ð13Þ
j2S
Theorem 5. (N, v) is supper-additive, i.e. vl ðSÞ þ vl ðT Þ þ vu ðSÞ þ vu ðT Þ 6 vl ðS [ T Þ þ vu ðS [ T Þ Proof
( l
l
u
u
v ðSÞ þ v ðT Þ þ v ðSÞ þ v ðT Þ ¼ ( ¼
X
) l
l
c ðjÞ c ðSÞ
j2S
X
for each S; T N and S \ T ¼ /: (
þ
u
c ðjÞ c ðSÞ
( þ
X
j2S[T
6
X
)
l
c ðjÞ c ðS [ T Þ
j2S[T
¼ vl ðS [ T Þ þ vu ðS [ T Þ:
u
X
cu ðjÞ fcu ðSÞ þ cu ðT Þg
j2S[T
! l
u
c ðjÞ c ðT Þ
cl ðjÞ fcl ðSÞ þ cl ðT Þg þ
X
l
c ðjÞ c ðT Þ
j2T
j2S
¼
) l
j2T
) u
X
þ
X j2S[T
! u
u
c ðjÞ c ðS [ T Þ
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
99
Also, we have n X
vl ðN Þ þ vu ðN Þ ¼
cl ðjÞ þ
j¼1
n X
cu ðjÞ fcl ðN Þ þ cu ðN Þg ¼
j¼1
n X
ðcl ðjÞ þ cu ðjÞÞ 1 P 0:
j¼1
Hence, the game (N, v) is 0-normalized. h 2.5. A minimum game The opposite side of the game can be constructed by (N, d) as follows: m X d l ðkÞ ¼ Min wki xlik i¼1
s:t:
m X
ð14Þ
wki ¼ 1;
i¼1
wki P 0; m X
d u ðkÞ ¼ Min
i¼1 m X
s:t:
i¼1 wki
i ¼ 1; . . . ; m:
wki xuik ð15Þ
wki ¼ 1; P 0;
l
i ¼ 1; . . . ; m:
u
The optimal value d (k) + d (k) assures the minimum division that player k can expect from the game. In this as a result of Theorem 1 we have Theorem 6 n X d l ðkÞ þ d u ðkÞ 6 1: k¼1
Proof n n n n X m n X m n n X X X X X X X ðd l ðkÞ þ d u ðkÞÞ ¼ d l ðkÞ þ d u ðkÞ ¼ wlik xlik þ wuik xuik ¼ xliðkÞk þ xuiðk0 Þk k¼1
k¼1
6
n X
k¼1
xl1k þ
k¼1
n X
k¼1
i¼1
k¼1
i¼1
k¼1
k¼1
xu1k ¼ 1:
k¼1
Analogously to the max game, for the coalition S N, we define m X d l ðSÞ ¼ Min wi xli ðSÞ i¼1
s:t:
m X
ð16Þ
wi ¼ 1;
i¼1
wi P 0; d u ðSÞ ¼ Min
m X
i ¼ 1; . . . ; m:
wi xui ðSÞ
i¼1
s:t:
m X
ð17Þ
wi ¼ 1;
i¼1
wi P 0; l
i ¼ 1; . . . ; m:
Apparently, it holds that d (N) + du(N) = 1.
h
100
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
Theorem 7. The min game (N, d) is supper-additive i.e. we have d l ðS [ T Þ þ d u ðS [ T Þ P d l ðSÞ þ d l ðT Þ þ d u ðSÞ þ d u ðT Þ
for each S; T N with S \ T ¼ /:
Proof. By renumbering the indexes, we have S = {1, . . . , h}, T = {h + 1, . . . , k} and S [ T = {1, . . . , k}. For these sets it holds that d l ðS [ T Þ þ d u ðS [ T Þ ¼ min
k X
i
xlij þ min i
j¼1
P min
h X
i
xuij
j¼1
xlij þ min i
j¼1
l
k X
l
k X
xlij þ min i
j¼hþ1 u
h X
xuij þ min i
j¼1
k X
xuij
j¼hþ1
u
¼ d ðSÞ þ d ðT Þ þ d ðSÞ þ d ðT Þ: Thus, this game starts from dl(k) > 0 and du(k) > 0 and enlarges the gains by the coalition until the grand coalition N with dl(N) + du(N) = 1 is reached. h Between the games (N, c) and (N, d) we have the following Theorem: Theorem 8 d l ðSÞ þ d u ðSÞ þ cl ðN n SÞ þ cu ðN n SÞ ¼ 1
for each S N ; S 6¼ N :
ð18Þ
Proof d l ðSÞ þ d u ðSÞ þ cl ðN n SÞ þ cu ðN n SÞ ! h h n n n n X X X X X X l u l u l l xij þ min xij þ max xij þ max xij ¼ min xij xij ¼ min j¼1
þ min min
j¼1 n X
j¼1 n X
xuij
j¼hþ1
!
n X
xuij
þ max
j¼hþ1
xlij min
j¼hþ1
n X
n X
j¼hþ1
xlij
j¼hþ1 n X
xuij þ max
j¼hþ1
j¼hþ1
þ max
n X
j¼1
xuij
j¼hþ1 n X
xlij þ max
¼ min
j¼hþ1
n X
xlij
þ
n X
j¼1
xuij ¼ 1:
! xuij
j¼1
j¼hþ1
Theorem 9 1 þ cl ðS \ T Þ þ cu ðS \ T Þ 6 cl ðSÞ þ cl ðT Þ þ cu ðSÞ þ cu ðT Þ for each S; T N with S [ T ¼ N : Proof. From the super-additivity of d( Æ ), we have the following equality: d l ðfS n ðS \ T Þg [ fT n ðS \ T ÞgÞ þ d u ðfS n ðS \ T Þg [ fT n ðS \ T ÞgÞ P d l ðS n ðS \ T ÞÞ þ d l ðT n ðS \ T ÞÞ þ d u ðS n ðS \ T ÞÞ þ d u ðT n ðS \ T ÞÞ: From Theorem 8, it holds that d l ðS n ðS \ T ÞÞ þ d u ðS n ðS \ T ÞÞ ¼ 1 fcl ðT Þ þ cu ðT Þg; d l ðT n ðS \ T ÞÞ þ d u ðT n ðS \ T ÞÞ ¼ 1 fcl ðSÞ þ cu ðSÞg; d l ðfS n ðS \ T Þg [ fT n ðS \ T ÞgÞ þ d u ðfS n ðS \ T Þg [ fT n ðS \ T ÞgÞ ¼ 1 fcl ðS \ T Þ þ cu ðS \ T Þg: Hence we have,
1 fcl ðS \ T Þ þ cu ðS \ T Þg P 1 fcl ðT Þ þ cu ðT Þg þ 1 fcl ðSÞ þ cu ðSÞg ; 1 þ cl ðS \ T Þ þ cu ðS \ T Þ 6 cl ðSÞ þ cl ðT Þ þ cu ðSÞ þ cu ðT Þ:
The other part will be proved similarly.
h
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
101
Theorem 10 1 þ d l ðS \ T Þ þ d u ðS \ T Þ P d l ðSÞ þ d l ðT Þ þ d u ðSÞ þ d u ðT Þ for each S; T N with S [ T ¼ N : 3. Extensions In this section, we consider a benefit–cost model to evaluate a kind of special case in which our data represent both superiority and inferiority of players. 3.1. A revenue-cost (R-C) game Up to now we consider only a special case in which our data represent only superiority of players. In this section, we represent a situation in which our data are supposed to consider both superiority and inferiority. To this end, consider s criteria for representing benefits and m criteria for costs. Let ½y lrj ; y urj ; r ¼ 1; . . . ; s and ½xlij ; xuij ; i ¼ 1; . . . ; m, be the benefits and costs of player j (j = 1, . . . , n), respectively. Now the upper and lower benefits of player j are evaluated by ðu1 y u1j þ þ us y usj Þ ðv1 xl1j þ þ vm xlmj Þ;
ð19Þ
ðu1 y lij þ þ us y lsj Þ ðv1 xu1j þ þ vm xumj Þ;
where u = (u1, . . . , us) and v = (v1, . . . , vm) are respectively, the virtual weights for benefits and costs. Analogous to expression (1), we define the relative scores of player j to the total scores as follows: Pm Ps u l r¼1 ur y rj i¼1 vi xij P Ps Pn P ; ð20Þ m n u i u l r¼1 ur k¼1 ðy rk þ y rk Þ i¼1 vi k¼1 ðxik þ xik Þ P Ps m l u r¼1 ur y ij i¼1 vi xij P Ps Pn P : ð21Þ m n u u l l r¼1 ur k¼1 ðy rk þ y rk Þ i¼1 vi k¼1 ðxik þ xik Þ Player j wishes to maximize his score, subject to the condition that the merit of all players is nonnegative. Therefore, we have s m X X ur y urk vi xlik P 0 ðk ¼ 1; . . . ; nÞ; ð22Þ r¼1
s X
i¼1
ur y lrk
m X
r¼1
vi xuik P 0
ðk ¼ 1; . . . ; nÞ:
ð23Þ
i¼1
We can express this situation by the linear programs below: s m X X ur y urj vi xlij cu ðkÞ ¼ max r¼1
s:t:
s X
i¼1
n s n X X X ur ðy urk þ y lrk Þ vi ðxuik þ xlik Þ ¼ 1;
r¼1
s X
k¼1
ur y urk
r¼1
s X s X
ur
n X
r¼1
s X
vi xlik P 0;
k¼1
k ¼ 1; . . . ; n;
m X
ð24Þ
vi xuij
ðy urk þ y lrk Þ
ur y lrk
ur P 0;
m X i¼1
k¼1
r¼1
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m:
vi P 0;
ur y lrj
r¼1
s:t:
i¼1
i¼1
ur P 0; cl ðkÞ ¼ max
m X
s X i¼1
vi xuik P 0;
vi
n X ðxuik þ xlik Þ ¼ 1; k¼1
k ¼ 1; . . . ; n;
i¼1
vi P 0;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m:
ð25Þ
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G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
Following the same manner as in the preceding sections, we can develop coalitions and imputations of this R-C game, although the row-wise normalization is not available in this game. That is, two characteristic function of the coalition S are defined by the linear programs below: s m X X X X cl ðSÞ ¼ max ur y lrj vi xuij s:t:
r¼1
j2S
s X
n X
r¼1 s X
ur
k¼1 m X
ðxlik þ xuik Þ ¼ 1;
k¼1
vi xuik P 0;
k ¼ 1; . . . ; n;
i¼1
s X
ur
y urj
m X
j2S
s X
i ¼ 1; . . . ; m; r ¼ 1; . . . ; s:
ur P 0; X
r¼1
ur
vi
X
ðy lrk þ y urk Þ
vi
n X ðxlik þ xuik Þ ¼ 1;
i¼1
ur y urk
r¼1
xlij ;
m X
k¼1 m X
ð26Þ
j2S
i¼1
n X
r¼1 s X
n X
vi
i¼1
ur y lrk
vi P 0;
s:t:
m X
ðy lrk þ y urk Þ
r¼1
cu ðSÞ ¼ max
j2S
i¼1
k¼1
vi xlik P 0;
k ¼ 1; . . . ; n;
i¼1
vi P 0;
i ¼ 1; . . . ; m; r ¼ 1; . . . ; s:
ur P 0;
ð27Þ
In the models (26) and (27) we kept the condition that the benefit of all players is nonnegative. Since the constraints of the programs (26) and (27) are the same for all coalitions, we have the following theorem: Theorem 11. The R-C max game satisfies a sub-additive property. Proof. For any S N and T N with S \ T = /,we have cl ðS [ T Þ þ cu ðS [ T Þ ¼ max
s X
v;u
X
ur
j2S[T
r¼1
¼ max
s X
v;u
ur
v;u
s X
ur
s X
v;u
ur
v;u
6 max v;u
ur
r¼1
þ max v;u
X
ur
s X r¼1
X
y lrj
y urj
y lrj
X j2S
X
m X
vi
u
vi
vi
y urj
m X
vi
xuij xlij
vi
þ
!
xuij
X
þ max
ur
X
!! xlij
m X
X
vi
j2T
i¼1
y urj
!! xuij
m X
vi
X
!! xlij
j2T
i¼1
j2T
s X
X
m X
X
þ max v;u
y lrj
X
ur
r¼1 s X r¼1
y lrj
j2T
ur
xlij
xuij
j2T
ur
!
j2S[T
r¼1
! xlij
X
s X
v;u
j2S
þ
vi
i¼1
!!
X
X
j2T
r¼1
!
j2S
i¼1
s X
þ
y urj
j2T
xlij
m X
j2S[T
j2S
j2T
X
xuij þ
X
!
X
ur
j2S
m X
j2S
¼ cl ðSÞ þ cl ðT Þ þ cu ðSÞ þ cu ðT Þ: l
X
m X
i¼1
vi
X
r¼1
X
i¼1
i¼1 m X
v;u
i¼1
! y urj
i¼1
X
j2S
ur
y lrj
s X
þ max
m X
j2T
j2S
r¼1 s X
þ
j2S
s X
þ max
y urj
j2S
r¼1
!
j2T
X
r¼1
¼ max
y lrj þ
xuij
j2S[T
X
j2S
!
X
vi
i¼1
X
r¼1
þ max
y lrj
m X
X j2T
vi
i¼1
y urj
m X i¼1
! xuij
j2T
vi
X
! xlij
j2T
It also holds that c (N) + c (N) = 1. We can also define the R-C minimum game, which satisfies a supper-additive property, and arrive at the following theorem:
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
103
Theorem 12. The R-C max game (N, c) and min game (N, d) are dual games, i.e. for any S N, we have d l ðSÞ þ d u ðSÞ þ cl ðN n SÞ þ cu ðN n SÞ ¼ 1: Proof cl ðN n SÞ þ cu ðN n SÞ ¼ max
s X
v;u
ur
s X
v;u
ur
s X
v;u
ur
s X
v;u
X
ur
v;u
¼ max v;u
ur
r¼1
s X
ur
ur
y lrj
max v;u
y urj
j2N
X
X
! y urj
m X
vi
v;u
m X
X
X
vi
vi
X
m X
vi
X j2N
m X X X ðy lrj þ y urj Þ vi ðxlij þ xuij Þ s X
ur
X
y lrj
j2S
j2N
m X
X
i¼1
vi
j2S
ur
y lrj
xlij
j2N nS
!! xlij
m X
vi
y uij
m X
j2S
X
vi
X
!! xlij
j2S
i¼1
!!
!! xuij
j2S
i¼1
ðxlij þ xuij Þ
max v;u
!
! xuij
X
X
ur
!
xuij
j2S
r¼1
i¼1
i¼1
X
s X
j2N
!
vi
j2S
r¼1
! xlij
xlij
X
i¼1
!!
X
j2N
y uij
j2S
s X
m X
j2N nS
xuij
X
! xuij
ur
j2N
m X
X
r¼1
i¼1
i¼1
ðy lrj þ y urj Þ
vi
i¼1
j2N
j2N
r¼1 l
i¼1
j2N
r¼1
y lrj
s X
þ max
m X
j2S
X
r¼1 s X
j2N
s X
þ max
y urj
j2N
r¼1
!
j2S
X
r¼1
¼ max
y lrj
xuij
j2N nS
X
j2N
!
X
vi
i¼1
X
r¼1
þ max
y lrj
m X
j2N nS
r¼1
¼ max
¼
X
max v;u
s X r¼1
ur
X
y urj
j2S
m X i¼1
vi
X
! xlij
j2S
u
¼ 1 ðd ðSÞ þ d ðSÞÞ: From Theorem 12, these two games are dual.
h
4. Numerical example We now apply this approach to some commercial bank of Iran. There are 20 branches in this district. Each branch uses 3 inferiority and 5 superiority criteria. Table 1 shows all kinds of these inferiority and superiority criteria. In Tables 2 and 3 the interval inferiority and interval superiority for these players are given, respectively. Also in Table 4, the results of approach are presented.
Table 1 Inferiority and superiority criteria Inferiority
Superiority
Payable interest Personnel Non-performing loans
The total sum of four main deposits Other deposits Loans granted Received interest Fee
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G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
Table 2 Inferiority criteria for the 20 branches of bank Playerj
xl1j
xu1j
xl2j
xu2j
xl3j
xu3j
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
5007.37 2926.81 8732.7 945.93 8487.07 13759.35 587.69 4646.39 1554.29 17528.31 2444.34 7303.27 9852.15 4540.75 3039.58 6585.81 4209.18 1015.52 5800.38 1445.68
9613.37 5961.55 17752.5 1966.39 17521.66 27359.36 1205.47 9559.61 3427.89 36297.54 4955.78 14178.11 19742.89 9312.24 6304.01 13453.58 8603.79 2037.82 11875.39 2922.15
36.29 18.8 25.74 20.81 14.16 19.46 27.29 24.52 20.47 14.84 20.42 22.87 18.47 22.83 39.32 25.57 27.59 13.63 27.12 28.96
36.86 2016 27.17 22.54 14.8 19.46 27.48 25.07 21.59 15.05 20.54 23.19 21.83 23.96 39.86 26.52 27.95 13.93 27.26 28.96
87,243 9945 47,575 19,292 3428 13,929 27,827 9070 412,036 8,638 500 16,148 17,163 17,918 51,582 20,975 41,960 18,641 19,500 31,700
87,243 12,120 50,013 19,753 3911 15,657 29,005 9983 413,902 10,229 937 21,353 17,290 17,964 55,136 23,992 43,103 19,354 19,569 32,061
Table 3 Superiority criteria for the 20 branches of bank Playerj
y l1j
y u1j
y l2j
y u2j
y l3j
y u3j
y l4j
y u4j
y l5j
y u5j
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2,696,995 340,377 1,027,546 1,145,235 390,902 988,115 144,906 408,163 335,070 700,842 641,680 453,170 553,167 309,670 286,149 321,435 618,105 248,125 640,890 119,948
3,126,798 440,355 1,061,260 1,213,541 395,241 1,087,392 165,818 416,416 410,427 768,593 696,338 481,943 574,989 342,598 317,186 347,848 835,839 320,974 679,916 120,208
263,643 95,978 37,911 2,29,646 4924 74,133 180,530 405,396 337,971 14,378 114,183 27,196 21,298 20,168 149,183 66,169 244,250 3063 490,508 14,943
382,545 117,659 503,089 268,460 12,136 111,324 180,617 486,431 449,336 15,192 241,081 29,553 23,043 26,172 270,708 80,453 404,579 6330 684,372 17,495
1,675,519 377,309 1,233,548 468,520 129,751 507,502 288,513 1,044,221 1,584,722 2,290,745 1,579,961 245,726 425,886 124,188 787,959 360,880 9,136,507 26,687 2,946,797 297,674
1,853,365 390,203 1,822,028 542,101 142,873 574,355 323,721 1,071,812 1,802,942 2,573,512 2,285,079 275,717 431,815 126,930 810,088 379,488 9,136,507 29,173 3,985,900 308,012
108634.76 32396.65 96842.33 32362.8 12662.71 53591.3 40507.97 56260.09 176436.81 662725.21 17527.58 35757.83 45652.24 8143.79 106798.63 89971.47 33036.79 9525.6 66097.16 21991.53
125740.28 37836.56 108080.01 39273.37 14165.44 72257.28 45847.48 73948.09 189006.12 791463.08 20773.91 42790.14 50255.75 11948.04 111962.3 165524.22 41826.51 10877.78 95329.87 27934.19
965.97 304.67 2285.03 207.98 63.32 480.16 176.58 4654.71 560.26 58.89 1070.81 375.07 438.43 936.62 1203.79 200.36 2781.24 240.04 961.56 282.73
6957.33 749.4 3174 510.93 92.3 869.52 370.81 5882.53 2506.67 86.86 2283.08 559.85 836.82 1468.45 4335.24 399.8 4555.42 274.7 1914.25 471.22
P P As we can see in example, nk¼1 ðcl ðkÞ þ cu ðkÞÞ > 1 and nk¼1 d l ðkÞ þ d u ðkÞ < 1.Therefore, in this example the ‘‘Egoist’s dilemma’’ has occurred and the players were not willing to negotiate with each other to attain a reasonable and fair division. 5. Conclusion In this paper, we have investigated a problem called ‘‘Egoist’s dilemma’’ with interval data in which some parts of optimization problem has been used. As we know in practical purposes, data are usually interval
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
105
Table 4 Sum of lower and upper results Playerj
c(j) = cl(j) + cu(j)
d(j) = dl(j) + du(j)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0.770328 0.139589 0.35728 0.285301 0.176518 0.37924 0.104281 0.498178 0.383411 0.973041 0.233868 0.185899 0.230978 0.160454 0.372076 0.223001 0.586103 0.08303 0.264289 0.083038
0.052617 0.013097 0.034481 0.013741 0.002732 0.019861 0.009084 0.030985 0.026538 0.005796 0.012743 0.008409 0.006985 0.004772 0.023202 0.010685 0.028867 0.001274 0.044835 0.004795
Sum
6.489905
0.355497
because of this the further study will be very useful in further cases. Also some properties of further problem have been extended in different Theorems in which some important qualities of dilemma have been introduced. An extension of problem, in which both superiority and inferiority criteria could be considered to players, has been discussed. Furthermore a numerical example, in which some commercial banks of Iran have been evaluated with proposed way, has been considered. Also, we propose to future researches, finding a practical algorithm to present a reasonable and fair division in all cases. Eventually, we hope that this study will be useful to future researches and will solve some practical difficulties in different fields. References [1] R. Baron, Jacques Durieu, Hans Haller, Philippe Solal, Complexity and stochastic evolution of dyadic networks, Computers and Operations Research 33 (2006) 312–327. [2] M. Davis, M. Maschler, The kernel of cooperative game, Naval Research Logistics Quarterly 12 (1965) 223–259. [3] G.R. Jahanshahloo, F. Hosseinzadeh Lotfi, M. Izadikhah, An algorithmic method to extend TOPSIS for decision-making problems with interval data, Applied Mathematics and Computation, Forth Coming. [4] I.V. Konnov, Splitting-type method for systems of variational inequalities, Computers and Operations Research 33 (2006) 520–534. [5] K. Nakabayashi, K. Tone, Egoist’s dilemma: a DEA game, Omega the International Journal of Management Science 34 (2006) 135–148. [6] M. Parlar, Z. Kevin Veng, Coordinating pricing and production decisions in the presence of price competition, European Journal of Operational Research 170 (2006) 211–227. [7] A. Shapley, On balanced sets and cores, Naval Research Logistics Quarterly 14 (1967) 453–460. [8] H. Young, N. Okada, T. Hashimoto, Cost allocation in water resources development, Water Resources Research 18 (1981) 463–475. [9] G. Zaccour, Special issue on game theory: Numerical methods and applications, Computers and Operations Research 33 (2006) 285– 297. [10] J. Zhu, Multidimensional quality-of-life measure with an application to fortune’s best cities, Socio-Economic Planning Sciences 35 (2001) 263–284.
Egoist’s dilemma with interval data G.R. Jahanshahloo, F. Hosseinzadeh Lotfi, S. Sohraiee
*
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
Abstract In this paper, we investigate the problems of consensus-making among individuals or organizations with multiple criteria for evaluating performance when the players are supposed to be egoistic and the score for each criterion for a player is supposed to be an interval. Egoistic means that each player sticks to his superiority regarding the criteria. The concept that is developed in optimization leads the problem to a dilemma called ‘egoist’s dilemma’. Some parts of game theory are considered to propose a solution. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Optimization; Game theory; Cooperative game
1. Introduction Consider n players each have m criteria for evaluating their competency or ability, which is represented by interval. For example, consider a usual classroom examination, the higher score for a criterion is, the better player is judged to perform that criterion [4,6,7]. For example let the players be three company A, B, C with three criteria, budget, number of employee and production of companies each of them are supposed to be variable in an interval (In special case when data are ratio the prove is considered in [5]). Now the problem is allocated a certain amount of fellowship in accordance to their score at these three criteria. All players are supposed to be selfish or egoistic in the sense that they insist on their own advantage on the scores [8–10]. However, they must reach a consensus in order to get the fellowship. This paper uses the deterministic situation [1–3,5] in order to propose a new scheme for allocating or imputing the given benefit under the framework of optimization. The sections of this paper are as follows. In Section 2, the basic models are considered in which our assumption on optimization problem are introduced. Also the meaning of ‘‘Egoist’s dilemma’’ is represented in Section 2 and some properties of problem are proved as Theorem. Section 3, investigates a special case of ‘‘Egoist’s dilemma’’ in which both superiority and inferiority of players are considered as criteria to evaluate the contribution of their fellowship. In Section 4, a numerical presentation of real data from a commercial bank of Iran is considered. At last, some concluding remarks follow in Section 5. *
Corresponding author. E-mail address: [email protected] (S. Sohraiee).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.058
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
95
2. Basic models of the game The basic models and structure of the game are introduced in this section. 2.1. Selfish behavior and the egoist’s dilemma Let ½xlij ; xuij be the score of player j in the criterion i for i = 1, . . . , m and j = 1, . . . , n and xlij > 0; xuij > 0. It is assumed that the higher the interval score for a criterion is, the better the player is judged to perform as regard to that criterion. Each player k has a right to choose two sets of nonnegative weights wk ¼ ðwk1 ; . . . ; wkm Þ to the criteria that are most preferable to the player. Using the weight wk, the relative scores of player k to the total score are defined as follows: Pm k l Pm k u i¼1 wi xik i¼1 wi xik P Pm k P P ; : ð1Þ n m n u k u l l w ð ðx þ x ÞÞ w ð ij i¼1 i j¼1 ij i¼1 i j¼1 ðxij þ xij ÞÞ The denominators represent the total score of all players as measured by players k weight selection. While the numerators indicates player k’s self-evaluation as a lower and upper bound and lower bound. Therefore, the statement (1) represents player’s k’s lower and upper relative importance in accordance to the xlik ; xuik . We assume that the weighted score are transferable. Player k wishes to maximize the ratios by selecting the most preferable weights wk for each of xlik ; xuik as follows: Pm k l i¼1 wi xik Max Pm k P ð2Þ n u l w i¼1 i j¼1 ðxij þ xij Þ s:t: Max s:t:
wki P 0; i ¼ 1; . . . ; m: Pm k u i¼1 wi xik Pm k P n u l i¼1 wi j¼1 ðxij þ xij Þ wki P 0;
ð3Þ
i ¼ 1; . . . ; m:
The motivation behind this problem is that the player k’s wishes to maximize his lower relative efficiency by (2), and upper relative efficiency by (3). As we can see in (2) the lower weighted sum of its record to the total weighted sum of all player records is maximized and in (3) the upper weighted sum of its record is maximized. Without loss of generality the following construct is added to the problem, in order to make the problem linear: n X ðxlij þ xuij Þ ¼ 1; i ¼ 1; . . . ; m: ð4Þ j¼1
We can do this with divide the xlij ; xuij to the will have the following linear problems: m X wki xlik cl ðkÞ ¼ Max
Pn
l j¼1 ðxij
þ xuij Þ. The problems (2) and (3) are changed by (4) and we
i¼1 m X
s:t:
wki ¼ 1;
ð5Þ
i¼1
wki
P 0;
cu ðkÞ ¼ Max s:t:
i ¼ 1; . . . ; m:
m X
wki xuik
i¼1 m X
wki ¼ 1;
i¼1
wki
P 0;
i ¼ 1; . . . ; m:
ð6Þ
96
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
After solving problems (5) and (6), if the optimal value of the problem (5) and (6) are cl(k) and cu(k) respectively, then cl(k) + cu(k) may be considered Pmas optimal value of the problem. Now, the problem is to maximize the objectives (5) and (6) on the simplex i¼1 wki ¼ 1. Apparently, the optimal solutions are given by assigning 0 1 to wkiðkÞ and wkiðk0 Þ for the criterion i(k) and i(k ), such that xliðkÞ ¼ Maxfxlik ji ¼ 1; . . . ; mg and xuiðk0 Þ ¼ Maxfxuik ji ¼ 1; . . . ; mg, respectively. Therefore, the optimal values will be as follows: cl ðkÞ ¼ xliðkÞ
and
cu ðkÞ ¼ xuiðk0 Þ ;
k ¼ 1; . . . ; n:
ð7Þ
The deterministic value cl(k) + cu(k) indicates the highest relative score of player k which is obtained by the optimal weight selecting behavior. The optimal weight wkiðkÞ and wkiðk0 Þ may differ from one player to another. Theorem 1 n X ðcl ðkÞ þ cu ðkÞÞ P 1:
ð8Þ
k¼1
Proof. Let the optimal weight for the player k be wlk ¼ ðwl1k ; . . . ; wlmk Þ and wuk ¼ ðwu1k ; . . . ; wumk Þ i.e. wliðkÞk ¼ 1 and wlik ¼ 0; i 6¼ iðkÞ and wuiðk0 Þk ¼ 1 and wuik ¼ 0; i 6¼ iðk 0 Þ. Then we have n n n n X m n X m n n X X X X X X X ðcl ðkÞ þ cu ðkÞÞ ¼ cl ðkÞ þ cu ðkÞ ¼ wlik xlik þ wuik xuik ¼ xliðkÞk þ xuiðk0 Þk k¼1
k¼1
P
n X
k¼1
xl1k þ
k¼1
n X
k¼1
i¼1
k¼1
i¼1
k¼1
k¼1
xulk ¼ 1:
k¼1
xliðkÞk
P xl1k and xuiðk0 Þk P xu1k and the last equality follows from the row-wise The inequality above follows from normalization. This proportion asserts that, if each player sticks to his egoistic sense of value and insists on getting the portion of the benefit as designated by cl(k) and cu(k), the sum of shares usually exceeds 1 and hence cl(k) + cu(k)cannot fulfill the role of division of the benefit. If eventually the sum of cl(k) and cu(k) turns out to be 1, all players will agree to accept the division cl(k) + cu(k), since this is obtained by most preferable weight selection. The latter case will occur when all players have the same and common optimal weight selection. More correctly we have the following theorem. h Pn l u Theorem 2. The equality k¼1 ðc ðkÞ þ c ðkÞÞ ¼ 1 holds if and only if our data satisfies the condition l l l u u x1k ¼ x2k ¼ ¼ xmk and x1k ¼ x2k ¼ ¼ xumk ; k ¼ 1; . . . ; n. That is, each player has the same score with respect to the m criteria. Proof. The ‘if’ part can be seen as follows. Since cl ðkÞ ¼ xl1k and cu ðkÞ ¼ xu1k for all k, we have n n n n n X X X X X ðcl ðkÞ þ cu ðkÞÞ ¼ cl ðkÞ þ cu ðkÞ ¼ xl1k þ xu1k ¼ 1: k¼1
k¼1
k¼1
k¼1
k¼1
xl11
> xl21 and xu11 > xu21 then there must be two columns The ‘only if’ part can be proved as follows. Suppose 0 l l u u h 5 1 and h 5 1such that x1h < x2h and x1h0 < x2h0 otherwise the second row sum can not attain 1. Thus we have cl ð1Þ P xl11
and
cl ðhÞ P xl2h > xl1h
cu ð1Þ P xu11 ; and
cu ðh0 Þ P xu2h0 > xu2h0 ;
cl ðjÞ P xl1j ; j 6¼ 1; h and
cu ðjÞ P xu2j ; j 6¼ 1; h0 :
Hence it holds that n n n n n n n X X X X X X X ðcl ðkÞ þ cu ðkÞÞ ¼ cl ðkÞ þ cu ðkÞ P xl1j þ xl2h þ xu2j þ xu2h0 > xl1j þ xu1j ¼ 1: k¼1
k¼1
k¼1
j¼1 j6¼h
j¼1 j6¼h0
j¼1
j¼1
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
97
This leads to a contradiction. Therefore, player 1 must have the same score in all criteria. The same relation must hold for the other players. In the above case, only one criterion is needed for describing the game and the division proportional to this score is a fair division. However, such situations might occur only in rare instances. In the majority of cases, we have n X ðcl ðkÞ þ cu ðkÞÞ > 1: k¼1
We may call this the ‘‘Egoist dilemma’’.
h
2.2. Assumption on the game and fair division The following statements are assumed among players in order to attain a fair division but we know that each player sticks to his superiority. Now the laws of game are as follows: (1) All players agree not to break off the game. (2) All players are willing to negotiate with each other to attain a reasonable and fair division z = (z1, . . . , zn). 2.3. Coalition with additive property Let the coalition S be a subset of players set N = (1, . . . , n). The record for coalition S is defined by X X xli ðSÞ ¼ xlij ; xui ðSÞ ¼ xuij ; i ¼ 1; . . . ; m: j2S
ð9Þ
j2S
These coalitions aim to maximize the outcomes cl(S), cu(S) as follows: m X wi xli ðSÞ; cl ðSÞ ¼ Max i¼1
s:t:
m X
wi ¼ 1;
i¼1
wi P 0; i ¼ 1; . . . ; m; m X cu ðSÞ ¼ Max wi xui ðSÞ;
ð10Þ
i¼1
s:t:
m X
wi ¼ 1;
i¼1
wi P 0;
i ¼ 1; . . . ; m:
ð11Þ
The cl(S) and cu(S), with cl(/) = 0 and cu(/) = 0, defines a characteristic function of the coalition S. Thus a game is represented by (N, c). Definition 1. A function f is called sub-additive, if for any S N and T N with S \ T = / the following statement holds: f ðS [ T Þ 6 f ðSÞ þ f ðT Þ: Definition 2. A function f is called super-additive, if for any S N and T N with S \ T = / the following statement holds: f ðS [ T Þ P f ðSÞ þ f ðT Þ:
98
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
Theorem 3. The characteristic function c is sub-additive, i.e. for any S N and T N with S \ T = / we have cl ðS [ T Þ þ cu ðS [ T Þ 6 cl ðSÞ þ cl ðT Þ þ cu ðSÞ þ cu ðT Þ:
ð12Þ
Proof. By renumbering the indexes, we have S ¼ f1; . . . ; hg;
T ¼ fh þ 1; . . . ; kg and
S [ T ¼ f1; . . . ; kg:
For these sets, considering the optimal solution as defined in (7), we have the following statement: cl ðS [ T Þ þ cu ðS [ T Þ ¼ max i
k X
xlij þ max i
j¼1
6 max i
h X
i
l
xuij
j¼1
xlij þ max
j¼1
l
k X
k X
xlij þ max i
j¼hþ1 u
h X
xuij þ max i
j¼1
u
¼ c ðSÞ þ c ðT Þ þ c ðSÞ þ c ðT Þ:
k X
xuij
j¼hþ1
Theorem 4 cl ðN Þ þ cu ðN Þ ¼ 1: Proof l
u
c ðN Þ þ c ðN Þ ¼
m X i¼1
wi
n X
xlij
þ
j¼1
m X
wi
i¼1
n X
xuij
m X
¼
j¼1
n X
wi
i¼1
! ðxlij
þ
xuij Þ
¼
j¼1
m X
wi ¼ 1:
i¼1
2.4. Another expression of the game Let us define another game (N, v) by X X vl ðSÞ ¼ cl ðjÞ cl ðSÞ and vu ðSÞ ¼ cl ðjÞ cu ðSÞ: j2S
ð13Þ
j2S
Theorem 5. (N, v) is supper-additive, i.e. vl ðSÞ þ vl ðT Þ þ vu ðSÞ þ vu ðT Þ 6 vl ðS [ T Þ þ vu ðS [ T Þ Proof
( l
l
u
u
v ðSÞ þ v ðT Þ þ v ðSÞ þ v ðT Þ ¼ ( ¼
X
) l
l
c ðjÞ c ðSÞ
j2S
X
for each S; T N and S \ T ¼ /: (
þ
u
c ðjÞ c ðSÞ
( þ
X
j2S[T
6
X
)
l
c ðjÞ c ðS [ T Þ
j2S[T
¼ vl ðS [ T Þ þ vu ðS [ T Þ:
u
X
cu ðjÞ fcu ðSÞ þ cu ðT Þg
j2S[T
! l
u
c ðjÞ c ðT Þ
cl ðjÞ fcl ðSÞ þ cl ðT Þg þ
X
l
c ðjÞ c ðT Þ
j2T
j2S
¼
) l
j2T
) u
X
þ
X j2S[T
! u
u
c ðjÞ c ðS [ T Þ
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
99
Also, we have n X
vl ðN Þ þ vu ðN Þ ¼
cl ðjÞ þ
j¼1
n X
cu ðjÞ fcl ðN Þ þ cu ðN Þg ¼
j¼1
n X
ðcl ðjÞ þ cu ðjÞÞ 1 P 0:
j¼1
Hence, the game (N, v) is 0-normalized. h 2.5. A minimum game The opposite side of the game can be constructed by (N, d) as follows: m X d l ðkÞ ¼ Min wki xlik i¼1
s:t:
m X
ð14Þ
wki ¼ 1;
i¼1
wki P 0; m X
d u ðkÞ ¼ Min
i¼1 m X
s:t:
i¼1 wki
i ¼ 1; . . . ; m:
wki xuik ð15Þ
wki ¼ 1; P 0;
l
i ¼ 1; . . . ; m:
u
The optimal value d (k) + d (k) assures the minimum division that player k can expect from the game. In this as a result of Theorem 1 we have Theorem 6 n X d l ðkÞ þ d u ðkÞ 6 1: k¼1
Proof n n n n X m n X m n n X X X X X X X ðd l ðkÞ þ d u ðkÞÞ ¼ d l ðkÞ þ d u ðkÞ ¼ wlik xlik þ wuik xuik ¼ xliðkÞk þ xuiðk0 Þk k¼1
k¼1
6
n X
k¼1
xl1k þ
k¼1
n X
k¼1
i¼1
k¼1
i¼1
k¼1
k¼1
xu1k ¼ 1:
k¼1
Analogously to the max game, for the coalition S N, we define m X d l ðSÞ ¼ Min wi xli ðSÞ i¼1
s:t:
m X
ð16Þ
wi ¼ 1;
i¼1
wi P 0; d u ðSÞ ¼ Min
m X
i ¼ 1; . . . ; m:
wi xui ðSÞ
i¼1
s:t:
m X
ð17Þ
wi ¼ 1;
i¼1
wi P 0; l
i ¼ 1; . . . ; m:
Apparently, it holds that d (N) + du(N) = 1.
h
100
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
Theorem 7. The min game (N, d) is supper-additive i.e. we have d l ðS [ T Þ þ d u ðS [ T Þ P d l ðSÞ þ d l ðT Þ þ d u ðSÞ þ d u ðT Þ
for each S; T N with S \ T ¼ /:
Proof. By renumbering the indexes, we have S = {1, . . . , h}, T = {h + 1, . . . , k} and S [ T = {1, . . . , k}. For these sets it holds that d l ðS [ T Þ þ d u ðS [ T Þ ¼ min
k X
i
xlij þ min i
j¼1
P min
h X
i
xuij
j¼1
xlij þ min i
j¼1
l
k X
l
k X
xlij þ min i
j¼hþ1 u
h X
xuij þ min i
j¼1
k X
xuij
j¼hþ1
u
¼ d ðSÞ þ d ðT Þ þ d ðSÞ þ d ðT Þ: Thus, this game starts from dl(k) > 0 and du(k) > 0 and enlarges the gains by the coalition until the grand coalition N with dl(N) + du(N) = 1 is reached. h Between the games (N, c) and (N, d) we have the following Theorem: Theorem 8 d l ðSÞ þ d u ðSÞ þ cl ðN n SÞ þ cu ðN n SÞ ¼ 1
for each S N ; S 6¼ N :
ð18Þ
Proof d l ðSÞ þ d u ðSÞ þ cl ðN n SÞ þ cu ðN n SÞ ! h h n n n n X X X X X X l u l u l l xij þ min xij þ max xij þ max xij ¼ min xij xij ¼ min j¼1
þ min min
j¼1 n X
j¼1 n X
xuij
j¼hþ1
!
n X
xuij
þ max
j¼hþ1
xlij min
j¼hþ1
n X
n X
j¼hþ1
xlij
j¼hþ1 n X
xuij þ max
j¼hþ1
j¼hþ1
þ max
n X
j¼1
xuij
j¼hþ1 n X
xlij þ max
¼ min
j¼hþ1
n X
xlij
þ
n X
j¼1
xuij ¼ 1:
! xuij
j¼1
j¼hþ1
Theorem 9 1 þ cl ðS \ T Þ þ cu ðS \ T Þ 6 cl ðSÞ þ cl ðT Þ þ cu ðSÞ þ cu ðT Þ for each S; T N with S [ T ¼ N : Proof. From the super-additivity of d( Æ ), we have the following equality: d l ðfS n ðS \ T Þg [ fT n ðS \ T ÞgÞ þ d u ðfS n ðS \ T Þg [ fT n ðS \ T ÞgÞ P d l ðS n ðS \ T ÞÞ þ d l ðT n ðS \ T ÞÞ þ d u ðS n ðS \ T ÞÞ þ d u ðT n ðS \ T ÞÞ: From Theorem 8, it holds that d l ðS n ðS \ T ÞÞ þ d u ðS n ðS \ T ÞÞ ¼ 1 fcl ðT Þ þ cu ðT Þg; d l ðT n ðS \ T ÞÞ þ d u ðT n ðS \ T ÞÞ ¼ 1 fcl ðSÞ þ cu ðSÞg; d l ðfS n ðS \ T Þg [ fT n ðS \ T ÞgÞ þ d u ðfS n ðS \ T Þg [ fT n ðS \ T ÞgÞ ¼ 1 fcl ðS \ T Þ þ cu ðS \ T Þg: Hence we have,
1 fcl ðS \ T Þ þ cu ðS \ T Þg P 1 fcl ðT Þ þ cu ðT Þg þ 1 fcl ðSÞ þ cu ðSÞg ; 1 þ cl ðS \ T Þ þ cu ðS \ T Þ 6 cl ðSÞ þ cl ðT Þ þ cu ðSÞ þ cu ðT Þ:
The other part will be proved similarly.
h
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
101
Theorem 10 1 þ d l ðS \ T Þ þ d u ðS \ T Þ P d l ðSÞ þ d l ðT Þ þ d u ðSÞ þ d u ðT Þ for each S; T N with S [ T ¼ N : 3. Extensions In this section, we consider a benefit–cost model to evaluate a kind of special case in which our data represent both superiority and inferiority of players. 3.1. A revenue-cost (R-C) game Up to now we consider only a special case in which our data represent only superiority of players. In this section, we represent a situation in which our data are supposed to consider both superiority and inferiority. To this end, consider s criteria for representing benefits and m criteria for costs. Let ½y lrj ; y urj ; r ¼ 1; . . . ; s and ½xlij ; xuij ; i ¼ 1; . . . ; m, be the benefits and costs of player j (j = 1, . . . , n), respectively. Now the upper and lower benefits of player j are evaluated by ðu1 y u1j þ þ us y usj Þ ðv1 xl1j þ þ vm xlmj Þ;
ð19Þ
ðu1 y lij þ þ us y lsj Þ ðv1 xu1j þ þ vm xumj Þ;
where u = (u1, . . . , us) and v = (v1, . . . , vm) are respectively, the virtual weights for benefits and costs. Analogous to expression (1), we define the relative scores of player j to the total scores as follows: Pm Ps u l r¼1 ur y rj i¼1 vi xij P Ps Pn P ; ð20Þ m n u i u l r¼1 ur k¼1 ðy rk þ y rk Þ i¼1 vi k¼1 ðxik þ xik Þ P Ps m l u r¼1 ur y ij i¼1 vi xij P Ps Pn P : ð21Þ m n u u l l r¼1 ur k¼1 ðy rk þ y rk Þ i¼1 vi k¼1 ðxik þ xik Þ Player j wishes to maximize his score, subject to the condition that the merit of all players is nonnegative. Therefore, we have s m X X ur y urk vi xlik P 0 ðk ¼ 1; . . . ; nÞ; ð22Þ r¼1
s X
i¼1
ur y lrk
m X
r¼1
vi xuik P 0
ðk ¼ 1; . . . ; nÞ:
ð23Þ
i¼1
We can express this situation by the linear programs below: s m X X ur y urj vi xlij cu ðkÞ ¼ max r¼1
s:t:
s X
i¼1
n s n X X X ur ðy urk þ y lrk Þ vi ðxuik þ xlik Þ ¼ 1;
r¼1
s X
k¼1
ur y urk
r¼1
s X s X
ur
n X
r¼1
s X
vi xlik P 0;
k¼1
k ¼ 1; . . . ; n;
m X
ð24Þ
vi xuij
ðy urk þ y lrk Þ
ur y lrk
ur P 0;
m X i¼1
k¼1
r¼1
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m:
vi P 0;
ur y lrj
r¼1
s:t:
i¼1
i¼1
ur P 0; cl ðkÞ ¼ max
m X
s X i¼1
vi xuik P 0;
vi
n X ðxuik þ xlik Þ ¼ 1; k¼1
k ¼ 1; . . . ; n;
i¼1
vi P 0;
r ¼ 1; . . . ; s; i ¼ 1; . . . ; m:
ð25Þ
102
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
Following the same manner as in the preceding sections, we can develop coalitions and imputations of this R-C game, although the row-wise normalization is not available in this game. That is, two characteristic function of the coalition S are defined by the linear programs below: s m X X X X cl ðSÞ ¼ max ur y lrj vi xuij s:t:
r¼1
j2S
s X
n X
r¼1 s X
ur
k¼1 m X
ðxlik þ xuik Þ ¼ 1;
k¼1
vi xuik P 0;
k ¼ 1; . . . ; n;
i¼1
s X
ur
y urj
m X
j2S
s X
i ¼ 1; . . . ; m; r ¼ 1; . . . ; s:
ur P 0; X
r¼1
ur
vi
X
ðy lrk þ y urk Þ
vi
n X ðxlik þ xuik Þ ¼ 1;
i¼1
ur y urk
r¼1
xlij ;
m X
k¼1 m X
ð26Þ
j2S
i¼1
n X
r¼1 s X
n X
vi
i¼1
ur y lrk
vi P 0;
s:t:
m X
ðy lrk þ y urk Þ
r¼1
cu ðSÞ ¼ max
j2S
i¼1
k¼1
vi xlik P 0;
k ¼ 1; . . . ; n;
i¼1
vi P 0;
i ¼ 1; . . . ; m; r ¼ 1; . . . ; s:
ur P 0;
ð27Þ
In the models (26) and (27) we kept the condition that the benefit of all players is nonnegative. Since the constraints of the programs (26) and (27) are the same for all coalitions, we have the following theorem: Theorem 11. The R-C max game satisfies a sub-additive property. Proof. For any S N and T N with S \ T = /,we have cl ðS [ T Þ þ cu ðS [ T Þ ¼ max
s X
v;u
X
ur
j2S[T
r¼1
¼ max
s X
v;u
ur
v;u
s X
ur
s X
v;u
ur
v;u
6 max v;u
ur
r¼1
þ max v;u
X
ur
s X r¼1
X
y lrj
y urj
y lrj
X j2S
X
m X
vi
u
vi
vi
y urj
m X
vi
xuij xlij
vi
þ
!
xuij
X
þ max
ur
X
!! xlij
m X
X
vi
j2T
i¼1
y urj
!! xuij
m X
vi
X
!! xlij
j2T
i¼1
j2T
s X
X
m X
X
þ max v;u
y lrj
X
ur
r¼1 s X r¼1
y lrj
j2T
ur
xlij
xuij
j2T
ur
!
j2S[T
r¼1
! xlij
X
s X
v;u
j2S
þ
vi
i¼1
!!
X
X
j2T
r¼1
!
j2S
i¼1
s X
þ
y urj
j2T
xlij
m X
j2S[T
j2S
j2T
X
xuij þ
X
!
X
ur
j2S
m X
j2S
¼ cl ðSÞ þ cl ðT Þ þ cu ðSÞ þ cu ðT Þ: l
X
m X
i¼1
vi
X
r¼1
X
i¼1
i¼1 m X
v;u
i¼1
! y urj
i¼1
X
j2S
ur
y lrj
s X
þ max
m X
j2T
j2S
r¼1 s X
þ
j2S
s X
þ max
y urj
j2S
r¼1
!
j2T
X
r¼1
¼ max
y lrj þ
xuij
j2S[T
X
j2S
!
X
vi
i¼1
X
r¼1
þ max
y lrj
m X
X j2T
vi
i¼1
y urj
m X i¼1
! xuij
j2T
vi
X
! xlij
j2T
It also holds that c (N) + c (N) = 1. We can also define the R-C minimum game, which satisfies a supper-additive property, and arrive at the following theorem:
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
103
Theorem 12. The R-C max game (N, c) and min game (N, d) are dual games, i.e. for any S N, we have d l ðSÞ þ d u ðSÞ þ cl ðN n SÞ þ cu ðN n SÞ ¼ 1: Proof cl ðN n SÞ þ cu ðN n SÞ ¼ max
s X
v;u
ur
s X
v;u
ur
s X
v;u
ur
s X
v;u
X
ur
v;u
¼ max v;u
ur
r¼1
s X
ur
ur
y lrj
max v;u
y urj
j2N
X
X
! y urj
m X
vi
v;u
m X
X
X
vi
vi
X
m X
vi
X j2N
m X X X ðy lrj þ y urj Þ vi ðxlij þ xuij Þ s X
ur
X
y lrj
j2S
j2N
m X
X
i¼1
vi
j2S
ur
y lrj
xlij
j2N nS
!! xlij
m X
vi
y uij
m X
j2S
X
vi
X
!! xlij
j2S
i¼1
!!
!! xuij
j2S
i¼1
ðxlij þ xuij Þ
max v;u
!
! xuij
X
X
ur
!
xuij
j2S
r¼1
i¼1
i¼1
X
s X
j2N
!
vi
j2S
r¼1
! xlij
xlij
X
i¼1
!!
X
j2N
y uij
j2S
s X
m X
j2N nS
xuij
X
! xuij
ur
j2N
m X
X
r¼1
i¼1
i¼1
ðy lrj þ y urj Þ
vi
i¼1
j2N
j2N
r¼1 l
i¼1
j2N
r¼1
y lrj
s X
þ max
m X
j2S
X
r¼1 s X
j2N
s X
þ max
y urj
j2N
r¼1
!
j2S
X
r¼1
¼ max
y lrj
xuij
j2N nS
X
j2N
!
X
vi
i¼1
X
r¼1
þ max
y lrj
m X
j2N nS
r¼1
¼ max
¼
X
max v;u
s X r¼1
ur
X
y urj
j2S
m X i¼1
vi
X
! xlij
j2S
u
¼ 1 ðd ðSÞ þ d ðSÞÞ: From Theorem 12, these two games are dual.
h
4. Numerical example We now apply this approach to some commercial bank of Iran. There are 20 branches in this district. Each branch uses 3 inferiority and 5 superiority criteria. Table 1 shows all kinds of these inferiority and superiority criteria. In Tables 2 and 3 the interval inferiority and interval superiority for these players are given, respectively. Also in Table 4, the results of approach are presented.
Table 1 Inferiority and superiority criteria Inferiority
Superiority
Payable interest Personnel Non-performing loans
The total sum of four main deposits Other deposits Loans granted Received interest Fee
104
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
Table 2 Inferiority criteria for the 20 branches of bank Playerj
xl1j
xu1j
xl2j
xu2j
xl3j
xu3j
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
5007.37 2926.81 8732.7 945.93 8487.07 13759.35 587.69 4646.39 1554.29 17528.31 2444.34 7303.27 9852.15 4540.75 3039.58 6585.81 4209.18 1015.52 5800.38 1445.68
9613.37 5961.55 17752.5 1966.39 17521.66 27359.36 1205.47 9559.61 3427.89 36297.54 4955.78 14178.11 19742.89 9312.24 6304.01 13453.58 8603.79 2037.82 11875.39 2922.15
36.29 18.8 25.74 20.81 14.16 19.46 27.29 24.52 20.47 14.84 20.42 22.87 18.47 22.83 39.32 25.57 27.59 13.63 27.12 28.96
36.86 2016 27.17 22.54 14.8 19.46 27.48 25.07 21.59 15.05 20.54 23.19 21.83 23.96 39.86 26.52 27.95 13.93 27.26 28.96
87,243 9945 47,575 19,292 3428 13,929 27,827 9070 412,036 8,638 500 16,148 17,163 17,918 51,582 20,975 41,960 18,641 19,500 31,700
87,243 12,120 50,013 19,753 3911 15,657 29,005 9983 413,902 10,229 937 21,353 17,290 17,964 55,136 23,992 43,103 19,354 19,569 32,061
Table 3 Superiority criteria for the 20 branches of bank Playerj
y l1j
y u1j
y l2j
y u2j
y l3j
y u3j
y l4j
y u4j
y l5j
y u5j
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2,696,995 340,377 1,027,546 1,145,235 390,902 988,115 144,906 408,163 335,070 700,842 641,680 453,170 553,167 309,670 286,149 321,435 618,105 248,125 640,890 119,948
3,126,798 440,355 1,061,260 1,213,541 395,241 1,087,392 165,818 416,416 410,427 768,593 696,338 481,943 574,989 342,598 317,186 347,848 835,839 320,974 679,916 120,208
263,643 95,978 37,911 2,29,646 4924 74,133 180,530 405,396 337,971 14,378 114,183 27,196 21,298 20,168 149,183 66,169 244,250 3063 490,508 14,943
382,545 117,659 503,089 268,460 12,136 111,324 180,617 486,431 449,336 15,192 241,081 29,553 23,043 26,172 270,708 80,453 404,579 6330 684,372 17,495
1,675,519 377,309 1,233,548 468,520 129,751 507,502 288,513 1,044,221 1,584,722 2,290,745 1,579,961 245,726 425,886 124,188 787,959 360,880 9,136,507 26,687 2,946,797 297,674
1,853,365 390,203 1,822,028 542,101 142,873 574,355 323,721 1,071,812 1,802,942 2,573,512 2,285,079 275,717 431,815 126,930 810,088 379,488 9,136,507 29,173 3,985,900 308,012
108634.76 32396.65 96842.33 32362.8 12662.71 53591.3 40507.97 56260.09 176436.81 662725.21 17527.58 35757.83 45652.24 8143.79 106798.63 89971.47 33036.79 9525.6 66097.16 21991.53
125740.28 37836.56 108080.01 39273.37 14165.44 72257.28 45847.48 73948.09 189006.12 791463.08 20773.91 42790.14 50255.75 11948.04 111962.3 165524.22 41826.51 10877.78 95329.87 27934.19
965.97 304.67 2285.03 207.98 63.32 480.16 176.58 4654.71 560.26 58.89 1070.81 375.07 438.43 936.62 1203.79 200.36 2781.24 240.04 961.56 282.73
6957.33 749.4 3174 510.93 92.3 869.52 370.81 5882.53 2506.67 86.86 2283.08 559.85 836.82 1468.45 4335.24 399.8 4555.42 274.7 1914.25 471.22
P P As we can see in example, nk¼1 ðcl ðkÞ þ cu ðkÞÞ > 1 and nk¼1 d l ðkÞ þ d u ðkÞ < 1.Therefore, in this example the ‘‘Egoist’s dilemma’’ has occurred and the players were not willing to negotiate with each other to attain a reasonable and fair division. 5. Conclusion In this paper, we have investigated a problem called ‘‘Egoist’s dilemma’’ with interval data in which some parts of optimization problem has been used. As we know in practical purposes, data are usually interval
G.R. Jahanshahloo et al. / Applied Mathematics and Computation 183 (2006) 94–105
105
Table 4 Sum of lower and upper results Playerj
c(j) = cl(j) + cu(j)
d(j) = dl(j) + du(j)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0.770328 0.139589 0.35728 0.285301 0.176518 0.37924 0.104281 0.498178 0.383411 0.973041 0.233868 0.185899 0.230978 0.160454 0.372076 0.223001 0.586103 0.08303 0.264289 0.083038
0.052617 0.013097 0.034481 0.013741 0.002732 0.019861 0.009084 0.030985 0.026538 0.005796 0.012743 0.008409 0.006985 0.004772 0.023202 0.010685 0.028867 0.001274 0.044835 0.004795
Sum
6.489905
0.355497
because of this the further study will be very useful in further cases. Also some properties of further problem have been extended in different Theorems in which some important qualities of dilemma have been introduced. An extension of problem, in which both superiority and inferiority criteria could be considered to players, has been discussed. Furthermore a numerical example, in which some commercial banks of Iran have been evaluated with proposed way, has been considered. Also, we propose to future researches, finding a practical algorithm to present a reasonable and fair division in all cases. Eventually, we hope that this study will be useful to future researches and will solve some practical difficulties in different fields. References [1] R. Baron, Jacques Durieu, Hans Haller, Philippe Solal, Complexity and stochastic evolution of dyadic networks, Computers and Operations Research 33 (2006) 312–327. [2] M. Davis, M. Maschler, The kernel of cooperative game, Naval Research Logistics Quarterly 12 (1965) 223–259. [3] G.R. Jahanshahloo, F. Hosseinzadeh Lotfi, M. Izadikhah, An algorithmic method to extend TOPSIS for decision-making problems with interval data, Applied Mathematics and Computation, Forth Coming. [4] I.V. Konnov, Splitting-type method for systems of variational inequalities, Computers and Operations Research 33 (2006) 520–534. [5] K. Nakabayashi, K. Tone, Egoist’s dilemma: a DEA game, Omega the International Journal of Management Science 34 (2006) 135–148. [6] M. Parlar, Z. Kevin Veng, Coordinating pricing and production decisions in the presence of price competition, European Journal of Operational Research 170 (2006) 211–227. [7] A. Shapley, On balanced sets and cores, Naval Research Logistics Quarterly 14 (1967) 453–460. [8] H. Young, N. Okada, T. Hashimoto, Cost allocation in water resources development, Water Resources Research 18 (1981) 463–475. [9] G. Zaccour, Special issue on game theory: Numerical methods and applications, Computers and Operations Research 33 (2006) 285– 297. [10] J. Zhu, Multidimensional quality-of-life measure with an application to fortune’s best cities, Socio-Economic Planning Sciences 35 (2001) 263–284.