On polytope of (0-1)-normal big boss games: redundancy and extreme points Текст научной статьи по специальности «Математика»

On Polytope of (0-1)-normal Big Boss Games: Redundancy and Extreme Points

Alexandra B. Zinchenko

Southern Federal University,

Faculty of Mathematics, Mechanics and Computer Science,

Milchakova, 8”a”, Rostoov-on Don, 344090, Russia E-mail: zinch46@gmail.com

Abstract The system of non redundant constraints for polytope of monotonic (0-1)-normal big boss games is obtained. The explicit representation of some types of extreme points of this polytope as well as the corresponding Shapley and consensus values formulas are given. We provide the characterization of extreme elements of set of such monotonic (0-1)-normal big boss games that all weak players are symmetric.

Keywords: cooperative game, big boss game, (0-1)-normal form, extreme points, Shapley value, consensus value.

1. Introduction

The class of big boss games was introduced to model economic, social and political situations in which one of the participants has a greater possibilities (power) than others (see, for example (Hubert and Ikonnikova, 2011)), (Tijs, et al., 2005) (O’Neill, 1982), (Tijs, 1990), (Aumann and Maschler, 1985), (Branzei, et al., 2006)). In (Muto, et al., 1988) the big boss games as well as strong big boss games were determined by means of three conditions: monotonicity, boss property and union property. Later appeared the work (Tijs, 1990) in which monotonicity condition was replaced by nonnegativity of characteristic function and marginal vector. The general (Branzei and Tijs, 2001) and total (Muto, et al., 1988) big boss games were also introduced. All types of big boss games are extensively studied. Moreover, the results received for clan games (Potters et al., 1989) are applicable to big boss games because the cone of each type of big boss games is a subset of cone of corresponding clan games. One of cooperative game theory problems is the characterization of extreme directions of polyhedral cones of various classes of games and description the behavior of solution concepts defined on these cones (Tijs and Branzei, 2005). The extreme directions of cone of non-monotonic clan games were described in (Potters et al., 1989). If the clan consists of one player these and only these directions define the cone of non-monotonic big boss games. To our knowledge the extreme elements of set of monotonic big boss games are not yet characterized.

Since big boss games can be converted to (0-1)-normal form without changing their essential structure and the most solution concepts satisfy on this class games the relative invariance with respect to strategic equivalence, this paper focus on (0-1)-normalized big boss games. At normalization the cone of monotonic big boss games will be transformed to (2n-1 — 2)-dimensional polytope Pn which can be described by its extreme points. From Theorem 4.1 in (Potters et al., 1989) it follows that only simple games are the extreme points of polytope of nonmonotonic (0-1)-normal big boss games. But for Pn this is hot true.

The paper has the following contents. Next section recall the facts of cooperative game theory which are useful later. The system of non-redundant constraints for Pn is described in Section 3. Section 4 is denoted to extreme points of Pn and their Shapley and consensus values. The characterization of extreme elements of set of monotonic (0-1)-normal big boss games with symmetric weak players is given in last section.

2. Preliminaries

A cooperative TU-game is a pair (N,v) where N = {1, 2,...,n} is a player set and v € Gn = {g : 2N ^ R | g(0) = 0} is a set function. Often v and (N, v) will be identified. A subset of N is called a coalition and v(S) is the worth of coalition S. A vector x € Rn is called an allocation. For any S € 2N and x € Rn let x(S) = ies Xi and x(0) = 0. Two players i,j € N are symmetric in (N,v) if

v(S U i) = v(S U j) for every S C N \{i, j}. We say that players of coalition S with IS I > 2 are symmetric in (N, v) if each pair of players of the coalition is symmetric in (N, v). Denote by M(v) € Rn the marginal vector (marginal) of game v € Gn,

i.e. Mi(v) = v (N) — v (N \ i), i € N .A TU-game is called:

• monotonic if v(T) < v(H) for all T C H C N,

• simple if v(S) € {0,1} for all S C N and v(N) = 1,

• (0-1 )-normal if v(N) = 1 and v(i) = 0 for all i € N,

• essential if ^n v(i) < v(N),

• clan game with nonempty coalition CLAN as clan (Potters et al., 1989) if: v > 0 and M(v) > 0,

v(S) = 0 if CLAN C S, v(N) — v(S) > £iEN\s Mi(v) if CLAN C S.

Later we need formulas for the Shapey value Sh (Shapley, 1953), the equal surplus division solution E and the consensus value K (Ju, et al., 2006). These values

are given by

Sh,M= n ps(HSu,)-HS)), w_PW»-W-m,

n!

S:i£S

17/ \ _ Ar JS, ^ E{v)+Sh{v)

Ei{v) = v{i) H------J-----, K{v) = ---.

n2

For simple game the Shapley value formula boils down to

Shi(v) = ^ ps, i € N,

SCRi

where Ri = {S C N \ i : v(S) = 0, v(S U i) = 1}. For (0-1)-normal game the consensus value is determined by

1 Shi(v)

KAv) =----1------, i <E N,

K ’ 2n 2 ’ ’

because Ei(v) = i G N.

For any set G C Gn a value on G is a function $ : G ^ Rn which assigns to

every v € G a vector $(v), where $i(v) represents the payoff to player i in v. We

shall use two axioms to be satisfied by $(v).

Efficiency: ieN $i(v) = v(N) for all v € G.

Symmetry: for all v € G and every symmetric players i,j € N, $i(v) = 4>j(v). Known that the Shapley value and the consensus value satisfy this axioms. The core (Gillies, 1953) of game v € Gn is a bounded polyhedral set (polytope) C(v) = {x € Rn : x(N) = v(N), x(S) > v(S), S C N}. The sets of integer and noninteger extreme points of polytope P will be denoted by extj(P) and extNI(P) respectively. The cardinality of set S is written as |S| . The rank of matrix A is denoted as rank(A).

3. Minimal test for big boss game

A game v € Gn (n > 3) is called a big boss game with player 1 as big boss (Muto, et al., 1988) if:

(a) v is monotonic,

(b) v(T) = 0 for all T C N with 1 € T (boss property),

(c) v(N) — v(T) > ieN\T Mi(v) for all T C N with 1 € T (union property). Inessential games are not interesting. Any essential game has the unique (0-1)-

normal form. Denote by Pn the polytope of all monotonic (0-1)-normal big boss games with player 1 as big boss. This set is determined by

v(N) = 1, v(T) = 0 whenever 1 / T ore T = {1}, (1)

v(H) > v(T), T C H C N, (2)

—v (T)+ ]T v (N \ i) > n —IT | —1, T 3 1, T C N. (3)

ieN\T

The following lemma shows that (1)-(3) is equivalent to a restricted system. Lemma 1. Let v € Gn . Then v € Pn iff it satisfies (1) and conditions

v(H) > v(T), T 3 1, T C H C N, 1 <ITI = IH| —1, IHI = n — 1, (4)

—v (t )+ E v (N \ i) > n —IT I —1, T 3 1, IT I<n — 2, T C N. (5)

ieN\T

Proof. If v Pn then it satisfies (1),(4),(5) because the system (2)-(3) contains all inequalities in (4)-(5). Now take v satisfying (1),(4),(5). The constraints in (3) corresponding to T = N and T = N \ i, i € N \ 1, are trivial. So, the systems (3) and (5) are equivalent. Obviously V satisfies (2) for coalitions T = 0 and H = {i},

i € N. Hence, it suffices to show that V satisfies (2) for following pairs of T and H.

1. T 3 1, T C H C N, IH I = n — 1, IT I = n — 2. The inequalities in (2)

corresponding to such coalitions are the form

v (N \ k) > v (N \{k,e}, k € N \ 1, e€N \ {1, k}. (6)

From (5) follows —V(N \ {k, e}) + V(N \ k) + V(N \ e) > 1. Due to (1) and (4) it

holds that 1 > V(N \ e). The summing of two last inequalities gives that V satisfies (6).

2. T 3 1, T C H C N, IH I > IT I + 1. The corresponding inequalities in (2) are satisfied for V since the binary relation ” > ” is transitive.

3. T 3 1, T C H C N. From case 1 we know that V satisfies the inequalities in (2) for such T and H that T 3 1, T C H C N, IT I = IHI — 1. Together with (1) this implies that V(S) > 0 for all S C N. Because V(S) = 0 for all S 3 {1}, we obtain that V satisfies (2). □

A constraint in a linear system is called redundant if the removal of this constraint from the system does not affect the feasible region. Next theorem provides the system of non-redundant conditions for Pn.

Theorem 1. The system (1),(4),(5) is non-redundant.

Proof. Let v € Gn be given by

r 5 3 1, \S\ <n-2,

i>(s) = m, S3i,\s\>n-i,

^ 0, otherwise,

for all S C N. Obviously v € Pn and v is the interior point of system (4),(5) feasible region. None of constraints in the system (1) is implied by the others because they are linear independent. The table 1 contains such games v € Gn that satisfy (1), (4), (5) except the unique constraint (corresponding to coalitions given in the first column). Vectors v 1-v3 do not satisfy one of inequalities in (4) and for v4-v5 one of inequalities in (5) is violated. It is assumed that T C H C N, IT I = IH I — 1 and S N.

Corollary 3.1. The polytope Pn is (2n 1 — 2)-dimensional.

Proof. The number of constraints in (1) is 2n-1 + 2. Using the fact that Pn C R2™ and non-redundancy the system (1),(4),(5) we obtain dim(Pn) = 2n-1 — 2. □

Corollary 3.2. P3 and P4 are the integral polytopes.

Proof. Every v € Pn is (0-1)-normal monotonic clan game with CLAN = 1. But in cases P3 and P4 the monotonicity conditions (4) are transformed in bounds on variables: v(1, i) > 0, v(N \ i) < 1, i € N \ 1. Theorem 4.1 in (Potters et al., 1989) implies that P3 and P4 have only integer extreme points. □

We have calculated all extreme points of P5 and partitioned the set extNI(P5) into seven equivalence classes. The representatives of these classes are given in Table 2. Each class contains such games that differ only the numbers of players from N \ 1. Note that (0-1)-normal form of 5-person game given in counterexample

4 in (Potters et al., 1989) coincides with &-1(V1).

4. Extreme points of polytope P”

Since Pn is contained in the unit hypercube, the simple games belonging to Pn are its integer extreme points. To make the following analysis simple, consider the polytope Pn determined by

v (T) > 0 if T € Q and IT I = 2, v (T) < 1 if T € Q and IT I = n — 1, (7)

v (H) > v (T) if T,H €Q, T C H, 2 <IT I = IH I — 1, IH I<n — 2,

(8)

Tablel: Non big boss games

Fixed coalitions

Games

T 3 1, \H \<n - 2, vl(S)

' 1, (T Ç S) A (S = H) V (\S\ = n - 1) A (S = N \ 1),

0, otherwise.

T3 1, H = N,

1, S = T, \S\ = n - 1, S = N \ 1 v2(S) = { 2, S = T,

0, otherwise.

T =1, \H\ = 2,

1, S = n - 1, S = N \ 1 3(S) = { -1, S = H,

0, otherwise.

T 3 1, 2 <\T \<n - 2, v4(S)

T = {1},

\S\

5(S) =

n- 1’

0,

|S1 -1

n- -1

1,

n- 3

n-2

0,

S = T,

S 31,

, otherwise.

\ S\ = n,

S 3 1, \S\ = otherwise.

Table2: Types of non integer extreme points of P5

{1,2} {1,3} {1,4} {1,5} {1 ,2,3} {1 ,2,4} {1 ,2, 5} {1,3,4} {1,3,5} {1 ,4,5} N\5 N\4 N\3 N\2

1 2 1 2 0 0 1 2 1 2 1 2 1 2 1 2 0 1 1 1 2 1 2

7/2 1 1 1 0 0 1 1 1 1 1 1 1 1 2 2 2

£/ 3 3 3 3 3 3 3 3 3 3 3 3

7/3 1 1 1 0 0 2 1 1 1 1 1 1 1 2 2 2

£/ 3 3 3 3 3 3 3 3 3 3 3 3

4 1 1 1 1 1 1 1 1 1 1 3 3 3 4

is 4 4 4 4 4 4 4 4 4 4 4 4 4 4

5 1 1 1 1 1 1 1 1 1 1 3 3 3 4

I/ 4 4 4 4 2 4 4 4 4 4 4 4 4 4

T/6 1 1 1 1 1 1 1 1 1 1 3 3 3 4

£/ 4 4 4 4 2 2 4 4 4 4 4 4 4 4

i/7 1 1 1 1 1 1 1 1 1 1 3 3 3 3

£/ 4 4 4 4 2 2 4 2 4 4 4 4 4 4

i/

y

v(N \ i) > n -\T | — 1), T e Q, \T \<n - 3, (9)

ieN\T

v(N \ i) > n — 2,

ieN\1

(10)

where Q = {5 e 2N : 2 < \S\ < n — 1, S 3 1}. The system (7)-(10) is obtained from (1),(4),(5) by elimination the variables which have constant value over Pn. Moreover, the monotonicity condition (4) are decomposed into three parts (in order to select upper and lower bounds on variables). The inequality corresponding to coalition T = {1} was selected from system (5). Thus, the polytope Pn is contained in the (2n-1 — 2)-dimensional Euclidean space whose coordinates refer to the coalitions S e Q. Theorem 1 and Corollary 1 imply that dim(Pn) = dim(Pn), i.e. the polytope Pn is full-dimensional. So, the system (7)-(10) is the unique non-redundant system which specifies Pn. The polytopes Pn and Pn are combinatorially equivalent since there is one-to-one map & : Pn ^ Pn saving the adjacency of faces. For any v e Pn the vector &(v) = (v(S))sen is the restriction of vector (v(S))Se2N to those coordinates which correspond to S e Q. Conversely, having v e Pn we obtain the game &-1(v ) = v e Pn by adding values v(N) and v(S), S e 2N \ Q, determined by (1). The following theorem describes some elements of extNI(Pn).

Theorem 2. Let n > 5; (i2,...,in) be an ordering on N \ 1, L = (1,i2,..., i;),

2 < l < n — 2 and for all S e Q

0(S ) =

n - 2 - ) n- 1’

n1

\S \ = n — 1,

otherwise,

vL (S) = <

1, \S\ = n — 1, L <£ S,

n~|L)~1, \S\=n-l, LCS, n —|L|

0, \S\ < n — 1, S C L,

otherwise.

n —|L|

Then v = & 1(v0) and v = & (v ) are the extreme points of P

Proof. Let us prove that v0 e ext(Pn). The system (7)-(10) contains d = 2n-1 — 2 variables. The subsystem (8) contains only (d — n + 1) of them and its matrix is the incidence matrix of connected graph in which the set of vertices equals the set of such coalitions S e Q that \S \<n — 2. The rank of this matrix is (d — n). Choose (d — n) linear independent constraints in (8) and denote by O the set of associated pairs of coalitions T and H. The system

v(H) = v(T), (T,H) e O, (11)

—v(T) +$3 v(N \ i) = n — 3,T e Q, \T\ = 2, (12)

ieN\T

^2 v(N \ i) = n — 2, (13)

ieN\1

contains d variable as much as equations. The elimination (d — n) variables from

(11) and substitution them in (12)-(13) gives the system Av = b where A is square

matrix of dimension n, b = (n — 3,..., n — 3, n — 2) e Rn. By transposition of columns and rows the matrix A can be represented in the form

eTn-1 D

0 en-1

where en-1 = (1,..., 1) is the (n — 1)-dimensional row vector and D is the square matrix of dimension (n — 1) with dij = 0 if i = j, dij = 1 if i = j. Vector v0 is the solution of system (11)-(13) and it is unique because rank(A) = rank(D) + 1 = n. It is easy to see that v0 e Pn. Hence,

v0 e ext(Pn) =^ v0 e ext(Pn).

Analogously one proves that vL(S) e ext(Pn). □

Propositions 1, 2 (below) provide the explicit Shapley and consensus values representation for some integer and noninteger extreme points of Pn.

Proposition 1. Let vk is determined for all k e {2, ...,n — 1} and S e Q by

vk (S) = J 1 \ S \ > k,

1 0, otherwise.

Then vk = &-1(vk) e exti(Pn) and

Kiiyk) = shM) =

SWS)- f"1,,, i€N\l.

2n(n — 1) n(n — 1)

Proof. Fix k e {2,...,n — 1}. The vector vk obviously satisfies (7)-(8). It also satisfies (9)-(10) since vk (N \ i) = 1, i e N \ 1. So, vk e Pn. This implies that vk e Pn. Further, vk e extl(Pn) because it is a simple game. Take i* e N \ 1. Then TU, = {S C N \ i* : |5| = k — 1} and ps = (fc-1)^w-fc)! for all S e Ri>. The substitution ps and R* in the Shapley value formula for simple game gives

Shi*{vk) = ps\Ri*\ = Ps (fc-2) = n(n-i) • ^ weak players are symmetric in vk. By Symmetry Shj(vk) = Shj** (vk), j e N \ {1,i*}. From Efficiency follows Shi(vk) = v(N) — J2ieN\i Shi(vk) = 1 — = ”~k+1. The consensus value of

game vk is defined by formula for (0-1)-normal games. □

Proposition 2. The Shapley and consensus values for game v0 e extNi(Pn) determined in Theorem 2 are

sw„°) = ^4, *-,(„«)= 4“-7

(n — 1) ’ 2n(n — 1) ’

o 2 n2 -6 n + 1

(n- l)2 ’ A J 2n(n — l)2

, 0. n2 — 4n + ^ ^ 2n2 — 6n + 7 .

Shiiy ) = — -—2-, Ki(v ) = —— --------------------------for ieN\ 1.

Proof. Take i* e N \ 1. Then for all S C N

-iT, S' ={1} or S':

z/°(SUi*) -z/°(S) = { *5?, S 3 1, |S| = n — 2

n-

0,

N \ i*,

S ^ i*,

otherwise.

The number of coalitions satisfying S 3 1, \ S \

PS

for such S. Further,

n(n— 1)

value formula we obtain Shi* (v0' Symmetry and Efficiency Sh1(v0)

P{t} - n(n-l) > Pn\

n — 2, S ^ i* is (n — 2) and —. By using the Shapley

’ n(n— l)2

i n2 —5n+7

n(n— 1)

in consensus value formula gives K (v0).

1 , (n—3)(n—2) , 1 __ n2 —4n+6 o-,.

“T n(n-l)2 “T n(n-l) — n(n-l)2 • ß-y

3n—6

%(n— 1)

□

The core C(v) = {x e Rn : x(N) = v(N), 0 < xi < Mi(v), i e N \ 1} of each game v e Pn is determined by marginal vector only (Muto, et al., 1988). So, all games in Pn having identical marginals have the same core. If C(v) is a singleton, i.e. C(v) = {xc}, then the bargaining set (Aumann and Maschler, 1964), kernel (Davis and Maschler, 1965) and lexicore (Funaki, et al., 2007) coincide with xc. Moreover, any core selector (for example, nucleolus (Schmeidler, 1969), t — value (Tijs, 1981), AL—value (Tijs, 2005)) coincides with xc. Thus, xc should reflect many principles of fairness. However, for games with zero Mi(v), i e N \ 1, we have xc = (1, 0,..., 0). According to xc the entire unit of surplus is allocated to player 1 (boss) that ignores the productive role of other players. Such games are in particular k determined in Proposition 1 and all games in their convex hull and also all games in Gn having corresponding (0-1)-form. At the same time, by formulas from Proposition 1 we obtain different consensus and Shapley values. For example, take two 6-person games v2 and v5. Then

,1 1

1 3

2 10

10

4 20

20

Thus, for v e co({vk}n=2) the consensus and Shapley values prescribes a rather natural outcomes.

1

5. ¿-symmetrical big boss games

We name a game v e Pn l-symmetric if each pair of powerless players i, j e N\ 1 is symmetric in v. Denote by SPn the class of l-symmetric games v e Pn. Let X be the set of all (0,1)-vectors x = (x2,... ,xn-2), s = \S\, J = {2, ...,n — 2} and v = &(v) whenever v e SPn. Next theorem characterizes extreme points of SPn. It shows also that all non-integer extreme points belongs to (2n — n — 1)-dimensional face

n2

{v G SP" : v(S) =-----, S 3 1, s = n — 1}

n1

and are in one-to-one correspondence with elements of X.

Theorem 3. Let v e GN. Then

(i) v e extNi(SPn) iff there is such x e X that v = vx = &-1(vx), where vx (S) = fX for all S e Q and

{o, s = 2, x2 = 0,

s = n-l,

fX-1, s > 3, xs = 0,

~—7, Xs = 1, n—1’ s ’

(ii) v e exti(SPn) iff there is such k e {2, . ..,n — 1} that v = vk, where vk was determined in Proposition 1.

Proof, (i) Suppose v e SPn and v = &(v). Then v(S) = f (\S\) = f s, S e Q. So, system (7)-(10) takes the form

/2>0, BEj < fn-1 < 1,

fs-i < fsi s G J \ 2,

fs - (n - s)f„-i < s + 1 - n, s G J.

Let Fn be the polytope specified by this system. Take f e Rn 2 with

s = n- 1,

n ’ ’

se J.

n ’ G

Since Fn C Rn 2 and f is the interior point of Fn then dim(Fn) = \ J\ = n — 2. For each x e X, fx e Fn and satisfies following (n — 2) equations

/2=0 if x2 = 0, fn—l =

fs = */ ^ /s-1 = /s */ S > 3 and Xs =0.

Obviously, system (14) has the unique solution. This implies

fx e extNi(Fn) =^ vx e extNi(Pn) =^ vx e extNi(Pn).

(14)

To prove inverse, take f' e extNl(Fn). We shall show in the beginning that f'n_ 1 Suppose

Suppose < fn_i < 1. For each s£ J \ 2 denote

ks = max{k G J : f < f's}

if there is such k G J that f'k < f's. From monotonicity conditions follows that ks < s. Obviously, exists 5 > 0 satisfying the inequalities

S < fn-1---------7, S <1 - fn-l,

n - 1

f ' f ' — f '

S<-^ for f' > 0, S<^-^ forf's>f'ks, seJ. n — s s — ks s

On Polytope of (0-1)-normal Big Boss Games Consider vectors f —, f+ determined by

f2-

0, f2 = 0,

f2 — (n — 2)S, f2 > °

f+ f0, f2 = 0,

f2 I f2 + (n — 2)S, f2 > 0,

fn

n-1,

n 1,

f — =< f' — (n — s)S, f' f 1, f+ =

fn — V s = n — 1,

f ' + (n — s)S f ' > f's-^

1,

f' = f J S j i

1,

s G J. From the definition of S and the fact that > n s 1 for all s G J, follows

n-1 n-

< fn-1 - se J. Moreover, /' - (n - s) fn—l < s + 1 - n if /' = /;_! > 0,

s e J \ 2, because otherwise we have

f ' — (n — s)fn—1 = s + 1 — n,

f — 1 — (n — s + 1)fL—1 < s — n f' = f' ,

J s J s — 1l

fn' 1 > 1,

which contradicts the assumption f^_ 1 < 1. Tables 3,4 show that f , f + e Fn.

The equality /' = ^ implies /' ^ ext(F"). Thus, all non-integer extreme points of Fn belongs to its facet determined by constraints

n 2 s 1

fn-1 = -----7, /2 > 0, /s_i < fs if s £ J\ {2}, /s < ---------- if s e J. (15)

n1

n1

Since the constraints matrix of system (15) is totally unimodular and f' e ext(F"), the values f's, s £ J\(n — 1), can be equal to 0 or for s = 2 and or fs—1 for s e J \ {2}, i.e. f' must be coincides with f% for some x e X.

Item (ii) is proved analogously.

Table3: Representation f through f'.

Cases

Is-1

fs - (n - S)fn-1

s = 2, f2 = 0,

0

s =2, f2 > 0, f2 — (n — 2)S,

s > 3,f = f — 1 = 0, 0

s > 3,f' = f'—1 > ° fks — (n — ks)S,

s > 3,f' > f's-1 = ° f' — (n — s)S,

f -f -

— (n — 2)(fn—1 — S),

f2 — (n — 2)f'n—l,

—(n — s)(fn—1 — S),

fks — (n — s)fn — 1 + (s — ks)S,

f' — (n — s)fn

n— 1,

> 3,f' > f1 > 0, f' — (n — s)S, fks — (n — ks)S,

f' — (n — s)fn

n1

f

□

Table4: Representation f + through f'.

Cases /+ ft i ft — (n — s)fnLi

* = 2, f2 = 0, 0 - -(n - 2)(fn_i + 5),

* = 2 f2 > 0 f2 + (n - ■ 2)5, - f2 - (n - 2) fn — 1,

* > 3,f = f— = 0, 0 f - (n - *)(fn-i +5),

* >3,fs = f's-1 > 0 fks +(n - ks )5, f - fks - (n - *)fii-1 - (* - ks)5,

* > 3,f > f 1-1 = 0 f I + (n - *)5, 0 fs - (n - s) fn—1,

S > 3Js > f's-1 >°, fs + (n~ s)6, f'k. + (n~ hs)5, f's - (n - s)fh_ 1

References

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