A generalized model of hierarchically controlled dynamical system Текст научной статьи по специальности «Математика»

Guennady A. Ougolnitsky

Southern Federal University,

Department of Applied Mathematics and Computer Sciences, Faculty of Mathematics, Mechanics, and Computer Sciences, Milchakov St., 8A, Rostov-on-Don, 344090, Russia E-mail: ougoln@mail.ru

Abstract The idea of the paper is to combine in the same model several concepts from game theory, graph theory, and controlled dynamical systems theory, namely: 1) a directed graph without contours and loops; 2) a game of n players in normal form; 3) a cooperative game; 4) a dynamical system controlled by several subjects. The combination permits to describe complex dynamical systems with hierarchical structure (particularly organizational and environmental systems) more completely and to take into consideration different interactive and interdependent aspects of the systems. Some examples are considered such as environmental control, corruption modeling, and linear multistage games with hierarchical matrices.

Keywords: hierarchical game theory, directed graphs, controlled dynamical systems.

1. Introduction

Management deals with organizational systems. The following characteristics (at least) are immanent for those systems.

1. Hierarchy. The hierarchical structure is generated by the relation of subordination between employees and departments in the organization.

2. Conflicts. The conflicts are conditioned by presence of many employees and departments with different interests using limited resources and aiming distinct goals.

3. Coalitions. The organizational coalitions are both official departments and other unions and different groups having common interests.

4. Dynamics. The state of organization changes in time due to managerial efforts and external impact.

These characteristics are studied by different disciplines and described by different mathematical models, as a rule separately. The idea of the paper is to combine in the same model several concepts from game theory, graph theory, and controlled dynamical systems theory, namely: 1) a directed graph without contours and loops; 2) a game of n players in normal form; 3) a cooperative game; 4) a dynamical system controlled by several subjects.

The combination permits to describe complex dynamical systems with hierarchical structure (particularly organizational and environmental systems) more completely and to take into consideration different interactive and interdependent aspects of the systems. Some examples are considered such as environmental control, corruption modeling, and linear multistage games with hierarchical matrices.

2. The Model

A generalized model of hierarchically controlled dynamical system may be represented as follows:

where N = is a set of players; A is a binary relation of hierar-

chy on N ; S is a set of coalition structures on N, s G S : s = {Ki,...,Km}, Ki U ... U Km = N, Ki n Kj = 0, Ki,..., Km C N ; C is a controlled dynamical system, C: x4 = xt-1 + f (x4, Uf,..., Ufn) , x0 = x0, t = 1,..., T; Ui is a set of strategies of the i-th player; Ji : U1 x U2 x ... x Un ^ IR is a payoff function of the i-th player.

The following properties are supposed to be fulfilled:

P1—hierarchy: the binary relation A is a strict order relation;

P2—stratification: each coalition structure s G S is ordered;

P3—economic rationality: each player i G N tends to maximize Ji.

Let’s define a Neumann-Morgenstern characteristic function:

where uK is a set of strategies of the players from K, uN\K is a set of strategies of the players from N \ K.

The model (1) together with the characteristic function (2) allows to combine four known concepts:

1. a directed graph without contours and loops D = (N, A) with additional set of ordered structures of the vertices S which characterizes a hierarchical structure of the system;

2. a game of n players in normal form G = (N, {U}iGN , {Jwhich represents the system as a set of independent rational individuals having conflict and searching for compromise;

3. a cooperative game rv = (N, v) which allows to describe coalitions, united actions and rational imputations;

4. a dynamical system controlled by several subjects inside and beyond the organization xl = + f (x4, Ut,..., ), xo = x0.

3. An Environmental Interpretation

The following interpretation of the model (1) for a problem of the sustainable development of a simple ecological-economic system is possible: N = {L, F}, where L (Leader) is an environmental protection agency, F (Follower) is an enterprise; A = {(L, F)} defines the administrative and economic dependence of F from L;

S = {Si,S2} ,S1 = {{L} , {F}} - isolated behavior, = {{L, F}} - cooperative behavior of L and F; C is defined by (5):

H = <N,A,S,C, , MW

(1)

v (K) = max

max mm

UrGUk un\k£Un\k

(2)

jl = Y1 [sL (pi,qi,ui,xi) - Mp(x*,XL)] ^ t g1 ax (3)

t—1 pt^Pt,qt EQt

T

JF = Y1 Sf (pi,ui,xi) ^ tmraxt, (4)

t—1 ut^U (q1)

x* = x*-1 + f (x*-1 , u*) , xo = x0, t = 1,...,T (5)

where Q is a set of Leader’s administrative strategies; P is a set of Leader’s economic strategies; U is a set of Follower’s strategies (environmental impacts); JL is Leader’s payoff function considering the sustainability requirement x* € XL; Mp (x*,XL) is a penalty function; JF is Follower’s payoff function; x* is a state vector of the ecological-economic system in the moment t; x0 is a known initial state. So, UL = {Q,P}; UF = U; JL, JF are defined above as (3), (4).

Methods of management (compulsion, impulsion, conviction) which permit Leader to provide sustainability x* € XL, t = 1,..., T, are formalized as Stackelberg equilibriums of a special form for a hierarchical game of L and F. The equilibriums formalize administrative, economic, and psychological methods in management respectively (Ougolnitsky, 2002; 2004).

4. Corruption Modeling

The phenomenon of corruption is intensively studied in the past two decades (Klitgaard, 1991; Bac, 1996; Bardhan, 1996; Rose-Ackerman, 1997). The principal-agent-client model may serve as a basic one for the corruption investigation (Levin and Zirik, 1998).

So, corruption is closely connected with hierarchy. In particular, corruption arises in the hierarchical ecological-economic systems. That’s why we introduce corruption in the model (3)-(5) in its derivative form with implicit description of dynamics (Ougolnitsky, 2002; 2004). We use a simplified version of the basic model with two hierarchically ordered players: Leader and Follower. This approach is also considered in (Denin, 2008).

The derivative model of a simple ecological-economic system with hierarchical structure considering corruption is represented as:

: ^2 Wv (p*, q*, u*, P*) - M p (u*, U^)] ^ max (6)

4=1

p* € P*, q* € Q*

T

Ju = E [sU (p‘, u*, P‘) - M p (u‘, UU)] ^ max (7)

*=1

u*

€ U* (q*) , P* € B*

The explications for the model (6)-(7) are given in Table 1.

Let’s classify the corruption phenomena in dependence with types of privileges given to Follower by Leader for bribe. Let’s distinguish three types of corruption:

J V

Tablel. Model notations

Notation Mathematical definition Environmental interpretation

ut (ui,...,ut„) G IR+ The set of Follower’s strategies in the year t: resource extraction, production of goods, pollution

ßt (ßt,...,ßn) G IR+ The set of Follower’s additional strate-

gies which may be interpreted as a bribe to Leader

pt (pi,..., pi) G IR+ The set of Leader’s strategies influencing Follower's control function: taxes, penalties, privileges

qt (qi,...,qi) G IR+ The set of Leader's strategies influencing Follower's set of admissible strategies: quotas, limits, restrictions

T 0 < T < to The period of consideration

P t,Qt Pv c ]R+, Qv C IR+ The sets of Leader's strategies which

t = 1,... ,T may vary in time

Ut (qt) Uv (qt ) C IR+ The set of Follower's strategies depend-

t = 1,... ,T ing on Leader’s strategies in the year t

Bt Bv c IR+ The set of admissible bribes from Fol-

t = 1,... ,T lower to Leader

gV gt : IRm x IR+ x IR+ x IR+ ^ IR Leader’s payoff function in the year t

gU gU : IR+ x ]Rm x IR+ ^ IR Follower's payoff function in the year t

p ( t v) [0, xv G X p(x ,X) = { > o, X G X A conditional function checking whether

the state vector belongs to a given set (for example, a set of sustainable development)

M M ^ to Penalty constant

Ut Ut ^ V 5 ^ u UV ,uU CIR+ The sets of Follower's strategies satis-

t = 1,... ,T fying the sustainable development conditions for Leader and Follower respectively

Jv , Ju Jv G IR, Ju G IR Leader and Follower's payoffs in the whole period T respectively

p-corruption. In this case Leader gives to Follower tax privileges for a bribe:

P = P (Pp) (8)

In the simplest case the dependence may be a linear function:

p* = Po - YPp, Y> 0 q-corruption. In this case Leader gives to Follower quota privileges for a bribe:

q* = q* (Pq) (9)

In the simplest variant the dependence may be a linear function as in the previous case:

q* = qo - ¿P*, 8 > 0

a-corruption. In this case Leader extends for a bribe a set of strategies satisfying the sustainable development conditions; in other words, Leader softens the conditions:

uV = U (p*) (10)

In fact, the a-corrupted Leader neglects the sustainable development conditions for a bribe. So this type of corruption is the most dangerous for environment and may results in critical implications.

Let’s consider the following problem as a model example:

T

[c*p* u* — g2 (p*,q*) — Mp (u*, UV)] ^ max

t=i

p* + ql

(1 -pt) (1 - qt)

0 < p* < 1, 0 < q* < 1

T ( )

c* (1 — p*) u* ^ max

i=i

0 < u* < 1 — q*

[0,a*] 0 < a* < 1 t = 1,...,T

The introduction of corruption factor in the basic model (Ougolnitsky, 2002; 2004) consists in that Follower gives a part of his payoff to Leader as a bribe. In exchange Leader provides to Follower some privileges in taxes, quotas, or sustainable development conditions.

Let’s consider the different types of corruption as defined above.

g2 (p* ,q^ =

U* =

^ V

p-corruption

T

Jv = ^2 [c*Pis (! - Pp) + c*PpM - g2 (pW) - Mp (V,UV)] — max (11)

t = 1 T

Ju = [c* (1 — ps) (i1 — Pp) u^] — max (12)

t=i

pS = p* - Y Pp + d*, Y > 0 (13)

0 < Ps < 1, 0 < q* < 1 (14)

0 < Pp < 1 (15)

0 < M < 1 - q* (16)

UV = [0,a‘] 0 < a* < 1 t = 1,... ,T (17)

Thus, a bribe factor Pp permits Follower to get some tax privileges. The parameter d* characterizes so-called “toughness” of corruption. Let’s consider that the value d* = 0 corresponds to a “tough” corruption, and the value d* = y to a “soft” one.

q-corruption

T

Jv ^ [c‘ p* (1 - Pg) u* + c* pq u* - g2 (p*, q‘) - M p (m*, UV)] —— max (18)

t=i T

Ju ^ [c* (1 - pi) (1 - P‘) w4] —— max (19)

i=1

0 < p* < 1, 0 < qS < 1 (20)

q* = q* - ô P* + d*, ô> 0 (21)

0 < Pq < 1 (22)

0 < M < 1 - q* (23)

UV = [0,ai] 0 < a* < 1 t = 1,...,T (24)

In this case Follower can get a bigger resource quota giving a bribe in exchange. As in the previous case, the value d* = 0 corresponds to a “tough” corruption, and the value d* = ô to a “soft” one.

a-corruption

T

[ct pt (l- — P«) ut + ct Pt* ut — g2 qt) —

t=1

—Mp (u4, UV (a4))] ^ max (25)

T

i1 — p4) i1—P^) u1 ^ max (26)

4=1

0 < p4 < 1, 0 < q4 < 1 (27)

A Pa + d4, A > 0 (28)

0 < Pa < 1 (29)

0 < u4 < 1 — q4 (30)

[0,a4 + a4] 0 < a4 + a4 < 1 t =1,...,T (31)

In this case Follower can use more resources due to an expansion of the domain of strategies satisfying to the relaxed (for a bribe in exchange) sustainable development conditions. As in two previous cases let’s consider that the value d4 = 0 corresponds to a “tough” corruption, and the value d4 = a4 to a “soft” one.

Let’s investigate the model for different types of corruption and different management methods (compulsion, impulsion, conviction) according to (Ougolnitsky, 2002; 2004) and compare the results with results for the model without corruption factor (a “pure” model). Here we give the results only for the case of compulsion. Index t is omitted because an optimal solution is chosen for the whole period of consideration.

q-corruption. According to the definition let’s consider that the value q^ = q — 6 Pq corresponds to a “tough” corruption, and the value q^ = q + 6 (1 — Pq) — to a “soft” one.

1. “Soft” corruption

The Follower problem is:

Ju = c (1 — p) (1 — Pp) u ^ max

q,Pp

where u € [0,1 — q^] , q^ = q — 6 Pq

It is evident that a quantity of resource used by Follower is equal exactly to the quota defined by Leader. Therefore u* = 1 — q^ = 1 — q + 6Pq. So the Follower’s problem may be represented as follows:

Ju = c (1 — p) (1 — Pp) (1 — q + 6 Pq) ^ max

q,Pp

From the equality (Ju)'p = 0 we get /3* = g+2<S(j~1, that is a point of local maximum for the function Ju (Pq). From the condition Pq > 0 this is true if 6 > 1 — q else P* = 0. Taking into consideration u < 1 we get an inequality 6 < 1 + q. Otherwise Follower chooses the strategy which provides him the absence of Leader’s quota, i. e. if S > 1 + q then Follower’s optimal strategy is equal to ¡3* = |.

V

u

t

So we get

(32)

c (1 - P) (1+4,sg) , ^£[l-q,l + q\ c (1 - p) (!-!)> 6 G [1 + q, +oo)

c (1 - p) (1 - q) , 6 G [0,1 - q]

.Cl — „

c (1 - p) (1+4,sg) , ô G [1 - q, 1

(33)

Now consider Leader’s payoff function. It depends on value of the parameter 5:

cP (1 - q),

c [p (1 -}) + }] ,

„ ,n (1+<S—g)2 : (<?+g—1 ) (<?+!—g) 6 P AS "T" AS

6 G [0,1 - q]

G [1 - q, 1 + q] (34)

6 G [1 + q, +ro)

Let’s consider the three cases separately.

(a) 5 G [0,1 — q]

From (34) it is evident that q* = 1 — a. Then Jv = cpa, otherwise, the equilibrium is the same as in the “pure” model (without corruption).

The sustainable development condition means that u G [0,a]. Taking into consideration Follower’s optimal reaction we get 5 < 2 a — 1 + q. From an

an optimal solution on 6 in the segment [1 - q, 2 a - 1 + q]. For the correct problem formulation it is necessary that 1 - q < 2 a - 1 + q, or 1 - q < a.

(c) 6 G [1 + q, +rc>)

In this case we get from (34) that q* = 1. Hence we will search for a Leader’s optimal strategy on q from the following conditions:

Then Leader’s optimal strategy is equal to: qß = 1 - 2 aßq. Besides,

By this means “soft” q-corruption permits Leader to increase his payoff essentially. As for Follower’s payoff, it diminishes twice in comparison with the “pure” model (Ougolnitsky, 2004).

2. “Tough” corruption

As in the case of “soft” corruption Follower’s optimal strategy is equal to u* = 1 — q^. Then Follower’s problem may be represented as

(b) 6 G [1 - q, 1 + q]

identical inequality 2 a — 1 + q < 1 + q follows that it is rational to search for

6 G [1 - q, 2 a - 1 + q] , q G [1 - a, 1]

Ju = c (1 - p) (1 - ) (1 - q - 6 (1 - )) —> max

q,ßq

Equating the first derivative on Pq of the function Ju to zero we get a point of local maximum {3* = 2 • From (22) this is true if 6 > Otherwise the

local maximum of the function Ju is reached in the point P* = 0. Hence

fO, ¿€[0,^]

[i_!;+00)

»_ fc(l-p) (l-q-6),S<= [0,^]

“ \c (! ~P) iirfL> ¿G[^,+oo)

So the “tough” corruption is also not profitable for Follower in any conditions. To find Leader’s optimal strategy it is necessary to consider two cases:

(a) ô <G [0, —, then Leader’s payoff function is equal to: Jv = cp (1 — q — 6).

It is evident that Leader’s optimal strategies satisfy the condition 6 + q = 1 — a, hen the payoff function has the form J* = cpa.

(b) ô <G then Leader’s payoff function is equal to:

J

V

,n (1—g) I (2 S+q—l) (1 —q)

P 46 "T" 46

. From the condition u < a we may deduce for this case the condition q G [1 — 2 a, 1]. Then the maximum is

reached when 5 is infinitely big or q = 1 — 2 a: lim Jv = c a.

6——+ ^

So,

q^ = 1 — 2 a + N (1 — Pq) ,

when N is infinitely big;

t*

Jv = ca

The investigation results may be represented as follows:

(p*,q*,u*,f3*) = (p, 1 - a, a, 1 -(N is infinitely big)

T* rs-'

Jv = ca

JU = 0

So the “tough” corruption permits Leader to control the whole resource and Follower’s payoff is close to zero.

a-corruption The set of sustainable development strategies has a form

Uv = [0, a + a (Pa)] , where a (Pa) = A Pa — d

1. “Soft” corruption

In this case a (Pa) = Apa. Then Follower’s problem may be represented as

Ju (p, q) = c (1 — p) (1 — Pa) u ^ max

It is evident that u* = 1 — q. Taking into consideration that Leader’s optimal strategy is equal to q* =1 — (a + a (Pa)) we get

Ju (p, q) = c (1 — p) (1 — Pa) (a + A Pa) ^ max

U,^a

The solution of the equation (Ju)^ gives for the function Ju (pa) the point of local maximum /?* = • But from (29) this statement is true only if A > a.

Otherwise Follower must choose a strategy without bribe, i. e. = 0. The condition a + a < 1 gives that if A > 2 — a then Follower must use the whole possible resource, or /?* = Hence

0, A G [0, a]

ß«=< TfAe[ß,2-a]

A G [2 - a, +oo)

c (1 - p) a c (1 -P)

A G [0, a]

A G [a, 2 — a]

Therefore Follower’s choice depends on the “wideness” of the set of sustainable development strategies. If the set is wide enough then Follower does not give any bribe to Leader and looses nothing in comparison with the “pure” case. If the set is not so wide then Follower can increase his payoff in comparison with the “pure” case using a bribe. It is evident that Follower’s payoff increases if a value of parameter A increases.

Hence

(pW,u*,ß* )= p, 0,1,

1a

2a

j* = rP+1~a

v 2 -a

JU = c (! - p)

2a

Respectively, Leader’s payoff considering Follower’s optimal strategy is

Jv

cpa,

A G [0, a]

A G [a, 2 — a]

P (! - ^ ^ J A G [2-a,+oo)

p

(A+a)2 (À —a) (A+a)

4 A

4 A

1-

The maximum of Leader’s payoff function on A is reached in the point A * = 2 —a. In this case J* = c (p + 1 — a)-

2. “Tough” corruption

Analogously to the previous cases we get the following results for Follower’s payoff function:

ßa

j

i A G [0,2]

^ A G [2, +oo)

c(l-p)f, A G [0,2]

c (1 ~P) (1 — x) ! ^ e [2, +oo)

1

In distinction to the “soft” corruption in this case to give a bribe is always profitable for Follower.

The maximum of Leader’s payoff function is reached when A = 2, from what

Let’s summarize the results of comparative analysis for different types of corruption. It is evident that corruption is profitable for Leader in all cases. Besides, the “tough” corruption is always more profitable for Leader and less profitable for Follower than the “soft” one.

In the case of compulsion the “tough” a-corruption is more profitable for Leader than the “tough” q-corruption if p > 2 a — 1. This condition is identically true if a < So, if the conditions of sustainable development are tough (a < ^) then a-corruption is economically profitable for Leader because it allows to avoid the conditions. In the same time a-corruption may be profitable for Follower too that makes the a-corruption even more dangerous. For example, in the case of compulsion the “soft” a-corruption is always more profitable for Follower than its absence in the “pure” case. If a < ^ then even the “tough” a-corruption gives Follower a bigger payoff than in the “pure” case.

In the case of impulsion the “tough” a-corruption is more profitable for Leader than the “tough” p-corruption if a < 1 — | (i.e. practically always).

5. Linear Multistage Games with Hierarchical Structure

Organizational systems have a hierarchical structure. The most adequate mathematical formalism for structural description is a directed graph (digraph) in which the vertices represent structural elements and the arcs represent directed relations between elements. It is also possible to take into consideration quantitative characteristics of the elements and relations by introducing weights (values) of the vertices and arcs respectively.

The hierarchy means that some structural elements have a priority in comparison with others. That’s why for description of the hierarchical structures it is necessary to use digraphs of a special type. It is the specific digraphs and their incidence matrices that are used to describe the hierarchical structures in the proposed approach.

Let’s call connected digraphs without contours and loops the strictly hierarchical ones (SH-digraphs). Their incidence matrices we also should call the strictly hierarchical ones.

It is known that a vertex without input arcs exists in every digraph without contours. We should say that all vertices without input arcs form the first layer in the digraph SH.

It is possible to use a digraph representing an organizational structure as a base for a game theoretic model formulation in a way that follows.

Let SH = (Y, Z) be a strictly hierarchical digraph with n vertices. Then we should call a strictly hierarchical non-cooperative two-person game generated by SH a linear multistage game in which payoff functions are the following:

follows:

1 +p 2

1 -P 2

- +1 JT+n

J1 = ('

+ E (r‘,x‘) + £ (s‘,

s4,«4)

t=1

T

t=0

T

J2 = (iT+1,xT+1)+Y, (i‘,x4) + Y, KV)

t=1

The equations of controlled process are

xt+1 = A£ x4 + B£ u4 + C v4,

The initial conditions are

4=0

t = 0,1,

(35)

(36)

(37)

: xo

The first player tends to maximize the payoff function (35) by choosing strate-

gies u with restrictions

Dt u4 < d4

't “ — “ (38) The second player tends to maximize the payoff function (36) by choosing strategies vt with restrictions

Gt vt < gt, t = 0,1,...,T

where xt, ut, vt, rt, st, lt, mt, dt,gt, x0 are vectors from IRn.

(39)

A set (xt, ut, vt, rt, st, lt, mt, /t, dt, gt, x0i) is a value of the vertex y € Y. For vertices from the first layer y € Li

dt = gt = 0

For vertices from the last layer y G Lm

A

u4 = v4 = 0

c4 -1 lcjl

Dt = ||dj||, Gt = ||gj|| are strictly hierarchical matrices. A set (aj , bj , cj , dj ,gj) is a value of the arc Zj € Z. Let’s introduce for each player a Hamilton function

(40)

(41)

H1 (p^yy) = (pt,At xt + U + Ct vt + (rt ,x^ + (st,«^ (42)

H2 (qt, xt, ut, vt) = |qt, At xt + Bj U + Ct vt + (lt, xt) + (mt, vt)) (43)

where pt, qt are conjugate variables according to (37) and defined by formulas

94-1 = r4 + Ap4, pT = 0

7i_1 = l4 + Aq4, qT = 0

(44)

(45)

The necessary conditions of optimality of strategies u*, v* in the game (35)-(41) are given by the following conditions (Gavrilov, 1969):

0

x

max Hi (pt,x *,ut,v *) = Hi (pt,x *,u*,v *) (46)

max H2 (qt,x *,u*,vt) = H2 (qt,x*,u*,v *) (47)

where x * is a trajectory of the system (37) generated by the optimal strategies u *, v * ; Ut,Vt are sets of admissible strategies given by conditions (38), (39) respectively.

To find maximums by (46), (47) it is sufficient to consider the Hamilton functions (42), (43)

Hu (pt,ut) = (st,ut) + (pt,B' ut) (48)

Hv (qt,vt) = (mt,vt) + (qt,Ct vt) (49)

Recurrent formulas (44), (45) permit to calculate pt,qt:

pt = rt+i + At rt+2 + ... + AT-t-i rT qt = lt+i + At lt+2 + ... + AT-t-i lT t = 0,1,...,T - 1

So the Hamilton functions (48), (49) are linear functions of the variables ut, vt respectively:

Hu (ut) = (KU,ut) , Hv (vt) = (KV ,vt) ,

where KU is a vector value which does not contain ut and depends on the parameters At, Bt, st, rt;

Kvt is a vector value which does not contain vt and depends on the parameters At, Ct, mt, lt.

The conditions of nonnegativity of the strategies arise naturally in applications:

ut > 0, vt > 0

To satisfy the necessary optimality conditions (46), (47) it is sufficient to solve the pair of linear programming problems

(KU ,ut ) ^ max, Dt ut < dt, ut > 0, t = 0,1,...,T

(KV,vt) ^ max, Gtvt < gt, vt > 0, t = 0,1,...,T

These linear programming problems have the same type as the initial game (35)-(41). That game is explicitly generated by an organizational structure given by the strictly hierarchical digraph SH.

6. Conclusion

We think it is worthwhile to combine several mathematical concepts in the same model for describing organizational dynamics and management processes. These concepts are: a directed graph without contours and loops, a game of n players in normal form, a cooperative game, a dynamical system controlled by several subjects. A generalized model (1) of hierarchically controlled dynamical system based on the idea is proposed in this paper and partly illustrated by some examples.

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