Dynamic consideration of compromise solutions in a many-sided conflict

Dynamic consideration of compromise solutions

77

The author thanks G. S. Lbov and S. M. Seitenov, who took part in seminars of the laboratory of pattern recognition of the IM Siberian Branch of the Academy of Sciences of the USSR and of the operations research dep~tment of Moscow State University, for discussion and comments, and also the journal editor. Translatedby D. E. Brown

1.

WILDE, D. J., Optimum-seeking methods, Prentice Hall, 1964.

2.

GERMEIER, Yu. B., Introduction to operations reseurch theop (Vvedenie v teoriyu issledovaniya opera&ii), Nauka, Moscow, 197 1.

3.

SUKHAREV, A. G., Optimal search for an extremum (Op~~‘nye Moscow, 1975.

4.

IVANOV, V. V., Optimal algorithms for minimizing functions of certain classes, Kibernetiko, No. 4, 81-94, 1972.

5.

CHERNOUS’KO, F. L., On optimal search for an extremum of a unimodal function, Zh. v_Zhisl.Mat. mat. Fiz., 10, No. 4, 922-933, 1970.

6.

TARASOVA, V. P., Optimal strategies for seeking the approximate global extremum for a certain class of functions, in collection, Computing systems (Vj&iil. sisremy), No. 67, 77-86, In-t matem. SO AN SSSR, Novosibirsk, 1976.

&%S.R. Comput. Maths Math. Phys. Vol. 18, pp. 77-88 0 Pergamon Press Ltd. 1979. Printed in Great Britain.

poisk ekstremuma), Izd-vo MCU.

0041-5553/78/0801-0077$07.50/O

DYNAMIC CONSIDERATION OF COMPROMISE SOLUTIONS IN A MANY-SIDED CONFLICT* A. V. BARKALOV and R. G. STRONGIN Gor’kii (Received 22 March 1977; revised27 April 1977) A DYNAMIC scheme for making decisions about the pay-offs in a three-sided conflict is proposed; the scheme is described as a sequence of plays of a three-person zero-sum game, the players’ interests being defined by the mean pay-off per play. The existence of stable solutions (of the equilibrium point type) is proved; they generate periodic motions of the players’ coalitions and ensure a mean share equal to Shapley’s vector. The stable solutions are possible both in the class of strategies that include information about the past plays, and in the class of strategies that only take account of the players’ running pay-offs. Considerable attention (see e.g. [l-9] ) has been paid to decision-making in many-sided conflicts, which simultaneously have utility for all participants and also stability (defined by the players’ attempt to increase their pay-offs). A feature of most studies is the description of the interaction between the players in game-theoretic language and the search for an optimal decision in the context of a static discussion. In certain problems, however, including those described by the classical models of coalitional zero-sum games ]I] , a static consideration implies that there are no stable solutions (e.g. of the C-kernel type) [6], while “valid” solutions (of the Shapley vector type [2]) in essence presuppose external stabilization. With regard to the Neum~-Morgenstern solutions, only the set of such solutions in the large, but not any individual solution, has stability. *Zh. vj%hisZ.Mat. mat. Fiz., 18, 4, 897-907, 1978.

78

A. V. Barkalov and R. G. Strongin

In the light of this, in the case of models of many-sided conflicts, in which there are no statically stable coalitions, ensuring equal or comparable pay-offs for the players, it is interesting to consider dynamically (see e.g. [9]) super-games, any play of which is a sequence of plays of the initial game, where the formation of the coalitions (and the share-outs) in each play of the sequence is carried out afresh in the light of the previous experience. The players’ different strategies in the super-game determine the different ways of utilizing the past experience when developing agreements about coalitions (or share-outs) at the next step of the super-game. If we bear in mind that the tolerance of a coalition of players in such a super-game in fact implies that certain players renounce a way of talking to one another that might be advantageous otherwise, then it is entirely natural to consider the case when the super-game is coalition-free. It is then possible to consider statically the super-game, describing the developing of the many-sided conflict, and to study the motion of the coalitions in the plays of the sub-game. (the initial coalition game), generated by the optimal behaviour of the players in the super-game. In the present paper (some results of which were presented in our paper [lo] ), a many-sided conflict, described by the model of a zero-sum three-person game, is discussed in this dynamic way. The relevant super game, including the model of the coalition formation in the plays of the sub-game, is described in Section 1. In Section 2 we establish the existence of equilibrium situations in the super-game and obtain an estimate for strategy efficiency, similar to Shapley’s vector. We indicate the connection of the strategies forming the solution, and the players’ information patterns.

1. Dynamic model of a constant-sum three-person game Let v be the characteristic function of the constant-sum three-person game, specified in (0, 1)reduced form; since the game is harmless (see [l] ), it can be described by the expressions vz=vs=l:

v)0=v,=0,

(1.1)

where the subscript indicates the number of players in the coalition. The outcome of such a for the coordinates of which we have the relation game is the share-out CI=(~,, a2, a,),

a*+az+a,=1,

al,

az,

a30,

(1.2)

where C.Y~ is the pay-off of player i= { 1, 2, 3). We shall take the following model of generation of the share-out (Y.Every player i proposes with share to another player j that they form the coalition {i, j}

0&=4.-&j,

Ctj=Gijy

an=O,

which we denote by cQii), and proposes independently to player k (we stipulate that (i, j, k) is either the sequence (1, 2, 3), or the first or second degree of cyclical substitution of the elements of this sequence) that they form a coalition {k, i} with share a(6jk) of the form

(l-3)

A. V, Barkalovand R. G. Strongin

80

i.e. I==={&j) holds when either both players i and i prefer this coahtion (case (1.9)); or when al1 players prefer different coalitions and the fmaI choice is determined by the fact that the player bearing least loss changes his preference (cases (1 .lO) and (1.11)); or when it is a matter of indifference to one of players i, j which coalition he takes part in, and the coalition is formed with the player having most at stake (cases (1.12) and (1.13)); or when two players have the same amount at stake (e.g. di = dk) and the coalition is formed with a third more interested player (cases (1.14) and (1 .lS)). The conditions for generating I= {fi, i} or I== (j, k) (1.15) by a cyclical change of subscripts i, i, k. In the case when

are obtained from (1.9) and

di=d>=dhT

(1.16)

i.e. the degree of interest of the players in the formation of their preferred coalitions is the same, the coalition I = 10 is formed, where b={{1,.%

{.a 31, (3, w7

(1.17)

while in the case when d;=-dk)dj=O,

(1.18)

the coalition I = 4 is formed, where

and the values of IO, 4 are not known to the players a priori From condition (1.18), by cyclical replacement of the subscripts, we can derive two further conditions, corresponding to the formation of a coalition according to the rule I = Ii, or according to the rule f = fk, where Ii, & are also not known a priori to the players, while the sets of their values are found from (1.19) by suitable cyclical replacement of the subscripts, Hence, I= {i, j}, if either one of conditions (1.9)-(1.15) is satisfied, or if (1.16) holds and 10={i, j), or if (1.18) hoids and I,= (i, jf , or if d,=-dj>di=O

(1.20)

and I,= (1, if. To demonstrate the correctness of the coalition generation rules, we introduce the quantities A,, AZ, Asr where 26ij-I,

(2fi*-1)

A* = i

(26ij-1)

1_26ji,

6ij>6ji,

y

Gij=8ji,

(1.21)

6ij<6ji,

and hence (see (1 S), (l-6)), (1.22)

Dynamic consideration of compromise solutions

81

It follows from (1.7) and (1.2 1) that the coordinates of the share-out tl ( {i, j} ) can be written as ai=‘/

aj=‘/z (l+Ak) t

(I-Ah) 7

c&=0,

(1.23)

whence, recalling (1.8), we find that di=-Aj-Ak. On substituting the expressions for di, dj, dk in (1.9)-(1.16), (1.18) (1.20), we obtain the conditions in the equivalent form

Aif AkGO,

(1.24)

Aj+Ak
Ai+Ak>O,

A,),

(1.25)

Aj+Ak>Ol

(1.26)

Aj
Ai>max (A,, A,), Ai+Ak=O,

Ai+AjZO,

A,
(1.27)

Aj+A,<=O,

AiSA,dO,

A,>O,

(1.28)

A,+A+O, Ai+Aj>O,

Al>Ak=Aj)

(1.29)

Aj
(1.30) (1.31)

A,=A;=Ah, A,+A,:=O,

A,=O,

Ai>07

(1.32)

AjfAk=O,

A<=07

AjcO.

(1.33)

Conditions (1.24)-( 1.33) isolate a connected domain in the cube (1.22), the domains corresponding to the three possible coalitions form a covering of cube (1.22) and the only common parts of these domains are the diagonal (1.3 1) and the three intervals defined by expressions of the type (1.32), (1.33). Our rule for forming coalitions thus covers all cases. For illustration, the domain corresponding to the coalition is shown in Fig. 1 (the {i. i), contour of the domain is represented by a heavy line, and the edges of the cube (1.22) by a thin line). The part of the boundary shaded by thin lines (including the bordering parts of the edges) refer to the domains for the coalition {j, k} (upper shading) and for the coalition {k, i} (lower shading).

FIG. 1

82

A. V. Barkalov and R G. Strongin

We now introduce the super-game, in which at each instant r, t = 0, 1, 2, . . . , a play of the above three-person game is realized; at every step of the super-game players i, j, k can change their proposals, depending on the outcome of the previous plays, according to the rule* o,j(t+l)=oij(a(Z(r)), Of(Tij(a,

z(t-l))*

(1.34)

I) &I,

(1.35)

where Z(t) and o(Z(r)) are the coalition at step t and the corresponding share-out, while u&r, Z) is a real function of the share-out and coalition, whose values are the modes of a uniform finite mesh in the interval [0, I] (notice that condition (1.35) is a consequence of conditions (1.5), (1.6)) which includes the points, 0, l/3 and 1. The expressions for the proposals &I,, &, 6jn, &,;, fiji as functions of time are found from (1.34), (1.35) by a suitable transposition of subscripts. Let us define the strategy si of player i, 1 Q i < 3, in the super-game as the ordered collection Si=(Sij(O),

6&(O)>

Oij(c1,

I>,

Oik(ay

1))

(1.36)

and denote the set of all such strategies by Si. (In accordance with (1.34), this concept of strategy presupposes that, after the end of a play, the share-outs in it become known to each player). We then specify the sequence of sets of seven (1.37) in which an element corresponding to a fured t serves to define the coefficients PI, Bz, i% in the expressions obtained from (1.7), and for defining the sets lo, II, I,, Is when conditions (1.16) and the conditions derived from (1 .18) are satisfied at the step t of the super-game. The first three elements of each set (1.37) is the number 0 or 1, the fourth element has values from the set (1.17), and the remaining elements have values from the sets into which (1.19) transforms by a suitable permutation of subscripts 1, 2, 3. Denote the set of all possible sequences 7 of (1.37) by r, and assume that an element 7 E r, appearing in the description of the super-game, is not known II priori to the players. As the i-th player’s pay-off function in the super-game we take

r

1 Mi (Si, Se, S,) = lim inf af (Z(t) 1, %-c.n‘6+1 z1 o TEr

(1.38)

where or(Z(t)) is the pay-off (share) of player i in the coalition Z(t) (dependent on 7). Since, with fixed strategies, the number of possible values of players’ proposals is finite, then only either periodic motion, or a fmed point, is possible for the 7, corresponding to the minimum in *Rule (1.34) assumes that the coalition 1(-l) is given. Let us stipulate that every player i can have his own +15(-l)), reflecting his assumption about value Ii of this quantity (i.e. it is possible e.g. that Ii the inter-relationships of the layers before the start of the game. Hence, the quantities 1,(--i), lGiS3, are given (they belong to the set (1.17)).

Dynamic consideration of compromise solutions

(1.38), in the space of values of six-dimensional function

vectors

83

(& (t) , . . . , b3, (t) )

Hence the

(1.38) exists.

2. Equilibrium situation in the super-game We introduce initial assumptions

the strategy sio of player

i, 1 <

i < 3, in the super-game,

defined by the

Si,”(0)=6ik” (0)=‘/S and by the functions

(2.1)

oij” (a, I), oik” (a, Z) of Table 1. According to this strategy, player i

proposes to another player a quantity

equal to 2/3, if the latter’s pay-off in the play just ended

of the sub-game was not more than l/3. If the player’s pay-off were not less than 2/3, then the player i proposes to him a quantity

equal to 1, etc. (see Table 1).

Theorem The triple of strategies

(sio, szo, s3’)

forms an equilibrium

situation

in the super-game,

where

szo, sgO) =Jfs @IO, GO,ho)=‘/s.

M, (S’O, szo,ssO)=M, (S’O,

Proofi The neatest proof is obtained by using are represented to which the shares (ai, aj, co,) height. The values of ai, ah ak are equal to the from the share-out point to the appropriate side of

FIG. 2

(2.2)

the geometric interpretation [l] , according as points of an equilateral triangle, of unit lengths of the perpendiculars dropped the triangle (barycentric coordinates).

84

A. V. Barkalov and R. G. Strongin

TABLE 1

I

=j

IIs <

aj

ai = 0

‘12

<

=i

(k, i)

aj = 0

0 < ai < VS aj

>

0

In accordance with (2.1) the initial proposals of all players are the same and equal to 2/3. Hence, by (1.7), to the coalition {i, 11 these corresponds the share-out (l/3, 2/3, 0), if ok(O) = 1, and the share-out (2/3, l/3, 0), if ok(O) = 0. These two share-outs are represented by the points on the base of the triangle (see Fig. 2). We also indicate the two possible initial share-outs in the coalition {k, i} (the points numbered 1 and 2, on the left-hand side of the triangle) and the two possible initial share-outs in the coalition {i, k} (points 3 and 4). It follows from (2.1) and (1.21) that 1A.iI= 1Aj1= 1Ak1=‘/s and hence either condition (1.31) must be satisfied, or else one of conditions (1.29), (1.30), or one of the conditions obtained from the last two by a suitable permutation of subscripts. Hence, to the coalitions formed at the instant t = 0 there can correspond any of six share-outs (they wiIl also be denoted by CL”,l 0. Then, using Table 1 and two other tables, derived by a suitable permutation of subscripts in accordance with (1.34), we obtain ~~j(t+l)~6~~(t+l)~6j~(t+l)~*/J~

6kj(t+l)=‘/3,

6*i(t+l)=6j&+l)=l, which, after substitution in (1.21), gives A,,=-I,

Aj=l,

Ai=l/s.

(2.3)

The formation of the coalition {i, k}, follows from (2.3) since condition (1.24) holds (after replacing i, j, k by i, k, i). The corresponding share-out (we use expression (1.23), rewritten for I= {i, k}) is the same as share-out 3. In short, realization of share-out 1 at the instant t > 0 leads (regardless of the value of y E I’) to realization of share-out 3 at the instant c t 1. Noting the symmetry of the problem, we see that one-place transitions exist, from share-out 3 to share-out 5, and from 5 to 1

85

Dynamic consideration of compromise solutions

(see Fig. 2, where the transitions are indicated by arrows), i.e., a periodic motion exists, passing through share-outs 1, 3, and 5. We can prove in the same way the existence of another periodic motion, passing through share-outs 2,4, and 6 (see Fig. 2). Hence, given any -y E I’, the triple (So”, szo, sjo) leads to one of two periodic motions, for each of which the statement (2.2) holds. Now consider the triple (s,“, s,, sk”). which differs from the above case in that player i uses an arbitrary strategy Sj E Sj, not the same as sju. Then, as may be seen from (2.1) (1.21). and conditions (1.24)-(1.33), the initial share-out a@(O)), depending on the proposals and on the value of r(O), is either the same as one of share-outs l-6 fi* (0) , &i (0) (e.g., c@(O)) with 6h (O), Sji(O) <*/s and p,(O) = 1 is the same as share-out 1, while if /3&O)= 0, it is the same as share-out 2) or else it belongs to one of the closed intervals A or B, indicated by shading in Fig. 3 (e.g., C@(O))E A if 6jh(0) >6fi(O) >2/5). Assume that, at some instant I > 0, we have a=a(Z(t)>d. In this case, a&‘/a,

a,,=0

(2.4)

and it follows from (1.34) and Table 1 that

6j,(t+l)=Gi~(t+~)=Glj(l+l)=‘/s,

6ki(t+l)

=I,

(2.5)

whence (see (1.21)) Ai = 1, and if oa(a, I)<‘/, or ul*(a, I)=‘/, and bi(tfl) =O, then 4 = -l/3. Under the conditions indicated (see (1.23)) &(t + 1)) is the same as share-out 4, since Z(t+i)={j,

k}

(2.6)

or (if 4. < l/3) by virtue of an expression of type (1.24), or (if 4 = l/3) by virtue of an expression of type (1.26). Notice that the case Ak > l/3 is impossible. If ua(a, I)>‘/, or up(ar Z) =‘is and bl(ti-l) =l, then (see (2.5) and (1.21)) L&> l/3 and (2.6) follows either (if f& t & < 0) from condition (1.24), or (if A,+Ana0 from condition (1.26), or (if Ai = 1) from condition (1.30). In all three cases, AiCl) a=a(Z(t+l))EB.

FIG. 3

and

(2.7)

FIG. 4

86

A. V. Barkulov and R. G. Strongin

Using the symmetry, we find that relation (2.7) implies either that a(Z(t + 2)) is the same as share-out 5 (if oj, (a, I) <‘/, or oji(a, I) = C/s and bk(t+2) =I ), or it belongs to the interval A (if oji (a, Z) >2/3 or o,i(a, Z) =“j3 and PA(t+2) =0 ). Hence, satisfaction of condition (2.4) or condition (2.7) leads either to a transition to one of share-outs 4 and 5 after a finite number of steps, or to the appearance of a periodic motion passing through the share-outs of intervals A and B (an example of this motion is shown in Fig. 3) where, in the latter case, Mj(s,‘, SjySk’) G

l/31

(2.8)

since, for a E A and a E B we have aj < l/3. Assume that, at some instant t > 0, the triple (s,‘, Sj, sko) leads to a share-out o(Z(r)), the same as share-out 1. Using Table 1 and conditions (1.21) (1.23)-(1.30), we find that, with ojk(a, Z)>‘/,, or G~~(cI,Z) (*/a or oji(a, Z)=‘/s and /31,(t+l)=O). Assume that

Z(t)={&

i}, p,(t+?)=O,

Ijlr(t+2)=1

a=a(z(t+l)>~D,

and (2.9)

here, if share-out a is the same as share-out 4, it is of no importance whether or not condition Z(t) = {k, i} holds. Then, given any functions o&(cc, I), Oji (a, I) share-out a(Z(tt 2)) is the same as share-out 2, since, under the conditions above indicated, it follows and hence (we use expressions from (1.21) and Table 1 that Ai=-I, Aj=-‘/3, Ak=I of type (1.23) and (1.24)), a(Z(t+2)) = a({& i}) =(‘/s, 0, ‘/s). It also follows from the symmetry of the problem that a(Z(t + 2)) is the same as share-out 1 when (2.9) is replaced by

FIG. 5

Dynamic consideration of compromire solutions

87

TABLE 2

#Ii. kl ii. kl

Yil”’ I,

liy kl ii. il

Ii, kl

(iv k)

(k, il ii, kl

(i+kl

{k, i)

In short, given any strategy SZE Sj, there is a sequence y E r such that a periodic motion of share-outs is established, either of the type (see Fig. 3) a (Z (t+&) ) =% a(Z(t+2p+l) >=B,

p=o, 1, . . . ,

or of the type (see Fig. 4) a (Z(t+4p) >=ar, a(Z(tf4Irfl) )ED, a (I (t+4p+2) >=a’, cc(Z(t+4p+3) >EC, p=o, 1,. . . , or else, similar to the motions illustrated in Fig. 5 (there are two further motions, symmetric with those shown in the first two diagrams of Fig. 5). In all these cases, assertion (2.8) is valid, since share-outs having the property ‘/s 2/3 cannot be satisfied for share-outs lying on the path of periodic motions realized with the triple (SF, Sj, Sk’) ) , belong only to domains C and D, from which a transition follows to share-out‘s 1 or 2, where CY~ = 0. Notes. It can be shown that the strategies obtained from siu by replacing conditions (2.1) by any other initial conditions, also form equilibrium situations. Equilibrium points also exist in the class of strategies in which the functions oij depend only on share-out cu(Z(t))and are independent of Z(f - 1). Strategies of this type can be. constructed on the basis of siu by replacing the rows of Table 1, corresponding to the case oi = 0, by conditions ((3i j, oih)

(1, 2/3) 9

aja’12,

(2/3r 1) 7

CZjC’l2.

=

A further interesting case is the conflict in which formation of the coalition share-out is based, not on condition (1.7), but on a condition of the type a({i, i}) =‘/2a(&j) +‘/2U(6ji). In this case the equilibrium point, having property (2.2), can be constructed in the class of strategies with functions oij(Z(t-I), Z(t)). One of the types of these functions is indicated in Table 2. i’hznslated by

D. E. Brown

K V. Podinovskii

88

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NEUMANN, J. von, and MORGENSTERN, O., Theory of games and economic behaviour, Princeton UP., 1955.

2.

VATEL’, I. A., and ERESHKO, F. I., Mathematics of con)7ict and cooperation i sotrudnichestva), Znanie, Moscow, 1973.

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ROZENMULLER, I., Cooperative games and markets (Russian translation),

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MOISEEV, N. N., Elements of the theory of optimal systems (Elementy teorii optimal’nykh sistem), Nauka, Moscow, 1975.

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GERMEIER, Yu. B., Games with unopposed interests (lgry s neprotivopolozhnymi Nauka, Moscow, 1976.

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BONDAREVA, 0. N., Some applications of methods of linear programming to the theory of cooperative games, in: Problems of cybernetics (Probl. kibernetiki), No. 10, Izd-vo Akad. Nauk SSSR, 119-139, Moscow, 1963.

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VOROB’EV, N. N., Coalitional games, Teotiya veroyatnostei i ee primeneniya

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VILKAS, E. I., Problem of optimality in the theory of games, III All-Union conference Abstracts, Izd-vo OGU, pp. 5-8, Odessa, 1974.

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KONONENKO, A. F., On equilibrium positional strategies in non-antagonistic Dokl. Akad. Nauk SSSR,

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interesami),

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10. BARKALOV, A. V., and STRONGIN. R. G., On a dynamic approach’to games with coalitions, III All-Union conference on game theory, Abstracts, lzd-vo OGU, pp. 108-110, Odessa, 1974. iJ.S.SR. Comput. Maths Math. phvs. Vol. 18, pp. 88-95 0 Pergamon Press Ltd. 1979. Printed in Great Britain.

0041-5553/78/0801-0088S(i7.50/0

CONSTRUCTION OF THE SET OF EFFECTIVE STRATEGIES IN A MULTI-CRITERION PROBLEM WITH IMPORTANCE-ORDERED CRITERIA* V. V. PODINOVSKII

Moscow (Received 1 November 1977; revised 22 November 1977) A CHARACTERISTIC property is established, for strategies which are unimprovable in the sense of the preference generated by information about the relative importance of the criteria. Using the property, a method of constructing the set of such unimprovable strategies is developed.

Introduction

In any multi-criterion problem of decision-making in conditions of determinacy [l] , it is assumed that at least two criteria K1, . . . , Km are specified. By criterion we mean a numerical function, defmed in the set of all strategies II. Hence every strategy u is characterized by a vector (vector estimate) K(u) = (K, (u) ! . . . , K, (u) ) . Henceforth it will be assumed that as high a value as possible is desired for each criterion.

*Zh. @hisI.

Mat. mat. Fiz., 18, 4, 908-915,

1978.