The theory of three-person games

THE THEORY OF THREE-PERSON GAMES*

Yu. B. GERMEIER

Moscow

(Received 12 December 1972)

THE principle of best guaranteed result is used to investigate three-person games with a hierarchical priority for the first player with regard to the sequence of moves and the information about them. We allow for the possibility of the other two players forming a coalition, for the first player having information about the principles of the coalition, and for the possible collapse of the coalition. The results obtained extend the similar research into two-person games.

In “classical” theory [ 1 ] , given the wide general definition of a game, it is still difficult to arrive at a conception of the rational behaviour of players which is both reasonably clear and corresponds to practical needs. It seems to us that this is particularly the result of an inadequate amount of attention being paid to the order in which the players move and the amount of information they have about each other. In addition, the excessive trend towards creating an “objective” concept of solution (i.e., a concept that treats all the players equally) makes the study of games unnecessarily difticult. In practice, clearly, the investigation is carried out individually for a given player, and not for all the players at once. It is therefore interesting to continue a study of what can be

achieved by applying the principle of best guaranteed result for the player who makes the first move; in this way we can continue the study of the operation [2] of this player. The approach was applied in [3-71 to the study of non-antagonistic two-person games; and in particular, results were obtained, suggesting the desirability of the first player transmitting information on his strategy to the second player. The same approach has proved fruitful in the theory of hierarchical systems [8] ; We may also mention the way it obviously matches up with the theory of “metagames” introduced in [9] ;

In the present paper we attempt to extend this approach to three-person games in specific conditions concerning the information the players have about each other and their sequence of moves. In fact, let Wi=fi (x1, x2, x3), i=l, 2, 3, be the expression for the efficiency criterion Wi of the i-th player in the absence of coalitions. When coalitions are formed the expression for Wi will clearly change, and this will be specially stipulated. We assume, therefore, that the i-th player wants to maximize Wi and that he has no other desire in the game.

*Zh. vjkhisl Mat. mat. Fiz., 13,6, 1459-1468, 1973.

112

The theory of three-person games 113

We shall also assume that the first player has exact information about x2 and x3

and at the same time makes the first move, and chooses his own strategy. This strategy is

the behaviour rule Ei =zl (G, x3) which, in the same way as e.g., in [4] , he communicates to the other players. As distinct from two-person games, the information available to the second and third players does not imply that their behaviour is clearly defined, since nothing has yet been stipulated about their interrelations and mutual information

(assuming that the values of their criteria are interdependent).

We shall consider several possible cases, without claiming to cover them all.

Case I: strict coalition between the second and third players, i.e., they act in complete accord (which can be exactly stipulated in advance) whatever the actions of the first player. This means [lo] that, from their criteria IV2 and W3, they form in some way a common (coalition) criterion W -23=f23 (xi, zz, x3) and they choose together the

vector (x2, x3). They thus amount in essence to a single player, and if the first player knows this (without precisely knowing E&ii;,), the present case is equivalent to a two-person game for him, such as has been discussed in the papers cited.

The simplest example of this situation is the formation of a Neumann coalition with collateral payoffs, when

Here, the behaviour and best guaranteed result of the first player are obviously independent of the way in which the second and third players divide their winnings. The results of the game are determined for them by the preassigned method of sharing the total result of the coalition

where x1’ (x2, z3) is the first player’s behaviour, rational in the sense of [4] .

Case 2: The second and third players may either act as a coalition known to the first player, or they may act individually, if a greater result can be obtained than that given by the coalition.

Suppose also that each has no information of his own regarding the other’s move (of course, other cases could be considered). It will be assumed that the order of moves of the second and third players is specified by the first player according to his judgement. The second and third players may or may not know f3 and f~.

We introduce

114 Yu. B. Germeier

L,= max min f3 (5r, z2, ta) x3 Xl, 9

and the sets E2 and Es, consisting respectively of the points x2 and x3 at which L2 and L3 are realized. The following natural conditions will be imposed on the coalition:

a) with an appropriate choice of (x2, x3), it ensures that, whatever x1 , the partners in the coalition obtain values of IV2 and IV3 greater than L2 and L3 respectively,

b) an increase in R 2s leads to an increase in IV2 and W3, the values of which, in the case of joint actions, are uniquely determined by W33 ; for instance, this is the situation in a coalition with collateral payoffs, if the sharing principle is fixed,

c) the coalition partners cannot obtain, by acting individually and independently of one another, results that are simultaneously as good as or better than those obtained by the coalition, when the first player acts in such a way as to minimize f23 ; this important condition implies that the coalition is good for its participants in difficult conditions.

We now put

L,=max min fz3 (XI, x2, 4, x2,x, Xl

and write L’a and L’s for the IV2 and Ws corresponding to L23 by virtue of b).

In this notation, condition c) means that the partners cannot hope to obtain more than L’2 and Lr3, if the first player realizes rn>%n fz3 (x,, x2, x3). If the second and third players adhere to the principle of the guaranteed result outside the coalition, without knowing each other’s moves, this contributes to the satisfaction of c).

Further, in the two-person game with criteriafi andfi3, let the best guaranteed result of the first player be Kr , and the corresponding strategy jr1 k. (This ensures a certain IT033 , to which correspond IV2 = L”2 and Ws = L”s .)

We now introduce the sets

D, N (4 = { (x,, xz) 1 fz (XI, xz, xd >Lz’>,

D,‘= { (xi, xz, x3) 1 fz (xi, xz, xs) >Lz’; fsh 52, 4 -a}

and the exactly similar sets D”3(~2) and D13.

The theory of three-peon games 115

Accordingly, we put

X,E& (XI, xz)~Dt”(~d K,“= _-oo

1 D;=@ for at any rate one

z&%3.

We define K’s and K”a similarly.

We also introduce the corresponding points, reahzing K2’, K, “, Ka’, KS N (possibly to an accuracy E ),

(2x1’, 22z’, 253’), (2%” (23)) 2x2n (53)) =Z12”,

( 3 5 I , 3 x2’, 3x3’), ( 3xi”(X2), %‘(r2) ) =&s;

for example, with any x3 E Es, x”12 realizes

SUP fl(% x2, += SUP fl @l, %, x3:3). (XI, X*)ax%) (x1, a

feh G2, Xa)>Lr’

Letzii,m realize min fz3 (x1, x2, 4, and Z,,“= [.rin(&, x2’(&) 1 rt4.b

mm f3 ( x1, x2, x3). We de&e Z1sn similarly. 111,Xz

Theorem 1

The maximum guaranteed result of the first player is

max [R,, K2’, Ka”, KS’, firs”] =Ka.

If Ke = Kr , then Ke is realized when the first player’s strategy, communicated

to the second and third players, is equal to Hi k, if the second and third players form a

coalition, or is Zr m, if they do not.

If Ke = Kf2, this result is realized when the first player’s strategy x’i is as follows:

-, -2 I 51 - Xl , X3=“X3’,

zt’=&“( if x3+293’~~x2=x2n (X3),

116 Yu, 3. Gemeier

the second value x3 being communicated by the first player at the opportune time.

Finally,

*, x, =z_lm, if x&2x21, x3=zx3’ or x,+-x2n(x3), x3+53’.

Here, the first player does not communicate x a. The complete?!r is communicated to the second and third players.

If& =K”z,Ko isrealizedwhen

S.?,‘=2X”( 53)) if XX=‘G”(G) and x:~EE,

(the first player then coruscates info~ation about x3 to the second), or when

iT~‘=xi”(x3) ( if xz=xzn(x3) and x#E3,

and Z,‘=Z,“’ otherwise.

The second and third players are again informed about S?, .

We can similarly define %’ 1 with KQ = K”3.

Pro08 1. The ?i, defined above guarantees that K. is obtained. For, if Ke = K I ,

this follows from [4], when the second and third players form a coalition. If they do not form a coalition, then, in accordance with a), b), and c), they lose, using%* m, as compared with the case when a coalition is formed. They therefore form a coalition,

and KO is obtained.

If Kc = K’, , the second player will choose x~=%~‘, and the third x~=‘x~‘, then

K,, will be obtained. For, if the second player refuses the behaviour preassigned to him, then, by virtue of 5?, m and c), he will not obtain more than ~5’~. But if the second player follows the prescribed behaviour, the third must choose x3=‘x3’, since otherwise,

as a result of the fust and second using Z, 2 n he will not obtain more than Ls , whereas ,

if he pays attention he can rigidly count on more (he receives information about the

first and second player’s strategies).

If KO = Kff2, it can be seen in exactly the same way that x3 must not leave Es, while the second player then adheres to 2x2 P (x3) .But then, the first player is obviously

guaranteed K”2. The remaining cases are proved shnllarly.

2. We will show that any strategy Z~=G (G, x3) guarantees the first player not more than Ko, regardless of what information is given to the first player.


For, if the second and third players then act in coalition, the first player cannot guarantee obtaining more than Kr <Ke, see [4] , If there is no coalition, this can only

happen with x2 and x3 such that at any rate one of these players obtains more than the coalition guarantees, i.e,, more than L’2 or L’ a. Suppose that this is achieved by the second player. Then, the third either obtains more than L3, or obtains exactly L3 (he cannot obtain less, since he can always take x+E~). In the first case [x1 (x2, x3), x2, x3] ells’,

and the result of the first player is then not greater than Kr2 < KO. In the second case, the third player, independently of the first, can choose an arbitrary x3 of E3. With this x3, the second player counts on obtaining more than Lf2. This means (with the first player possibly obtaining information about x3) that x2 is such that

but then, (x1 (G, 1~~) , x2) =D2 N (x3), and hence the first player cannot obtain more than x” r2 gives, while in view of the arbitrariness of x&Z~, he cannot be guaranteed more than K”2 < KO . The state of affairs is similar if the third player obtains more than in the coalition, i.e., more than L’3 ..The theorem is proved.

Notes, 1. The first player’s guaranteed result is in general smaller in the case of a rigid coalition, since it is then equal to K I *

2. If we assume that, in the absence of a coalition, the second and third players adhere to the principle of the guaranteed result while having no independent information on each other’s moves, the condition c) can be dropped, as can the requirement that the first player establishes the order in which the other two players move.

For then, taking, say, the second player, his penalty (in the absence of a coalition) can consist in his not being told about the behaviour of the other player, while being

told the 5?r minimizingfa(xl, x2, x3) with arbitrary x2 and x3. Since x3 is unknown, the second player can only be guaranteed the result L2 < L12. This penalty will apply whatever the order of the moves, whether fixed or not. But if the players form a coalition, they are penalized as before by means of x1 m. In this case it would seem that they might

also be penalized by indifference or antagonism to the communicated target function

The case when it is known a priori that the players form no coalitions at all can be considered under the same assumptions. Then, we only have to omit K1 and replace Lr2 and L’3 by L2 and L3 throughout, which leads to a widening of the corresponding regions, and hence to an increase in the relevant K. Common sense would suggest that an increase is to be expected in the first player’s guaranteed result, but this actually requires examination in view of the absence of the term K, in the expression for Ko.

118 Yu. B. Germeier

Case 3. The first player does not know what precise coalition can be formed by the second and third players. We shah assume a loose coalition, which only satisfies the following conditions.

a. The coalition is without collateral payoffs and ensures that, given reasonable moves of its part, the partners will obtain not less than L2 and L3. This means that the coalition always chooses x2 and x3 such that f2 (x,, x2, z3) >L2, f3 ( xi, x2, x3) >L3. In view of this, we distinguish below the region

D*={(s,. 5a) 1 min f&L,, min f3>L3), 21 21

in which the first player cannot have a guaranteed action on the other players.

b. It is also known that the coalition guarantees the second player (given the worst behaviour of the first) a result not greater than L*2, and the third a result not greater than L*3. Clearly, since there are no collateral payoffs,

L,*< max min f3, I,,*< max min f3. (-Tz, 23) x1 (%,%) 21

Hence, if there is no information at alI about the coalition (except for the fact that there are no collateral payoffs), L*2 and L *3 will in fact be equal to the maximins indicated (in the case of sufficiently accurate information, L,‘=L,‘, L,*=L,') .

c. As above, we shall assume that the second and third players, even outside the coalition, cannot count on obtaining more than L *2 and L*3, if the first player does not

want them to. In the case of Lz*= max min fz and of the similar L*3, this is obvious, (X2,53) Xl

while in the other cases this condition will certainly be satisfied if the second and third

players are careful, even when they have no information on each other’s moves.

For the rest, the first player knows nothing about the coalition and may therefore encounter any mode of action of the other players within the limits of conditions a and b.

We now introduce the quantity

and the “absolute” optimal strategy ?ra realizing max f, (xi, x2, a). As “penalizing”

strategies we shall use Zi m, realizing min min [ f,TL,; f3-LB] ; 21 2, realizing

min fz; Ei3, realizing min f3, and fini;, the Zi2’ and?,, ’ indicated above. Xl Xl

On the basis of L*? and L*3 (instead of L’? and Lf3) we introduce the regions

02*,D2**(23),D~*)I;)S**( 9)) al g 5 an o ous to 0’2, @‘2(x3), etc. Into these regions we also introduce KS*, KZ**, K,‘, K3** and the points realizing’them: otherwise


(2x1*, 2x2*, 2x3*)) Xl,2**= (2xi*’ (x3) ) 2x2** (x3) ) )

(3xi*, 3x2*, 3x3*), X13 - **.

Theorem 2

In the conditions stated, the first player’s maximum guaranteed result is

K,,=max CM, FL*, &**, KS*, &**I.

If Ke = M, Ke is realized with

If K,, = K*2, KO is realized by the strategy

- 2 xi= x1*, if x2=252*, x3=2x3*,

?“1=5,” (x3), if x+~x~*, but xz=xz”( x3),

f,‘$i2 otherwise

If K,, = K**2, K. is realized by the strategy

LiTi=2xlr* (X3)) if z~=~x~** (x3) and xSEES (with suitable

exchange of information). &=Lc~“(x~), if x2=xzn,but x#Es, ii?I=x",2 otherwise

Similar situations hold for KO = K*3 and KO = K**3, In all these cases, the

strategies are communicated to the second and third players.

Proof 1. If KO = M, and the second and third players do not adhere to D*, application of 2 1 m will imply minimization of whichever of fi (~1, .x2, .c) --Li, i=2, 3, is the smaller. Hence, with fixed x2 and x3, the result is

min I min fz (xl, x2, z&-L; min f3 (XI, 52, x3)--L31. XI 21

Since at least one of these differences is negative outside D*, at least one of the players

will obtain less than L i.

On communicating to one* of them (say the second), information about he move of the third, the first gives him the possibility of obtaining, with any xs ,

*When M is realized, the or&r in which the second and third players move can be fired. Then, the first player gives the information in the required way. For instance, if the second player in fact makes the second move, he ia only told that information about his move will be communicated to the third player, and the information is in fact transmitted.

120 Yu. B. Germeier

max min fz (x,, x2, zcs) by the appropriate strategy x2 (xa). Since, when x+E2,

in>epe:dently of xa ,

we have in general

and hence, if (az (x,) , x3) eD’,the third player’s payoff is less than L3 ; if he does not agree with this he can always choose x3~E3, and this, while guaranteeing him L3, will

at the same time “pen” (x2(x3), xs) in D*.

The second and third players will thus not go outside D*. But, with (x2, x3) ED*,

the first player, on using?i’, will be guaranteed M. In the case Ke = K*2, the penalty of the third, jointly with the second, will ensure that the third obtains x3 = 2x3 *. If, after

this, the second does not take 2x2 * or will not participate in the penalty of the third, then, by virtue of condition c), the application of Zii, 2 will not enable him to obtain more than L*2, on which he can fully count when he acts jointly with the first player (here, of course, we make the assumption that the second player knows that the first is exactly aware of the interests of all the players).

In other cases, we can prove in a similar way (see Theorem 1) that the Xi indicated are reasonable.

2. Suppose we are given any strategy x, (x2, x3). If the second and third players act as a coalition (according to the first player’s idea), we must have, in accordance with

condition a), fi(zi (x2, x3), x2, x3) >L, with i = 2,3. Since the players’ actions are not otherwise predictable, the guaranteed result for the first player is not greater than

since the region of points (~2, x3), at which ji (zl (x2, x3), x2, x3) >Li, certainly contains D*.

The players can then destroy the coalition, if at any rate one of them (say the second) can do better than the coalition. Of course the other then obtains not less than L3, since he can always take z+EQ. If, however, the second here obtains not more than L*2, the actions of the second and third players fall, from the point of view of the first, within the coalition framework (i.e., a coalition exists for which such results are achievable) and the first cannot distinguish between the x2 and x3 which satisfy conditions a) and b), regardless of whether there is actually a coalition or not. On gathering all the a priori possible coalitions within the indicated conditions and counting


on his guaranteed result, the first player here obtains M < Ke . On the other hand, if the (actual) coalition is destroyed so that e.g., the second player obtains more than L*s ,

arguments come into force exactly similar to those in Theorem 1, and the fust player cannot count on more than max [ K2*, Kz**] GK,.

The situation is similar if the third player obtains more than L*3. The theorem is proved.

Case 4. Theorem 2 may easily be extended to the case when the unknown

coalition is a coalition with collateral payoffs (or may be such a coalition). Then, condition a) is replaced by the wider condition

fi (51, x2, &) +fS (Xi, zz, ZS) 2L,fL,.

The set D* is correspondingly replaced by

~={@z~ 4 I min U2+f3)>L2+L31.

In condition b), we only require tha:

L”O$xjrn~ vz+fd

Then, ;Si m is best defined as realizing

min [f2tfsl. x1

For the rest, the statement and proof of the theorem should remain unchanged. But it needs to be noted that, if

then the guaranteed breakdown of the coalition is in fact impossible, i.e., D2 * = a. This situation is quite realistic, if fa = f3 and depends strongly on x3 but not strongly on x1 . In these circumstances, the first player also should use collateral payoffs for control

(i.e., increase the dependence of f2 on his actions).

Notice that the remarks made on Theorem 1 naturally also apply to Theorem 2.

It is also clear that the possibility of the second players using a coalition with collateral payoffs decreases Ke , since it increases D* and decreases the remaining regions, due to the increase in L*i.

The last note on Theorem 2 refers to the possibility of the first player obtaining

a revised value Of L*i from the player who wants to gain by leaving the coalition, but does not count on obtaining an amount greater than the L*i known a priori to the first player. In practice, this can mean the first player asking about the actual L*i for which a revised value can be given to him (if this is convenient to the player who is asked).

122 Ytr. %. Genneier

The mo~~cations to the statements of problems noted above should undoubtedly be taken into account in a deeper study of the rational choice of strategies in three-person

games.

Translated by D. E. Brown

REFERENCES

1. OWEN, G., Game theory, Saunders, 1968.

2. GERMEIER, Yu. B., Introduction to theory of operational research (Vvedenie v teoriyu issledovaniya operatsiil, Nauka, Moscow, 1971.

3. GORELIK, V. A,, Principle of the guaranteed result in non-~~ga~stic two-person games with exchan8e of information, in: operahcmai research (Issl. opera&ii), 2, VTs Akad. Nauk SSSR, Moscow, 10%118,197l.

4. GERMEIER, Yu. B,, Two-person games with a fued sequence of moves, Dokl. Akad. Nauk SSSR, 198,5, 1001-1004, 1971.

5. ERESHKO, F. I., and KONONENKO, A. F., Solution of a game with a right to the first move and inexact information on the partner’s objective, Zh. vjVhis1. Mat. mat, Fiz., 13,1,217-221, 1973.

6. KUKUSHKIN, N. S,, The role of mutual information of the sides in games of two players with non-antagonistic interests, Zh. @hi& Mat, mizt. Fir., 12,4, 1029-1034, 1972.

7. KONO~NKO, A. F., The role of ~formation on the opponent’s target function in two-person games with a fvied sequence of moves, Zh, vj%hisl. Mat. mat. Fir., 13,2,311-317, 1973.

8. GERMEIER, Yu. B., and MOISEEV, N. N., Some problems in the theory of hierarchical control systems, in: Problems of applied mathematics and mechanics (Probl. p&l. matem. i mekhan.), N&a, Moscow, 30-43,1971.

9. HOWARD, N., Theory of metagames, Management Sci. Center, Univ. of Pennsylvania, 1966.

IO. GERMEIER, Yu. B., Game concepts in systems research, Zzv. Akad. Nauk SSSR, Tekhn. kibernetika, 2,.25-33, 1970.

11. GERMEIER, Yu. B., and KONONENKO, A. F., Games with auxiliary efficiency criteria, Zzv. Akad. Nauk SSSR. Tekhn kibemetika, 1,3-10, 1973.

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The theory of three-person games