• Nie Znaleziono Wyników

On the structure of two-person, finite, zero-sum games

N/A
N/A
Protected

Academic year: 2022

Share "On the structure of two-person, finite, zero-sum games"

Copied!
5
0
0

Pełen tekst

(1)

ZD ZISŁA W W YD ERK A*

ON THE STRUCTURE OF TWO-PERSON, FINITE, ZERO-SUM GAMES

Abstract. The algebraical and topological structure of the set S? resp. of all n x m matrix games with saddle points (resp. with unique saddle point) in the space R"*m of all such games is studied. It has been shown that S f is a closed cone with vertex zero and includes the origin. Moreover, it is neither convex nor dense subset of R" *m. The s e t i s a non-convex cone which does not include the origin. It is neither closed nor open.

The concept of “reserve of non-saddlexity” has been also introduced.

1. Introduction. In his papers [8, pp. 43—44], [9, p. 9] N .N . Vorobyov is regretted th at the authors of the papers from game theory consider only some p articular games and the facts related to those games but the general classes or spaces of the games and some particular subsets of these spaces are n ot in consideration. In this note we study some properties of the subsets of the space

'SC of all m atrix games which contains:

a) the games with saddle points (the set S ’),

b) the games with unique saddle point (the set S j).

This is proved th at S ’ and S ’ v are cones with vertex 0 (0 e S ’ while 0 ^ S ' ^ which are not convex sets. S ’ is a closed subset of SC which is not dense in 3C and — in general — is now here dense in SC. In general, Sf x is not closed no r open subset of 9C. The concept of “reserve of non-saddlexity” of the game A S' is introduced as the distance between A and the set S ’.

2. Algebraical and topological structure of the sets S and S ’ x. Let us consider the space SC of all two-person, finite, zero-sum games r with the fixed sets of pure strategies: {1, . . . , n} for the first and {1, . . . , m} for the second player, n , m > 1. Let A = [Oj.j], i = 1, . . . , n , j = 1, . . . , m be the payoff m atrix of the game r ; then we identify the game r with the m atrix A (we will write r and A exchangeable), so, the spaces SC and R "x m are isom orphic. The space SC with the usual operations and with the norm

(1) ||/1|| = max max Ifljjl

1 j

is a Banach space. Let us denote by S ’ (resp. S ’1) the set of all games with a saddle point (resp. with the unique saddle point). M ore precisely, A e S ’ iff max m in aU} = m in max aitj = a io jo and A e S ' t iff there is a unique pair (i0, j0)

i j j i

with above property. Obviously, S ' 1 c S ’.

In this paper the algebraical and topological structure of the sets S ’ and Sf x will be studied. F rom the well-known properties of the m ax min and min max operations it follows the following

R eceived Septem ber 15, 1983.

A M S (M O S) Subject classification (1980). Primary 90D 05. Secondary 49A45.

*Instytut M atem atyki U niw ersytetu Śląskiego, K atow ice, ul. B ankow a 14, Poland.

(2)

T H E O R E M 1. The set i f is a cone with vertex 0 which includes the origin while i fj is a cone which does not includes the origin.

Let A e i f (resp. A e i f x), A ^ 0 be an arbitrary game. Then the line i f such th at A e i f which is parallel to the line jV ( T = [ b ,j] e yK iff b\ ,\ = ^1,2 = ••• = bn m) is also included in the set i f , i f c i f (resp. i f c: i f x), so, by Theorem 1

(2) C o n { if ) c= i f (resp. C o n (if) c i f t),

where Con(&) denotes the conical hull of the set <&. In general Con ( if ) is a proper subset of the tw o-dim ensional subspace (plane) determ ined by i f and 0. The sets i f and i f x are set-theoretical sums of such a parts of those planes.

EX A M PLE 1. Let n = 2, m = 3, A = 1 2 3 4 5 6 e i f .

Then —A = -1 - 2 - 3

-4 - 5 - 6 g i f also.

By Theorem 1 there exist a one-dim entional subspace of R6 which is included in

i f , but by (2) there exists a tw o-dim ensional subspace with this property.

In the case n = m A. I. Sobolev proved [7] that the m aximal dimension of the subspace of 9C included in i f is equal (n —1)2 + 1 for n ^ 3 and is equal 3 for n = m — 2.

R E MA R K 1. The sets i f and i f x are not convex.

To prove this rem ark let us consider the following

EX A M PLE 2. Let n = m, A l = diag (al5 . . . , a i_1, 0 , a i+1, . . . , an), A 2 =

= diag (bl , . . . , bk- i , 0 , b k+x, . . . , bn) where i ^ k, a; , b, > 0 for j =£ i, I ^ k. We have A x, A 2 e i f y a i f but for X g (0 , 1), XAx +(1 X)A2 $ i f .

T H E O R E M 2. i f is a closed subset o f 3C.

P r o o f. It suffices to prove th at 3C \ i f is open in 3C. Let T = [a ,/J $ i f . Denote k = m ax min a, j and K = min max Then K — k > 0.

i j j i

Define e = ^ ( K — k) and let

2, t

W ( r , e ) = {[aUj + cifl]; \cUJ\ < e for i = 1, . . . , n, j = 1, . . . , m}

be an open heighbourhood of the game r . Let r x = [ h ,j] g ^ ( / \ e ) be an arbitrary game. Then

max min bLj < max m in(a, j + e) = k + e,

j > j

min max bt j > min m ax (a, j — e) = K — e.

j j >

therefore T, $ i f , so, ^ l ( T, s) n i f = 0 .

(3)

C O R O LLA R Y . Sf is not a dense subset o f SC.

In general, Sf is not now here dense in 3C. We prove this fact by contradiction.

F o r to be a nowhere dense subset of SC it suffices to prove by [5, Ch. XI, §4, Theorem 3] th at in an arb itrary ball ^ ( r , e ) there is a ball aU'{T',t!'),

< r ( r ,£ ') c <%(r,s) such th a t < r ( r , s ’)0 ^ = 0. Let us consider the following EX A M PLE 3. Let n = m = 2, T = 2

1,9 e < 0,45. Then but in the ball ° il {r , e) there is no ball aW with desirable property.

In general the set ^ is not closed nor open. Let us consider the following EX A M PLE 4. Let n = m = 2, r = e and let 0 < e < e < 1

Then J \ 1 1 — s

0 2 e ^ ( r , e ) but J \ so, Sf x is n ot open set. But the interior of Sf x is non-em pty by Exam ple 3.

3. The concept of “ reserve of non-saddlexity” . Now we introduce one new idea which is related to some ideas know n from stability theory and controllability theory of linear autonom uous dynam ical systems.

The space of all linear hom ogeneous autonom uous systems x = A x , x e R",

is isom orphic with the space R" *" of all quadratic matrices A. In this space the set of all stable systems is an open cone with vertex 0 (0 £ HT). F o r an arb itrary system A e W we m ay define some num ber d > 0 (which is called “reserve o f stability”) as the distance between A and the nearest unstable system. Therefore all systems {A + C\ \ct J\ < 5 for all i , j ) are stable still.

Similarly, the space of all linear au tonom uous control systems x = A x + Bu, x e R " , u e R m,

m ay be identified with the space R" *(n + m) of all pairs of m atrices ( A , B) of the dim ensions n x n and n x m respectively. In this space the set of all controllable (in K alm an sense) systems is a cone with vertex 0 (0 ^ 3£) and is an open set. F o r the system ( A , B) it is defined [3] some num ber y > 0 (which is called “reserve o f controllability”) as the distance between the system ( A , B ) and the nearest uncontrollable system. Then all perturbed systems {(A , B) + (C, D)} such th at the norm of p ertu rb ation is smaller th an y are controllable.

By an analogical way we define the idea of “reserve of non-saddlexity”. Let F $ be an arbitrary game.

D E F IN IT IO N . Reserve o f non-saddlexity of the game r iś the distance a (F) (in the norm (1)) between the game r and the set Sf.

(4)

If r then a ( r ) > 0 and for T e 9 we have a ( r ) = 0. If we perturbe the m atrix A by some m atrix G such th a t |gUj\ < a ( r ) for all i , j then the perturbed game F — A + G $ 6 f .

Those three ideas are very im p o rtan t if we know the elements of the corresponding m atrices with some given accuracy (not exactly) or by some experim ental data. In practice, calculation of the num bers d,y,<x(F) is very difficult.

R E M A R K 2. The idea “reserve o f saddlexity” which m ay be define in the sim ilar way is n ot well-defined: there are some games from £f with positive reserve of saddlexity (as in the Exam ple 4) an d some other games with null reserve of saddlexity (as in the Exam ple 3).

4. Few words about the games without saddle points. N ow we give a few words ab o u t those games from which have a solution in mixed strategies (denote this set by 9 >). Let denotes the set of all games from 3C with the unique solution in mixed strategies. Obviously, Zf c= Żf, Ł <=. x, Ż?x c 9*. By the well-known J.

von N eum ann theorem we have Ś? = SC. The set is a cone with vertex 0 (0<£ and, by the next example, this is not convex set.

E X A M PL E 5. Let n = 2, m = 3, A l = 100 170 80 120 1000 280

A 2 —

100 10 20

90 30 110 , A = 0,1. We have A 1, A 2 e 6 f 1 c Sf x and denote by A th e game

A = XA1 + { l - X ) A 2 = 100 26 26 93 127 127 The optim al mixed strategies are: 34 74

108’ 108 for the first player and

101 7

108’ ’ 108 or 0 ). for the second one. Therefore A & S f , .

108 108 ' 1

Topological structure of the set was studied in [1], This is proved th at P x is an open, dense subset of the space SC.

5. Some other classes of games. Some similar problem s for other classes of games were studied in [2], [4], [6].

In [2] this is proved th a t in the space of all continuous games over unit square the set of all games with the unique solution is a dense set of G a-type.

In [4] some m axim al subspace of the linear space of all continuous games over unit square is constructed such th at every game from this subspace has a saddle point.

(5)

By using of some m odification of the well-known Lucas’ example the a u th o r of the paper [6] gives one hypothesis concerning the set of all cooperative n-person games which have a solution is a dense subset of the space of all cooperative n-person games.

REFERENCES

[1] H. F. BOHNENBLUST, S. KARLIN, L. S. SHAPLEY, Solutions o f discrete two-person games, in Contribution to the theory o f games I, Princeton 1950, 51— 72.

[2 ] G. N. D YU BIN . Concerning the set o f the games over unit square with the unique solution, Dokl.

Akad. N auk, SSSR , 184, N o. 2 (1969), 267— 269 (in Russian).

[3] J. K L A M K A ,Estimation o f controllability and observability by canonical Jordan form , Podstawy sterowania, vol. 4, N o. 4 (1974), 349— 369 (in Polish).

[4 ] V. W. KREPS. On m axim ality o f some linear spaces o f continuous functions with saddle points, in Achievements o f game theory, Vilnius 1973, 47— 50 (in Russian).

[5 ] K. KURATOW SKI, An introduction to set theory and topology, Warszawa 1966 (in Polish).

[6] N. Y. N AU M O W A, On the existence o f solutions fo r cooperative games, in Game theory, Erewan 1973, 253 (in Russian).

[7 ] A. I. SOBOLEV, On dimension o f linear space o f matrices with saddle points, in Achievements o f game theory, Vilnius 1973, 66— 68 (in Russian).

[8] N .N . VOROBYEV, Present-day state o f game theory, in Game theory, Erewan 1973, 5— 57 (in Russian).

[9 ] , Scientific results o f conference, in Achievements o f game theory, Vilnius 1973, 7— 13 (in Russian).

Cytaty

Powiązane dokumenty

Application of a linear Padé approximation In a similar way as for standard linear systems Kaczorek, 2013, it can be easily shown that if sampling is applied to the

Henning, Trees with equal average domination and independent domina- tion numbers, Ars Combin. Sun, On total restrained domination in graphs,

(For the case q = 1, this proof was also given in [11].) In fact, it shows that certain cases of Theorem (3.1) are equivalent to Doob’s results.. We end the section by deriving the

In the proof of this theorem, the key role is played by an effective interpretation of the well-known fact that an irreducible polynomial which is reducible over the algebraic

Beginning in the seventies the multivalued Cauchy problem in abstract spaces has been studied by many authors; we mention the existence theorems obtained by Chow and Schuur

In this paper we use recent results on the Lyapunov spec- trum of families of time varying matrices [11] in order to characterize the domain of null controllability of bilinear

Within this approach, in [2]–[7], [13], [18], [21] qualified decay was obtained for the first boundary value problem for the wave equation in an exterior domain for which the

Following the spectacular result of Drury (“the union of two Sidon sets is a Sidon set”), a lot of improvements were achieved in the 70’s about such sets Λ.. Rider, in