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II Some Geometric Aspects of Quantitative Games
II
Some Geometric Aspects o j Quantitative Games
2.1 TRANSFER OF STATE AND PERFORMANCE INDEX
As we have seen in Chapter I , given an initial state xi in G, the rules of the game, the strategies r p and rE chosen by players J,. and .IE from prescribed sets R, and RE,respectively, and a function x E ?(xi,rI,, ra2), the state of the game moves along describing curve rX(xi, r,, r E ) . Here we shall consider the paths which are generated by all strategy pairs. In particular, we shall be concerned with transferring the state along 9 path from an initial state xi to any one in prescribed set of states 0 in G. In general, some paths will correspond to a transfer of the state from xi to some terminal point xf in 8, while others will correspond to a transfer from xi to a terminal point which does not belong to 8. Definition 2.1. Next let us adopt a functional which assigns a unique real number to each transfer effected by a strategy pair along a path. Such a functional will be called a performance index, and the number which it assigns to a transfer from initial point xi to end point xs,along a path risin G , will be called the cost of the transfer.
We shall admit a performance index such that the cost of transfer from xito xsalong path risdepends on the generating strategies r p and rR. We shall denote the cost of such a transfer by V(xi, xs;r I , , r E , ris). Now we introduce Assumption 2.1. The cost which is associated with a null path no is zero; that is Y ( x i , x i ; r p , rE, TO) = 0 V r , E R,, V r , E R, (2.1)
12 2.2
I1
SOME GEOMETRIC ASPECTS OF QUANTITATIVE GAMES
OPTIMALITY
I t will be convenient to consider a quantitative game as a game with two targets O,, and O,, for which 8,. = 0, = 8
Then, for any initial state xi, a strategy pair (rl+ Y E ) which is playable for
J,, at point xi is also playable for J E at that point. Accordingly, in that case, we shall say that (rI+ re) is a playable strategy pair at point xi, without specifying the player who is concerned, and we shall let ~F(xi) denote the set of all playable pairs ( r p , re) at point xi. We shall let V ( x i ,0 ; rz,, r E ) denote the cost of transfer from xi to 8 along a path rifgenerated by ( r p , r E ) E .F(xi). Since, for given xi and (r,,, rTJ € . F ( x i ) , rif need not be unique, V ( x i ,8; r p , r,) need not be unique. Along nif we have V ( x i , 8 ; rl,, rE) = %'-(xi, xf; r,,, r E , .riff>
(2.2)
Definition 2.2. Let there be given a fixed set X c G . We shall say that a strategy pair (rl,*, r E * ) is optimal on X , if (i) it is playable at all initial points xi E X ; and (ii) V ( x i , 8; rz,*, rB;*) is defined for all xi E X ; that is, if there is more than one path starting at point x i , generated by ( r p * , rE*) and terminating on 8, then all the values of V(xi, 8; rI,*, r E * ) are equal; (iii) the saddle-point condition V ( s i , 8; rI,*, rI.:)
< V(xi,0 ; r p * , r R * ) Q
V ( x i , 8 ; r p , r E * ) (2.3)
holds for all x i E X and for all strategy pairs ( r p * , r E ) E Y ( x 2 ) and (rt+ rrc*)E 7(xi).
We emphasize that (2.3) must hold for all values of V ( x i , 8; rp*, r E ) and all values of V ( x i ,0 ; r P , rh;*). Definition 2.3. Since V(x', 8 ; rp*, rE*) is defined for all xi s' V*(xi),where V*(xi) L V ( x i , 8 ; rp*, r E * )
E
X, V * :
(2.4)
is a (single-valued) function on X . We shall call V * ( x i ) the Value of the game for play starting at xi.
If (i)-(iii) are satisfied for all xi
E
X , then, by Assumption 2.1, (i)-(iii)
2.3
13
GAME AND ISOVALUE SURFACES
are satisfied for all xi
E
X U 6; and, of course, for xi E 6 we have V*(XZ) =
Let X *
0.
A X u 6.
2.3 GAME AND ISOVALUE SURFACES
+
Let us introduce another variable, xo, and consider an (n 1)-dimensional Euclidean space En+l of points x , where x = (xo, x) = (x,,, x,, . . . , x,) is the vector which defines a point relative to a rectangular coordinate system in En+l, the augmented state space. Definition 2.4. Since V * : xz+ V*(xi)is defined on X * , we can define a game surface X(C) in X* A X* x {xo} by
C(C)
A {x:
xo
+ V * ( x ) = C}
(2-5)
where C is a constant parameter. The intersection of E(C) with X * will be called an isovalue surface7 S(C)
& {x:
V * ( x ) = C}
(2.6)
Surface S(C) is the locus of all initial states for which the Value of the game is the same, namely C ; hence the name isovalue surface. As the value of parameter C is varied, Eqs. (2.5) and (2.6) define one-parameter families of surfaces {C(C)) and {S(C)). For each given C, Eq. (2.5) defines a single-sheeted surface in %*; that is, X(C) is a set of points which are in one-to-one correspondence with the points of X * . Consider now two game surfaces C(C,) and X(C2), corresponding to two different parameter values C, and C,, respectively. Let xol and xO2 denote the values of xo on C(C,) and C(C2), respectively, for the same state. Then it follows from (2.5) that xol - x o 2 =
c, - c,
vx E X ”
Consequently, the members of the one-parameter family {C(C)> may be deduced from one another by translation parallel to the xo-axis. Furthermore, these surfaces are ordered along the xo-axis in the same way as the parameter value C ; that is, the “higher” a surface in the family the f Note that S(C) may be an empty set.
14
11
SOME GEOMETRIC ASPECTS OF QUANTITATIVE GAMES
greater is the parameter value C. Thus one and only one game surface passes through a given point x in X*. Now let us introduce some nomenclature. Definition 2.5. A given game surface C( C ) separates the set %* into two disjoint sets. We shall denote these sets by A / C ( C ) [“above” C(C)] and B/C(C) [“below C(C)”], respectively. For a game surface X(C)
Frci. 2.1. Game and isovalue surfaces.
corresponding to parameter value C , we have A/C(C) & ( x :
B/C(C) 4 { x :
> c - Y*(x)> xo < c - V * ( x ) } xo
(2.7)
(2.8)
A point x E A/C(C) will be called an A-point relative to X(C), and a point x E B/C(C) a B-point relative to C(C).
2.4
I
15
PATHS I N AUGMENTED STATE SPACE
\\
I
' \ \ I
II
XE
S/C(C)
"
I I
FIG.2.2. A-point and B-point relative to C(C).
Hence, the equation of the game surface through a point xi E X *is
xo + V * ( x ) = xgi and
+ V*(xi) = c
(2.9)
> V*(xi)- V*(x)} - xgi < V*(xi)- V*(x))
A / C ( C ) = { x : xo - xgi
(2.10)
B/C(C) = ( x : x,
(2.11)
2.4 PATHS IN AUGMENTED STATE SPACE
Definition 2.6.
Paths n i s ( C )in 3 & G x {x,} are defined by
niS(C)& { x ' :
xgk
+ V ( Xxs; " , r p , rE, nks)= C , rkS c nis} (2.12)
where rris is a path in G from xi to xsgenerated by strategies r , rE E RE,and C is a constant parameter.
E
R, and
Thus, path nisis the projection on G of paths nis(C)in 9. Let 0 A 8 x {xo} be the target in augmented state space. One can
16
SOME GEOMETRIC ASPECTS OF QUANTITATIVE GAMES
I1
X’
Xi
FIG.2.3. Path in augniented state space.
easily verify that the above definition of a path in 3 agrees with the general definition of a path given in Section 1.3. By varying the value of parameter C in (2.12) one generates a oneparameter family of paths {IIis(C)}. This family belongs to an x,,cylindrical surface whose intersection with G is path ris. Definition 2.7. Three families of paths emanating from points in %*, (n;i,(C)},{np(C)},and (II;&(C)>, are defined by TI$(C) = { x k :
xp + ~ ( x ” x, f ; rIJ*,TI.,.,
= C, r;;c
n-2) (2.13)
n ” , ” ~A) { x k :
x:
+ y ( x k , xm; r I J ,rR;* , r R ) = C, rp c r p > km
(2.14)
nrE(C)= { x k : x:, + Y ( x k ,xQ;rlJ*,rP*, r:yI;) A
1 j = C , n-PE c r nq PE
(2.15)
2.6
17
SOME PROPERTIES OF GAME SURFACES
Paths T$, T:? and T;% are paths in G, emanating from points xi,xz,and xn of X * , generated by the strategy pairs (rp*, rE), (rl>,rE*)and (rp*,rE*) and ending at points x3,9" and xQ,respectively. Definition 2.8. Since (rE,*,r$;*) is playable at all points xn E X * , there exists a path n;f,(C) which reaches 0 at point x f . It is called an optimal path. Hereafter we shall denote an optimal path by n*(C). Definition 2.9. If (rI,*, rE) and (rp, rE*) are playable strategy pairs at points xz and xl,respectively, there also exist paths n$(C) and ng(C) which reach @.t They are called P-optimal and E-optimal paths, respectively. Hereafter we shall write simply lI,(C) and IT,(C) to indicate these paths. 2.5
JOINING OF PATHS
Let ni3be a path emanating from xi,generated by ( r p , rE), and ending at x3; and let xis be a path emanating from xi,generated by (Ep, i E ) and , ending at xs. Tn addition to Assumption 1.1, according to which rii u xis is a path riS generated by ( p r , p6), we shall introduce Assumption 2.2
+ V ( x j ,xs; E,,
F,,
= Y ( d ,xs; p p ,
PE,
Y ( x i , x3; rl,, r E , TP)
nlS) T i S )
(2.16)
We shall call this assumption an additivity property of the cost. 2.6 SOME PROPERTIES OF GAME SURFACES
In preparation for a fundamental theorem, let us now prove Lemma 2.1. No point of a path rIp(C') which emanates from x i is an A-point relative to the game surface through x i . iOf course, superscript f means that the terminal point of the path belongs to 0. However the terminal point need not be the same for paths IIYfC), rIL!(C), and IIFL (C).
18
11
SOME GEOMETRIC ASPECTS OF QUANTITATIVE GAMES
\
\
Frci. 2.4. Paths
I
I
I
I
n;(C’), n;(C”), n$B(C”) and
\
game surface through xz.
Lemma 2.2. No point of a path n$(C”)which emanates from x i is a B-point relative to the game surface through xi.
Let x 3 = (so3,xj) be a point of a path rlg(C’) generated by (rI,*, r E ) , where . K ~is its projection on G. Let r i j c r Ii,sbe the path in G emanating from s’,generated by ( r p * , r E ) and ending at xj, which has the same supporting curve as T:. First note that, since the family of game surfaces is defined on X * , xj cannot be an A-point relative to any member of the family if xj 6X * . Hence let us suppose that xj E X*. By condition (i) in the definition of an optimal strategy pair, there exists a path ryE which emanates from xj, is generated by ( r p * , r E * ) and reaches 8 at point sf. From Assumption 1.1 it follows directly that T$ U rjpfEis a path r);. Let ( r p * , pE) be the strategy pair that generates this path. Path r;: reaches 8 at point x f , and associated with it is a value of the cost of transfer ~ ( x ’ ,6 ; r p * , pE) = V(x2,
x’; rp*, p E ,
rg)
(2.17)
2.6
19
SOME PROPERTIES OF GAME SURFACES
In view of the saddle-point condition we see that V ( x 7 ,8; rp*, p E )
< V ( x Z 0;,
V*(xz)
rp*, YE*)
(2.18)
From the additivity property of cost (2.16) we have Y ( x z ,x f ;rI.*, pE,
TI?)
= Y ( x z ,x3;rp*, rE, +)
+ V ( x 3 ,x f ;r,,*, rE*, n;fE)
(2.19)
where
V ( x ’ , xf;rI,*, rE*, + ),
n
= V*(x’)
(2.20)
From (2.17), (2.18), (2.19) and (2.20) we obtain Y ( x ’ , x’; rf,*, r E ,
+> V*(x’) Q
*(x3)
Then it follows from (2.13) and from the additivity property of cost that, along path IIY(C’), x,,’ - x:
and hence X”’
= Y ( x z ,x 3 ; rp*, rE, rg)
- xo” < V*(XZ) - V*(X?)
(2.21)
Finally, it follows from (2.21), (2.9) and (2.10) that x 3 I S not an A-point relative to the game surface through x’. Lemma 2.2 can be established by similar arguments. Lemmas 2.1 and 2.2 have the following corollaries. Corollary 2.1. All points in %* of a path nyE(C“’)whicfz emanates from x’ belong to the game surface through x’.
This corollary is a direct consequence of Lemmas 2.1 and 2.2 together with the definitions of A- and B-points relative to a game surface. Corollary 2.2. A path II$(C’) whose initial point is a B-point relative to C(C) has no A-point relative to C(C) and, of course, no point in C(C). Corollary 2.3. A path ng(C”) whose initial point is an A-point relative to C(C) has no B-point relative to C(C) and, of course, no point in C(C).
These corollaries are direct consequences of Lemmas 2.1 and 2.2, respectively, together with the translation property of game surfaces.
20 2.7
11
SOME GEOMETRIC ASPECTS OF QUANTITATIVE GAMES
UNIQUENESS OF THE VALUE OF THE GAME
Suppose there exist two optimal strategy pairs ( r l , * , rF;*)and (tp*,t E * ) ; that is, both satisfy conditions (i)-(iii) of Definition 2.2. Upon applying these conditions we arrive at once at Lemma 2.3. If and i j
XI
E
(r,>*,
X * , both (rl.*, rfi;*) and (FI,*, FK*) are optimal on X * , E .F(x') and
( F P * , r g * ) E F(.Y'),
then
V ( s ' , 8 ; r I , * ,rR;*) = V ( Y , 0;
FE*)
I n other \c~orrls,the Value of the game V * ( s' ) is independent of the choice of optimal straregy pair. We also have Corollary 2.4. If both (rl,*, r E * ) and (tl,*,Fx*) are optimal on X * , antlif(rp*, i,.*) and (F,,*, Y E * ) areplayable for allxi E X , then ( r I > *tE*) , and ( F T , * , rr:.*)are also optimal on X * .
Some Geometric Aspects o j Quantitative Games
2.1 TRANSFER OF STATE AND PERFORMANCE INDEX
As we have seen in Chapter I , given an initial state xi in G, the rules of the game, the strategies r p and rE chosen by players J,. and .IE from prescribed sets R, and RE,respectively, and a function x E ?(xi,rI,, ra2), the state of the game moves along describing curve rX(xi, r,, r E ) . Here we shall consider the paths which are generated by all strategy pairs. In particular, we shall be concerned with transferring the state along 9 path from an initial state xi to any one in prescribed set of states 0 in G. In general, some paths will correspond to a transfer of the state from xi to some terminal point xf in 8, while others will correspond to a transfer from xi to a terminal point which does not belong to 8. Definition 2.1. Next let us adopt a functional which assigns a unique real number to each transfer effected by a strategy pair along a path. Such a functional will be called a performance index, and the number which it assigns to a transfer from initial point xi to end point xs,along a path risin G , will be called the cost of the transfer.
We shall admit a performance index such that the cost of transfer from xito xsalong path risdepends on the generating strategies r p and rR. We shall denote the cost of such a transfer by V(xi, xs;r I , , r E , ris). Now we introduce Assumption 2.1. The cost which is associated with a null path no is zero; that is Y ( x i , x i ; r p , rE, TO) = 0 V r , E R,, V r , E R, (2.1)
12 2.2
I1
SOME GEOMETRIC ASPECTS OF QUANTITATIVE GAMES
OPTIMALITY
I t will be convenient to consider a quantitative game as a game with two targets O,, and O,, for which 8,. = 0, = 8
Then, for any initial state xi, a strategy pair (rl+ Y E ) which is playable for
J,, at point xi is also playable for J E at that point. Accordingly, in that case, we shall say that (rI+ re) is a playable strategy pair at point xi, without specifying the player who is concerned, and we shall let ~F(xi) denote the set of all playable pairs ( r p , re) at point xi. We shall let V ( x i ,0 ; rz,, r E ) denote the cost of transfer from xi to 8 along a path rifgenerated by ( r p , r E ) E .F(xi). Since, for given xi and (r,,, rTJ € . F ( x i ) , rif need not be unique, V ( x i ,8; r p , r,) need not be unique. Along nif we have V ( x i , 8 ; rl,, rE) = %'-(xi, xf; r,,, r E , .riff>
(2.2)
Definition 2.2. Let there be given a fixed set X c G . We shall say that a strategy pair (rl,*, r E * ) is optimal on X , if (i) it is playable at all initial points xi E X ; and (ii) V ( x i , 8; rz,*, rB;*) is defined for all xi E X ; that is, if there is more than one path starting at point x i , generated by ( r p * , rE*) and terminating on 8, then all the values of V(xi, 8; rI,*, r E * ) are equal; (iii) the saddle-point condition V ( s i , 8; rI,*, rI.:)
< V(xi,0 ; r p * , r R * ) Q
V ( x i , 8 ; r p , r E * ) (2.3)
holds for all x i E X and for all strategy pairs ( r p * , r E ) E Y ( x 2 ) and (rt+ rrc*)E 7(xi).
We emphasize that (2.3) must hold for all values of V ( x i , 8; rp*, r E ) and all values of V ( x i ,0 ; r P , rh;*). Definition 2.3. Since V(x', 8 ; rp*, rE*) is defined for all xi s' V*(xi),where V*(xi) L V ( x i , 8 ; rp*, r E * )
E
X, V * :
(2.4)
is a (single-valued) function on X . We shall call V * ( x i ) the Value of the game for play starting at xi.
If (i)-(iii) are satisfied for all xi
E
X , then, by Assumption 2.1, (i)-(iii)
2.3
13
GAME AND ISOVALUE SURFACES
are satisfied for all xi
E
X U 6; and, of course, for xi E 6 we have V*(XZ) =
Let X *
0.
A X u 6.
2.3 GAME AND ISOVALUE SURFACES
+
Let us introduce another variable, xo, and consider an (n 1)-dimensional Euclidean space En+l of points x , where x = (xo, x) = (x,,, x,, . . . , x,) is the vector which defines a point relative to a rectangular coordinate system in En+l, the augmented state space. Definition 2.4. Since V * : xz+ V*(xi)is defined on X * , we can define a game surface X(C) in X* A X* x {xo} by
C(C)
A {x:
xo
+ V * ( x ) = C}
(2-5)
where C is a constant parameter. The intersection of E(C) with X * will be called an isovalue surface7 S(C)
& {x:
V * ( x ) = C}
(2.6)
Surface S(C) is the locus of all initial states for which the Value of the game is the same, namely C ; hence the name isovalue surface. As the value of parameter C is varied, Eqs. (2.5) and (2.6) define one-parameter families of surfaces {C(C)) and {S(C)). For each given C, Eq. (2.5) defines a single-sheeted surface in %*; that is, X(C) is a set of points which are in one-to-one correspondence with the points of X * . Consider now two game surfaces C(C,) and X(C2), corresponding to two different parameter values C, and C,, respectively. Let xol and xO2 denote the values of xo on C(C,) and C(C2), respectively, for the same state. Then it follows from (2.5) that xol - x o 2 =
c, - c,
vx E X ”
Consequently, the members of the one-parameter family {C(C)> may be deduced from one another by translation parallel to the xo-axis. Furthermore, these surfaces are ordered along the xo-axis in the same way as the parameter value C ; that is, the “higher” a surface in the family the f Note that S(C) may be an empty set.
14
11
SOME GEOMETRIC ASPECTS OF QUANTITATIVE GAMES
greater is the parameter value C. Thus one and only one game surface passes through a given point x in X*. Now let us introduce some nomenclature. Definition 2.5. A given game surface C( C ) separates the set %* into two disjoint sets. We shall denote these sets by A / C ( C ) [“above” C(C)] and B/C(C) [“below C(C)”], respectively. For a game surface X(C)
Frci. 2.1. Game and isovalue surfaces.
corresponding to parameter value C , we have A/C(C) & ( x :
B/C(C) 4 { x :
> c - Y*(x)> xo < c - V * ( x ) } xo
(2.7)
(2.8)
A point x E A/C(C) will be called an A-point relative to X(C), and a point x E B/C(C) a B-point relative to C(C).
2.4
I
15
PATHS I N AUGMENTED STATE SPACE
\\
I
' \ \ I
II
XE
S/C(C)
"
I I
FIG.2.2. A-point and B-point relative to C(C).
Hence, the equation of the game surface through a point xi E X *is
xo + V * ( x ) = xgi and
+ V*(xi) = c
(2.9)
> V*(xi)- V*(x)} - xgi < V*(xi)- V*(x))
A / C ( C ) = { x : xo - xgi
(2.10)
B/C(C) = ( x : x,
(2.11)
2.4 PATHS IN AUGMENTED STATE SPACE
Definition 2.6.
Paths n i s ( C )in 3 & G x {x,} are defined by
niS(C)& { x ' :
xgk
+ V ( Xxs; " , r p , rE, nks)= C , rkS c nis} (2.12)
where rris is a path in G from xi to xsgenerated by strategies r , rE E RE,and C is a constant parameter.
E
R, and
Thus, path nisis the projection on G of paths nis(C)in 9. Let 0 A 8 x {xo} be the target in augmented state space. One can
16
SOME GEOMETRIC ASPECTS OF QUANTITATIVE GAMES
I1
X’
Xi
FIG.2.3. Path in augniented state space.
easily verify that the above definition of a path in 3 agrees with the general definition of a path given in Section 1.3. By varying the value of parameter C in (2.12) one generates a oneparameter family of paths {IIis(C)}. This family belongs to an x,,cylindrical surface whose intersection with G is path ris. Definition 2.7. Three families of paths emanating from points in %*, (n;i,(C)},{np(C)},and (II;&(C)>, are defined by TI$(C) = { x k :
xp + ~ ( x ” x, f ; rIJ*,TI.,.,
= C, r;;c
n-2) (2.13)
n ” , ” ~A) { x k :
x:
+ y ( x k , xm; r I J ,rR;* , r R ) = C, rp c r p > km
(2.14)
nrE(C)= { x k : x:, + Y ( x k ,xQ;rlJ*,rP*, r:yI;) A
1 j = C , n-PE c r nq PE
(2.15)
2.6
17
SOME PROPERTIES OF GAME SURFACES
Paths T$, T:? and T;% are paths in G, emanating from points xi,xz,and xn of X * , generated by the strategy pairs (rp*, rE), (rl>,rE*)and (rp*,rE*) and ending at points x3,9" and xQ,respectively. Definition 2.8. Since (rE,*,r$;*) is playable at all points xn E X * , there exists a path n;f,(C) which reaches 0 at point x f . It is called an optimal path. Hereafter we shall denote an optimal path by n*(C). Definition 2.9. If (rI,*, rE) and (rp, rE*) are playable strategy pairs at points xz and xl,respectively, there also exist paths n$(C) and ng(C) which reach @.t They are called P-optimal and E-optimal paths, respectively. Hereafter we shall write simply lI,(C) and IT,(C) to indicate these paths. 2.5
JOINING OF PATHS
Let ni3be a path emanating from xi,generated by ( r p , rE), and ending at x3; and let xis be a path emanating from xi,generated by (Ep, i E ) and , ending at xs. Tn addition to Assumption 1.1, according to which rii u xis is a path riS generated by ( p r , p6), we shall introduce Assumption 2.2
+ V ( x j ,xs; E,,
F,,
= Y ( d ,xs; p p ,
PE,
Y ( x i , x3; rl,, r E , TP)
nlS) T i S )
(2.16)
We shall call this assumption an additivity property of the cost. 2.6 SOME PROPERTIES OF GAME SURFACES
In preparation for a fundamental theorem, let us now prove Lemma 2.1. No point of a path rIp(C') which emanates from x i is an A-point relative to the game surface through x i . iOf course, superscript f means that the terminal point of the path belongs to 0. However the terminal point need not be the same for paths IIYfC), rIL!(C), and IIFL (C).
18
11
SOME GEOMETRIC ASPECTS OF QUANTITATIVE GAMES
\
\
Frci. 2.4. Paths
I
I
I
I
n;(C’), n;(C”), n$B(C”) and
\
game surface through xz.
Lemma 2.2. No point of a path n$(C”)which emanates from x i is a B-point relative to the game surface through xi.
Let x 3 = (so3,xj) be a point of a path rlg(C’) generated by (rI,*, r E ) , where . K ~is its projection on G. Let r i j c r Ii,sbe the path in G emanating from s’,generated by ( r p * , r E ) and ending at xj, which has the same supporting curve as T:. First note that, since the family of game surfaces is defined on X * , xj cannot be an A-point relative to any member of the family if xj 6X * . Hence let us suppose that xj E X*. By condition (i) in the definition of an optimal strategy pair, there exists a path ryE which emanates from xj, is generated by ( r p * , r E * ) and reaches 8 at point sf. From Assumption 1.1 it follows directly that T$ U rjpfEis a path r);. Let ( r p * , pE) be the strategy pair that generates this path. Path r;: reaches 8 at point x f , and associated with it is a value of the cost of transfer ~ ( x ’ ,6 ; r p * , pE) = V(x2,
x’; rp*, p E ,
rg)
(2.17)
2.6
19
SOME PROPERTIES OF GAME SURFACES
In view of the saddle-point condition we see that V ( x 7 ,8; rp*, p E )
< V ( x Z 0;,
V*(xz)
rp*, YE*)
(2.18)
From the additivity property of cost (2.16) we have Y ( x z ,x f ;rI.*, pE,
TI?)
= Y ( x z ,x3;rp*, rE, +)
+ V ( x 3 ,x f ;r,,*, rE*, n;fE)
(2.19)
where
V ( x ’ , xf;rI,*, rE*, + ),
n
= V*(x’)
(2.20)
From (2.17), (2.18), (2.19) and (2.20) we obtain Y ( x ’ , x’; rf,*, r E ,
+> V*(x’) Q
*(x3)
Then it follows from (2.13) and from the additivity property of cost that, along path IIY(C’), x,,’ - x:
and hence X”’
= Y ( x z ,x 3 ; rp*, rE, rg)
- xo” < V*(XZ) - V*(X?)
(2.21)
Finally, it follows from (2.21), (2.9) and (2.10) that x 3 I S not an A-point relative to the game surface through x’. Lemma 2.2 can be established by similar arguments. Lemmas 2.1 and 2.2 have the following corollaries. Corollary 2.1. All points in %* of a path nyE(C“’)whicfz emanates from x’ belong to the game surface through x’.
This corollary is a direct consequence of Lemmas 2.1 and 2.2 together with the definitions of A- and B-points relative to a game surface. Corollary 2.2. A path II$(C’) whose initial point is a B-point relative to C(C) has no A-point relative to C(C) and, of course, no point in C(C). Corollary 2.3. A path ng(C”) whose initial point is an A-point relative to C(C) has no B-point relative to C(C) and, of course, no point in C(C).
These corollaries are direct consequences of Lemmas 2.1 and 2.2, respectively, together with the translation property of game surfaces.
20 2.7
11
SOME GEOMETRIC ASPECTS OF QUANTITATIVE GAMES
UNIQUENESS OF THE VALUE OF THE GAME
Suppose there exist two optimal strategy pairs ( r l , * , rF;*)and (tp*,t E * ) ; that is, both satisfy conditions (i)-(iii) of Definition 2.2. Upon applying these conditions we arrive at once at Lemma 2.3. If and i j
XI
E
(r,>*,
X * , both (rl.*, rfi;*) and (FI,*, FK*) are optimal on X * , E .F(x') and
( F P * , r g * ) E F(.Y'),
then
V ( s ' , 8 ; r I , * ,rR;*) = V ( Y , 0;
FE*)
I n other \c~orrls,the Value of the game V * ( s' ) is independent of the choice of optimal straregy pair. We also have Corollary 2.4. If both (rl,*, r E * ) and (tl,*,Fx*) are optimal on X * , antlif(rp*, i,.*) and (F,,*, Y E * ) areplayable for allxi E X , then ( r I > *tE*) , and ( F T , * , rr:.*)are also optimal on X * .