D, who are engaged in a game. We shall follow the fortunes of R,
Who is attempting to score an a. The rules are as follows. They
Have before them Table 11/3/1, which can be seen by both:
Table 11/3/l
R
α β γ
1
D 2
3
b
a
c
202
a
c
b
c
b
a
D must play first, by selecting a number, and thus a particular row.
R, knowing this number, then selects a Greek letter, and thus a
Particular column. The italic letter specified by the intersection of
The row and column is the outcome. If it is an a, R wins; if not, R
Loses.
Examination of the table soon shows that with this particular
Table R can win always. Whatever value D selects first, R can
Always select a Greek letter that will give the desired outcome.
Thus if D selects 1, R selects β; if D selects 2, R selects α; and so
On. In fact, if R acts according to the transformation
1 2 3
β α γ
Then he can always force the outcome to be a.
R’s position, with this particular table, is peculiarly favourable,
For not only can R always force a as the outcome, but he can as
Readily force, if desired, b or c as the outcome. R has, in fact, com-
Plete control of the outcome.
Ex. 1: What transformation should R use to force c as outcome?
Ex. 2: If both R’s and D’s values are integers, and the outcome E is also an inte-
Ger, given by
E = R – 2D,
Find an expression to give R in terms of D when the desired outcome is 37.
Ex. 3: A car’s back wheels are skidding. D is the variable “Side to which the tail
Is moving”, with two values, Right and Left. R is the driver’s action “Direc-
Tion in which he turns the steering wheel” with two values, Right and Left.
Form the 2 x 2 table and fill in the outcomes.
Ex. 4: If R’s play is determined by D’s in accordance with the transformation
↓
↓
1
β
2
α
3
γ
And many games are observed, what will be the variety in the many outcomes?
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Ex. 5: Has R complete control of the outcome if the table is triunique?
The Table used above is, of course, peculiarly favourable to
R. Other Tables are, however, possible. Thus, suppose D and R,
Playing on the same rules, are now given Table 11/4/1 in which D
Now has a choice of five, and R a choice of four moves.
If a is the target, R can always win. In fact, if D selects 3, R has
Several ways of winning. As every row has at least one a, R can
Always force the appearance of a as the outcome. On the other
Hand, if the target is b he cannot always win. For if D selects 3,
There is no move by R that will give b as the outcome. And if the
Target is c, R is quite helpless, for D wins always.
203
A N I N T R O D UC T I O N T O C Y B E R NE T I C S
REQ U ISI TE V A RI ETY
It will be seen that different arrangements within the table, and
Different numbers of states available to D and R, can give rise to
A variety of situations from the point of view of R.
Table 11/4/l
R
α β γ
1
2
D 3
4
5
b
a
d
d
d
d
d
a
b
a
a
a
a
a
b
Table 11/5/l
R
α β
γ
δ
a
d
a
b
d
Ex. 1: With Table I l /4/ l, can R always win if the target is d?
Ex. 2: (Continued.) What transformation should R use?
Ex. 3: (Continued.) If a is the target and D, for some reason, never plays 5, how
Can R simplify his method of play?
Ex. 4: A guest is coming to dinner, but the butler does not know who. He knows
Only that it may be Mr. A, who drinks only sherry or wine, Mrs. B, who
Drinks only gin or brandy, or Mr. C, who drinks only red wine, brandy or
Sherry. In the cellar he finds he has only whisky, gin, and sherry. Can he find
Something acceptable to the guest, whoever comes ?
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F f k
K e f
M k a
B b b
D 5c q c
H h m
J d d
A p j
L n h
Only one move in response to each possible move of D. His spec-
Ification, or “strategy” as it might be called, might appear:
If D selects 1, I shall select γ
,, ,, ,, 2, ,, ,,,, α
,, ,, ,, 3, ,, ,,,, β
……
,, ,, ,, 9, ,, ,,,, α
He is, of course, specifying a transformation (which must be sin-
Glevalued, as R may not make two moves simultaneously):
Can any general statement be made about R’s modes of play
And prospects of success ?
If full generality is allowed in the Table, the possibilities are so
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