The necessary condition for using the theory of a "trick" is that
players are to have insufficient information about each other. In this
case the "trick" consists in guessing the intentions of the adversary on
condition of hiding one's own intentions:
A positive "trick" and a
negative "trick". The tactics of every player is to be very flexible,
and one and the same "trick" should not be used for many times,
otherwise it will become "tactics" and will revert, like a boomerang,
back to its user.
The player should strive to modify his game according
to the reaction of his adversary by making the most successful choice
for this situation: hence comes the probability of probabilities.
Oscar
Morgenstern (born in 1902) gave an example of a successful choice in
the situation unsuccessful by itself. The example was based on one of
the stories about Sherlock Holmes. Being chased by professor Moriarty,
he took a train heading from London to Dover through Canterbury.
But
while getting on the train, he noticed that Moriarty has also taken this
train. Holmes knew that if he set off simultaneously with Moriarty, he
would certainly be killed. He had to get to Dover alone, in order to
embark on the steamer that crossed the channel. This was his objective.
The following variants are possible:
a) Holmes gets off in Dover;
b) Holmes gets off in Canterbury;
c) Moriarty gets off in Canterbury;
d) Moriarty gets off in Dover. The outcome, in Holmes' opinion, can be:
1) complete success: ас
2) partial success: bd
3) failure: ad or bс.
These
three outcomes, in the point of view of Holmes' preferences,
sequentially decrease as those worthy of the choice, the last one being
the worst. Moriarty's system of preferences is opposite to that of
Holmes. Immediately evident is the difficulty of choice due to lack of
information. The decision both for Holmes and Moriarty is the result of a
random choice that plays the role of a defensive tactics.
Both of them
are well-prepared, and both alertly wait for the smallest neglect of the
adversary in order to attack at once. But apart from this possible
(accidental) mistake, the chance rules the game. Thus we get what G. von
Neuman (born in 1903) revealed.
We can mathematically express the
game before its beginning by introducing probabilistic preferences of
both players: for instance, Pr(a) = p; Pr(b) = l - p Pr(c) = q; Pr(d) = l
- q.
Then the probabilities of various outcomes (moves) are calculated with the help of rules of compound possibilities: Рr(ас) = р * q; Pr(bc) = (1 - р) * q;
Pr(ad) = р(1 - q); Pr(bd) = (1 - р) * (1 - q),where: Pr(ad или bc) = р(1 - q) + q(1 - р) = p + q - 2pq.
But these probabilities are initially unknown to players. For example, Holmes does not know q, but even if he knew q, his choice would not become less probabilistic. Every player acts, meditating on the possible move of the adversary, and at the moment previous calculations represent the problem well, making an instant estimation of probabilities for p and q.
The
practical value of the threshold d, for which the alternative of
"success" with probability d and death with probability 1 - d is
preferred to "certain defeat", depends on the boldness of the famous
English detective.
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