Theory of a “Trick”

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 ones 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 aim heading from London to Dover by Canterbury. But while getting on the aim, he noticed that Moriarty has also taken this aim. 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) uncompletely success: bd

3) failure: ad or bс.

These three outcomes, in the point of view of Holmes preferences, sequentially decline as those worthy of the choice, the last one being the worst. Moriartys system of preferences is opposite to that of Holmes. closest apparent is the difficulty of choice due to without of information. The decision both for Holmes and Moriarty is the consequence 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 except this possible (accidental) mistake, the chance rules the game. consequently 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 example, 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 already 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 past calculations represent the problem well, making an moment estimation of probabilities for p and q.

The functional value of the threshold d, for which the different 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.

The game theory finds application also in the economic life for strategic calculations. But the problems that arise in this case are quite difficult…

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