Telepathy strategy

^{One of the editions of this mechanics}

There is a deck of 100 cards with natural numbers from 1 to 100, each one by one. You and the second player draw 3 cards from this deck. The task - one by one, put them on the table in ascending order. But you can not change the information among themselves and show the map until it is placed on the table. That is, you cannot talk, knock on the table, wink and so on - nothing is impossible.

If at the end of the game 6 cards are laid out correctly, both players win. At the moment when the first card goes wrong - both players lose.

Your task is to win in this game. A person you have not seen before is playing with you. He may not even speak languages ​​you know. But he is definitely earthling. And he probably already tried to play this game with other people today. What is your strategy?

Do not rush under the cat, first think a little, please.

First we look at transactions and available information:

1. There is an act of laying out a card - this is, in fact, the transfer of information that "this is the minimum card of those on my hand, and I consider it minimal in principle."
2. There is a non-laying out of the map : this means that "I hesitate about the minimal value hypothesis of the map". Unleashing a card is a function of time, that is, a pause between actions also generates a significant event.
3. There is a visual arrangement of the card on the hand - from where exactly from these three cards does it drag. Most likely, this information is still noise, but it is.

Now we have to think about how to act correctly in this game. Suppose you have cards:

11, 47, 93

On the table, in fact, there is a card 0 (only it is not visible), which allows us to equalize the first move with the others according to the conditions of the matmodel.

We do not know the opponent's cards.

First approach

The first strategy is to lay out cards in turn with the second player. As it grows. In general, given the distribution, it will make some sense (at least, better than random actions).

But if the second player has a set like 5, 33, 41 in his hands, naturally, this will not work.

Second approximation

We divide the interval from 1 to 100 into 6 parts by the number of cards, we get 16, (6). We believe that each next card should fall into this interval and put it if it lies, and do not put if not. If we have two cards fall in this interval - we put both in a row.

Third approximation

We start counting the expectation of the next card. This means that the longer the interval between our cards in hand, the less likely it is to lay out the next card on our side. We do not lay out our card if the probability indicates that the action must be made by the second player.

All three methods suffer from an obvious disadvantage: they do not guarantee victory.

Fourth approximation

Here we first begin to need the help of a second player. We should get into his head and try to understand how he thinks. Most likely, he is semi-intuitively or consciously comes to something similar - if it is unlikely that the card should be spread, then he will wait. If it seems, then lay out immediately.

For example, if there is a card on the table 45, and he has a card 46 - it is obvious that he will do it almost instantly.
If we put 45, and he has 47, it is also obvious that he will do it instantly, because if we had 46, we would have immediately put it behind 45.
If we have 48, and he did not post 47 at once, it means that you can put 48.
If he sees that we have not posted 48, and he has 49, then he puts it.

That is, further the task turns into the measurement of the function of time between the laying of the cards. But there is an important transition to why this is so.

Go to the interaction of players

When players cannot change information, an interesting part of game theory comes into play. Game theory is generally a strange thing, which assumes that there are only rational people around you. And the earthling who plays with you is also rational.

Game Theory offers the following option: Suppose you could agree before the game. How would you both act optimally?

And the joke is that rational people with the same basic data do not need a dialogue in order to agree. They can accurately predict each other's actions.

That is, the question “what would we agree on” is equal in this situation to the question “how would you both act optimally in principle if you knew that the other player would also act optimally?”

The goal is the players one.

Most likely, we would start counting seconds from the calculation of the last card. Each new would mean that our next cards are quite further and further from the last one laid out. In the ideal case, if our card were 15 units more than the last, and the second player card was 20 units, then we would lay out at the 15th second, and he would at the 20th second.

Which brings us to the model, when you can play blindly and without any players at all. Since we have already decided to count the seconds between the display of cards to give another player a chance to go, then why stop? The model in the finals for any number of players looks like this: everyone silently counts the time and puts the card on the current number of seconds. At the 17th second, you need to lay out a map of 17, if it exists. And so on. To do this, do not even need to see any other actions of the players. And even know how many of them.

So you will always win and guaranteed. There are other optimal strategies, but they are generally weaker.

Why is beat a second? Because with some stretch we use the common framework for earthlings. This may be a day or a minute, but, you see, it is much less likely in the general situation.

See what just happened? We created a second player in the head, agreed with him - and if he is rational, then he really understood it and acts exactly as agreed. Because he did the same thing - he created us in his head, agreed with us and began to act.

That is, the knowledge that everything in the game is rational allows you to negotiate without transferring information. Because there is a clear criterion of which strategy to go.

Of course, there is an obvious problem.

Slightly away from rationalism

The problem is that the players are irrational. People are generally irrational, accept this.

We played in the company in the game "Make a number from 1 to 100, and if it will be equal to half of the average, you win." Freeze on the second and guess the number, assuming that you play the whole of Habr, including marketers who publish posts in corporate blogs.

Is it done?


But the joke is that we don’t know whether someone will think so in principle among the players. We need a forecast of the proportion of players who will use this strategy. Well, we do not know how many iterations in the player’s thoughts will pass. And we do not even know how an intelligent player evaluates the ability of other players to think. That is, we need an amendment to the fact that even fully rational people have amended the irrational.

This game was played several times in newspapers. Usually in practice, the result (the number that had to be thought) is around 4-15. Our average was 16, that is, it was necessary to think 8 to win. Our first player made 92. I relied on him and relied, knowing the results of his last two talks.

I acted wrong in this situation from the point of view of theory, but I won. The reason - I had a more realistic prediction than the one that gives a rational game theory.

Therefore, the next layer of rationalism in the game of telepathy is to assume how much our second player is able to repeat the same calculations and adjust his strategy for this. For example, most likely, it may happen that he intuitively understands the time difference, but will not be able to separate cases with an interval of 20 or 30 units from the laid out map. We'll have to adjust to the fly during the game - in his moves. Most likely, we will have to level out under its time scale (nonlinear, similar to quadratic scale). His irrational actions will reduce our overall chances of winning, but our actions will help straighten out what happens.

Of course, if we had explained to him about the seconds, he would have understood and agreed (in the overwhelming number of cases). But he just might not understand. Or consider tacting something else. Or in its cultural code there is no concept of time. Or it is not the same as ours. That is life. But we did what we could, and couples with us will win a little more than couples of two irrational people.

Perhaps, if we realize that it is completely inadequate, we will need to switch the strategy to the calculation of the expectation of his card and his card.

Mechanics has several practical applications. For example, if the lights on the game store are cut off, the animators have an emergency chip: turn on the flashlight, tell everyone that the flashlight is now off, and you need to count up to one hundred in turn (about half of the participants). Everyone can shout only one number once. If two voices got off or shouted at the same time, you need to start from the very beginning. Then turn off the lantern, shout "one" and go find out what's going on. Usually, the power can be restored much earlier than the players calculate. And since they were told how to play, few people get scared.