Card games

Card games are interesting not only because they are often associated with cash rates, but also for many other reasons. The number of different variants of card games is extremely large, but in this chapter we will focus on options in which cards are randomly shuffled at the beginning of the game and each player receives cards in his hands that he does not show to other players. These games include bridge, whist, "Hearts" and some types of poker.

At first glance, it may seem that card games are completely similar to the games with the draw: the cards are distributed randomly, and the cards themselves determine which moves can be made by each player. However, in the cards all the draw comes from the very beginning! This remark will be discussed in more detail below and it will be shown that such a feature of the card games in question is very useful in practice. At the same time, this remark is at the same time completely wrong for very interesting reasons.

Imagine that two players, MAX and MIN, play some real hand with four cards in a bridge with two players, in which all the cards are open. There are the following cards on hand and the MAX player must go first:

MAX: ♥ 6 ♦ 6 ♣ 9 8 MIN: ♥ 4 2 ♣ 10 5

Suppose the player MAX is walking from the card ♣ 9. Now the player MIN has to walk, who can roll a card ♣ 10 or ♣ 5. The player MIN puts the card 10 and takes the bribe. Then the turn turns to the player MIN, who walks with a card ♣ 2. The player MAX has no peak (and therefore cannot take this bribe), therefore, he must throw some other card. An obvious option is the ♦ 6 card, since the other two remaining cards are older. Now, no matter what card the MIN player went on during the next draw,
player MAX will take the two remaining bribes and the game will end in a draw, with two bribes from each player. Using the appropriate minimax search option, you can easily show that the actual move of the No. 9 card made by the MAX player was optimal. Now let's replace the cards in the hands of the MIN player and, instead of the ♥ 4 card, we introduce a ♦ 4 card:

MAX: ♥ 6 ♦ 6 ♣ 9 8 MIN: ♥ 4 2 ♣ 10 5

The two cases considered are completely symmetrical: the course of the game will be the same, except that when playing the second bribe, MAX player rolls a card ♥ 6. The game ends in a draw with two bribes from each player, and the turn of card ♣ 9 is optimal.

Until now, everything went well. Now let's hide one of the MIN player's cards and assume that the MAX player knows that the MIN player either has the first hand (with a ♥ 4 card) or the second hand (with a ♦ 4 card), but he doesn’t know which of the of them. Player MAH reasons as follows.

The turn of the card ♣ 9 is the optimal solution in the game against the first and second distribution in the hands of the MIN player, so now this move should be optimal, since it is known that the MIN player has one of these two distribution options.

At a more generalized level, we can say that the player max uses an approach that can be called "predicted averaging." Its idea is to evaluate each possible course of action when there are cards in the hands of the enemy that are not visible to the player, first calculating the minimax value of this action for each possible card deal, and then calculating the expected value for all hands using probability each distribution.

If the reader thinks such an approach is reasonable (or if he cannot judge him because he is not familiar with the bridge), then he should reflect on the story below.

• First day Road A leads to a pile of gold bars; the road leads to a fork. If you go from the fork to the left, you will find a mountain of jewels, and if you go to the right, you will fall under the bus.
• Second day Road A leads to a pile of gold bars; Road B leads to a fork. If you go from the fork to the right, you will find a mountain of jewels, and if you go to the left, you will fall under the bus.
• Third day Road A leads to a pile of gold bars; the road leads to a fork. If you choose the right direction from the fork, you will find a mountain of jewels, and if it is wrong, you will fall under the bus.

Obviously, in the first two days, the decision to choose the road in no sense, but on the third day, no one in their right mind would choose the road to. However, it is precisely this variant that suggests the averaging method according to forecasts: road B is the optimal situations that arise on the first and second days, so it is optimal on the third day, since one of the two previous situations must occur. Let's return to the card game: after player MAX goes with card ♣ 9, player MIN will take a bribe with card ♣ 10. As before, MIN will go from card ♣ 2, but now the player will be in front of a fork in the road without any instructions. If the MAX player rolls a ♥ 6 card, and the MIN player still has a ♥ 4 card, this ♥ 4 card will become a trump and the MAX player will lose the game. Similarly, if player MAX rolls a ♦ 6 card and player MIN still has a card ♦ 4, player MAX will also lose. Therefore, the game with the first move of the card * 9 leads to a situation in which the player max has a 50% chance of losing. (It would have been much better for him to play cards ♥ 6 and ♦ 6 first, guaranteeing a draw game for himself.)

From all this you can learn a lesson that with a lack of information the player must take into account what information he will have at each moment of the game. The disadvantage of the algorithm used by the MAX player is that it assumes that with each possible distribution the game will develop as if all the cards remained visible. As this example shows, this forces the MAX player to act in such a way as if uncertainty will be resolved when the time comes.

In addition, in the MAX player algorithm, the decision is never made to collect information (or to provide information to a partner), since this is not required to be done within each individual distribution; nevertheless, in such games as bridge, it is often advisable to play such a card that would help to learn something about the cards of the enemy or inform the partner with your own cards.

Such behavioral stereotypes are formed automatically by an optimal algorithm for playing games with incomplete information. Such an algorithm does not search in the state space of the world (by this means the cards that are in the hands of the players), but in the space of trusting states (ideas about who has what cards and with what probabilities). Authors will be able to explain this algorithm properly in Chapter 17 after developing all the necessary probabilistic tools.

And on this page it is also necessary to dwell on one final and very important remark: in games with incomplete information it is best to give the enemy as little information as possible, and the best way to do this is often to act unpredictably. That is why sanitary doctors, visiting for an audit catering, do not report in advance about their visits.