Piles: Discover a Rare and Intriguing Card Game of Chance
Piles is a simple gambling card game. Piles is usually played by three or more players, one of whom acts as the banker. What are the rules of the game and what mathematical advantage does the banker have over other players (bettors)?
Rules
Piles is played with a deck of 52 or 32 cards, depending on the players' preference or what's available. The basic rules of the game are simple. The banker makes piles, i.e., divides the cards into several piles. This can be done in two ways.
First, the banker makes as many piles as there are players in the game (including the banker). Each player then chooses one pile to bet on, and one pile remains for the banker.
Second, the banker makes any number of piles and each player can bet on any pile, even multiple players on the same pile. The banker can then turn over any pile that doesn't have money bet on it. If all piles are occupied, the banker takes any pile for themselves.
Image 1: Deck of 52 cards (French, whist cards)
Image 2: Deck of 32 cards (piquet cards)
Goal of the Game
The outcome of the game is decided by the bottom card of the pile. The players' goal is to have a higher card than the banker; in this case, the banker pays out even money (1:1). If a player has a card of equal or lower value than the banker, they lose their bet. The suit of the card has no influence (Hearts are not trump etc.).
Both players and the banker can strategize a bit. The banker can make just one pile and then take from it. Or, if they make more piles, players can bet on all but one, leaving it for the banker, thus deciding their own fate. A skilled banker might try to cheat and attempt to shuffle cards of the same value to the bottom of all piles, then they would have nothing to lose...
Banker's Advantage
It's not hard to guess that the banker's advantage comes from winning when the cards are equal. This is well illustrated in the following image, which schematically captures all the possibilities that can occur in Piles. For simplicity, let's first assume we're playing only with piquet cards (32 cards in the deck).
Table 1: Piles - Number of Winning and Losing Possibilities for the Player and Banker
| All Possibilities 1 = The Player Wins |
Banker's Card | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| 7 | 8 | 9 | 10 | J | Q | K | A | ||
| Player's Card | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 9 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 10 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | |
| J | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | |
| Q | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | |
| K | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | |
| A | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | |
The suit of the card has no effect on the outcome. The first row of the table shows that if a player turns over a pile with a seven (7) at the bottom, they lose in any case. If they have an eight (8), see the second row, they can only win if the banker has a seven (7), and so on.
Zero (0) indicates possibilities where the player loses, one (1) where they win. We can notice that the player and the banker have the same probability that their card will have a lower/higher value than the opponent's card, but equal cards also play in favor of the banker, which is marked in darker red on the diagonal.
Using the classic method - number of favorable possibilities divided by number of all possibilities - we can determine the probability of the player winning (and thus the banker). The player wins in 28 cases out of 64, so the probability of the player winning is 28/64 = 0.4375 (43.75%). The probability of the banker winning is the complement to one (or 100%), thus 0.5625 (56.25%).
Since the win is paid out as even money, the difference between these probabilities forms the banker's advantage: 0.5625 - 43.75 = 0.125 (12.5%). If a player bets $1, for example, in each game they can win $1 with a probability of 0.4375 and lose $1 with a probability of 0.5625.
The expected long-term (average) result for the player is then:
0.4375 × $1 + 0.5625 × (-$1) = $-0.125.
In other words, in the long-term average, the player will lose (the banker will gain) about $0.125 from each dollar invested in the card game Piles.
For a deck of 52 cards:
This time we can simplify the calculations a bit. With a deck of 52 cards (4 × 13), the cards of the player and banker can have 13 different values (card suits again don't play a role), giving a total of 13 × 13 = 169 different possibilities. The banker's advantage is formed by the zeros on the diagonal, and there are logically 13 of these (i.e., 13 possibilities for the banker and player to have cards of the same value).
The banker's advantage is then 13/169, after reduction 1/13, thus 0.0769 (7.69%). In long-term play, the player will on average lose (the banker will earn) about $0.08 from each dollar bet in the game. Conclusion? A larger number of cards in Piles favors the player, as the banker's advantage in case of equal cards is more diluted.
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Based on the original Czech article: Kopky (Hromádky) – karetní hra, pravidla.