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Chemin de Fer
My history is rather murky, but I believe that Chemin de Fer is a precursor or modern-day baccarat. Some early James Bond movies feature the game. It is similar to baccarat, except the there was some room for decision making in drawing a third card. I am not aware of any remaining Chemin de Fer tables, especially in the United States.
Despite having been to Europe three times, I still have never knowingly seen a game of Chemin de Fer.
To avoid confusion, when addressing the hands of Player and Banker, I will use capital letters. When referring to a human player or banker, I will use lower case. The hand scoring rules in Chemin de Fer are the same as baccarat. Otherwise, the rules of Chemin de Fer are as follows.
- Six 52-card decks are used.
- Instead of the casino banking the game, a single player does. The turn to be the banker rotates around the table.
- The banker always has the Banker hand.
- The banker would indicate the maximum he was willing to lose.
- From the banker's left, the other players would indicate how much they wanted to bet. If a player wanted to bet the maximum he could, subject to the banker's stated maximum loss, he would declare "banco," and no other players could bet.
- The dealer would give deal two cards each as the Banker and Player hands. The Banker hand would be given to the banker, and the Player hand to the player with the largest wager.
- The players with control of the two hands would check their cards. If either had an 8 or 9 total (known as a "natural"), he would turn them over immediately. The other hand would be turned over as well.
- Otherwise, unlike baccarat, the player with the largest wager could choose whether to take a third card. I have read it was traditional to always hit on 4 or less, and stand on 6 or more, with the five the only Player total that was seen going either way.
- If the player with control of the Player hand took a third card then he must turn it face up after examining it.
- After the player with the Player hand either stood pat or took a third card the banking player then had the option to take a third card.
- All other cards would be turned over, the hands scored, and wagers resolved.
- I do not think there was a tie bet, but my analysis below shows what the tie bet would pay anyway at a win of 8 to 1.
Most of the time strategy in Chemin de Fer will be the same, as follows:
- Player should hit on 0 to 4 and stand on 6 and 7.
- If the Player stood, then the Banker should hit on 0 to 5 and stand on 7.
- If the Player hit, then the Banker should follow the following strategy, according to the Player's third card and the Banker's two-card total.
The only situations that depend on the other player's strategy are:
- Player total of 5.
- Banker total of 6 when the Player stood.
- Banker total of 5 when the Player hit and got a 4.
- Banker total of 3 when the Player hit and got a 9.
If somehow the Banker knows that the Player tends to stand on 5, then he should be more passive and play these strategy-dependent situations as follows:
- Stand on 6 when the Player stood.
- Stand on 5 when the Player hit and got a 4.
- Stand on 3 when the Player hit and got a 9.
If somehow the Banker knows the Player will hit on a 5, then he should be more aggressive and hit in the situations above.
If somehow the Player knows the Banker will follow the strategy above, then he should hit a total of five. Likewise, if somehow the Player knows the Banker will hit on the plays above, then he should stand on a five.
In other words, the Banker should mimic the Player in his standing/hitting strategy, and the Player should try to be a contrarian and do the opposite of the Banker's standing/hitting strategy.
This all makes for a perfect game theory problem!
Let's call the strategies as follows:
- Player strategy A = Stand on 5.
- Player strategy B = Hit on 5.
- Banker strategy A = Stand on 6 when Player stands, stand on 5 when Player draws a 4, and stand on 3 when Player draws a 9.
- Banker strategy B = Hit on 6 when Player stands, hit on 5 when Player draws a 4, and hit on 3 when Player draws a 9.
The following table shows the Player's expected value according to the two sets of strategies used. To get the Banker's expected value, just change the minus sign to a plus.
Player Expected Value by Strategy Combination
Without the ability to read your opponent's mind, what the optimal solution? The answer is to choose your strategy randomly, as follows:
- The Player should follow A with probability 22.0503% and B with probability 77.9497%.
- The Banker should follow A with probability 57.2805% and B with probability 42.7195%.
The following table shows the probability of each final outcome, assuming two perfect logicians were playing.
Assuming both the Player and Banker follow the randomization strategy above, the overall expected value will be -0.012766 for the Player and +0.012766 for the Banker, assuming the house doesn't take any commission.
At the BetVoyager Internet casino you can play Chemin de Fer. The casino charges a 3% commission on Banker wins. The would make the expected values on each side as follows:
3% Banker Commission
As you can see, the expected value on the Banker is less, so the player should invoke his right to bank when it is his turn. The indifference point would be at a commission of 5.5481%. Anything less and it is better to be the Banker.
Everything above is based on a six-deck game. If you should ever find Chemin de Fer with eight decks the strategies are the same, except change probabilities as follows:
- The Player should follow A with probability 21.9405% and B with probability 78.0595%.
- The Banker should follow A with probability 57.3650% and B with probability 42.6350%.
In the game of baccarat they take all the guessing game out of Chemin de Fer and force the Player to hit on 5 and have the Banker follow the optimal strategy given the Player's strategy.
I get into a simplified version of this kind of game theory problem on my MathProblems.info site, problem #192.
Banker has Less Free Will Rule
According to The Doctrine of Chances: Probabilistic Aspects of Gambling by Stewart N. Ethier, in modern Chemin de Fer the Banker must stand on 6 if the Player stands. If this is the case, then the Player should always hit on 5, and the Banker should always follow the conventional baccarat rules for drawing after the player draws, specifically drawing to a 3 against a Player 9, and 5 against a Player 4.
- Wikipedia page on Chemin de Fer.
- The Doctrine of Chances: Probabilistic Aspects of Gambling by Stewart N. Ethier. This college text has several pages devoted to the mathematicis of Chemin de Fer, although based on an infinite deck, which does affect the odds and strategy.
Written by: Michael Shackleford