590PR Final Project

Title: Monte Carlo Method for Baccarat (card game)

Team Member(s):

Xiaoliang Jiang and Te Lin

Monte Carlo Simulation Scenario & Purpose:

Scenario:

Baccarat is a popular card game in casinos, especially among Asian gamblers. It is a card game played between two hands, the "player" and the "banker". Each baccarat coup (round of play) has three possible outcomes: "player" (player has the higher score), "banker", and "tie". In baccarat, cards have a point value: cards two through nine are worth face value (in points); tens, jacks, queens and kings have no point value (i.e. are worth zero); aces are worth 1 point; jokers are not used. Hands are valued according to the rightmost digit of the sum of their constituent cards. For example, a hand consisting of 2 and 3 is worth 5, but a hand consisting of 6 and 7 is worth 3 (i.e., the 3 being the rightmost digit in the combined points total of 13). The highest possible hand value in baccarat is therefore nine.

Rules of Baccarat:

If neither the player nor the banker is dealt a total of 8 or 9 in the first two cards (known as a "natural"), the tableau is consulted, first for the Player's rules, then the Banker's.

Player's rules

If the Player has an initial total of 0–5, Player draws a third card. If player has an initial total of 6 or 7, Player stands.

Banker's rules

If the player stood pat (i.e., has only two cards), the banker regards only his own hand and acts according to the same rule as the player. That means the banker draws a third card with hands 0–5 and stands with 6 or 7.

Hands: the "player" and the "banker."

Possible outcomes each round of play: "player" (player has the higher score), "banker" (banker has the higher score), and "tie."

Rules: After assigning the first four cards to player and banker (the order is player-banker-player-banker), if neither the player nor the banker is dealt a total of 8 or 9 in the first two cards (the single-digit of the sum of two cards on hand), the tableau of drawing rules is consulted, first for the player's rules, then the banker's.

Tableu of Drawing Rules:

If the player drew a third card, the banker acts according to the following more complex rules:

• If the banker total is 2 or less, then the banker draws a card, regardless of what the player's third card is.
• If the banker total is 3, then the banker draws a third card unless the player's third card was an 8.
• If the banker total is 4, then the banker draws a third card if the player's third card was 2, 3, 4, 5, 6, 7.
• If the banker total is 5, then the banker draws a third card if the player's third card was 4, 5, 6, or 7.
• If the banker total is 6, then the banker draws a third card if the player's third card was a 6 or 7.
• If the banker total is 7, then the banker stands.

Payoffs:

Traditional Payoff Rules

1. Gamblers place bets on Bankers and Banker wins, the ratio of payoff to stake is 1:1, while the casino takes 5% commission, so the overall payoff is 0.95:1.
2. Gamblers bet on Player and Player wins, the ratio of payoff to stake is 1:1.
3. Gamblers bet on tie and tie happens, then the ratio of payoff to stake is 8:1.

Side	Stake	Payoff
Banker	1	0.95
Player	1	1
Tie	1	8

Besides the Traditional Payoff Rules above, we invented two new payoff rules to help a casino to attract more customers (we will exam the outcomes and compare each bundle of new rules to the traditional payoff rules only later):

New Payoff Rule 1

If the side a gambler bet on has a pair (2 Aces, 2 eights, etc.), the gambler gains additional 6-time payoffs, the total payoff now is 6.95:1 (Banker), 7:1 (Player), and 14:1 (tie);

if the side a gambler bets on has three of a kind (3 Aces, 3 eights, etc.), the gambler gains additional 36-time payoffs, the total payoff now is 36.95:1 (Banker), 37:1 (Player), and 44:1 (tie).

A pair shows up on the side a gambler bets on:

Side	Stake	Payoff
Banker	1	6.95
Player	1	7
Tie	1	14

Three of a kind shows up on the side a gambler bets on:

Side	Stake	Payoff
Banker	1	36.95
Player	1	37
Tie	1	44

New Payoff Rule 2

If a pair show up on the table, disregarding which side the gambler bets on, the gambler gains additional 2-time payoffs, the ratio of payoff to stake is now 2.95:1 (Banker), 3:1 (Player), and 11:1 (tie) ;

if the side a gambler bet on has three of a kind (3 Aces, 3 eights, etc.), the gambler gain additional 100-time payoffs, the total payoff now is 100.95:1 (Banker), 101:1 (Player), and 108:1 (tie).

A pair shows up on the table (disregarding which side a gambler bets on):

Side	Stake	Payoff
Banker	1	2.95
Player	1	3
Tie	1	11

Three of a kind shows up on the table (disregarding which side a gamblber bets on):

Side	Stake	Payoff
Banker	1	100.95
Player	1	101
Tie	1	108

Purposes:

With Monte Carlo sampling method, we want to:

1. Test the theoretical expectation of each strategy (gamblers always bet on bankers, gamblers always bet on players and gamblers always bet on tie) .
2. Compare different ratios of bets to initial balances while controlling the strategies .
3. Explore the possibilities of gamblers earning money from this game.
4. Compare the performances of three different rule bundles: 1) Traditional Payoff Rules only; 2) Traditional Payoff Rules + New Payoff Rule 1; and 3) Traditional Payoff Rules + New Payoff Rule 2

Simulation's variables of uncertainty

1. Strategies: a total four kinds of strategies, which are
   1. gamblers always bet on Player;
   2. gamblers always bet on Banker;
   3. gamblers always bet on Tie;
   4. gamblers place bets randomly (Player, Banker or Tie).
2. Ratios of bets to initial balances (abbreviation: bet ratios): 1%, 2%, 5%, 10%, 20%, 50%.
3. rounds: not tested individually, but were shown on the charts.
4. Rule Bundles (abbreviation: rules):
   1. Traditional Payoff Rules only;
   2. Traditional Payoff Rules + New Payoff Rule 1; and
   3. Traditional Payoff Rules + New Payoff Rule 2.

Hypothesis or hypotheses before running the simulation:

Analytical variables: Strategies, Goals and Bets

Suppose gamblers carry the same balances before starting games, and the casino's balance is unlimited. Gamblers would have different strategies, goals, or bets (variables).

On our analytical stage, we want to exam the outcomes with two of the variables as controlled variables, and the rest one as a test. After we gather the outcome, we will plot them accordingly.

	Control 1	Control 2	Control 3	Test
Simulation 1	Bet ratios	Rounds	Rules	Strategies
Simulation 2	Strategies	Rounds	Rules	Bet ratios
Simulation 3	Bet ratios	Rounds	Strategies	Rules

1. In the simulation 1, we conducted 1000 times of simulations with 2000 rounds per simulation, a bet to initial balance ratio, Traditional Payoff Rules only, and returned outcomes with the following strategies:

   1. gamblers always bet on Player;
   2. gamblers always bet on Banker;
   3. gamblers always bet on Tie;
   4. gamblers place bets randomly (Player, Banker or Tie).

2. In the simulation 2, we conducted 1000 times of simulations with 2000 rounds per simulation, Traditional Payoff Rules only, random strategy and returned outcomes with the following bet ratios: 1%, 2%, 5%, 10%, 20%, and 50%.

3. In simulation 3, we conducted 1000 times of simulations with 2000 rounds per simulation, random strategy, a bet to initial balance ratio of and returned outcomes with the following Rule Bundles (abbreviation: rules):

   1. Traditional Payoff Rules only;
   2. Traditional Payoff Rules + New Payoff Rule 1; and
   3. Traditional Payoff Rules + New Payoff Rule 2.

Analytical Summary of your findings: (e.g. Did you adjust the scenario based on previous simulation outcomes? What are the management decisions one could make from your simulation's output, etc.)

Simulation 1

From Simulation 1, we can conclude that in the short run, betting on tie only performs the worst, betting on Bankers performs the best; however, in the long run, the odds of winning betting on tie only would delay a gambler losing all its balance to the casino, betting on Banker or Player are indifferent -- they all leads to a 0 in the balance. Please refer to our slides for more images/results.

Simulation 2

From Simulation 2, we can conclude that the smaller the ratios of bets to initial balances, the later a gambler would lose all its balance and the smoother the lines are.

Simulation 3

Ratio of bets to initial balances: 1%

1. Traditional Payoff Rules only;

2. Traditional Payoff Rules + New Payoff Rule 1;

3. Traditional Payoff Rules + New Payoff Rule 2.

From simulation 3 with a bet ration of 1%, we can conclude that rule bundle 2 and rule bundle 3 significantly improved gamblers' odds of winning. For example, at the point of fininshing 2000 rounds in a game, with traditional rules only, a gambler are very likely losing all its balance, however, with New Payoff Rule 1, the average of balance on gamblers' hands are $400, and with New Payoff Rule 1, the average of balance on gamblers' hands are around $300 to $600 using different strategies. In another words, it might encourage travlers or guests to participate in Baccarat. More importantly, a casino is still earning money with new bundles of rules.

Ratio of bets to initial balances: 50%

1. Traditional Payoff Rules only;

2. Traditional Payoff Rules + New Payoff Rule 1;

3. Traditional Payoff Rules + New Payoff Rule 2.

From simulation 3 with a bet ration of 50%, we can see that the line of tie still performs the best after 1000 rounds of games even with new bundles of rules. With the new payoff rule 1, the treads are just similar to the traditional rule, however, we can see the death of a gambler (losing all the balances) is significantly delayed. But with the new payoff rule 2, betting on tie is much better than other strategy. It is partially because of betting on tie relies on balance more than other two strategis.

Possibilities of A Gambler Reaches Its Goal (10,000 games, 100 rounds per game)

1) gamblers always bet on Player

2) gamblers always bet on Banker

3) gamblers always bet on Tie

4) gamblers place bets randomly (Player, Banker or Tie)

The upper parts of graphs above represent the possiblities of reaching goals. The first rows represent the times of balances a gambler wish to reach. For example, 1.2 means when a gambler has $1000 of initial balance, his or her balance has ever reached $1200. The first column represents different bet ratios a gambler would put on to reach its goal.

The lower parts of graphs above represent the expected earnings, which are calculated by the value in the same position of the upper table by the times of balances a gambler wish to reach (first row). For example, in the chart 1), the top left value 0.12474 in the lower part of the chart is calculated by 11.34% (the same position in the upper chart) times 1.1.

In the upper parts, the greener a block is, the higher the possibility a gambler would reach its goal; In the lower parts, the greener a block is, the higher the expected earning. However, even the highest expected earning in the lower parts among these four charts (it happend in the chart 1, row: 50%, col: 1.5) are lower than 1, which means a gambler can not expect to earn money from Baccarat.

5) gamblers place bets randomly (Player, Banker or Tie) with the Rule Bundle 2 (Traditional Payoff Rules + New Payoff Rule 1)

6) gamblers place bets randomly (Player, Banker or Tie) with the Rule Bundle 3 (Traditional Payoff Rules + New Payoff Rule 2)

Generally speaking, two new rule bundles perform better than the Rule Bundle 1 (Traditional Payoff Rules only), however, the expected earning is still lower than 1, which means a gambler can not expect to earn money from Baccarat, in another word, a casino could still beat the gamblers in the long run.

Instructions on how to use the program:

Please run Baccarat_v2.0.py in the home directory folder.

In the main function, we provided some example about how to use our python scripts to do the simulation. In general it can return three different types of results based on a customized simulation processing.

1. The first one focuses on the detailed information on each rounds in a game. It can return a list of lists which contains all gamblers balance information in each rounds. Usually, you can set show_results as True in rounds function to show the detailed information about the card type in each round.
2. The second example is about comparing the differences between different strategy or chips in each round. It will run the simulation ab times in total, a is the rounds in each game, and b is the number of games the function simulated It will return a nm csv file, n is the number of gamblers, and m are the number of games divided by a given step. By using the output of this file, users could generate several plot to see the different trends of different strategy via Monte Carlo simulation.
3. The last one aims to answer a question that in what possibility and how much money we expected to earn from this game. In this part, it will return a table with one single line which contains the probability in each earning ratio from 1.1, 1.2 to 7.5, 10. Users can run this example with different values to generate several lines of results and to check the differences.

Our presentation slides:

https://docs.google.com/presentation/d/1yqReaaVZrzMYm1ZRf6JNUTihplAXr3Z5myLCViKRvLE/edit?usp=sharing

All Sources Used:

Wikipedia (2018, Dec 02). Baccarat (card game). Retrieved from https://en.wikipedia.org/wiki/Baccarat_(card_game) WikiHow. (2018, November 11). How to Play Baccarat. Retrieved from https://www.wikihow.com/Play-Baccarat Kelly criterion. (2018, October 22). Retrieved from https://en.wikipedia.org/wiki/Kelly_criterion