baccarat card counting

[ragroller]

I know this has been discussed before but i havent read it thoroughly,
i mean the only difference in "cards " with blackjack is that the player and banker gets dealt 3 cards max and BJ can deal as many till bust or stand.

whats the quick overall reason y bacca cant be counted? why do they also consider the tie when counting? would it make a difference if you didnt count tie chances?

 

[SleightOfHand]

http://wizardofodds.com/baccarat/baccaratapx2.html

 

[Katweezel]

Rare event?

I know this has been discussed before but i havent read it thoroughly,
i mean the only difference in "cards " with blackjack is that the player and banker gets dealt 3 cards max and BJ can deal as many till bust or stand.

whats the quick overall reason y bacca cant be counted? why do they also consider the tie when counting? would it make a difference if you didnt count tie chances?

One situation when counting cards at Baccarat could be profitable at 8:1 for a tie, is when/if the last remaining cards in the shoe must all be of 0-value. But that is going to be a fairly rare event...I guess. :cat:

 

[SuperTrump]

I answered this question in a previous thread:

http://www.blackjackinfo.com/bb/showthread.php?t=12356&page=2

 

[callipygian]

whats the quick overall reason y bacca cant be counted?

The effect of removal is low. The countability of a game is the effect of removal divided by the house edge.

 

[taipafan]

In some places dealer only cuts 7 cards fixed and would play the game until 6 or less cards remain. Sometimes we can see the bottom card.
In this case, tie bets will create +20+% EV, which happens once in every two shoes,... on average.

 

[moo321]

In some places dealer only cuts 7 cards fixed and would play the game until 6 or less cards remain. Sometimes we can see the bottom card.
In this case, tie bets will create +20+% EV, which happens once in every two shoes,... on average.

Well, you've found the holy grail, my friend. Baccarat is quite beatable with 7 cards cut off. Do some research, because you'll find more than 20% advantages. There are some 800% advantages available.

 

[FLASH1296]

Interesting Reading part #1

Baccarat Card Counting​

Michael Hall ([email protected])
Wed, 24 Aug 1994 21:46:35 GMT

In article <[email protected]>, Rob Jaeger <[email protected]> wrote: >Has anyone come up with a working card counting system for Baccarat?

Short Answer:

Although there are card counting systems to beat baccarat, I've shown that none are worthwhile.

Very Long Answer:

(I prepared this a few months ago, to counter Rex's report, which never surfaced.)

The Bullshitless Baccarat Report
Copyright 1994, Michael Hall

This Version Dated: 8/24/94

I. INTRODUCTION
II. THE BASICS

i. The Rules
ii. The Odds
III. WHAT TO EXPECT
i. The Multinomial Distribution
ii. Expected Value
iii. Variance
iv. The Normal Distribution
IV. THEOREMS AND SYSTEMS
i. Theorems
ii. Systems
V. CARD COUNTING
i. The Kelly Criterion
ii. Perfect Non-Linear Card Counting
iii. Ultimate Linear Counting System
VI. CONCLUSIONS

I. INTRODUCTION

Much has been written on baccarat, but much of the published literature has been worthless or even harmful. "Money management" and progression betting systems do not work on baccarat any more than they work on craps or roulette; however, because baccarat is played with cards that are not shuffled between each trial, the trials are not truly independent, and so it may not be immediately apparent that such systems are doomed. This report debunks the systems by showing what happens with perfect card counting and by explaining the enormous variability in results, which sometimes confuses systems players into thinking they are playing with a winning system.This report debunks the systems by showing what happens with perfect card counting and by explaining the enormous variability in results, which sometimes confuses systems players into thinking they are playing with a winning system.

II. THE BASICS

Baccarat is a relatively simple game to describe how to play and to analyze the odds.

i. The Rules

[This subsection was lifted from a rec.gambling post, but I have lost the attribution. I did receive permission to use the text, however.]

Your only choice in Baccarat is whether to bet on the bank, the player, the tie, or some combination, and how much to bet. There is a 5% commission on winning bank hands and ties pay 9 for 1, which is 8 to 1.

The "banker" and the "player" are each dealt a hand of two cards; the value of each hand is determined by adding the two cards' value aces count for 1, tens and face cards for 0 -- and taking the last digit of the sum. A hand holding the 3 of clubs and 4 of diamonds has the value 7; a hand with the ace of hearts and jack of spades is worth 1.

Now, if either or both side holds a score of 8 or 9 right off the top, it's called a natural, there is no draw, and the high value wins -- the hand is a tie if both have 8 or if both have 9. If there is no natural, play begins with the player hand. If the player hand holds a score of 0 through 5, the player draws a third card; if the player hand holds a score of 6 or 7, the player stands. Now turn to the bank hand. The bank hand always draws with a 0, 1, or 2. With a three, the bank draws unless the player drew and its third card was an 8. With a four, the bank draws unless the player drew and its third card was outside the range of 2 through 7. With a five, the bank draws unless the player drew and its third card was outside the range of 4 through 7. With a six, the bank stands unless the player drew and its third card was either a 6 or a 7. With a seven, the bank stands.

After either, neither or both sides have drawn a third card according to the above rules, the final value of the hand is determined and the high hand wins -- or a tie is declared. A score of 0 is commonly called baccarat; the good scores, 8 and 9, are often called la petite and la grande, respectively. Baccarat is a negative expectation game which makes up a disporportionately high percentage of the casinos' intake, given the relatively low number of tables and players it attracts. The reason is that it is, as Silberstang says, the casino's only real "glamorous game," and the shills and tuxedoed dealers lend it an aura of elegance that attracts big, and sometimes huge, bettors.

ii. The Odds

I wrote a combinatorial analyzer that, not too surprisingly, output figures consistent with published figures:

BACCARAT COMBINATORIAL ANALYSIS
(8 DECKS)

WIN EXPECTED
BET PROBABILITY VALUE
(%) (%)
Bank 45.859742 -1.057906
Player 44.624661 -1.235081
Tie 9.515597 -14.359629
When bank pays only 4% commission instead of 5%, the expected value on bank is -0.599308%, and when tie pays 9 *to* 1 instead of 9 *for* 1, the expected value on tie is -4.844032%.

As you can see, the bank wins more often than it loses (i.e., than the player wins.) To make up for this, a commission is charged on winning bank bets.

"Expected value" doesn't mean we expect to lose 1.057906% of our bet on any particular bank bet, for example - that's impossible - we either win 0.95 units, lose 1.0 units, or push on any particular bank bet. Instead, expected value means we expect to lose that much on *average*. More on that in the next section.

III. WHAT TO EXPECT

Somebody played 2392 hands of real baccarat and recorded the following outcomes:

1103 bank hands
1075 player hands
214 tie hands

How likely is this outcome?

i. The Multinomial Distribution

The multinomial distribution will tell us exactly (plus or minus a bit because of the very slight correlation between hands) how probable this particular outcome was:

n!
P(r1,r2,r3) = ---------------(p1^r1)(p2^r2)(p3^r3)

(r1)!(r2)!(r3)!

Where r1, r2, r3 is the number of bank, player, and tie outcomes,

p1, p2, p3 is the probability of bank, player, and tie and n = r1+r2+r3, the total number of hands

Here r1=1103, r2=1075, r3=214, p1=.4596, p2=.4462, p3=.0952, and n=2392, so

2392!
P(1103,1075,214)= --------------(.4596^1103)(.4462^1075)(.0952^214)

1103!1075!214!

.003387102761933
The point is simply that we can compute how likely a particular outcome, or set of outcomes, was. However, such computations are a bit awkward, so you can forget about that equation... I'm going to shift to a normal distribution approximation in a second here, discussing expected value and variance along the way.

ii. Expected Value

Expected value is the sum of the products of each possible outcome and its probability. For baccarat, the expected values are as follows for betting $20 a hand for 2392 hands:

EV(banker) = 2392*$20*(.4596*.95 + .4462*(-1)) = -$458

^^^^^^^^^^^^^^^^^^^^^^^^
(-1.057906%)
EV(player) = 2392*$20*(.4462*1.0 + .4596*(-1)) = -$641

^^^^^^^^^^^^^^^^^^^^^^^^
(-1.235081%)
EV(tie) = 2392*$20*(.0952*8.0 + .9048*(-1)) = -$6851

^^^^^^^^^^^^^^^^^^^^^^^^
(-14.359629%)
If you mix up your betting between banker and player, you will on average lose somewhere in between those amounts. If you raise and lower your bets, recompute your average bet in place of $20 and that's how much you will lose. For every $100 you bet on bank, $1.06 goes to the house on average, for every $100 bet on player, $1.24 goes to the house on average, and for every $100 bet on tie, $14.36 goes to the house on average. Period. Think of things that way, and you will soon be cured of any gambling problem you might have.

iii. Variance

Variance is the mean of the square of a random variable minus the square of the mean. It measures the variability of the outcome.

The variances per hand for flat betting $20 a hand for 2392 hands are as follows:

VAR(banker) = (2392)($20^2)(0.860969 - .0106^2) = $$822,848

^^^^^^^^^^^^^^^^^^^^
0.8608766
VAR(player) = (2392)($20^2)(0.905800 - .0124^2) = $$865,904

^^^^^^^^^^^^^^^^^^^^
0.9056462
VAR(tie) = (2392)($20^2)(6.9976 - .1436^2) = $$6,675,573

^^^^^^^^^^^^^^^^^^^^

If you're ranging your bets with an average of $20, that will increase your variance relative to flat betting $20.

The standard deviation is the square root of the variance.

Here:

STD(banker) = sqrt($$822,848) = $907.11 STD(player) = sqrt($$865,904) = $930.54 STD(tie) = sqrt($$6,675,573) = $2,583.71

iv. The Normal Distribution

Because the sample size is more than 100 or so and the trials are very nearly independent, the Central Limit Theorem takes hold and the winnings will fall very nearly exactly into a normal distribution, that is a bell-shaped curve. Sometimes you do very well, sometimes very poorly, but most of the time you fall in the big bulge somewhere, and the center of this bulge is on the expected value. The normal distribution can be used to find out just how likely or unlikely it was for you to win at least as much as you did. In this case, the normal distribution is easier to use than the multinomial or binomial distributions, which are the exact models for coin-tossing type games.

About 2/3's of the time, your normally-distributed result will be within +- one standard deviation of your expected value. So the $20/hand banker after 2392 hands will 2/3's of the time be within $907.11 (see the previous subsection) of his expected value of -$458 (see a couple subsections ago.)

In general, don't get excited unless your results are outside 1.96 times the standard deviation relative to your expected value; values fall outside this range 5% of the time (loosely speaking). You can make a 95% confidence interval on your results by adding and subtracting 1.96 times the standard deviation of your result. So a 95% confidence interval on the $20/hand banker after 2392 hands is [-$458-$907.11, -$458+907.11] = [-$1365.11, $449.11]. So, loosely speaking, 95% of the time the results would be between losing $1365.11 and winning $449.11.

The standardized normal variable z, which I need to consult the probability charts for a normal distribution, is:

z = (x-EV)/sqrt(VAR)

Here x is the point of interest. Suppose you won $401 during those 2392 hands, and you wanted to know how unusual that was.

z(banker) = ($401-(-$468))/sqrt($$822,848) = 0.958 z(player) = ($401-(-$641))/sqrt($$865,904) = 1.120 z(tie) = ($401-(-$6851))/sqrt($$6,675,573) = 2.803

Those z's are the number of standard deviations from the norm.

Consulting a normal distribution chart, I find that the probability of such a win or better when betting banker, player, or tie while never saying "presto" (or else all mathematical probability is null and void) is:

Banker 17%
Player 13%
Tie 0.2%

So, this win of $401 would not be unusual, unless you were betting on the tie every time.

~ continued ~

 

[FLASH1296]

Baccarat Card Counting part #2

IV. THEOREMS AND SYSTEMS

Your results won't necessarily follow a normal distribution if you make your bets a function of your previous results. However, the expected value stuff still holds. You can use a system such as Martingale to give you a high chance of walking away a winner (up $1), but this is always exactly balanced out by some small chance of having a big loss - the average of all the possible outcomes produces the same house percentage as always. In a house game such as baccarat, you will lose a fixed percentage of the amount you put on the table, on average. For betting any one of bank, player, or tie, you can neither turn this house percentage into your favor, nor lessen it, nor worsen it.

i. Theorems

In _Theory of Gambling and Statistical Logic_, Epstein writes:

| Theorem I: If a gambler risks finite capital over a large number of plays
| in a game with constant single-trial probability of winning, losing, and
| tying, then any and all betting systems lead ultimately to the same value
| of mathematical expectation of gain per unit amount wagered.

| It follows that an unfavorable game remains unfavorable regardless of the
| variation in bets.

| The number of "guaranteed" betting systems, the proliferation of myths
| and fallacies concerning such systems, and the countless people believing,
| propagating, venerating, protecting, and swearing by such systems are
| legion. Betting systems constitute one of the oldest delusions of
| gambling history. Betting system votaries are spiritually akin to the
| proponents of perpetual motion machines, butting their heads against the
| second law of thermodynamics.

| Theorem II: No advantage accrues to the process of betting only on
| some subsequence of a number of independent repeated trials forming
| a complete sequence.

| Corollary: No advantage in terms of mathematical expectation accrues
| to the gambler who possesses the option of discontinuing the game
| after each play.

| Theorem III: For n plays of the general game, the mean or mathematical
| expectation is [as I defined it above] and the variance is [as I defined
| it above].

[Theorem IV omitted.]

| Theorem V: A gambler with initial fortune z, playing a game with the
| fixed objective of increasing his fortune by the amount a-z, has an
| expected gain that is a function of his probability of ruin (or success).
| Moreover, the probability of ruin is a function of the betting system.
| For equitable or unfair games, a "maximum boldness" strategy is
| optimal - that is, the gambler should wager the maximum amount
| permissible consistent with his objective and current fortune...

ii. Systems

Given the theorems, for the baccarat player who does not count cards, the following are the *optimal* systems for the specified goals:

If you want to maximize your expected value, your bankroll size after N hands, or anything like that, then DON'T PLAY.
If you want to win a goal amount, bet it all (or less if that would suffice to reach the goal if you win) on bank.
If you want to play for the longest time while losing the least, bet the minimum on bank (and don't bet many hands as possible.)
If you want to lose as much as you can as quickly as possible, bet it all on tie and keep parlaying until you lose it all.
One might argue that one could do better by keeping track of the number of player, bank, and tie wins in a shoe. The casino gives you scorecards for this purpose. Hmmm, they wouldn't encourage you to do it if it actually were worthwhile, now would they? No, they would not. The number of player, bank, and tie wins is extremely weakly correlated to the cards that have been removed, and as you'll see in the next section, even perfect card counting reveals depressingly few favorable situations.

V. CARD COUNTING

Card counters use knowledge of the cards that have been seen and discarded from play to alter their betting for the next round. I will present the results of the exhaustive combinatorial analysis program applied to various subsets of remaining cards in baccarat. I am not the first to do this. Griffin published figures in _Theory of Blackjack_, but his figures were in terms of average advantage in positive expected value situations, and this may have been disceiving.

Because I will for the first time be talking about positive expected value for baccarat, I will first make a detour through the Kelly criterion.

i. The Kelly Criterion

Many card counters bet according to the Kelly criterion. The Kelly criterion maximizes the size of your bankroll in the limit, which I think you'll agree is a Good Thing if you live forever, and it works pretty well for us mortals too. It turns out that that maximizing the size of your bankroll in the limit is equivalent to maximizing the logarithm of your bankroll at each step.

If you can buy into that, then you can use a neat little formula that computes the "Certainty Equivalent" or CE, which is an approximation for the amount of hard cash that a Kelly criterion believer would be ambivalent about exchanging for a given gamble:

CE = EV - VAR/2B

where EV is the expected value (described previously)
VAR is the variance (described previously)
and B is the bankroll
The Kelly criterion dictates that one should bet a fraction of one's bankroll equal to EV'/VAR', approximately, where EV' and VAR' are in terms of abstract "units", whereas EV and VAR above must be in the same units as B. When EV is negative, don't bet, or bet the minimum, of course. In the following computer analyses, I use this betting heuristic in the computation of the certainty equivalent, normalizing to a bankroll of B=1. If you bet according to this heuristic and know your EV and VAR, then the above formula turns into EV^2/2VAR, but when you are betting according to an estimate of your EV and VAR, this simplification doesn't hold.

ii. Perfect (Non-Linear) Card Counting

I fed the combinatorial analyzer random subsets of N cards drawn from 8 decks. It determined in each case the probabilities of bank, player, and tie winning, and from this computed the expected value. Any time the expected value was positive, this was counted as an "opportuntity to bet" and the certainty equivalent was computed. Overall statistics were then output for the percent opportunties to bet on each of the possible bets and the certainty equivalent in terms of the percent bankroll won per 100 observations at the given number of cards. One of the reasons for multiplying the results by 100 is to bring the results into the same magnitude as "advantage". Another reason is that it makes it easy to estimate hourly win rates, if you assume 100 hands are dealt per hour.

COMBINATORIAL ANALYSIS OF BACCARAT KELLY CARD COUNTER
PERFECT (NON-LINEAR) COUNTING
(Shoe composition sample size per deck level: 2000)
Tie pays 8 to 1; 8 deck shoe.

Percent Opportunities Mean Percent Bankroll Won/100 Obs.
================================== ==================================
cards bank 4% bank 5% player tie bank 4% bank 5% player tie
----- ------- ------- ------- ------- ------- ------- ------- -------
6 31.800 26.900 22.650 34.900 9.293 8.299 3.841 155.149
11 26.750 13.600 8.800 14.000 0.302 0.159 0.427 5.496
16 17.900 7.100 6.150 6.700 0.116 0.049 0.089 1.806
21 15.350 4.250 4.350 3.800 0.054 0.019 0.051 0.255
26 12.800 2.600 2.750 2.050 0.025 0.004 0.021 0.257
31 10.250 1.550 2.150 0.950 0.016 0.002 0.006 0.018
36 7.550 1.400 1.050 0.600 0.011 0.002 0.002 0.015
41 6.550 0.400 1.050 0.200 0.004 0.000 0.003 0.008
46 4.750 0.350 0.750 0.200 0.004 0.000 0.001 0.002
51 4.450 0.400 0.450 0.000 0.003 0.000 0.001 0.000
56 3.550 0.150 0.450 0.000 0.001 0.000 0.001 0.000
61 3.500 0.150 0.350 0.000 0.002 0.000 0.000 0.000
66 2.050 0.000 0.150 0.000 0.001 0.000 0.000 0.000
71 1.450 0.000 0.150 0.000 0.000 0.000 0.000 0.000
76 1.750 0.000 0.050 0.000 0.000 0.000 0.000 0.000
81 1.050 0.000 0.000 0.000 0.000 0.000 0.000 0.000
86 0.400 0.000 0.050 0.000 0.000 0.000 0.000 0.000
91 0.750 0.000 0.000 0.000 0.000 0.000 0.000 0.000
96 0.600 0.000 0.000 0.000 0.000 0.000 0.000 0.000
101 0.900 0.000 0.000 0.000 0.000 0.000 0.000 0.000
106 0.450 0.000 0.000 0.000 0.000 0.000 0.000 0.000
111 0.250 0.000 0.000 0.000 0.000 0.000 0.000 0.000
116 0.250 0.000 0.000 0.000 0.000 0.000 0.000 0.000
121 0.200 0.000 0.000 0.000 0.000 0.000 0.000 0.000
126 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
131 0.050 0.000 0.000 0.000 0.000 0.000 0.000 0.000
136 0.050 0.000 0.000 0.000 0.000 0.000 0.000 0.000
141 0.050 0.000 0.000 0.000 0.000 0.000 0.000 0.000
146 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
My deepest thanks to Tony Glenning for supplying the immense CPU power necessary to perform these experiments.

4% bank commission baccarat is available at Binion's Horseshoe in Las Vegas. Suppose in one shoe you play your first round after seeing one card and your last round after you have seen all but 25 of the cards. You will average .00163% bankroll per 100 rounds round, or about $16.30 per hour if you have a million dollar bankroll. Remember this is for absolutely *perfect* card counting. Only a computer is capable of computing the exact probabilities of winning and losing given the cards that have been seen and discarded from the shoe.

iii. Ultimate (Non-Linear) Counting System

In _Theory of Blackjack_ Peter Griffin presents the following table describing the "ultimate" point count values for baccarat:

Denomination `Player Bet' `Bank Bet' `Tie Bet'

A -1.86 1.82 5.37
2 -2.25 2.28 -9.93
3 -2.79 2.69 -8.88
4 -4.96 4.80 -12.13
5 3.49 -3.43 -10.97
6 4.69 -4.70 -48.12
7 3.39 -3.44 -45.29
8 2.21 -2.08 27.15
9 1.04 -.96 17.68
T,J,Q,K -.74 .78 21.28

Full Shoe % -1.23508 -1.05791 -14.3596
These are the best linear estimates of the effects of removal from a full shoe. (Actually, Griffin made a small error: 2's should be weighted as 2.17, not 2.28.)

Griffin gives an example of how to use this:

Suppose the first hand out of the shoe uses a 3 and 4 for the Player and a 9 and Jack for the Bank. Our running count is 2.69+4.80-.96+.78

+7.31. Now don't pludge into the `Bank' bet just because you have a positive count! Rather, divide it by the number of remaining cards, which is 416-4=412. You estimate the `Bank' expectation to be
-1.05791 + 7.31/412 = -1.04016%

so the shoe is not quite ready for us.

I repeated the previous combinatorial analysis, but this time computed the opportunity and certainty equivalent for the above Ultimate Linear Count System for baccarat.

COMBINATORIAL ANALYSIS OF BACCARAT KELLY CARD COUNTER
ULTIMATE (LINEAR) COUNT SYSTEM
(Shoe composition sample size per deck level: 25000)
Tie pays 8 to 1; 8 deck shoe.

Percent Opportunities to Bet Certainty Equivalent, %BR/100 Obs.
================================== ==================================
cards bank 4% bank 5% player tie bank 4% bank 5% player tie
----- ------- ------- ------- ------- ------- ------- ------- -------
10 23.616 10.224 7.372 3.500 0.1498 0.0467 0.0576 -0.0162
15 18.664 5.924 3.628 1.284 0.0741 0.0163 0.0172 -0.0010
20 15.012 3.340 1.832 0.444 0.0369 0.0051 0.0054 -0.0014
25 12.612 2.020 0.884 0.172 0.0226 0.0022 0.0016 -0.0015
30 10.600 1.340 0.384 0.068 0.0157 0.0014 0.0008 -0.0004
35 8.340 0.756 0.228 0.028 0.0087 0.0003 0.0003 0.0000
40 6.400 0.348 0.116 0.008 0.0053 0.0001 0.0001 -0.0000
45 5.404 0.200 0.064 0.000 0.0035 0.0000 0.0001 0.0000
50 4.488 0.080 0.020 0.000 0.0022 0.0000 0.0000 0.0000
55 3.580 0.028 0.004 0.000 0.0014 0.0000 0.0000 0.0000
60 2.860 0.012 0.008 0.000 0.0009 -0.0000 0.0000 0.0000
65 1.932 0.024 0.004 0.000 0.0006 0.0000 0.0000 0.0000
70 1.708 0.004 0.000 0.000 0.0004 0.0000 0.0000 0.0000
75 1.308 0.000 0.000 0.000 0.0003 0.0000 0.0000 0.0000
80 1.120 0.000 0.000 0.000 0.0002 0.0000 0.0000 0.0000
85 0.804 0.000 0.000 0.000 0.0001 0.0000 0.0000 0.0000
90 0.688 0.004 0.000 0.000 0.0001 0.0000 0.0000 0.0000
95 0.496 0.004 0.000 0.000 0.0001 0.0000 0.0000 0.0000
100 0.364 0.000 0.004 0.000 0.0000 0.0000 0.0000 0.0000
105 0.260 0.000 0.000 0.000 0.0000 0.0000 0.0000 0.0000
110 0.224 0.000 0.000 0.000 0.0000 0.0000 0.0000 0.0000
115 0.152 0.000 0.000 0.000 0.0000 0.0000 0.0000 0.0000
120 0.092 0.000 0.000 0.000 0.0000 0.0000 0.0000 0.0000
125 0.100 0.000 0.000 0.000 0.0000 0.0000 0.0000 0.0000
130 0.060 0.000 0.000 0.000 0.0000 0.0000 0.0000 0.0000
135 0.040 0.000 0.000 0.000 0.0000 0.0000 0.0000 0.0000
140 0.056 0.000 0.000 0.000 0.0000 0.0000 0.0000 0.0000
145 0.012 0.000 0.000 0.000 0.0000 0.0000 0.0000 0.0000
150 0.024 0.000 0.000 0.000 0.0000 0.0000 0.0000 0.0000
My deepest thanks to Tony Glenning for supplying the immense CPU power necessary to perform these experiments.

Note that the tie bet is often producing a negative certainty equivalent. This is because the linear count is either saying we have an advantage when we don't or vastly overestimating our advantage when we do have an advantage. To me, this says in black and white that the best linear estimate isn't the best estimate to use. First, one should use the linear estimate that maximizes one's profits, probably by erring on the side of conservatism; there is no excuse for having a *negative* certainty equivalent, as we could do better by not counting (and thus not betting.) Second, the tie bet is inherently nonlinear when you get down to small card subsets, so probably no linear count would do well on it.

For the previous case of the Binion's Horseshoe 4% bank bet dealt down to less than 25 cards unseen, the ultimate linear baccarat count would produce a certainty equivalent of .00109% bankroll per 100 rounds, down considerably from the .00163% of the perfect counter case. A millionaire ultimate linear baccarat counter would earn just $10.90 per hour, roughly.

VI. CONCLUSIONS

Sadly, baccarat is a tough nut to crack. Even perfect card counting yields pathetic results, and linear counting systems and progressive systems must by definition do worse. You may be able to use the information here to play baccarat intelligently for the comps, or perhaps you'll think of some "angle" to beat the game that violates the assumptions here (for example, I assumed no errors in payoffs of bets), but otherwise if you want to play with an advantage, you should stick to blackjack, poker, certain video poker machines, or any of a handful of other beatable games. Baccarat may be "beatable" with a huge bankroll, but it just isn't worth it.