The evolution of the game of baccarat

arXiv:1308.1481v2 [math.OC] 28 Apr 2015

S. N. Ethier∗ and Jiyeon Lee†

Abstract
The game of baccarat has evolved from a parlor game played by French
aristocrats in the first half of the 19th century to a casino game that
generated over US$41 billion in revenue for the casinos of Macau in 2013.
The parlor game was originally a three-person zero-sum game. Later in
the 19th century it was simplified to a two-person zero-sum game. Early
in the 20th century the parlor game became a casino game, no longer
zero-sum. In the mid 20th century, the strategic casino game became a
nonstrategic game, with players competing against the house instead of
against each other. We argue that this evolution was motivated by both
economic and game-theoretic considerations.
Key words and phrases: baccarat, banque, chemin de fer, punto banco,
zero-sum game, nonzero-sum game, best response, Nash equilibrium.

1 Introduction
The game of baccarat, which generated over US$41 billion in revenue for the
casinos of Macau in 2013, is a descendant of a 19th-century French parlor game.
But games evolve and a number of changes have been made to baccarat over
the more than 170 years it is known to have been played. Our aim in this paper
is to explain the likely motivation for these changes. In brief, the primary aim
was to make the game more profitable for casino operators. The identities of
the people responsible for this evolution are, with one exception, lost to history,
but there can be no doubt that they were remarkably successful in achieving
their goal.
In Section 2 we provide the historical background for the game of baccarat,
emphasizing the three principal forms of the game, baccarat banque (a three-
person game), baccarat chemin de fer (a two-person game), and baccarat punto
banco (a nonstrategic game). Section 3 treats the zero-sum parlor game of
∗ Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT

84112, USA. e-mail: ethier@math.utah.edu. Partially supported by a grant from the Simons
Foundation (209632).
† Department of Statistics, Yeungnam University, 214-1 Daedong, Kyeongsan, Kyeongbuk

712-749, South Korea. e-mail: leejy@yu.ac.kr. Supported by the Basic Science Research
Program through the National Research Foundation of Korea (NRF) funded by the Ministry
of Education (2013R1A1A3A04007670).

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baccarat chemin de fer in detail. Section 4 considers the nonzero-sum casino
game of baccarat chemin de fer, both in its original form and in its modern form.
Section 5 discusses baccarat punto banco, which is the form of baccarat most
widely played today. Finally, we summarize the reasons for game’s evolution in
Section 6.

2 Baccarat history
A number of authors have claimed that the game of baccarat (usually spelled
“baccara” in French) dates back to 15th-century Italy, but this claim is ques-
tionable because the earliest known mention of the game appears in Van-Tenac
(1847, pp. 84–97); on the other hand, the game was played exclusively in aristo-
cratic circles, which may explain why it “does not grace the realms of recorded
history before the nineteenth century” (Parlett 1991, pp. 81–82).
The Van-Tenac reference deals with baccarat banque, a three-person zero-
sum parlor game, which is likely the earliest form of baccarat. A simpler ver-
sion of the parlor game, baccarat chemin de fer, is a two-person zero-sum game
that appeared later. The first reliable mathematical study is due to Dormoy
(1873), who referred to the latter as baccarat tournant (rotating baccarat).
The term “baccarat chemin de fer” (baccarat railway) came into vogue by 1880
(Badoureau 1881). Baccarat chemin de fer was studied by Bertrand (1889,
pp. 38–42), who was unaware of Dormoy’s (1873) monograph, and although
Borel (1924) correctly characterized Bertrand’s work as “extremely incomplete,”
it motivated Borel to begin the development of game theory in the 1920s (Di-
mand and Dimand 1996, p. 132). The first game-theoretic analysis of baccarat
chemin de fer was carried out by Kemeny and Snell (1957).
Well before the legalization of casino gambling in France in 1907, baccarat
was played in clubs, and to pay the expenses a commission was levied. At
baccarat chemin de fer, the amount of the commission may have varied to some
extent (Villiod 1906, p. 165), but a five percent commission on Banker wins
became standard. This makes the game a two-person nonzero-sum game, or a
bimatrix game, that has a unique Nash equilibrium, at least under the Kemeny–
Snell “with replacement” assumption.
Eventually, Banker’s strategic options at baccarat chemin de fer were severely
constrained, with the goal of ensuring optimal play, and perhaps reducing the
level of skill needed to play the role of Banker. Early descriptions of casino
baccarat chemin de fer (e.g., Beresford 1926, p. 74) included the five percent
commission on Banker wins, but Banker’s strategy was still unconstrained. It
was also unconstrained in the descriptions of Le Myre (1935) and Boll (1936,
1944). The constraints were present in Scarne’s (1949, p. 206) description of the
game, though he was referring not to French casinos but to (illegal) US casinos.
In pre-revolutionary Havana, a third form of baccarat, sometimes called
punto banco but more often just baccarat, was introduced. In baccarat punto
banco the casino banks the game, offering bets on Player and on Banker. More-
over, the strategic options of baccarat chemin de fer are eliminated, resulting

2
in a nonstrategic game, which has become the most widely played form of bac-
carat. It currently enjoys its greatest popularity in the territory of Macau, where
the 35 casinos generated gambling revenue of about US$45 billion in 2013, of
which over 91% came from baccarat (Schwartz 2014). Baccarat punto banco
(the Spanish term is “punto y banca”) is said to have first appeared in Ar-
gentina, but the Argentinian game was different: Although Banker’s strategy
was fixed, Player still had the option of drawing or standing on 5 and the game
was not house-banked (Chapitre 1954, pp. 25–26). The Cuban innovations were
to make the game house-banked, allowing bets on Player and on Banker, as well
as to mandate Player drawing on 5. It seems likely that the former modifica-
tion occurred during the 1940s, and the latter during the 1950s. Indeed, Scarne
(1949, p. 213) referred to “baccarat” as a house-banked version of chemin de fer,
though without mentioning where it was played. And Tommy Renzoni wrote,
“I opened the Baccarat game at the Capri [in Havana, c. 1956] with a somewhat
revised set of rules that made the game more automatic, more a matter of pure
chance.” (Renzoni and Knapp 1973, pp. 49–50.) This suggests that it was he
who mandated Player drawing on 5. It is well known that this form of baccarat
came to the Las Vegas Sands in November 1959 (Renzoni and Knapp 1973,
p. 69), and to Macau in 1962.
Thus, over a lifetime of at least 170 years, the game of baccarat has under-
gone five significant rules changes (the three-person game of baccarat banque
was simplified to the two-person game of baccarat chemin de fer; a five percent
commission was imposed on Banker wins at baccarat chemin de fer; Banker’s
strategic options at baccarat chemin de fer were severely constrained; baccarat
punto banco was introduced as a house-banked game, with bets offered on Player
and on Banker; and all remaining strategic options were eliminated at baccarat
punto banco), yet in most respects it is the same game. Our aim here is to
argue that the game’s evolution was motivated by economic and game-theoretic
considerations. See Section 6 for a summary of our conclusions in this regard.

3 The parlor game of baccarat chemin de fer

The rules of the parlor game of baccarat chemin de fer are as follows. The role
of Banker rotates among the players (counter-clockwise), changing hands after
a Banker loss or when Banker chooses to relinquish his role. Banker announces
the amount he is willing to risk, and the total amount bet on Player’s hand
cannot exceed that amount. After a Banker win, all winnings must be added to
the bank unless Banker chooses to withdraw. Denominations A, 2–9, 10, J, Q,
K have values 1, 2–9, 0, 0, 0, 0, respectively, and suits are irrelevant. The total
of a hand, comprising two or three cards, is the sum of the values of the cards,
modulo 10. In other words, only the final digit of the sum is used to evaluate a
hand. Two cards are dealt face down to Player and two face down to Banker,
and each looks only at his own hand. The object of the game is to have the
higher total (closer to 9) at the end of play. Winning bets on Player’s hand
are paid by Banker at even odds. Losing bets on Player’s hand are collected by

3
Banker. Hands of equal total result in a tie or a push (no money is exchanged).
A two-card total of 8 or 9 is a natural. If either hand is a natural, play ends.
If neither hand is a natural, Player then has the option of drawing a third card
(but this option is heavily constrained; see below). If he exercises this option,
his third card is dealt face up. Next, Banker, observing Player’s third card, if
any, has the option of drawing a third card. This completes the game. Since
several players can bet on Player’s hand, Player’s strategy is restricted. He must
draw on a two-card total of 4 or less and stand on a two-card total of 6 or 7.
When his two-card total is 5, he is free to draw or stand as he chooses. (The
decision is usually made by the player with the largest bet.) Banker, on whose
hand no one can bet, has no constraints on his strategy.
This is the original form of the game. If we assume, as did Kemeny and Snell
(1957), that cards are dealt with replacement from a single deck (often expressed
by assuming a shoe with infinitely many decks), and that each of Player and
Banker sees the total of his own two-card hand but not its composition, then
baccarat is a 2×288 matrix game. Let us reduce the size of the game considerably
by eliminating strictly dominated columns of the payoff matrix A. We find that
Banker’s optimal move does not depend on Player’s strategy in 84 of the 88
strategic situations, the exceptions being (3, 9) (Banker total 3, Player third
card value 9), (4, 1), (5, 4), and (6, ∅) (Banker total 6, Player stands). See
Table 1.

Table 1: Banker’s optimal move at baccarat chemin de fer, indicated by D

(draw) or S (stand), except in the four cases indicated by ∗ in which it depends
on Player’s strategy. The shading of D entries is for improved readability.

Banker’s Player’s third card value (∅ if Player stands)

two-card
total 0 1 2 3 4 5 6 7 8 9 ∅
0 D D D D D D D D D D D
1 D D D D D D D D D D D
2 D D D D D D D D D D D
3 D D D D D D D D S ∗ D
4 S ∗ D D D D D D S S D
5 S S S S ∗ D D D S S D
6 S S S S S S D D S S ∗
7 S S S S S S S S S S S

The case of (3, 9) occurred in the climactic hand in Ian Fleming’s (1953)
novel Casino Royale (see Ethier 2010, pp. 616–617, for the full quotation), with
James Bond in the role of Player, Le Chiffre in the role of Banker, and 32 million
francs at stake. As Fleming noted (Chap. 13), “Holding a three and giving nine
is one of the moot situations at the game. The odds are so nearly divided

4
between to draw or not to draw.” Le Chiffre chose to draw (which happens to
be the correct move from the game-theoretic perspective), drawing 5 for a total
of 8. Then Bond’s hand was turned over to reveal two queens for a total of 9
and the win. Technically, the game they played was not quite baccarat chemin
de fer because the role of Banker did not rotate among the players. Instead, Le
Chiffre purchased the role of Banker for one million francs (Chap. 9).
Using Table 1 we can reduce the matrix game to 2 × 24 , so we now regard
A as being a 2 × 16 matrix. We can further reduce this 2 × 16 matrix to a 2 × 5
matrix using strict dominance. Specifically, A is replaced by the matrix (again
denoted by A)

SSSS SSDS DSDS DSDD DDDD

 
S on 5 −4636 −4635 −4564 −2692 −2585 16
A= , (1)
D on 5 −3585 −3600 −3705 −4121 −4126 (13)6

where, for example, the Banker pure strategy DSDS means draw on (3, 9), stand
on (4, 1), draw on (5, 4), and stand on (6, ∅). It is easy to check that the kernel is
specified by columns DSDS and DSDD, and that leads to the following theorem.
Theorem 1 (Kemeny–Snell 1957). The parlor game of baccarat chemin de fer,
a 2 × 288 matrix game, has a unique solution. Player’s optimal mixed strategy
is to draw on 5 with probability p := 9/11. Banker’s optimal mixed strategy is
as in Table 1, except that he draws on (3, 9), stands on (4, 1), draws on (5, 4),
and mixes on (6, ∅), drawing with probability q := 859/2288. The value of the
game (to Player) is vP := −679568/[11(13)6] ≈ −0.0127991.
See Ethier (2010, Chap. 5) for a detailed proof and Deloche and Oguer
(2007b) for an alternative derivation. Other analyses of the parlor game of
baccarat chemin de fer have been given by Foster (1964), Downton and Lock-
wood (1975), and Ethier and Gámez (2013). These papers treated increasingly
realistic models of the game.

4 The casino game of baccarat chemin de fer

Initially, the casino game differed from the parlor game in only one respect, the
imposition of a five percent commission on Banker wins. Let us consider, more
generally, a 100α percent commission on Banker wins, where 0 ≤ α < 1/15. If
we continue to assume that cards are dealt with replacement from a single deck,
and that each of Player and Banker sees the total of his own two-card hand
but not its composition, then baccarat is a 2 × 288 bimatrix game, with payoff
matrix A to Player as before, and payoff matrix Bα to Banker with Banker
winning 1 − α per unit lost by Player and losing 1 per unit won by Player.
Let us reduce the size of the game considerably by eliminating strictly dom-
inated columns of Bα . Assuming 0 ≤ α < 1/15, we find that Banker’s optimal
move does not depend on Player’s strategy in 84 of the 88 strategic situations,
the exceptions again being (3, 9), (4, 1), (5, 4), and (6, ∅). Table 1 still applies.

5
This reduces the game to a 2 × 24 bimatrix game. We can further reduce it to
2 × 5, with A as in (1) and Bα equal to the matrix whose transpose is

S on 5 D on 5
 
SSSS 9272 − 278353α 7170 − 276363α
SSDS 
 9270 − 278423α 7200 − 276433α 
 8
BαT = DSDS 
 9128 − 278423α 7410 − 276593α 
 (13)6 . (2)
DSDD  5384 − 278007α 8242 − 278673α 
DDDD 5170 − 277971α 8252 − 278733α
We know that there exists a Nash equilibrium, and we can use the support
enumeration algorithm to find it and show it is unique. This suffices, provided
the game is nondegenerate, which means that no mixed strategy of support size
s ≥ 1 has more than s pure best responses. This hypothesis can easily be
verified.
Theorem 2. The classic casino game of baccarat chemin de fer with a 100α
percent commission on Banker wins, a 2 × 288 bimatrix game parameterized by
α, has a unique Nash equilibrium for 0 ≤ α < 1/15. Player’s equilibrium mixed
strategy is to draw on 5 with probability
9−α
p := . (3)
11 − 6α
Banker’s equilibrium mixed strategy is as in Table 1, except that he draws on
(3, 9), stands on (4, 1), draws on (5, 4), and mixes on (6, ∅), drawing with prob-
ability q := 859/2288. The safety level for Player is vP of Theorem 1 and that
for Banker is
8(84946 − 3099233α + 1668708α2)
vB := .
(11 − 6α)(13)6
If α = 0.05, then vB ≈ −0.0101991. These are also the expected payoffs if both
Player and Banker use their equilibrium strategies.
By safety level we mean the amount a player can guarantee himself on av-
erage, regardless of his opponent’s strategy.
Banker’s safety level vB is greater than Player’s
√ safety level vP if and only
if 0 ≤ α < α′ , where α′ := (34601239 − 1060031672799697)/36711576 ≈
0.0556531. A commission on Banker wins greater than or equal to α′ would be
counter-productive because there would then be no incentive for players to take
the role of Banker. (We are assuming, in effect, that all players are rational and
knowledgeable, which may be unrealistic.) From the casino’s perspective then,
the commission α should be maximized subject to the constraints that α < α′
and α is simple enough to permit mental calculations. 5.5% was presumably
deemed too complicated, so five percent became the accepted figure.
The reason for assuming 0 ≤ α < 1/15 in Theorem 2 is that this is the
maximal interval over which Table 1 applies without change.
As mentioned previously, in modern baccarat chemin de fer, Banker’s strat-
egy is highly constrained. Specifically, the 84 of the 88 strategic situations that

6
require a draw or stand decision in Table 1 are all part of the modern Banker
strategy. In addition, Banker stands on (4, 1) (Banker total 4, Player third card
value 1), perhaps because the improvement in expected gain from drawing in-
stead of standing when Player draws on 5 is very small (about 0.0025641 when
α = 0.05). Finally, Banker stands on (6, ∅) (Banker total 6, Player stands),
despite the fact that the improvement in expected gain from drawing instead of
standing when Player draws on 5 is rather substantial (about 0.0673077 when
α = 0.05). Under these rules, which predate 1949 (Scarne 1949, p. 206) the only
optional cases for Banker are (3, 9) and (5, 4).1 Thus, we have a 2 × 4 bimatrix
game with payoff bimatrix (A, Bα ), where

SS SD DS DD
 
S on 5 −4636 −4635 −4565 −4564 16
A=
D on 5 −3585 −3600 −3690 −3705 (13)6

and
S on 5 D on 5
 
SS 9272 − 278353α 7170 − 276363α
BαT =
SD 
 9270 − 278423α  8 ;
7200 − 276433α 
DS  9130 − 278353α 7380 − 276523α  (13)6
DD 9128 − 278423α 7410 − 276593α
here, for example, the Banker pure strategy DS means draw on (3, 9) and stand
on (5, 4).
Theorem 3. The modern casino game of baccarat chemin de fer with a 100α
percent commission on Banker wins, a 2 × 4 bimatrix game parameterized by
α, has a unique Nash equilibrium for 0 ≤ α < 2/5, and it is a pure Nash
equilibrium. Player’s equilibrium strategy is to draw on 5. Banker’s equilibrium
strategy is as in Table 1, except that he draws on (3, 9), stands on (4, 1), draws
on (5, 4), and stands on (6, ∅). The safety levels are vP := −3705(16)/(13)6 ≈
−0.0122814 and vB := 8(7410 − 276593α)/(13)6. If α = 0.05, then vB ≈
−0.0106400. These are also the expected payoffs if both Player and Banker use
their equilibrium strategies.
Because the casino levies a commission on Banker wins, it is to the casino’s
advantage to have Banker play well. Perhaps this explains why Banker’s strat-
egy is so severely constrained in the modern game. A more convincing explana-
tion is that casinos may have wanted to reduce the level of skill required to take
the role of Banker. There has been speculation that Banker’s strategy evolved
gradually by trial and error, but that seems unlikely. The essence of Table 1
dates back to the 19th century. Specifically, Dormoy (1873), Billard (1883), and
Hoffman (1891) had versions of Table 1, but with errors in two or three entries.
Finally, by the 20th century, the exact values had been computed, for example
1 We also find these rules in a 1920 Spanish book (Chapitre 1920), though they appear to

be recommendations rather than requirements. At (3, 9) and (5, 4) the recommendation is

“indifferent”, not “optional”.

7
by Le Myre (1935). Thus, it is not surprising that, in the 84 entries that do not
depend on Player’s strategy, all appear correctly in the modern Banker strategy.
The only surprise is the stipulation that Banker stand on (6, ∅) because this
decision depends on Player’s strategy. (The same is true in the case of (4, 1), but
we have noted that a mandated Banker stand makes sense in that case.) Some
authors have attributed this to a mistake, while others have suggested that it
was an attempt to equalize the Player and Banker safety levels. Interestingly, at
Crockford’s Club in London in the early 1960s, Banker was allowed to draw or
stand on (6, ∅) (Kendall and Murchland 1964) as well as on (3, 9) and (5, 4), and
this was more than a decade after the mandatory stand on (6, ∅) had become
conventional wisdom. Perhaps Crockford’s had a mathematician advising them.

5 The casino game of baccarat punto banco

Although baccarat chemin de fer and baccarat banque are still available at the
Casino de Monte-Carlo, the most widely played form of baccarat today is called
baccarat punto banco (or just baccarat) and is not a strategic game because
both Player and Banker have mandated strategies: Player draws on 5 or less
and stands on 6 and 7, and Banker uses Table 1 and draws on (3, 9), stands on
(4, 1), draws on (5, 4), and stands on (6, ∅). An even more important distinction
between baccarat chemin de fer and baccarat punto banco is that, in the latter,
one can bet either on Player (paid at even odds) or on Banker (paid at 19 to
20, equivalent to even odds with a five percent commission on winning Banker
bets). Both bets are banked by the casino.
To better understand the flow of money at baccarat, let P and B be the
probabilities that Player wins and Banker wins. Then, under the usual “with
replacement” assumption, P = 2153464/(13)6 and B = 2212744/(13)6, so
296400
B−P = ≈ 0.0122814, (4)
5(13)6
256786
P − (19/20)B = ≈ 0.0106400, (5)
5(13)6
553186
(1/20)B = ≈ 0.0229214. (6)
5(13)6

Eq. (4) is the casino’s expected gain per unit bet on Player, (5) is the casino’s
expected gain per unit bet on Banker, and (6) is the casino’s expected gain per
unit bet at baccarat chemin de fer. Notice that (6) is the sum of (4) and (5).
This is consistent if we regard the casino’s expected gain at baccarat chemin
de fer as coming not just from Banker but from Player via Banker and from
Banker directly.
In baccarat chemin de fer, the total amount bet on a hand is limited by the
amount Banker is willing to risk; thus, the more bettors, the more likely that
there will be unfulfilled demand. To make this more explicit, let us suppose
there are n players who would be willing to bet x1 , x2 , . . . , xn > 0 units on

8
Player. In baccarat chemin de fer, if Banker announces a bet of y > 0 units,
then the amount bet on Player (as well as the amount bet by Banker) would
be min(s, y) units, where s := x1 + x2 + · · · + xn ; whereas in baccarat punto
banco there would be s units bet on Player and y units bet on Banker. Thus,
s + y − 2 min(s, y) = |s − y| units is the amount of unfulfilled demand. Another
advantage of having the casino bank the bets is that a game can begin with
as few as one player; in baccarat chemin de fer, the game is delayed until the
required number of players are available.
There is one attractive feature (from the casino’s perspective) of baccarat
chemin de fer: The players may win or lose but the casino only wins; there is
no risk to the house. But as Lucas and Kilby (2012, p. 266) pointed out (again,
from the casino’s perspective), “the feature that makes [baccarat chemin de fer]
attractive is also its greatest weakness.” Much more money is to be made by
banking the game than by “raking” it, even if there is a resulting increase in
risk.
There are at least two advantages to mandating the strategies of Player and
Banker. First, it avoids the unpleasantness of one player making a decision that
negatively impacts other players. Second, it speeds up the game when there are
no decisions to be made. The main issue we want to address here is why the
mandatory drawing rules were chosen. There are three potential explanations.

1. The Player and Banker strategies mandated by the rules of baccarat punto
banco are the closest pair of pure strategies to the optimal mixed strategies
of Kemeny and Snell (1957) for the parlor game of baccarat chemin de fer.
2. The Player and Banker strategies mandated by the rules of baccarat punto
banco coincide with the pure Nash equilibrium at modern baccarat chemin
de fer, in which Banker has options only on (3, 9) and (5, 4); see Section 4.
(When α = 0, the pure Nash equilibrium is a saddle point.)
3. The Banker strategy mandated by the rules of baccarat punto banco co-
incides with Banker’s best response to Player’s (1/2, 1/2) mixed strat-
egy (that is, Player draws on 5 with probability 1/2), often expressed in
the baccarat literature as Banker’s best response when he does not know
whether Player is in the habit of drawing or standing on 5. Furthermore,
the Player strategy mandated by the rules of baccarat punto banco coin-
cides with Player’s best response to Banker’s mandated strategy.

Since baccarat punto banco almost certainly predates the Kemeny–Snell

paper, the first explanation is not credible and is merely a coincidence. The
second explanation is also doubtful because game-theoretic concepts such as
saddle points were not well known in the casino industry in the 1950s and
earlier. Further, there is no mention of this property in the baccarat literature.
The third explanation is the most plausible. Banker’s best response to (1/2, 1/2)
was implicitly noted in a well-known book by Le Myre (1935)2 and explicitly
2 Although Le Myre’s mixture was intended to be (1/2, 1/2) (p. 37), it was actually

(89/194, 105/194) when Player draws (p. 85) and (3/5, 2/5) when Player stands (p. 51).

9
noted by Boll (1936, §187) in another widely circulated book. In a subsequent
book, Boll (1944, Fig. 13) argued that a (1/3, 2/3) mixed strategy (that is,
Player draws on 5 with probability 2/3) is more realistic, and this too has the
same Banker best response. This book was published the same year as von
Neumann and Morgenstern’s Theory of Games and Economic Behavior, so it is
not surprising that its author was unaware of the subject of game theory. As a
best response to a Player mixture of (1/2, 1/2), this Banker strategy had already
been published, albeit slightly inaccurately, by Billard (1883) and Hoffmann
(1891). Finally, as is clear from Chapitre (1954, pp. 25–26), “punto y banca”
had a mandated Banker strategy in Argentina, whereas Player still had a choice.
So when that game was modified for use in Havana, it would have been easy
to check that Player’s best response is to draw on 5. This is consistent with
Tommy Renzoni’s statement mentioned earlier. Thus, we can arrive logically
at the mandated strategies at baccarat punto banco without having to assume
that mistakes were made at some point in the distant past.

6 Summary
We mentioned that the game of baccarat has undergone five significant rules
changes. Here we review these changes and the likely reasons for them.
The three-person game of baccarat banque was simplified to the two-person
game of baccarat chemin de fer in the 19th century. A likely reason is that
baccarat banque required more players and was more difficult logistically. Later,
it would prove to be profitable in the very exclusive environment of high-stakes
gambling (Graves 1963).
A five percent commission was imposed on Banker wins at baccarat chemin
de fer to make it possible to offer the game in clubs and casinos. The amount
of the commission was presumably chosen as large as possible consistent with
the goals of permitting simple mental calculations and not discouraging players
from taking the role of Banker.
Later, Banker’s strategy was severely constrained, leaving only two draw-or-
stand options, at (3, 9) and (5, 4), perhaps partly because it was in the interest
of the casino to have Banker play well. A more important reason is that less
skill would be necessary to take the role of Banker. The actual strategy chosen
is very nearly optimal in the game-theoretic sense. (Only the mandatory stands
at (4, 1) and (6, ∅) can be questioned, and as we have noted, at least the first
of these has a logical justification.)
Then, in Argentina, Banker’s strategy became fixed, with mandatory draws
at (3, 9) and (5, 4). Perhaps the reason was to make baccarat more like black-
jack, in which players do not have to fear a skillful dealer. The game-theoretic
justification for this fixed Banker strategy is that it is the best response to an
equally weighted mixture of Player’s two pure strategies, for which the concept
(if not the implementation) dates back to the 19th century.
The most important change to baccarat seems to have been made in Havana
in the 1940s. Baccarat punto banco became a house-banked game, offering bets

10
on Player and on Banker. This made the game vastly more profitable for the
casinos (at the cost of additional risk), eliminating the need for offsetting bets,
and allowing any number of players to bet as much as they want. Further, the
ability to bet on either Player or Banker gives the perception that the game is
fair.
Finally, in baccarat punto banco as played in Havana in the 1950s, Player’s
strategy was also fixed by mandating a draw on 5, which is the best response
to Banker’s fixed strategy. Eliminating strategy decisions speeds up the game
and makes it purely a game of chance rather than one of chance and skill.
All of these changes were motivated, at least in part, by a desire to make the
game of baccarat more profitable for casino operators. However, before a game
can be profitable to casino operators it must be attractive to casino customers,
and perceived fairness plays an important role in attractiveness. Baccarat punto
banco is almost unique among casino games in term of its perceived fairness and
its profitability.

Acknowledgment
We thank Régis Deloche for valuable advice concerning an earlier draft of this
paper.

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