Computer Studies of Baccarat, II: Baccarat-Banque

Computer Studies of Baccarat, II: Baccarat-BanqueAuthor(s): F. Downton and Carmen LockwoodSource: Journal of the Royal Statistical Society. Series A (General), Vol. 139, No. 3 (1976), pp.356-364Published by: Wiley for the Royal Statistical SocietyStable URL: http://www.jstor.org/stable/2344840 .

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J. R. Statist. Soc. A, 356 (1976), 139, Part 3, p. 356

Computer Studies of Baccarat, II: Baccarat-banque

By F. DOWNTON and CARMEN LOCKWOOD

University of Birmingham

SUMMARY Baccarat-banque is a game of chance in which a banker plays chemin-de-fer simultaneously against two players. It is permitted by the Gaming Clubs (Banker's Games) Regulations, 1970. It may be treated as either a two-person or a three-person zero-sum game. This paper describes various optimum strategies for the banker and the players on the assumption that the deck of cards with which the game is played is infinite (or, equivalently, that a finite deck is sampled with replacement).

Keywords: BANKER'S GAMES; GAMING LAWS; BACCARAT; CHEMIN-DE-FER; BACCARAT-BANQUE; OPTIMUM STRATEGIES; TWO-PERSON GAMES; THREE-PERSON GAMES

1. INTRODUCTION AND DISCUSSION OF THE GAME IN an earlier paper (Downton and Lockwood, 1975, referred to hereafter as I) the form of baccarat known as chemin-de-fer was described in detail, and strategies appropriate to that game for a variety of finite deck structures were obtained. The other main form of baccarat permitted under the Gaming Act of 1968 and its subsequent regulations is baccarat-banque, in which a banker plays chemin-de-fer simultaneously against two players. Since the rules of chemin-de-fer were given in detail in I they will not be recapitulated here. Suffice it to say that the aim of all players in both chemin-de-fer and baccarat-banque is to obtain with either two or three cards a total as close to 9 as possible, where the cards other than court cards have their natural value. The court cards each have value zero and the total is the sum of the card values (mod 10). All bets are paid at evens and the order of merit of hands are totals of 0, 1, 2, 3, 4, 5, 6, 7, 8 (three cards), 9 (three cards), natural 8 (two cards) and natural 9 (two cards). Equal hands result in a void bet. Players must draw a third card if their first two cards total 4 or less and cannot draw a third card if the first two cards total 6 or more. Only when the first two cards total 5 have the players any choice as to whether a third card may or may not be drawn. As in chemin-de-fer the banker is constrained by a table of play such as that given in Table 1

TABLE 1

Banker's upper limit for mandatory drawing and lower limit for mandatory standing, respectively (Casino Association of Great Britain Rules, 1972). With a total between (but not including) these

limits banker has option of the two strategies

Player 2 given

0-1 2-3 4 5 6-7 8 9 No card Natural

0-1 3,4 3,5 3,6 3,6 3,7 2,4 2,4 3,6 3,4 2-3 3,5 4,5 4,6 4,6 4,7 2,5 2,5 4,6 4,5 4 3,6 4,6 4,6 4,6 4,7 2,6 2,6 4,6 4,6 5 3,6 4,6 4,6 5,6 5,7 2,6 2,6 5,6 5,6

Player 1 6-7 3,7 4,7 4,7 5,7 6,7 2,7 2,7 5,7 6,7 given 8 2,4 2,5 2,6 2,6 2,7 2,3 2,4 2,6 2,3

9 2,4 2,5 2,6 2,6 2,7 2,4 2,4 2,6 2,4 No card 3,6 4,6 4,6 5,6 5,7 2,6 2,6 5,6 5,6 Natural 3, 4 4,5 4,6 5, 6 6, 7 2,3 2,4 5, 6



1976] DOWNTON AND LOCKWOOD - Computer Studies of Baccarat 357

of I, except that since in baccarat-banque the banker has two opponents the table of play is only binding on the banker if there is no conflict between the constraints implied by the cards he has given to each player. The constraints on the banker in baccarat-banque implied by Table 1 of I for chemin-de-fer are given in Table 1 of the present paper. The difficulty for the banker in baccarat-banque, regarded as requiring the use of the most experienced and skilful croupiers, arises from the fact that the play allows the banker so many options.

The extent to which the banker should employ these options will depend on the amount bet by and on each of the two players, which is not under the banker's control. Nor is differential betting under the control of the players; not only is co-operation as to staking policy explicitly excluded by the rules but, in addition, spectators who are not actually playing the hands may also bet on those hands. The game has therefore to be examined essentially from the point of view of the strategy appropriate to a banker faced with a particular configuration of bets; the players' roles have to be regarded as defensive, attempting only to minimize their losses against a banker's optimum strategy. These roles may be contrasted with those in the other major banker's game, blackjack, where the players have control over strategy and have an attacking role.

Published work on baccarat-banque (Foster, 1964; Kendall and Murchland, 1964; Downton and Holder, 1972) has assumed that both players have equal status; in fact, the second player to receive his cards (Player 2) had additional information not available to the first player (Player 1). The extent of this additional information depends on custom; in the traditional form of the game (see, for example, Lawson, 1950) Player 2 knows the value of the card, if any, given to Player 1 before he is required to decide whether he should take an additional card. Current practice in many British clubs is that Player 2 only has knowledge of whether Player 1 proposes to draw or stand with his first two cards. In this paper the traditional form of the game, which is slightly less favourable to the bank, has been assumed.

If Player 2 is to use this additional information, his strategy must depend on the action of Player 1 and its consequences, while the banker's action depends on that of both players. On the other hand, the strategy of Player 1 depends implicitly on the future behaviour of both Player 2 and the banker. Baccarat-banque is thus a three-person game. Even what constitutes a "solution" to such a game is by no means clear in general (see, for example, Lucas, 1971). In this paper three types of "solution" have been considered. These are:

(i) Fixed player strategy solutions; given the players' strategies the banker's strategy is determined which maximizes the banker's expected gain per unit total stake (from both players).

(ii) The co-operative optimum strategy; for this strategy the two players use the information available in the play co-operatively in order to minimize the banker's maximum expected gain per unit total stake. The corresponding banker's strategy, which maximizes his minimum expected gain per unit total stake, to counter this type of play is also found. This approach is in effect reducing the three-person game to a two-person one. This solution is probably the one of greatest interest to a banker, since it provides a safety-first strategy which guarantees a return to the bank, whatever strategy the players actually adopt.

(iii) The optimum equilibrium strategy; an equilibrium strategy in a multi-person game means here one such that if any of the players involved deviates unilaterally from the strategy he (and usually at least one other player) will reduce his expected gain per unit personal stake. It is by no means clear whether in a general multi-person game such solutions exist or whether if one is found it is unique. For the game of baccarat-banque the algorithm which has been used itself proves the existence of at least one such strategy and this has been derived. It can be argued, however, that its existence and properties are academic since it implies an attitude to the game by all three participants, which is unlikely to be realized in practice.



358 DOWNTON AND LOCKWOOD - Computer Studies of Baccarat [Part 3, All of these different types of strategy have been determined on the assumption that the

actions of all participants are determined by considering which action gives the best result in terms of expected gain per unit stake. No psychological considerations are involved; for example, it may be that in actual play a perceptive banker may be able to judge from a player's reactions whether he has a good hand or not, and this would influence that banker's strategy. Such considerations have not been taken into account here.

2. THE COMPUTER PROGRAM

It would have been possible to use the chemin-de-fer program by which the results in I were obtained to develop a baccarat-banque program for a finite deck of cards. There were two reasons for not doing this. First, the computing time for such a program would have been large; second, the number of parameters involved in describing strategies for baccarat-banque means that to try to present results for finite decks with different structures would be impracticable. A program using the experience gained with the chemin-de-fer program of I was therefore written supposing that an infinite deck was in use; that is, the same assumption on which the original solution for chemin-de-fer by Kemeny and Snell (1967) was based.

Only an outline of the program will be given here, since it involves no new principles from the program described in I. For reasons that have already been explained, the game is looked at from the point of view of the banker. Suppose the banker holds two cards totalling t (since the sampling is with replacement the cards which go to make up this total are irrelevant) and has given cards g1 and g2 to Players 1 and 2, respectively. Here g1 and g2 may take values 0-9 or have coded values 10 (denoting no card has been given) or 11 (denoting that the player has a "natural" 8 or 9). For fixed probabilityp, that Player 1 draws on a total of 5 the expected gain to the banker from unit stake by Player 1 may be evaluated by computing the probabilities of all possible hands arising from that player having a card g1 and two other cards. Suppose the expected gain when the banker stands is E(1) (gl,g2; p3) and when he draws is F(l) (g1,g2; ps). Similarly if the probability that Player 2 draws on total 5 is P2 the corresponding expected gains to the banker from unit stake by Player 2 will be E(2) (gp,g2; p2) and F12) (g1,g2; P2) when the banker stands and draws respectively. The total expected gains per unit stake to the banker when the stakes are in the proportion 0: (1- 0), 0 < 0@ 1, are then

and E~(g1,g2; PP2)l = 0E()(g1,g2; p,)+(I - 0)E2)(gpg2;P) = ( Fl(gl,g2; P1,P2) = 0 F(1) (gl,g2; p) +(l -o) F(21 (gl,g2; p2 J(

according to whether the banker stands or draws, respectively. If

Ej(g1, g2; Pl'P2) > F(g', g2; P1,P2) (2.2) then the banker should stand against the players' strategy (Pl,P2) and if

Et(g1,g2; Pl P'2) <JI(gl, g2; P P2 (2.3) he should draw holding t and giving (gl,g2). In fact, if equality should occur the banker's behaviour is irrelevant; it has been included in (2.2) simply for convenience and to make the strategy unique. It follows that by accumulating the appropriate expected gains, both the banker's strategy and his overall expected gain againstfixed player strategy may be determined.

The co-operative optimum strategy assumes that the players select values of p, and of P2, which depends on gl, in such a way that the maximum expected gain to the banker per total unit stake is minimized. For a fixed value of Pi the banker's strategy would be to stand if (2.2) holds for both P2 = 0 and P2 = 1 and to draw if (2.3) holds. When (2.2) holds for P2= 0 and (2.3) for P2 = 1 or vice versa the banker would be required to adopt a mixed strategy of sometimes standing and sometimes drawing. In such a situation the players' strategy would be to draw with probability p*, where p* satisfies the equation

p2* Ei(gl,g2; Pi, 1)+(1-p9*)Ei(gl,g2; Pl' 0) = p* Fi(gl,g2; Pi' 1)+(1-p2*)F1(gl,g2; Pi, 0). (2.4)




This equation is a consequence of the fact that with Pl, g1 and g2 fixed we have a simple two-person game situation whose optimum mixed strategy is given by the intersection of the line joining the values of Et with P2 = 0 and 1 with the line joining the corresponding values of Ft. This is equivalent to determining p* so that E,(g, g2; Pl,Pp*) = F1(gl, g2; Pi,P*). Over the whole set of values of (t, g2) for g1 fixed, there will be a limited number of values p* which, with 0 and 1, are possible optimum values of P2 p2(g1) for unknown (t,g2). That is, when the expected gains are averaged over the set of values (t, g2) these averages will remain piece-wise linear with respect to P2 and the optimum mixed strategy for Player 2 (for fixed Pi and g1) must be represented by one of these calculated values of p*. The computer program worked through these values to determine which of them gave the minimum expected gain per unit stake to the banker and also determined the banker's strategy corresponding to A2(g1), the parameter of the optimum strategy by Player 2 corresponding to Player 1 being given g1. By repeating this for the possible values of g, the optimum strategy and the corresponding total mimimax expected gain per unit stake for a given value of Pi, Player l's probability of drawing on total 5, was determined. These computations were repeated for a range of values of Pi; an interpolation process based on exploiting the piece-wise linearity of the total expected gain with respect to Pi was used to determine the optimum value A1 for Player l's strategy. The complete results, A1 and P2 (i), i = 0, 1, .. ., 11, together with the corresponding banker's strategy and the minimax expected gain could then be printed out. The results will be discussed later.

An alternative "solution", here called the equilibrium solution, was also determined. This required a slight modification of the co-operative optimum program so that instead of finding the values of A2(gl) to minimize the banker's maximum expected gain per total unit stake, N2(g) was found which minimized the banker's maximum expected gain per unit stake from Player 2. Similarly, ft1 minimized the maximum expected gain per unit stake from Player 1. Thus each player and the banker optimized his own expected gain per unit stake. Any unilateral deviation by any player (regarding the banker as a player) from this strategy would result in that player doing worse, but would also in general result in one of the other players doing worse as well.

RESULTS A summary of some of the results obtained is given in Tables 2-5. Five strategies have been

considered; three of these are fixed player strategies (a) Pi = P2 = 0, (b) Pi = 0, P2 = 1 and (c) P1 = P2 = 1, while the other two are the co-operative optimum (d) and the equilibrium (e) strategies. Results for the fixed player strategy Pi = 1, P2 = 0 may be obtained from (b) by symmetry. The banker's strategy for each of these five player strategies are given in Table 2(a)-(e), respectively, for stake distributions corresponding to 0 = 0 3, 0 5 and 0 7. When 0 = 0 or 1 the banker should, of course, concentrate on the player on whom all the money is staked and his strategy would be that of chemin-de-fer given by Kemeny and Snell (1957). These three values of 0 give only an indication of those card configurations to which the banker's strategy is sensitive. In no sense is interpolation possible between these values, since for a particular pair of cards given to Players 1 and 2 the discontinuities in the game may result in a small change in 0 causing the banker's minimum total for standing to change from, say, 7 to 3 without passing through intermediate values.

For the co-operative optimum strategy (d) the strategy given is not quite the optimum. This is because for this strategy the banker should, in approximately 8 combinations of given cards for each value of 0, mix the strategies of standing and drawing on one of his totals. Where the probability of drawing a card in the banker's mixture of strategies is less than 2 this has been recorded a "stand", while if it is greater than I it has been recorded as "draw". The effect of this on the overall performance of the strategy will be small. This co-operative strategy (d) is probably the one which is of greatest interest to the banker since it corresponds to the worst that can happen to him; in practice, where the players do not (or cannot because

15



360 DOWNTON AND LOCKWOOD - Computer Studies of Baccarat [Part 3,

TABLE 2(a) Minimum total on which banker should stand when both players are standing on a total of 5

(pI = 0, P2 = O) for values of 0 = 0 3, 0 5 and 0 7, respectively

Player 2 given

No 0 1 2 3 4 5 6 7 8 9 card Natural

0 4,4,44,4,44,4,45,4,45,5,46,5,46,5,45,4,44,4,43,4,46,5,54,4,4 1 4,4,44,4,44,4,45,5,45,5,56,5,56,6,55,5,44,4,44,4,46,5,54,4,4 2 4,4,44,4,45,5,55,5,55,5,56,5,56,6,56,5,54,4,54,4,46,6,55,5,5 3 4,4,54,5,55,5,55,5,55,5,56,6,56,6,56,5,55,5,54,4,56,6,55,5,5 4 4,5,55,5,55,5,55,5,55,5,56,6,66,6,66,6,65,5,54,5,56,6,65,5,5

Playerl 5 4,5,65,5,65,5,65,6,66,6,66,6,66,6,67,6,66,6,64,6,66,6,66,6,6 given 6 4,5,65,6,65,6,65,6,66,6,66,6,67,7,77,7,76,6,74,6,66,6,67,7,7

7 4,4,54,5,55,5,65,5,66,6,66,6,77,7,77,7,77,7,73,4,76,6,77,7,7 8 4,4,44,4,45,4,45,5,55,5,56,6,67,6,67,7,73,3,33,3,36,6,63,3,3 9 4,4,34,4,44,4,45,4,45,5,46,6,46,6,47,4,33,3,33,3,36,6,53,3,3

Nocard 5,5,65,5,65,6,65,6,66,6,66,6,66,6,67,6,66,6,65,6,66,6,66,6,6 Natural 4,4,44,4,45,5,55,5,55,5,56,6,67,7,77,7,73,3,33,3,36,6,6

TABLE 2(b)

Minimum total on which banker should stand when Player 1 stands and Player 2 draws on a total of 5 (p, = 0, P2 = 1) for values of 0 = 0 3, 0 5 and 0 7, respectively

Player 2 given

No O 1 2 3 4 5 6 7 8 9 card Natural

0 4,4,44,4,44,4,45,4,45,5,46,5,46,4,44,4,44,4,44,4,46,6,44,4,4 1 4,4,44,4,45,4,45,5,45,5,55,5,56,5,44,4,44,4,44,4,46,6,54,4,4 2 4,4,45,5,55,5,55,5,55,5,56,5,56,5,55,5,54,4,4 4,4,46,6,55,5,5 3 4,5,55,5,55,5,55,5,55,5,56,6,56,6,55,5,54,5,54,4,56,6,55,5,5 4 5,5,55,5,55,5,55,5,56,6,66,6,66,6,66,6,65,5,54,5,56,6,65,5,5

Playerl 5 5,5,65,5,65,6,66,6,66,6,66,6,66,6,66,6,65,6,65,5,67,6,66,6,6 given 6 5,5,65,5,65,6,66,6,66,6,66,6,67,7,77,7,74,6,64,5,67,6,77,7,7

7 4,5,55,5,65,5,65,6,66,6,76,6,77,7,77,7,74,4,74,4,57,7,77,7,7 8 4,4,44,4,45,5,45,5,55,5,56,6,66,6,67,3,33,3,34,4,37,7,63,3,3 9 4,4,44,4,44,4,45,4,45,5,46,5,46,4,44,3,33,3,34,4,36,6,43,3,3

Nocard 5,5,65,6,65,6,66,6,66,6,66,6,66,6,66,6,66,6,65,6,66,6,66,6,6 Natural 4,4,4 5,5,5 5,5,5 5,5,5 6,6,6 6,6,6 7,7,7 7,7,7 3,3,3 4,4,4 7,7,7




TABLE 2(c)

Minimum total on which banker should stand when both players are drawing on a total of 5 (P1 1b P2 = I) for values of 6 = 0-3, 0-5 and 0 7, respectively

Player 2 given

No 0 1 2 3 4 5 6 7 8 9 card Natural

O 4,4,4 4,4,4 5,5,4 5,5,4 5,5,5 6,5,5 6,5,5 4,4,4 4,4,4 4,4,4 6,6,5 4,4,4 1 4,4,4 5,5,5 5,5,5 5,5,5 5,5,5 6,5,5 6,6,5 5,5,5 4,4,4 4,4,4 6,6,5 5,5,5 2 4,5,55,5,55,5,55,5,55,5,56,6,56,6,55,5,54,4,54,4,56,6,65,5,5 3 4,5,55,5,55,5,55,5,55,5,56,6,56,6,66,5,54,5,54,5,56,6,65,5,5 4 5,5,55,5,55,5,55,5,56,6,66,6,66,6,66,6,65,5,54,5,56,6,66,6,6

Playerl 5 5,5,65,5,65,6,65,6,66,6,66,6,66,6,66,6,65,6,64,5,67,6,66,6,6 given 6 5,5,65,6,65,6,66,6,66,6,66,6,67,7,77,7,74,6,64,5,67,7,77,7,7

7 4,4,455,5,55,5,55,5,66,6,66,6,67,7,77,7,74,4,44,4,47,7,77,7,7 8 4,4,44,4,45,4,45,5,45,5,56,6,56,6,44,4,43,3,34,4,47,6,63,3,3 9 4,4,44,4,45,4,45,5,45,5,46,5,46,5,44,4,44,4,44,4,46,6,54,4,4

Nocard 5,6,65,6,66,6,66,6,66,6,66,6,77,7,77,7,76,6,75,6,67,7,77,7,7 Natural 4,4,4 5,5,5 5,5,5 5,5,5 6,6,6 6,6,6 7,7,7 7,7,7 3,3,3 4,4,4 7,7,7

TABLE 2(d)

Minimum total on which banker should stand when co-operative optimum strategy is being played by the players, for values of 0 = 0 3, 0-5 and 0-7, respectively

Player 2 given

No O 1 2 3 4 5 6 7 8 9 card Natural

0 4,4,44,4,45,4,45,5,45,5,56,5,56,5,44,4,44,4,44,4,46,6,54,4,4 1 4,4,44,4,45,5,55,5,55,5,56,5,56,6,55,5,54,4,44,4,46,6,54,4,4 2 4,4,55,5,55,5,55,5,55,5,56,5,56,6,55,5,54,4,54,4,56,6,55,5,5 3 4,5,55,5,55,5,55,5,55,5,56,6,66,6,65,5,54,5,54,4,56,6,65,5,5 4 5,5,55,5,55,5,55,5,56,6,66,6,66,6,66,6,65,5,54,5,56,6,66,6,6

Playerl 5 5,5,65,5,65,6,66,6,66,6,66,6,66,6,66,6,65,6,64,5,66,6,66,6,6 given 6 4,5,65,6,65,6,66,6,66,6,66,6,67,7,77,7,74,6,64,5,66,6,67,7,7

7 4,4,45,5,55,5,55,5,66,6,66,6,67,7,77,7,74,4,74,4,46,6,67,7,7 8 4,4,44,4,45,4,45,5,45,5,56,6,56,6,64,4,43,3,34,4,36,6,63,3,3 9 4,4,44,4,45,4,45,4,45,5,46,5,56,6,44,4,44,4,44,3,46,6,54,4,4

Nocard 5,6,65,6,66,6,66,6,66,6,6,6,6,67,6,77,7,76,6,65,6,66,6,66,6,6 Natural 4,4,44,4,45,5,55,5,56,6,66,6,677,7,77,7,73,3,34,4,46,6,6



362 DOWNTON AND LoCKWOOD - Computer Studies of Baccarat [Part 3,

TABLE 2(e)

Minimum total on which banker should stand in equilibrium strategy for 0 = 0-3, 0.5 and 0-7, respectively

Player 2 given

No 0 1 2 3 4 5 6 7 8 9 card Natural

0 4,4,4 4,4,4 5,4,4 5,5,4 5,5,5 6,5,5 6,5,4 4,5,4 4,4,4 4,4,4 6,5,5 4,4,4 1 4,4,4 4,4,4 5,5,55,5,5 5,5,5 6,5,5 6,5,5 5,5,5 4,4,4 4,4,4 6,6,5 4,4,4 2 4,4,5 5,5,5 5,5,5 5,5,5 5,5,5 6,5,5 6,6,5 5,5,5 4,4,5 4,4,5 6,6,4 5,5,5 3 4,5,5 5,5,5 5,5,5,5, 5,55,5, 56,6,5 6,6,6 5,5,5 4,5,5 4,5,5 6,6,6 5,5,5 4 5,5,5 5,5,5 5,5,5 5,5,5 6,6,6 6,6,6 6,6,6 6,6,6 5,5,5 4,5,5 6,6,6 6,6,6

Playerl 5 5,5,6 5,5,6 5,6,6 5,6,6 6,6,6 6,6,6 6,6,6 6,6,6 5,6,6 4,5,6 6,6,6 6,6,6 given 6 5,5,6 5,6,6 5,6,6 6,6,6 6,6,6 6,6,6 7,7,7 7,7,7 4,6,6 4,5,6 7,7,7 7,7,7

7 4,4,5 5,5,5 5,5,5 5,5,6 6,6,6 6,6,6 7,7,7 7,7,7 4,4,4 4,4,4 7,7,7 7,7,7 8 4,4,4 4,4,4 5,4,4 5,5,4 5,5,5 6,6,5 6,6,4 4,4,4 3,3,3 4,4,4 6,6,6 3,3,3 9 4,4,4 4,4,4 5,4,4 5,4,4 5,5,4 6,5,4 6,5,4 4,4,4 4,4,4 4,4,4 6,6,5 4,4,4

Nocard 5,6,6 5,6,6 6,6,6 6,6,6 6,6,6 6,6,6 6,7,7 7,7,7 6,6,6 5,6,6 7,7,7 6,6,6 Natural 4,4,4 4,4,4 5,5,5 5,5,5 6,6,6 6,6,6 7,7,7 7,7,7 3,3,3 4,4,4 6,6,6 -

TABLE 3

Players' co-operative optimum strategy: probabilities that players draw with total 5

P2 when Player 1 is given

No 0 P1 0 1 2 3 4 5 6 7 8 9 card Natural

0 - 0*82 0-82 0.82 0-82 0-82 0*82 0-82 0-82 0*82 0-82 0*82 0*82 0.1 0 0-91 0-95 0-96 *093 0-89 0*86 0-80 0-79 0*83 0-87 0*88 0-82 0.2 0-72 1 1 1 1 0-96 0.91 0-81 0-78 0*88 0-97 0*84 0*82 0.3 0-76 1 1 1 1 1 0.97 0-80 0-76 0*93 1 0*84 0*82 0*4 0-78 0-48 1 1 0-74 1 1 0-80 0-73 0-99 0-58 0'84 0*82 0.5 0-79 0*28 0 1 0-34 1 1 0-79 0-69 1 0 0*85 0*82 0-6 0.81 1 0-65 0-15 0 1 1 0*78 0*63 0*79 0-71 0*83 0*82 0*7 0*82 1 1 0-67 0 1 1 0*75 0.51 0 0 0.81 0*82 0*8 0.82 0 1 1 0-23 1 1 0-71 0.23 0 0 0X81 0 82 0*9 0-84 0 0 1 1 1 1 0-57 0 0 0 0-60 0-82 1.0 0-82



1976] DOWNTON AND LOCKWOOD - Computer Studies of Baraccat 363

of local rules) adopt the corresponding optimum strategy the banker will do rather better than predicted. It should be added that local cards of play (for example, that given in Table 1) may prevent the banker from adopting the strategy advocated in the tables; in such cases the banker should adopt the strategy closest to that given. The player strategies corresponding to the banker's strategies (d) and (e) are given in Tables 3 and 4.

TABLE 4

Players' equilibrium strategy: probabilities that players draw with total 5

P2 when Player 1 is given

No 0 Pi 0 1 2 3 4 5 6 7 8 9 card Natural

0 0-82 0.82 0.82 0.82 0-82 0.82 0-82 0-82 082 0-82 082 0-82 0.1 0O55 091 091 091 0.82 082 0-82 0-82 0-82 0-82 0.82 0-82 0.82 0.2 0.82 1 1 1 1 1 0-91 0-82 0-82 082 091 0.82 0-82 0.3 0-63 1 1 1 1 1 0.91 0.91 0.82 0.91 1 091 082 0-4 0-70 0-82 1 1 0.82 1 082 0-82 082 091 1 0-82 082 0 5 0.82 0 1 1 1 1 1 0-82 0.82 1 0-91 0-82 0.82 0-6 0-80 1 0-49 0-82 1 1 0.91 0-82 0-82 1 0-49 0.82 0-82 0 7 0-82 1 1 0-49 1 1 1 0-82 0-82 1 1 0-82 0-82 0-8 0-82 1 1 1 0 1 1 082 0.35 0-82 0 0.82 0-82 0 9 0.82 0 1 1 1 1 1 049 0 0 0 082 0*82 1.0 0 82

The apparently erratic behaviour of the results given in Tables 3 and 4 illustrates further the difficulty of viewing this game from the point of view of the two players. These players are essentially defending their position against the banker. Because of the discontinuities in the expected gains they may have to switch their strategies quite radically for small changes in the differential stake proportion 0, even though the banker's strategy may not change. As a result no interpolation can be made in these tables; it would require tabulation at very much smaller intervals of 0 to specify these discontinuities and make interpolation possible.

The banker's maximin expected gains per unit stake are given, for the various strategies and values of 0, in Table 5. It may be noted that the smallest of these is for the co-operative optimum strategy (d) when 0 = 0 5 giving an overall expected gain to the bank of 0-85 per cent of the stakes. In fact, because of the asymmetry of the game the worst (from the banker's point of view) distribution of stake money corresponds to a value of 0 = 0 474, when the banker's maximin expected gain is 0-852 per cent of the stakes. This might be regarded as the "value" of baccarat-banque, being the minimum expected gain to a "skilful" banker. The expected gains for the fixed strategies in Table 5 are consistent with those given by Foster (1964), and the banker's strategies with some obtained by Foster (private communication).

It may be added that all these calculations have been made on the assumption that the deck of cards is infinite. In I it was shown that using a finite deck of cards in the simpler game of chemin-de-fer increased the advantage to the bank. This is likely also to be the case for baccarat-banque.

REFERENCES DAWSON, L. H. (1950). Hoyle's Games Modernised, 20th ed. London: Routledge and Kegan Paul. DOWNTON, F. and HOLDER, R. L. (1972). Banker's games and the Gaming Act 1968. J. R. Statist. Soc. A,

135, 336-364. DOWNTON, F. and LOCKWOOD, CARMEN (1975). Computer studies of baccarat, I: Chemin-de-fer, J. R. Statist.

Soc. A, 138, 228-238.



364 DOWNTON AND LOCKWOOD - Computer Studies of Baccarat [Part 3, FOSTER, F. G. (1964). Contribution to the discussion of Kendall and Murchland. J. R. Statist. Soc. A, 127,

387-389. KEENY, J. G. and SNELL, J. L. (1957). Game-theoretic solution of baccarat. Amer. Math. Monthly, 64,

465-469. KENDALL, M. G. and MURcHLAND, J. D. (1964). Statistical aspects of the legality of gambling. J. R. Statist.

Soc. A, 127, 359-383. LucAs, W. F. (1971). Some recent developments in n-person game theory, SIAM Rev., 13, 491-523.

TABLE 5 Percentage expected gains per unit stake to banker from Player 1, Player 2 and both players for

different player strategies. (Banker playing the appropriate maximin strategy in each case)

(a) (b) (c) (d) (e) Player r P1=O P=0 pi=l pI=l Co-operative

0 strategy P2=O P2=1 P2=O P2=1 optimum Equilibrium

Player 1 - - 0 Player 2 1P54 1P37 1-54 1P37 1.28 1.28

Both 1P54 1-37 1-54 1-37 1-28 1*28

Player 1 -0'25 -0-42 -0-16 -0-25 -0-14 -0-07 0.1 Player 2 1P52 1P36 1P52 1P35 1.25 1*26

Both 1-34 1-18 1-35 1P19 1-12 1.13

Player 1 005 003 001 0-23 030 030 0.2 Player 2 1P47 1P23 1P49 1.27 1*18 1*18

Both 1P19 1-04 1-20 1-06 lP00 1P01

Player 1 0 43 0 70 0-18 0 50 0 49 0 50 0 3 Player 2 1P33 1.09 1.44 1.18 1*12 1*12

Both 1P06 0-97 1-06 0-98 0-93 0*94

Player 1 0-61 0'79 050 0-72 0.63 0.67 0 4 Player 2 1P24 1P04 1P28 1-07 1*04 1*02

Both 0.99 0-94 0-97 0-93 0-87 0*88

Player 1 0'96 0 94 0-91 0-92 0-87 0-98 0X5 Player 2 0.96 0.91 0 94 0-92 0.84 0 75

Both 0.96 0-93 0-93 0-92 0-85 0.86

Player 1 1.24 1P28 1-04 1-07 1P03 1P07 0.6 Player 2 0'61 0 50 0-79 0-72 0-63 0*59

Both 0.98 0 97 0 94 0-93 0-87 0-88

Player 1 133 1-44 1-13 1P19 1X13 1X16 0 7 Player 2 0 53 0-18 0-61 0 50 0-46 0X42

Both 1P06 1P06 0 97 0-98 0 93 0 93

Player 1 1P47 1'49 1-23 1P27 1-18 1-21 0-8 Player 2 0-05 0 01 0-03 0-23 0-29 0.19

Both 1P19 1'20 1P04 1P06 1P00 1P01

Player 1 1'52 1P52 1P36 1P36 1P27 1P27 0 9 Player 2 -0-25 -0-16 -0-42 -0-27 -0 19 -0 19

Both 1'34 1P35 1P18 1P19 1P12 1P12

Player 1 1P54 1P54 1P37 1P37 1P28 1P28 1P0 Player2 - - - -

Both 1-54 1P54 1P37 1P37 1-28 1*28



Documents

Computer Studies of Baccarat, II: Baccarat-Banque