A game-theoretic analysis of baccara chemin de fer

Assuming 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, baccara is a 2 x 2^88 matrix game, which was solved by Kemeny and Snell in 1957. Assuming that cards are dealt without replacement from a d-deck shoe and that Banker sees the composition of his own two-card hand while Player sees only his own total, baccara is a 2 x 2^484 matrix game, which was solved by Downton and Lockwood in 1975 for d=1,2,...,8. Assuming that cards are dealt without replacement from a d-deck shoe and that each of Player and Banker sees the composition of his own two-card hand, baccara is a 2^5 x 2^484 matrix game, which is solved herein for every positive integer d.


Introduction
The game of baccara chemin de fer (briefly, baccara) played a key role in the development of game theory. Bertrand's (1889, pp. 38-42) analysis of whether Player should draw or stand on a two-card total of 5 was the starting point of Borel's investigation of strategic games (Dimand and Dimand 1996, p. 132). Borel (1924) described Bertrand's study as "extremely incomplete" but did not himself contribute to baccara. It is unfortunate that Borel was unaware of Dormoy's (1873) work, which was less incomplete. Von Neumann (1928), after proving the minimax theorem, remarked that he would analyze baccara in a subsequent paper. But a solution of the game would have to wait until the dawn of the computer age. Kemeny and Snell (1957), assuming 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, found the unique solution of the resulting 2 × 2 88 matrix game. In practice, cards are dealt without replacement from a sabot, or shoe, containing six 52-card decks. Downton and Lockwood (1975), allowing a d-deck shoe dealt without replacement and assuming that Banker sees the composition of his own twocard hand while Player sees only his own total, found the unique solution of the resulting 2 × 2 484 matrix game for d = 1, 2, . . . , 8. They used an algorithm of Foster (1964).
Our aim in this paper is to solve the game without simplifying assumptions. We allow a d-deck shoe dealt without replacement and allow each of Player and Banker to see the composition of his own two-card hand, making baccara a 2 5 × 2 484 matrix game. We derive optimal Player and Banker strategies and determine the value of the game, doing so for every positive integer d. We too make use of Foster's (1964) algorithm. We suspect that these optimal strategies are uniquely optimal, but we do not have a proof of uniqueness.
It will be convenient for what follows to classify the game-theoretic models of baccara in two ways. First, we classify them according to how the cards are dealt.
• Model A. Cards are dealt with replacement from a single deck.
• Model B. Cards are dealt without replacement from a d-deck shoe.
Second, we classify them according to the information available to Player and Banker about their own two-card hands.
• Model 1. Each of Player and Banker sees the total of his own two-card hand but not its composition.
• Model 2. Banker sees the composition of his own two-card hand while Player sees only his own total.
• Model 3. Each of Player and Banker sees the composition of his own two-card hand.
(We do not consider the fourth possibility.) Thus, Model A1 is the model of Kemeny and Snell (1957), Model B2 is the model of Downton and Lockwood (1975), and Model B3 is our primary focus here. Model A2 was discussed by Downton and Holder (1972), but Models A3, B1, and B3 have not been considered before, as far as we know. Like others before us, we restrict our attention to the classical parlor game of baccara chemin de fer, in contrast to the modern casino game. (Following Deloche and Oguer 2007, we use the authentic French spelling "baccara" rather than the more conventional "baccarat" to emphasize this.) The rules are as follows. Denominations A, 2-9, 10, J, Q, K have values 1, 2-9, 0, 0, 0, 0, respectively. The total of a hand, consisting of 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. A two-card total of 8 or 9 is a natural. If either hand is a natural, the game is over and the higher total wins. Hands of equal total result in a push (no money is exchanged). If neither hand is a natural, Player then has the option of drawing a third card. 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, and the higher total wins. Winning bets on Player's hand are paid even money, with Banker, as the name suggests, playing the role of the bank. Again, hands of equal total result in a push. Since bystanders 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. Banker, on whose hand no one can bet, has no restrictions on his strategy.
In the modern casino game, not only is Banker's strategy highly constrained but the casino collects a five percent commission on Banker wins.
In Section 3, we show how to evaluate the payoff matrix. We emphasize Model B3 but treat the other models as well. In Section 4, we use strict dominance to reduce the payoff matrix under Model B3 to 2 5 ×2 n d , where n d depends on the number of decks d and satisfies 18 ≤ n d ≤ 23. We get similar reductions of the other models. To proceed further, in Section 5 we re-examine the unique solution of Kemeny and Snell (1957) under Model A1 and notice that there are multiple solutions under Models A2 and A3. In Section 6 we derive the unique solution under Model B1 for every positive integer d. Model B1 is of interest because it shows the price of the "with replacement" assumption more clearly than do Models B2 and B3. In Section 7 we re-derive the unique solution of Downton and Lockwood (1975) under Model B2, extending it to every positive integer d. These results lead us in Section 8 to a solution under Model B3 for every positive integer d. The feature of the game that allows this is that, under Model B3, the kernel is 2 × 2. The two Banker pure strategies specified by the kernel are dependent on d, while the two Player pure strategies specified by the kernel are independent of d. Optimality proofs are computer-assisted, with all computations carried out in infinite precision using Mathematica.
Remark. The game can be thought of as follows. Player I chooses a pure strategy i ∈ {0, 1, . . . , m − 1}. Let Z i be a random variable with distribution P(Z i = l) = p i (l) for l = 0, 1, . . . , n. Given that Z i = 0, the game is over and player I's conditional expected gain is a i (0). If Z i ∈ {1, 2, . . . , n}, then player II observes Z i (but not i) and based on this information chooses a "move" j ∈ {0, 1}. Given that Z i = l and player II chooses move 1 (resp., move 0), player I's conditional expected gain is a i,1 (l) (resp., a i,0 (l)). Thus, player II's pure strategies can be identified with subsets T ⊂ {1, 2, . . . , n}, with player II choosing move 1 if Z i ∈ T and move 0 if Z i ∈ T c . The lemma implies that, regardless of player I's strategy choice, it is optimal for player II to choose move 1 if Z i ∈ T 1 and move 0 if Z i ∈ T 0 . We now formalize Foster's (1964) algorithm for solving 2 × 2 n matrix games as described in the special case of Lemma 1 in which m = 2. (See also Foster's discussion of Kendall and Murchland 1964.) The method is purely algebraic, so we simplify the notation slightly by defining for i, j = 0, 1 and l = 1, 2, . . . , n.
This player I mixed strategy and player II best response leads to a player I expected gain of The function p → V (p) is the lower envelope of the family of linear functions p → (1 − p)a 0,T + p a 1,T , where T ranges over T 11 ⊂ T ⊂ T 11 ∪ T 01 ∪ T 10 . Therefore, the value of the game is If the last equality uniquely determines p * and if p * = p(l * ) for a unique l * ∈ T 01 ∪ T 10 , then player I's unique optimal strategy is (1 − p * , p * ) and the two columns of the kernel are uniquely specified as T (p * ) and T (p * ) ∪ {l * }. Their unique optimal mixture (1 − q * , q * ) is obtained by solving the 2 × 2 kernel.
Proof. Notice that l belongs to T (p) if l ∈ T 11 or if both l ∈ T 01 ∪ T 10 and (1 − p)e 0,1 (l) + p e 1,1 (l) < (1 − p)e 0,0 (l) + p e 1,0 (l), implying (2). The function (3) is continuous and piecewise linear, hence is maximized at 0, 1, or one of the points p(l) at which its slope changes. The remaining conclusions of the lemma follow easily.
On the other hand, Banker's pure strategies can be indexed by the sets T satisfying Assuming X ≤ 7 and Y ≤ 7, Banker draws if (Y 1 , Y 2 , X 3 ) ∈ T and stands otherwise. Since there are 44 pairs (j 1 , j 2 ) satisfying 0 ≤ j 1 ≤ j 2 ≤ 9 and M (j 1 + j 2 ) ≤ 7, and since 44 × 11 = 484, it follows that Banker has 2 484 pure strategies. Thus, baccara is a 2 5 × 2 484 matrix game. Let us denote by G S,T Player's profit from a one-unit bet when he adopts pure strategy S and Banker adopts pure strategy T , so that a S,T := E[G S,T ] is the (S, T ) entry in the payoff matrix. Then Let us now define, for S and T , for (j 1 , j 2 ) satisfying 0 ≤ j 1 ≤ j 2 ≤ 9 and M (j 1 + j 2 ) ≤ 7, and for k ∈ {0, 1, . . . , 9}, where l = 1 if (j 1 , j 2 , k) (resp., (j 1 , j 2 , ∅)) belongs to T ; and l = 0 if (j 1 , j 2 , k) (resp., (j 1 , j 2 , ∅)) belongs to T c (the complement of T relative to (5)). Defining also we have, from (6), To evaluate the conditional expectations in (7), we condition on (X 1 , X 2 ): if k = ∅ and To evaluate the conditional expectations in (9) and (10), there are four cases to consider: Case 1. (i 1 , i 2 ) ∈ S, (j 1 , j 2 , k) ∈ T with k = ∅ (both Player and Banker draw). Here, for d decks, which becomes, as d → ∞, Case 2. (i 1 , i 2 ) ∈ S, (j 1 , j 2 , k) ∈ T c with k = ∅ (Player draws, Banker stands). Regardless of the number of decks, which becomes, as d → ∞, Case 4. (i 1 , i 2 ) ∈ S c , (j 1 , j 2 , ∅) ∈ T c (both Player and Banker stand). Regardless of the number of decks, Finally, to evaluate we begin with a full d-deck shoe except for three cards, one j 1 , one j 2 , and one k, removed. It will comprise m 0 0s, m 1 1s, . . . , and m 9 9s, where The number of equally likely two-card hands is 52d−3 2 , and the number of those that belong to S is and Also, to evaluate we begin with a full d-deck shoe except for two cards, one j 1 and one j 2 , removed. It will comprise m ′ 0 0s, m ′ 1 1s, . . . , and m ′ 9 9s, where The number of equally likely two-card hands is 52d−2 2 , and the number of those that belong to S c is and This suffices to complete the evaluation of (9) and (10) when cards are dealt without replacement from a d-deck shoe.
The assumption that cards are dealt with replacement from a single deck can be modeled by letting d → ∞ in the assumption that cards are dealt without replacement from a d-deck shoe. The formulas are simpler in this case: Here 89 comes from 25 + 16 + 16 + 16 + 16, where the summands correspond to totals 0, 1, 2, 3, 4; 32 is 16 + 16, corresponding to totals 6 and 7. In summary, we can evaluate (8) under Model B3 or A3. Restricting S to the two extremes in (4), we obtain (8) under Model B2 or A2 as a special case. Finally, as for Models B1 and A1, we can derive the analogue of (8) from results already obtained. Specifically,

Banker's strictly dominated pure strategies
Our next step is to show that Lemma 1 applies (with Player and Banker playing the roles of player I and player II, respectively), allowing us to reduce the game to a more manageable size. The payoff matrix (8) has the form (1) with m = 32, n = 484, p i (0) = P(X ∈ {8, 9} or Y ∈ {8, 9}), and a i (0) = 0. It remains to evaluate T 0 , T 1 , and T * of the lemma.
Results are summarized in Table 1. T 1 (resp., T 0 ) is the set of triples (j 1 , j 2 , k) for which a S,1 (j 1 , j 2 , k) < a S,0 (j 1 , j 2 , k) (resp., >) for each of Player's 2 5 pure strategies S, indicated by a D (resp., S) in the corresponding entry of the table. T * is the remaining set of triples (j 1 , j 2 , k), indicated by a * in the corresponding entry of the table. Of particular interest is n d := |T * |.
For example, one of the 19 is (3, 3, 6). Indeed,  Table 1 is * if d ≤ 10 and D if d ≥ 11. The other 483 cases are analyzed similarly.
Requiring that Banker make the optimal move in each of the cases that do not depend on Player's strategy, we have reduced the game, under Model B3 (resp., B2), to a 2 5 × 2 n d (resp., 2 × 2 n d ) matrix game, where 18 ≤ n d ≤ 23.
We have similar results for Model B1. Again, n d := |T * |.
Theorem 4. (a) Under Model B1 with the number of decks being a positive integer d, Lemma 1 applies. The sets T 0 , T 1 , and T * of the lemma can be inferred from Table 2, with entries S, D, and * located at elements of T 0 , T 1 , and T * , respectively. In particular, n 1 = 4, n 2 = 3, and, n d = 4 for all d ≥ 3.

Solutions under Models A1, A2, and A3
We recall Kemeny and Snell's (1957) solution of the 2 × 2 88 matrix game that assumes Model A1. (See Deloche and Oguer 2007 for an alternative approach based on the extensive, rather than the strategic, form of the game.) Implicitly using Lemma 1, they reduced the number of Banker pure strategies from 2 88 to just 2 4 (see Theorem 4). The 2 × 2 kernel of the game was determined to be B: S on 6, ∅ B: D on 6, ∅ P: S on 5 −4564 −2692 P: D on 5 −3705 −4121 implying that the following Player and Banker strategies are uniquely optimal. Player draws on a two-card total of 5 with probability and Banker draws on a two-card total of 6, when Player stands, with probability The value of the game (to Player) is v = − 679 568 53 094 899 ≈ −0.0127991.
The fully specified optimal strategy for Banker is given in Table 4 with d ≥ 4 and q as in (25). Let us extend this analysis from Model A1 to Model A3. Again we have a 2 5 × 2 484 matrix game, and the payoff matrix can be evaluated as in Section 4, using (12), (14), and (19)-(22) in place of (11), (13), and (15)-(18). We can apply Lemma 1 and reduce the game to a 2 5 × 2 22 matrix game. We obtain the special case of Table 1 in which d ≥ 11.
When we evaluate this 2 5 × 2 22 payoff matrix, we find that a number of rows are identical. When several rows are identical, we eliminate duplicates. When we make this reduction and rearrange the remaining rows in a more natural order, we are left with 9 rows, labeled by 0-8, which have a special structure. Specifically, row i corresponds to Player's mixed strategy (under Model A1) of drawing on a two-card total of 5 with probability i/8. The reason that multiples of 1/8 appear is that, given that Player has a two-card total of 5, he has (0, 5), (1, 4), (2, 3), (6, 9), or (7, 8) with probabilities 4/8, 1/8, 1/8, 1/8, and 1/8, respectively.
Next, we observe that column (j 1 , j 2 , j 3 , j 4 ) is a mixture of the 2 4 (Model A1) pure strategies of Banker that remain after application of Lemma 1. By the results for Model A1, optimal strategies for Banker must satisfy j 1 = 8, j 2 = 0, and j 3 = 8. This reduces the game to a 9 × 17 matrix game, whose columns we relabel as 0-16. Specifically, column j corresponds to Banker's mixed strategy (under Model A1) of drawing on a two-card total of 6, when Player stands, with probability j/16.
Finally, what is the solution of the 9 × 17 game? We have seen that rows 1-7 (resp., columns 1-15) are mixtures of rows 0 and 8 (resp., columns 0 and 16). In particular, rows 1-7 and columns 1-15 are dominated, but not strictly dominated. Eliminating these rows and columns results in a 2 × 2 matrix game, namely the kernel (23). But eliminating dominated, but not strictly dominated, rows and columns may result in a loss of solutions, and it does so in this case. Indeed, there are many solutions. For Player, given two pure strategies i, i ′ ∈ {0, 1, 2, . . . , 8} with there is a unique p ∈ (0, 1) such that and the (1 − p, p)-mixture of pure strategies i and i ′ is optimal for Player. There are 7 choices of i and 2 choices of i ′ that satisfy (27), hence 14 pairs (i, i ′ ) that meet this condition. For Banker, given two pure strategies j, j ′ ∈ {0, 1, 2, . . . , 16} with there is a unique q ∈ (0, 1) such that (1 − q) j 16 + q j ′ 16 = 859 2288 , and the (1 − q, q)-mixture of pure strategies j and j ′ is optimal for Banker. There are 7 choices of j and 10 choices of j ′ that satisfy (28), hence 70 pairs (j, j ′ ) that meet this condition. Each such pair (i, i ′ ) can be combined with each such pair (j, j ′ ), so there are 14 × 70 = 980 pairs of optimal strategies of this form. These are the extreme points of the convex set of equilibria. All 980 of them appear when the game solver at http://banach.lse.ac.uk/ is applied.
Let us single out one of them. Take i = 6 and i ′ = 7, getting p = 6/11, and take j = 1 and j ′ = 9, getting q = 179/286. This does not uniquely determine a pair of optimal strategies because of the duplicate rows and columns that were eliminated, but one pair of optimal mixed strategies to which this corresponds is shown in Table 3. As we will see, this pair of optimal mixed strategies is the Table 3: A pair of optimal mixed strategies in Model A3. For Banker's fully specified optimal strategy, see Table 5 with d ≥ 10.
Player's two-card total is 5  Foster (1964) remarked, "It is an interesting fact that this [optimal Player mixed] strategy is often attained approximately in practice by standing on the pair 2, 3 and calling [i.e., drawing] on any other combination adding to 5; this gives approximately the right frequency of calling [i.e., drawing]." Actually, it gives a drawing probability of 7/8, not the required 9/11. But, as Table 3 suggests, Player should stand also on (1, 4) with probability 5/11. Then the probability of Player drawing on a two-card total of 5 is 3 4 + 1 8 6 11 = 9 11 .
A similar analysis applies under Model A2. Lemma 1 reduces the 2 × 2 484 matrix game to 2 × 2 22 . Eliminating duplicate columns reduces the game to 2 × 9 2 (17) 2 , and finally using the optimal solution under Model A1, we are left with a 2 × 17 matrix game. This is the game identified by Downton and Holder (1972). As above, there are 70 extremal solutions, and they all appear when the game solver at http://banach.lse.ac.uk/ is applied.
Again, we single out one of them. Player draws on a two-card total of 5 with probability 9/11, and Banker follows Table 3. As we will see, this pair of optimal mixed strategies is the limiting pair of optimal mixed strategies under Model B2 as d → ∞.

Solution under Model B1
Recall that Table 2 and Banker draws on a two-card total of 6, when Player stands, with probability The fully specified optimal strategy for Banker is given in Table 4. Comparing the solution under Model A1 with that under Model B1 reveals the effect of the "with replacement" assumption. The solutions are identical except for the three parameters [(24)-(26) vs. (29)-(31)]. For each parameter, the relative error is less than one percent. This analysis extends to every positive integer d.
Theorem 5. Under Model B1 with d being a positive integer, the following Player and Banker strategies are uniquely optimal. Player draws on a two-card total of 5 with probability and Banker draws on on a two-card total of 6, when Player stands, with probability The fully specified optimal strategy for Banker is given in Table 4.
To confirm this, we must show that, with A denoting the 2 × 2 4 payoff matrix, we have This involves checking 16 inequalities (of which two are automatic). For example, the eighth and ninth components of (1 A similar analysis can then be carried out for 1 ≤ d ≤ 3, in which case the kernel is given by columns 8 and 9 (of 0-15).
We notice that the above kernel converges, as d → ∞, to the kernel (23).

Solution under Model B2
The result that Table 1 is identical under Models B2 and B3 is less surprising than it may first appear to be. In Section 5 we saw that, under Model A3 with Player's pure strategies labeled from 0 to 31, pure strategy u ∈ {1, 2, . . . , 30} is a (1 − p, p) mixture of pure strategies 0 and 31, where here u 1 u 2 u 3 u 4 u 5 is the binary form of u, that is, u 1 , u 2 , u 3 , u 4 , u 5 ∈ {0, 1} and u = 16u 1 + 8u 2 + 4u 3 + 2u 4 + u 5 . Consequently, a u,l (j 1 , j 2 , k) lies between a 0,l (j 1 , j 2 , k) and a 31,l (j 1 , j 2 , k) for all u ∈ {1, 2, . . . , 30}, l = 0, 1, 0 ≤ j 1 ≤ j 2 ≤ 9 with M (j 1 + j 2 ) ≤ 7, and k = 0, 1, . . . , 9, ∅. Under Model B3, the conditional expectations in (37) should not differ much from their Model A3 counterparts, especially for large d, hence we would expect that (37) holds with few exceptions. In fact, the only exceptions occur when l = 1, M (j 1 + j 2 ) = 2, k = 8, and d ≤ 7 because in these cases, a 0,l (j 1 , j 2 , k) and a 31,l (j 1 , j 2 , k) are very close. When we consider the differences b u (j 1 , j 2 , k), there are even fewer exceptions, as noted previously. One might ask why Model B2 was even considered by Downton and Lockwood (1975), inasmuch as its asymmetric assumption about the available information (beyond the asymmetry inherent in the rules) may seem contrived. The answer, we believe, is that there already existed an algorithm, due to Foster (1964), for solving such games. That algorithm was formalized in Lemma 2 of Section 2.
Let us recall Downton and Lockwood's (1975) solution of the 2 × 2 484 matrix game that assumes Model B2. In the case d = 6, the 2 × 2 kernel of the game is found to be B: S on (0, 6), ∅ B: D on (0, 6), which is less than (31) because Banker has additional options while Player's options are unchanged. The fully specified optimal strategy for Banker is given in Table 5. This analysis extends to every positive integer d. We do not display the kernel, only the solution. Table 5: Banker's optimal move (final version) under Model B2 or B3 with d being a positive integer, indicated by D (draw), S (stand), or (S, D) (stand with probability 1 − q, draw with probability q). In Model B2, q is as in (39)-(42), and in Model B3, q is as in (49)

Solution under Model B3
We have reduced the game to a 2 5 × 2 n d matrix game, where 18 ≤ n d ≤ 23. The fact that Table 1 is identical under Models B2 and B3 suggests the existence of a 2 × 2 kernel under Model B3 whose columns are the same as those of the 2 × 2 kernel under Model B2 as described in Section 7. Then the resulting 2 5 × 2 matrix game will of course have a 2 × 2 kernel, which is easy to find by graphical or other methods, and it will remain to confirm that this kernel corresponds to a solution of the 2 5 × 2 n d matrix game.
As we will see, this approach works for all positive integers d except 1, 2, and 9. These last three cases can be treated separately.
The kernel is easily found to be given by rows 19 and 27, so it is equal to and Banker draws on (0, 6), when Player stands, with probability q 6 = 18 885 571 36 781 056 ≈ 0.513459.
The value of the game (to Player) is which is greater than (38) because Player has additional options while Banker's options are unchanged. The fully specified optimal strategies for Player and Banker are given in Tables 6 and 5. This analysis extends to every positive integer d. We do not display the kernel, only the solution. Table 6: Player's optimal move under Model B3, indicated by D (draw), S (stand), or (S, D) (stand with probability 1 − p, draw with probability p). Here p is as in (48).
We hasten to add that, just as in the case d = 6, there is a more efficient way. The two Player pure strategies specified by the kernel do not vary with d and are rows 19 and 27 (of 0-31). We again apply Lemma 2 to establish (54), saving time and avoiding the issue that occurred in the cases d = 1, 2, 9. We find that, for d = 1, 2, . . . , 9, n d = 7, 9, 8, 9, 6, 6, 6, 7, 7, respectively, and, if d ≥ 10, n d = 6. In each case, T 10 is empty.