A game-theoretic analysis of baccara chemin de fer, II

In a previous paper, we considered several models of the parlor game baccara chemin de fer, including Model B2 (a $2\times2^{484}$ matrix game) and Model B3 (a $2^5\times2^{484}$ matrix game), both of which depend on a positive-integer parameter $d$, the number of decks. The key to solving the game under Model B2 was what we called Foster's algorithm, which applies to additive $2\times2^n$ matrix games. Here"additive"means that the payoffs are additive in the $n$ binary choices that comprise a player II pure strategy. In the present paper, we consider analogous models of the casino game baccara chemin de fer that take into account the $100\,\alpha$ percent commission on Banker (player II) wins, where $0\le\alpha\le1/10$. Thus, the game now depends not just on the discrete parameter $d$ but also on a continuous parameter $\alpha$. Moreover, the game is no longer zero sum. To find all Nash equilibria under Model B2, we generalize Foster's algorithm to additive $2\times2^n$ bimatrix games. We find that, with rare exceptions, the Nash equilibrium is unique. We also obtain a Nash equilibrium under Model B3, based on Model B2 results, but here we are unable to prove uniqueness.


Introduction
The parlor game baccara chemin de fer was one of the motivating examples that led to the development of noncooperative two-person game theory (Borel, 1924).We can classify game-theoretic models of baccara in two ways.First according to how the cards are dealt: • Model A (with replacement).Cards are dealt with replacement from a single deck.
• Model B (without replacement).Cards are dealt without replacement from a d-deck shoe.
And second according to the information available to Player and Banker about their own two-card hands: • Model 1 (Player total, Banker total).Each of Player and Banker sees the total of his own two-card hand but not its composition.
• Model 2 (Player total, Banker composition).Banker sees the composition of his own two-card hand while Player sees only his own total.
• Model 3 (Player composition, Banker composition).Each of Player and Banker sees the composition of his own two-card hand.
(We do not consider the fourth possibility.)Under Model A1 baccara is a 2×2 88 matrix game, which was solved by Kemeny and Snell (1957).Under Model B2 baccara is a 2 × 2 484 matrix game, which was solved in part by Downton and Lockwood (1975) and in full by Ethier and Gámez (2013).Under Model B3 baccara is a 2 5 × 2 484 matrix game, which was solved in part by Ethier and Gámez (2013).Each of these works was concerned with the parlor game baccara chemin de fer, in contrast to the casino game.The rules of the parlor game, which also apply to the casino game, are as in Ethier and Gámez (2013): 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.The game is played with six standard 52-card decks mixed together.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.A two-card total of 8 or 9 is a natural.If either hand is a natural, the game is over.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 by Banker at even odds.Losing bets on Player's hand are collected by Banker.Hands of equal total result in a tie or a push (no money is exchanged).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 stand or draw 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 under classical rules.
There is one important additional rule in the casino game: The house collects a five percent commission on Banker wins.(This commission has been known to be as high as ten percent; see Villiod (1906).)Thus, our aim in the present paper is to generalize the aforementioned results to allow for a 100 α percent commission on Banker wins.We will assume that 0 ≤ α ≤ 1/10.This makes baccara chemin de fer a bimatrix game instead of a matrix game, one that depends on a positive integer parameter d (under Model B), the number of decks, as well as a continuous parameter α (under Model A or B), the commission on Banker wins.In the case of Model A1 all Nash equilibria were identified in an unpublished paper by the authors (Ethier and Lee, 2013), assuming only 0 ≤ α < 2/5.Under Model A1 and the present assumption (0 ≤ α ≤ 1/10), the Nash equilibrium is unique for each α.
There are also unimportant additional rules in the casino game.Specifically, in modern casino baccara chemin de fer, Banker's strategy is severely restricted.With a few exceptions, these restrictions are benign, but because of the exceptions we ignore them entirely.Ethier and Gámez (2013) studied Models A2, A3, B1, B2, and B3 in the special case α = 0.That was part I, and the present paper, with 0 ≤ α ≤ 1/10, is part II.
To keep the paper from becoming unduly long, we will focus our attention on Models B2 and B3, leaving the simpler models A2, A3, and B1 to the interested reader.The key to solving the parlor game under Model B2 was what we called Foster's algorithm, which applies to additive 2 × 2 n matrix games.Foster (1964) called it a computer technique.Here "additive" means that the payoffs are additive in the n binary choices that comprise a player II pure strategy.
In Section 2 we generalize Foster's algorithm to additive 2 × 2 n bimatrix games.The generalization is not straightforward.In Section 3 we show that, with rare exceptions, the Nash equilibrium is unique under Model B2.Uniqueness is important because it ensures that optimal strategies are unambiguous.The proof of uniqueness is computer assisted, with computations carried out in infinite precision using Mathematica.In Section 4 we obtain a Nash equilibrium under Model B3, based on Model B2 results, but here, just as for the parlor game, we are unable to prove uniqueness.

Two Lemmas for Additive Bimatrix Games
A reduction lemma for additive m × 2 n matrix games was stated by Ethier and Gámez (2013).It had already been used implicitly by Kemeny and Snell (1957), Foster (1964), and Downton and Lockwood (1975).Here we generalize to additive m × 2 n bimatrix games.
Then, given T ⊂ {1, 2, . . ., n}, player II's pure strategy T is strictly dominated unless T 1 ⊂ T ⊂ T 1 ∪ T * .Therefore, the m × 2 n bimatrix game can be reduced to an m × 2 n * bimatrix game with no loss of Nash equilibria.
Remark.The game can be thought of as follows.Player I chooses a pure strategy u ∈ {0, 1, . . ., m − 1}.Then Nature chooses a random variable Z u with distribution P (Z u = l) = p u (l) for l = 0, 1, . . ., n.Given that Z u = 0, the game is over and player II's conditional expected payoff is b u (0).If Z u ∈ {1, 2, . . ., n}, then player II observes Z u (but not u) and based on this information chooses a "move" j ∈ {0, 1}.Given that Z u = l and player II chooses move 0 (resp., move 1), player II's conditional expected payoff is b u,0 (l) (resp., b u,1 (l)).Thus, player II's pure strategies can be identified with subsets T ⊂ {1, 2, . . ., n}, with player II choosing move 0 if Z u ∈ T c and move 1 if Z u ∈ T .The lemma implies that, regardless of player I's strategy choice, it is optimal for player II to choose move 0 if Z u ∈ T 0 and move 1 if Proof.Suppose that the condition T 1 ⊂ T ⊂ T 1 ∪ T * fails.There are two cases.In case 1, there exists l 0 ∈ T 1 with l 0 / ∈ T .Here define where the ± sign is a plus sign in case 1 and a minus sign in case 2. This tells us that player II's pure strategy T is strictly dominated by pure strategy T ′ , as required.
(c) Under the assumptions of part (b), if p = p(l) for exactly one choice of l ∈ T 0,1 ∪ T 1,0 , namely l ′ , then every Nash equilibrium (p, q) must have p = (1 − p, p) and q with entries 1 − q and q ∈ [0, 1] at the coordinates corresponding to player II pure strategies T (p) and T (p) ∪ {l ′ } (0s elsewhere), where (2) q is called an equalizing strategy.
Again, q is called an equalizing strategy.
(b) The reason for referring to this lemma as an algorithm is that it gives straightforward conditions for determining all Nash equilibria.These conditions primarily involve checking for equalizing strategies in a limited number of cases.
Proof.(a) For T (p) to be player II's unique best response, it must be the case that T → (1 − p)b 0,T + p b 1,T is uniquely maximized at T = T (p).Now, for arbitrary T that excludes l ′ , the additivity of player II's payoffs implies that if and only if which holds if and only if if and only if If we assume that p / ∈ {p(l) : l ∈ T 0,1 ∪ T 1,0 }, then Inclusion ( 6) is equivalent to l ′ ∈ T (p) c .The first conclusion of part (a) follows.For the second conclusion, notice that Inequalities (3)-( 5), with the inequalities replaced by equalities, are equivalent to each other and to p(l ′ ) = p.This suffices.The third conclusion follows similarly.
(b) For p ∈ [0, 1] with p / ∈ {p(l) : l ∈ T 0,1 ∪ T 1,0 }, we have seen that the pure strategy T (p) is the unique best response to p = (1 − p, p).However, for 0 < p < 1, the mixed strategy p cannot be a best response to the pure strategy T (p) unless a 0,T (p) = a 1,T (p) , which has been ruled out.To extend this to p = 0 and p = 1, we note that neither (0, T (0)) nor (1, T (1)) is a pure Nash equilibrium, by virtue of the other assumptions of part (b).
(c) We assume that p = p(l ′ ) for a unique l ′ ∈ T 0,1 ∪ T 1,0 .By part (a), any mixture of the pure strategies T (p) and T (p) ∪ {l ′ } will be a best response to the mixed strategy p = (1 − p, p), but at most one such mixture, namely the equalizing strategy that chooses T (p) with probability 1 − q and T (p) ∪ {l ′ } with probability q, where q satisfies Equation (2), will result in a Nash equilibrium.
(d) The proof is similar to that of part (c).

Model B2
In this section we study Model B2.Here cards are dealt without replacement from a d-deck shoe, and Player sees only the total of his two-card hand, while Banker sees the composition of his two-card hand.Player has a stand-or-draw decision at two-card totals of 5, and Banker has a stand-or-draw decision in 44 × 11 = 484 situations (44 compositions corresponding to Banker totals of 0-7, and 11 Player third-card values, 0-9 and ∅), so baccara chemin de fer is a 2 × 2 484 bimatrix game.We denote Player's two-card hand by (X 1 , X 2 ), where 0 and Player's and Banker's third-card values (possibly ∅) by X 3 and Y 3 .Note, for example, that X 1 and X 2 are not Player's first-and second-card values; rather, they are the smaller and larger values of Player's first two cards.We define the function M on the set of nonnegative integers by that is, M (i) is the remainder when i is divided by 10.We define Player's twocard total by X := M (X 1 + X 2 ).We further denote by G 0 and G 1 Banker's profit in the parlor game from standing and from drawing, respectively, assuming a one-unit bet.More precisely, for v = 0 (Banker stands) and v = 1 (Banker draws), We next define the relevant probabilities when cards are dealt without replacement.Let be the probability that Player's two-card hand is (i 1 , i 2 ), where 0 ≤ i 1 ≤ i 2 ≤ 9, and Banker's two-card hand is (j 1 , j 2 ), where 0 ≤ j 1 ≤ j 2 ≤ 9. To elaborate on this formula, we note that the order of the cards within a two-card hand is irrelevant, so the hand comprising i 1 and i 2 can be written as (min(i 1 , i 2 ), max(i 1 , i 2 )), and the factor (2 − δ i1,i2 ) adjusts the probability accordingly.The factors of the form (1 + 3δ i1,0 ) take into account the fact that cards valued as 0 are four times as frequent as cards valued as 1, for example.Finally, the terms of the form − δ i2,i1 are the effects of previously dealt cards.In practice, the order in which the first four cards are dealt is Player, Banker, Player, Banker.But it is mathematically equivalent, and slightly more convenient, to assume that the order is Player, Player, Banker, Banker.Second, is the probability that Player's two-card hand is (i 1 , i 2 ), where 0 ≤ i 1 ≤ i 2 ≤ 9 and M (i 1 + i 2 ) ≤ 7, Banker's two-card hand is (j 1 , j 2 ), where 0 ≤ j 1 ≤ j 2 ≤ 9 and M (j 1 + j 2 ) ≤ 7, and the fifth card dealt has value k ∈ {0, 1, . . ., 9}.Note is the probability that Player's two-card hand is (i 1 , i 2 ), where 0 ≤ i 1 ≤ i 2 ≤ 9 and M (i 1 +i 2 ) ≤ 7, Banker's two-card hand is (j 1 , j 2 ), where 0 ≤ j 1 ≤ j 2 ≤ 9 and M (j 1 +j 2 ) ≤ 7, the fifth card dealt has value k ∈ {0, 1, . . ., 9}, and the sixth card dealt has value l ∈ {0, 1, . . ., 9}.Note that 0≤l≤9 p 6 ((i 1 , i 2 ), (j 1 , j 2 ), k, l) = p 5 ((i 1 , i 2 ), (j 1 , j 2 ), k).
Notice that the denominators of Equation ( 11) and Equation ( 12) are equal; we denote their common value by p u ((j 1 , j 2 ), k).
We want to find all Nash equilibria of the casino game baccara chemin de fer under Model B2, for all positive integers d and for 0 ≤ α ≤ 1/10.We apply Lemma 2, Foster's algorithm.Lemma 1 also applies, reducing the game to 2 × 2 20 , but that is not needed.We demonstrate the method by treating the case d = 6 and 0 ≤ α ≤ 1/10 in detail.Then we state results for all d.
The first step is to derive a preliminary version of Banker's optimal move at each information set for α = 0 and for α = 1/10.At only three of the 44 × 11 = 484 information sets does Banker's optimal move differ at α = 0 and α = 1/10.Because b u,v ((j 1 , j 2 ), k) is linear in α, if the optimal move at ((j 1 , j 2 ), k) is the same for α = 0 and α = 1/10, then it is also the same for 0 ≤ α ≤ 1/10.Results are shown in Table 1.
We are now ready to identify the cases that must be checked for equalizing strategies.If r is the number of best-response-discontinuity curves and s is the number of points of intersection of these curves, then there are r +2s α-intervals and s α-values that must be checked for equalizing strategies.When d = 6 we have seen that r = 20 and s = 23, hence there are 66 α-intervals and 23 α-values that require attention.We have summarized these 89 cases in Tables 2 and 3.
Let us provide more detail on Table 2.For each best-response-discontinuity curve, if it is intersected by m other best-response-discontinuity curves, that divides the interval [0, 1/10] into m + 1 subintervals, each of which contributes a row to  No two curves of the same color intersect each other.The labels on the red, blue, and black curves are listed from largest p to smallest p.For example, p((6, 9), 4) > p((0, 5), 4) > p((7, 8), 4).
Table 3: The 23 intersections that must be checked for an equalizing strategy, under Model B2 with d = 6 and 0 ≤ α ≤ 1/10.The meaning of the Banker strategies is as in Table 2.Only in case 18 are there equalizing strategies.
case intersecting curves approx.α Banker strategy at (3, 9), (4, 1), (5, 4), (6, ∅), (6,6) Notice that the Nash equilibrium with p as in Equation ( 25) and q as in Equation ( 21) coincides with the one from row 45 of Table 2, the Nash equilibrium with p as in Equation ( 25) and q as in Equation ( 24) coincides with the one from row 44 of Table 2.The two others, with p as in Equation ( 25) and q as in Equation ( 22) or Equation ( 23), are new.
The next step is to verify the three conditions in part (b) of Lemma 2. The first condition is easy because the work has already been done in checking for equalizing strategies.Consider [0, 1/10] × [0, 1] minus the union of the 20 bestresponse-discontinuity curves, as shown in Figure 1.It is the union of 43 disjoint connected open regions.The best response T α (p) is constant on each of these regions, so we can see that the entries of A corresponding to column T α (p) have already been computed in analyzing the 66 cases of Table 2.
The second condition is easiest because the strategy is the same for p = 0 and all α.(The case d = 1 is an exception, and it can be checked separately.) The third condition is a little more involved because of the three bestresponse-discontinuity curves that intersect p = 1.They divide [0, 1/10] into four intervals, and the third condition can be confirmed for each.
This completes the analysis of the case d = 6.Statistics for other values of d are shown in Table 4.
Table 4: Dependence on d of various quantities associated with the casino game under Model B2 with 0 ≤ α ≤ 1/10.Column (a) contains the number of best-response-discontinuity curves; column (b) contains the number of points of intersection of these curves; column (c) contains the number of these curves that intersect p = 1 or p = 0; column (d) contains the number of α-intervals that must be checked for equalizing strategies; and column (e) contains the number of α-values at which the Nash equilibrium is nonunique.Next, we summarize results under Model B2 for all d ≥ 1. See Table 5.First, all Nash equilibria (p, q) = ((1 − p, p), q) have the same p, namely which generalizes Equation ( 25).
Table 6 indicates the strategies on which Banker mixes, with drawing probability q.For d = 1, q = 288,499/450,072 if α ∈ (α 1 , α 2 ), ( 28) 591,845/4,119,192 We can obtain the uniqueness of the Nash equilibrium for each d = 1, 2, . . ., 76.For d ≥ 77, we observe that the best-response-discontinuity curves are ordered in a way that does not depend on d.The six curves corresponding to (4, 1) intersect the six partial curves corresponding to (6,6) , and p((1, 5), 6) intersects  p((2, 4), 6).Thus, there are 37 points of intersection for all d ≥ 77.With this information we can apply Foster's algorithm with a variable d to get the desired uniqueness.
We have established the following theorem.
Theorem 1.Consider the casino game baccara chemin de fer under Model B2 with d a positive integer and 0 ≤ α ≤ 1/10.With rare exceptions, there is a unique Nash equilibrium.Player's equilibrium strategy is to draw at 5 with probability as in Equation (26).Banker's equilibrium strategy is as in Tables 5 and 6.The number of exceptions is two if d ∈ {1, 8, 9, 10, 11}, one if d ∈ {2, 4, 5, 6, 7}, and none otherwise.For each of these exceptional values of α, there are four Banker equilibrium strategies of support size 2.
Let us briefly compare the Nash equilibrium of the casino game (Theorem 1) with that of the parlor game (Ethier and Gámez, 2013), under Model B2 in both cases.We also compare them in the limit as d → ∞.
In the casino game, Player's mixing probability (i.e., Player's probability of drawing at two-card totals of 5) is as in Equation ( 26), which depends explicitly on d and α.Banker's mixing probability (i.e., Banker's probability of drawing at ((0, 6), ∅)) depends on d and is a step function in α with zero, one, or two discontinuities (zero, hence no α dependence, if d = 3 or d ≥ 12).In the limit as d → ∞, Player's mixing probability converges to while Banker's mixing probability converges to 179/286.It follows that Banker's limiting probability of drawing at (6, ∅), including ((0, 6), ∅) and ((8, 8), ∅), is and we recognize Equations ( 37) and ( 38) as the parameters of the Model A1 Nash equilibrium.
In the parlor game, the results of the preceding paragraph apply with α = 0.
and define S u to be the set of two-card hands at which Player, using pure strategy u, draws: The complement of S u with respect to is the set of two-card hands at which Player, using pure strategy u, stands.
The random variables (X 1 , X 2 ), (Y 1 , Y 2 ), X 3 , Y 3 , G 0 , and G 1 , as well as the function M , have the same meanings as in Section 3.
Let d = 6.We begin by applying Lemma 1, both with α = 0 and α = 1/10, reducing the game to 2 5 × 2 20 , where 20 refers to the same 20 information sets we identified in Model B2.In fact, if we attempt to derive the analogue of Table 1 under Model B3, we find that it is identical to Table 1.But here an S entry, for example, means that S is optimal versus each of Player's 2 5 pure strategies.An S/D entry, for example, means that S is optimal versus Player's pure strategy SSSSS (u = 0) and D is optimal versus Player's pure strategy DDDDD (u = 31).

Table 1 :
Banker's optimal move (preliminary version) in the casino game baccara chemin de fer under Model B2 with d = 6 and with α = 0 and α = 1/10, indicated by S (stand) or D (draw), except in the 20 cases indicated by S/D (stand if Player always stands at two-card totals of 5, draw if Player always draws at two-card totals of 5) or D/S (draw if Player always stands at two-card totals of 5, stand if Player always draws at two-card totals of 5) in which it depends on Player's pure strategy.

Table 2 .
The Banker strategy for a given row can be deduced from Lemma 2. Let us consider row 44.The Banker strategy DDDDD-SSS-DDD-MSSSSD-DDD, together with Table1, allows us to evaluate Player's 2×2 payoff matrix, which is
A, B) if and only if entries 19 and 27 (of 0-31) of Aq T are equal and maximal, (52) entries 1,019,407 and 1,019,663 (of 0-1,048,575) of pB are equal and maximal (53) Now Condition (52) is automatic by virtue of how (p, q) was chosen, so it remains to verify Condition (53), which concerns only rows 19 and 27 (of 0-31) of B. Let B • be the 2 × 2 20 submatrix of B comprising rows 19 and 27, so that Condition (53) is equivalent to entries 1,019,407 and 1,019,663 (of 0-1,048,575) of (1 − p, p)B •