Expectation of the largest bet size in the Labouchere system

For the Labouchere system with winning probability p at each coup, we prove that the expectation of the largest bet size under any initial list is ﬁnite if p > 12 , and is inﬁnite if p ≤ 12 , solving the open conjecture in [6]. The same result holds for a general family of betting systems, and the proof builds upon a recursive representation of the optimal betting system in the larger family.


Introduction
The Labouchere system, also known as the cancellation system, is one of the most well-known betting systems used in roulette. It was popularized by Henry Du Pré Labouchere, an English politician, writer and journalist. Before the betting, the bettor chooses an initial list L 0 of positive real numbers (e.g., L 0 = (1, 2, 3, 4)). During each bet, the bet size equals the sum of the first and last numbers on the list (if only one number remains on the list, then the bet size equals that number). After a win, the first and last terms are canceled from the list; after a loss, the amount just lost is appended to the last term of the list. This system is continued until the list is empty. Table 1 illustrates an example of the Labouchere system.
We introduce the following notations. Let L n be the list after the n-th coup, l n be the corresponding list length, B n be the bet size at the n-th coup, T n be the remaining target profit (i.e., the sum of the numbers in the list) after the n-th coup, and N be the stopping time that the list first becomes empty, i.e., L N = ∅. In this paper, we investigate the behavior of the largest bet size B max 1≤n≤N B n (or sup n≥1 B n if N = ∞) in the Labouchere system, and in particular, whether or not B has a finite expectation.
There is very limited literature on analyzing the Labouchere system. Let p ∈ [0, 1] be  [7] shows that a sufficient condition of E[B ] < ∞ is that p > p 0 ≈ 0.613763, while matching necessary conditions are still missing. Hence, it remains an open conjecture for more than a decade if the largest bet size B also has an infinite expectation when 1 3 < p ≤ 1 2 , which is the main focus of this paper.
There is also another betting system which is similar to the Labouchere system, i.e., the Fibonacci system. Instead of considering the first and last numbers in the list at each coup, the last two numbers are added or canceled in the Fibonacci system. Ethier [5] showed that E[B ] = +∞ in Fibonacci system if and only if p ≤ 1 2 . However, the proof heavily relies on the fact that any list in a Fibonacci system is uniquely determined by its length, which does not hold for the Labouchere system where the list evolves in a more complicated "history dependent" manner.

Main results
To study the Labouchere system, we first introduce a larger family of betting systems called (a, b)-list systems: Definition 2.1 ((a, b)-List System). Let a < 0 ≤ b be integers. An (a, b)-list system consists of a target sequence {T n }, a bet sequence {B n } and a length sequence {l n }, which evolve as follows: 1. At the beginning, T 0 > 0 and l 0 ∈ {1, 2, · · · }; 2. At the n-th coup, the system makes a bet size B n ∈ [0, T n−1 ] which may depend on the entire history. Then the target and length sequences evolve as  3. Termination condition: let N = inf{n : l n = 0} be the stopping time that the length becomes zero, we must have T n = l n = 0 for any n ≥ N and B n = 0 for any n > N .
In such a list system, the target T n represents the remaining amount of money one would like to earn at the end of the n-th coup; consequently, T n shrinks after a win, and increases after a loss. The length l n represents the length of the "list" at the n-th coup, where it may be some real/virtual list which governs the betting process. For example, the well-known martingale system (where the bet is doubled after each loss) belongs to the (−1, 0)-list system with l 0 = 1 and B n = T n−1 , and both the Labouchere and Fibonacci systems fall into the category of (−2, 1)-list systems. The termination condition ensures that, as long as the list length l n hits zero, the target must be fulfilled as well (i.e., T n = 0), and the betting process terminates.
In this paper, we only consider (−2, 1)-list systems where the Labouchere system is included, but our results and proof techniques are generalizable to general (a, b)-list systems. Our first result characterizes the behavior of the largest bet size B under general list systems: For any (−2, 1)-list system, the following holds: Theorem 2.2 shows that for any (−2, 1)-list systems, the expectation E[B ] of the largest bet size B has a phase transition at p = 1 2 : the expectation is finite if the player is favored, and is infinite if the house takes the advantage. Consequently, we have the following corollary: The fair-game case p = 1 2 requires more delicate analysis, and is summarized in the following theorem: be two sequences taking value in [0, 1]. Suppose that some (−2, 1)-list system satisfies that T n−1 b ln−1 ≤ B n ≤ T n−1 b ln−1 for any n, and one of the following conditions holds: Note that B n /T n−1 is the bet proportion at the n-th coup, and general (−2, 1)-list systems correspond to the case where b l = 1, b l = 0 for any l. Theorem 2.4 shows that, if the bet proportion either vanishes or is lower bounded from below as the list length l grows, the largest bet size still has an infinite expectation in a fair game. The following corollary follows from Theorem 2.4: Based on Theorem 2.4, a natural question would be that whether E[B ] = ∞ holds in any (−2, 1)-list systems. We have the following partial result: Theorem 2.6. For any (−2, 1)-list system and ε > 0, the following holds under p = 1 2 : Theorem 2.6 shows that, the moment E[(B ) α ] always has a phase transition at α = 1 in a fair game. However, the exact answer for α = 1 is still unknown, and we leave it as a conjecture:

Proof of Theorems 2.2 and 2.6
In this section, we first prove Theorem 2.6, and then apply Theorem 2.6 to proving Theorem 2.2.

Proof of Theorem 2.6
We make use of the asymptotic tail behavior of the stopping time N in the (−2, 1)-list system.
where l 0 is the length of the initial list, D l0 (n) is a constant only depending on l 0 and n (mod 3), and κ 27 4 p(1 − p) 2 < 1.
Based on Lemma 3.1, we are about to prove Theorem 2.6. We first show that Note that B n ≤ T n−1 , for λ ≥ 2 we therefore have where C > 0 is some universal constant. As a result, where in the last step we have used that By Lemma 3.1, for c > 0 sufficiently small we have E[e cN ] < ∞. Moreover, [6] shows that A combination of the previous two inequalities yields E[B (1 ∨ log B )] = ∞.

Proof of Theorem 2.2 and Corollary 2.3
Now we prove Theorem 2.2 using Theorem 2.6 and a change of measure.
Fix any p > 1 2 , let P be the probability measure over the betting process under winning probability p, and Q be the counterpart under winning probability 1 2 . Note that for any sample path ω with stopping time N = n, there must be n 3 + c wins and 2n 3 − c losses, where c is a constant depending only on the initial length l 0 and n (mod 3). As a result, the likelihood ratio is where C > 0, ρ ∈ (0, 1) are numerical constants independent of n, and we have used that the function p → p(1 − p) 2 is strictly decreasing in p ∈ [ 1 3 , 1]. As a result, Since T n ≤ T n−1 +B n ≤ 2T n−1 in any list system, we have B ≤ max 0≤n≤N T n ≤ T 0 ·2 N , and therefore for any ε > 0. Choosing ε > 0 small enough such that ρ2 ε < 1, Theorem 2.6 implies that E P [B ] < ∞. For p ∈ ( 1 3 , 1 2 ), we use the same argument to obtain dP dQ ≥ Cρ N for some ρ > 1. Then l n ] + c 2 = ∞ as desired. The proof of Theorem 2.2 is completed.
As for Corollary 2.3, it suffices to verify that the condition B n ≥ c 1 l n−1 + c 2 holds for the Labouchere system. Let a > 0 be the minimum number in the initial list L 0 , a simple induction on n yields that B n ≥ a(l n−1 − l 0 ) + , which shows that the condition is fulfilled with c 1 = a > 0, c 2 = −al 0 .

Proof of Theorem 2.and Corollary 2.5
In this section, we first use a recursive representation of the optimal list system to prove Theorem 2.4. Then we investigate the specific properties of the Labouchere system and show that the condition in Theorem 2.4 holds, thereby proving Corollary 2.5.

Proof of Theorem 2.4
If inf l b l ≥ c > 0, we have B ≥ c max 0≤n≤N T n , which has an infinite expectation [6]. Now we assume that lim l→∞ b l = 0 and prove Theorem 2.4 by contradiction. We first introduce the following definition: For any x > 0 and l ∈ {1, 2, · · · }, we define f (x, l) to be the infimum of E[B ] over all possible (−2, 1)-list systems with initial target x and initial length l, such that B n ≤ b ln−1 T n−1 for any n. Definition 4.1 considers an optimal (−2, 1)-list system with initial target x and initial length l, where optimality is measured in terms of a smallest expectation of the largest bet size B . The quantity f (x, l) ∈ R + ∪ {+∞} is the corresponding expectation, and it is well-defined even if the optimal list system does not exist. The next lemma presents recursive relations between f (x, l) with different l.

Lemma 4.2.
There exists some sequence (a l ) ∞ l=1 taking value in R + ∪ {+∞} such that f (x, l) = xa l for any x > 0. Moreover, the sequence (a l ) ∞ l=1 satisfies the following inequalities: Proof. When the initial target x is scaled by λ > 0, we may always scale all bet sizes by λ to arrive at a new list system with the initial target λx, and vice versa. Hence, f (x, l) is proportional to x, and f (x, l) = xa l . For l ≥ 3 and any (−2, 1)-list system, let b ∈ [0, b l ] be any bet size at the first coup with initial target T 0 = 1 and initial length l. Let B 1 , B 2 be the largest bet sizes (excluding the first bet) after winning/losing the first coup, respectively. Then by definition of f (x, l), we have Note that B is either max{b, B 1 } or max{b, B 2 }, we have where the first inequality is due to the convexity of x → max{b, x}. Note that this inequality holds for any list systems, taking infimum over the LHS gives the desired inequality for l ≥ 3. The other inequalities for l ≤ 2 can be established analogously.
Based on Lemma 4.2, we may investigate more properties of a l . If a 1 = ∞, it is obvious that a l = ∞ for any l ∈ N (since any initial list may evolve into length one with a non-zero probability), and Theorem 2.4 holds. Next we show that a 1 < ∞ is impossible. Assume by contradiction that a 1 < ∞, we will have the following lemma. Lemma 4.3. If a 1 < ∞, the sequence {a l } will be strictly decreasing, i.e., a 1 > a 2 > a 3 > · · · .
Proof. For l ≥ 3, by Lemma 4.2 we have where in the last step we have used the fact that an affine function attains its minimum at the boundary. Consequently, if we already know that a 1 ≥ a 2 ≥ · · · ≥ a l , we must also have a l ≥ a l+1 . Hence, by induction on l, the sequence {a l } is decreasing.
If we have a l = a l+1 , we will also have a l−2 = a l+1 based on the previous inequality. Due to the decreasing property of {a l }, a l−1 = a l also holds, and repeating this process yields a 2 = a 3 , a contradiction to Lemma 4.2. Hence a l > a l+1 for any l.
Based on Lemmas 4.2 and 4.3, we are about to arrive at the desired contradiction.
Fix any ε > 0 such that ρ 1−ε 1+ε + ( 1−ε 1+ε ) 2 > 1. Since lim l→∞ b l = 0, we take l 0 > 0 large enough such that b l < ε for any l > l 0 . Then for l > l 0 , Lemma 4.2 yields where in the last step we have used a l+1 ≤ a l−2 by Lemma 4.3. A rearrangement of the previous inequality gives for any l > l 0 . Similarly, Adding them together yields Our choice of ε implies ρ > 1, and therefore a l+2k−2 − a l+2k ≥ ρ k (a l−2 − a l ) for any k ∈ N and l > l 0 . Since a l+2k−2 − a l+2k ≤ a 1 , and a l−2 > a l by Lemma 4.3, this inequality implies that a 1 ≥ ρ k (a l−2 − a l ) for any k = 1, 2, · · · , a contradiction to the assumption a 1 < ∞. The proof of Theorem 2.4 is complete.

Proof of Corollary 2.5
First we observe that it suffices to prove the case where the initial list consists of a single positive number. This observation is due to that there is a positive probability to reduce the list length to l n = 1 after finitely many coups for any initial list L 0 .
To study the combinatorial properties of the Labouchere system, we introduce the following definition: Definition 4.4. A list of positive real numbers (a 1 , a 2 , a 3 , · · · , a n ) is called good if it satisfies the following conditions: • Every element in the list is positive, i.e., a i > 0 for any i; • The list is non-decreasing, i.e., a 1 ≤ a 2 ≤ · · · ≤ a n ; • The difference of the list is non-decreasing with difference at most a 1 , i.e., a 2 − a 1 ≤ a 3 − a 2 ≤ · · · ≤ a n − a n−1 ≤ a 1 .
The key properties of a good list are summarized in the following lemmas.

Lemma 4.5.
If the initial list L 0 is good, the list L n after n-th coup is also good for any n.
Proof. It suffices to prove that, if L n−1 = (a 1 , · · · , a l ) is a good list, so is L n . Based on the outcome at n-th coup, there are only two possibilities: • L n = (a 1 , a 2 , · · · , a l , a 1 + a l ), or • L n = (a 2 , a 3 , · · · , a l−1 ).
In either case, one can check from Definition 4.4 directly that L n is a good list, as desired.