Monotonous betting strategies in warped casinos☆
Introduction
A bet in a game of chance is usually determined by two values: the favorable outcome and the wager x one bets on that outcome. If the outcome turns out to be the one chosen, the player gains profit x; otherwise the player loses the wager x. Many gambling systems for repeated betting are based on elaborate choices for the wager x, while leaving the choice of outcome constant. In this work we are interested in such ‘monotonous’ strategies, which we also call single-sided, and their linear combinations (mixtures). Consider the game of roulette, for example, and the binary outcome of red/black.1 Perhaps the most infamous roulette system is the martingale,2 where one constantly bets on a fixed color, say red, starts with an initial wager x and doubles the wager after each loss. At the first winning stage all losses are then recovered and an additional profit x is achieved. Such systems rely on the fairness of the game, in the form of a law of large numbers that has to be obeyed in the limit (and, of course, require unbounded initial resources in order to guarantee success with probability 1). In the example of the martingale the relevant law is that, with probability 1, there must be a round where the outcome is red. Many other systems have been developed that use more tame series of wagers (compared to the exponential increase of the martingale), and which appeal to various forms of the law of large numbers.3
When the casino is biased, i.e. the outcomes are not entirely random, we ought to be able to produce more successful strategies. Suppose that we bet on repeated coin-tosses, and that we are given the information that the coin has a bias. In this case it is well known that we can define an effective strategy that, independent of the bias of the coin (i.e. which side the coin is biased on, or even any lower bounds on the bias), is guaranteed to gain unbounded capital, starting from any non-zero initial capital. This strategy, as we explain in §2.3, is the mixture of two single-sided strategies, where the first one always bets on heads and the second one always bets on tails. A slightly modified strategy is successful on every coin-toss sequence X except for the case that the limit of the relative frequency of heads exists and is 1/2. The same kind of strategy exists for the case where the relative frequency of heads is 1/2, but beyond some point the number of tails is never smaller than the number of heads (or vice-versa). These examples show that many typical betting strategies are separable in the sense that they can be expressed as the sum of two single-sided strategies. In the following we refer to any binary sequence which is produced by a (potentially partially) random process, as a casino sequence. Note that if a separable strategy succeeds along a casino sequence, one of its single-sided parts has to succeed. The only case where separability is stronger than single-sidedness is when we consider success with respect to classes of casino sequences.
A casino sequence may have a (more subtle) bias while satisfying several known laws of large numbers, such as the relative frequency of 0s tending to 1/2. Formally, we can say that a casino sequence X is biased if there is an ‘effective’ (as in ‘constructive’ or ‘definable’) betting strategy which succeeds on X, i.e. produces an unbounded capital, starting from a finite initial capital. By adopting stronger or weaker formalizations of the term ‘effective’ one obtains different strengths of bias, or as we usually say, non-randomness of X. In general, ‘effective’ means that the strategy is definable in a simple way, such as being programmable in a Turing machine. Suppose that we know that the casino sequence X has a bias in this more general sense, i.e. there exists some ‘effective’ betting strategy which succeeds on it. The starting point of the present article is the following question: In other words, can any ‘effective’ betting strategy be replaced by a single-sided ‘effective’ betting strategy without sacrificing success? An equivalent way to ask this question is as follows. We will see that, depending on the way we formalize the term ‘effective’, and especially the term effective monotonous betting these questions can have a positive or negative (or even unknown) answer.
Our results. A straightforward interpretation of ‘effective’ is computable, in the sense that there is a Turing machine that decides, given each initial segment of the casino sequence: These choices, in combination with the revelation of the outcome, determine the capital at the beginning of the next betting stage. In §3 we show that in this case questions (1) and (2) have a positive answer. Another formalization of ‘effective’ which is very standard in computability and algorithmic information theory (and used in the standard definition of algorithmic randomness) is ‘computably enumerable’. When applied to betting strategies this gives a notion which is equivalent to infinite mixtures of strategies which are generated by a single Turing machine, see the introductory part of §2. There are two very different ways that one can define computably enumerable monotonous strategies:
- (i)
Uniform way: as the mixture (linear combination) of a computable family of monotonous strategies with bounded total initial capital;
- (ii)
Non-uniform way: as a monotonous strategy that can be expressed as the mixture of a computable family of strategies with bounded total initial capital.
Monotonous strategies under the non-uniform case (ii) are intuitively more powerful, as we explain in §2.2, and our arguments do not appear to be adequate for answering questions (1) and (2) in this case. The study of the power of strategies in (ii) is quite interesting from the point of view of stochastic processes, as it relates to key concepts such as martingale decompositions, variation and various forms of boundedness or integrability. (1), (2) under (ii) are also directly relevant to a question about the separation of two randomness notions in algorithmic information theory, asked by Kastermans (see [3] and [4, §7.9]). As we point out in §5, a positive answer of (1) or (2) for the case of strategies under (ii) would give a very simple and elegant positive answer to Kasterman's question.
Outline of the presentation and common notation. The concept of a betting strategy in terms of martingale functions is formalized in the first part of §2. Monotonous strategies are formalized in §2.1 and effective versions of mixtures of monotonous strategies are given in §2.2, along with relevant characterizations in terms of computable enumerability. In §2.3 we show that many types of betting are monotonous and in §2.4, after recalling that Hausdorff dimension is expressible in terms of speed of martingale success, we use these facts in order to show that there exists a separable strategy which succeeds in all casino sequences of effective Hausdorff dimension . In §3 we first describe a decomposition of computable martingales into two single-sided (orthogonal) martingales, which provides the positive answer to questions (1) and (2) stated in the introductory discussion, for the case of computable strategies. We then give a detailed argument establishing a strong negative answer of the same questions for the special case of a single separable strategy. This argument is then used in a modular way in §4 in order to obtain a proof of the full result, with respect to every possible strategy that is expressible as a mixture of a computable family of separable martingales. Finally in §4.4 we generalize this result to the more general class of decidably-sided strategies. Concluding remarks and a critical discussion of our results, along with open problems and directions for future investigations are given in §5.
The numbering of the bits of a sequence X starts from 0. Hence the first n bits of X, denoted by , are bits . The concatenation of strings is denoted by and if p is a parameter in a construction, we let denote the statement that p is defined or undefined respectively, at the particular stage of reference.
Section snippets
Monotonous betting strategies and their mixtures
Betting strategies are formalized by martingales4 which are used in order to express the capital after each betting stage and each casino outcome. Formally, a martingale in the space of binary outcomes is a function
The power of single-sided martingales and their mixtures
We show that if a computable martingale succeeds on some casino sequence X, then there exists a computable single-sided martingale which succeeds on X. This is a consequence of the following decomposition, which was also noticed independently by Frank Stephan. Lemma 3.1 Single-sided decomposition Every martingale M is the product of a 0-sided martingale N and a 1-sided martingale T. Moreover are computable from M. Proof The proof idea is to let N simulate the 0-bets (i.e. bets on outcome 0) of M, while ignoring the 1-bets of M and
Proof of Theorem 3.3 and generalizations
It is possible to adapt the proof of Lemma (18) into an effective construction for the proof of Theorem 3.3, which also gives that the real X is left-c.e. For simplicity, we opt for a less constructive initial segment argument for the proof of Theorem 3.3, which uses the facts we obtained in §3 in a modular way. The price we pay is that the constructed X is no-longer left-c.e. as in (18). The following is the main tool for the proof of Theorem 3.3, where has the same value as in §3. Lemma 4.1 Inductive property There
Conclusion and some questions
We have studied the strength of monotonous strategies, which bet constantly on the same outcome (single-sided martingales) or bet on a computable outcome (decidably-sided martingales). In the case of computable strategies we have seen that they are as strong as the unrestricted strategies, while in the case of uniform effective mixtures of strategies (strongly left-c.e. martingales) they are significantly weaker. On the other hand, for casino sequences of effective Hausdorff dimension less than
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Barmpalias was supported by the 1000 Talents Program for Young Scholars from the Chinese Government No. D1101130, NSFC grant No. 11750110425 and Grant No. ISCAS-2015-07 from the Institute of Software. Fang was supported by the China Scholarship Council of China. Partial support was received by the Jiangsu Provincial Advantage Fund, during a visit to the University of Nanjing.