Monotonous betting strategies in warped casinos

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Abstract

Suppose that the outcomes of a roulette table are not entirely random, in the sense that there exists a successful betting strategy. Is there a successful ‘separable’ strategy, in the sense that it does not use the winnings from betting on red in order to bet on black, and vice-versa? We study this question from an algorithmic point of view and observe that every strategy M can be replaced by a separable strategy which is computable from M and successful on any outcome-sequence where M is successful. We then consider the case of mixtures and show: (a) there exists an effective mixture of separable strategies which succeeds on every casino sequence with effective Hausdorff dimension less than 1/2; (b) there exists a casino sequence of effective Hausdorff dimension 1/2 on which no effective mixture of separable strategies succeeds. Finally we extend (b) to a more general class of strategies.

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:Is it possible to succeed on any such warped casino sequence with a single-sided ‘effective’ betting strategy, i.e.one that can only place bets on 0 or only on 1? 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.Suppose that we are betting with the restriction that we cannot use our earnings from the successful bets on 0s in order to bet on 1s, and vice-versa. Can we win on any casino-sequence X which is ‘biased’ in the sense that there is an (unrestricted) strategy which wins on X? 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:(a) how much of the current capital to bet;(b) which outcome to bet on. 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.

In the uniform case we show that questions (1) and (2) have negative answers. In fact, we show that there are casino sequences X on which mixtures of computable families of strategies generate infinite capital exponentially fast, in the sense thatlimsupnM(Xn)αn=whereα(1,2)andMis the capital after the firstnbets onX, where Xn denotes the first n bits of X, but no strategy under (i) succeeds. We also show the converse, i.e. that if a computably enumerable strategy (i.e. a mixture of computable family of strategies) M exists such that limsupnM(Xn)/αn= for some α>2, then there exists a single-sided computably enumerable strategy N which succeeds on X, in the sense that limnN(Xn)=. We will see that these results can also be stated in terms of the effective Hausdorff dimension of the casino sequence. Under the uniform case we also consider a more general class of strategies, which we call decidably-sided, and which are not necessarily monotonous, but there is a computable prediction (or choice) function which indicates the favorable outcome at each state. We then generalize our previous arguments and show that there is a casino sequence and a computably enumerable betting strategy M that strongly succeeds on it as before, in the sense of (3), but such that no decidably-sided computably enumerable strategy succeeds on it.

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 <1/2. 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 X(0),X(1), of a sequence X starts from 0. Hence the first n bits of X, denoted by Xn, are bits X(0),,X(n1). The concatenation of strings σ,τ is denoted by στ and if p is a parameter in a construction, we let p,p 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 M:2<ω

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 N,T 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 qn 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.

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