Adaptive Bet-Hedging Revisited: Considerations of Risk and Time Horizon

Models of adaptive bet-hedging commonly adopt insights from Kelly's famous work on optimal gambling strategies and the financial value of information. In particular, such models seek evolutionary solutions that maximize long term average growth rate of lineages, even in the face of highly stochastic growth trajectories. Here, we argue for extensive departures from the standard approach to better account for evolutionary contingencies. Crucially, we incorporate considerations of volatility minimization, motivated by interim extinction risk in finite populations, within a finite time horizon approach to growth maximization. We find that a game-theoretic competitive-optimality approach best captures these additional constraints, and derive the equilibria solutions under straightforward fitness payoff functions and extinction risks. We show that for both maximal growth and minimal time relative payoffs the log-optimal strategy is a unique pure-strategy symmetric equilibrium, invariant with evolutionary time horizon and robust to low extinction risks.Comment: Accepted for publication in Bulletin of Mathematical Biolog

On the Benefit of Nonlinear Control for Robust Logarithmic Growth: Coin Flipping Games as a Demonstration Case

The takeoff point for this letter is the voluminous body of literature addressing recursive betting games with expected logarithmic growth of wealth being the performance criterion. Whereas almost all existing papers involve use of linear feedback, the use of nonlinear control is conspicuously absent. This is epitomized by the large subset of this literature dealing with Kelly Betting. With this as the high-level motivation, we study the potential for use of nonlinear control in this framework. To this end, we consider a “demonstration case” which is one of the simplest scenarios encountered in this line of research: repeated flips of a biased coin with probability of heads p , and even-money payoff on each flip. First, we formulate a new robust nonlinear control problem which we believe is both simple to understand and apropos for dealing with concerns about distributional robustness; i.e., instead of assuming that p is perfectly known as in the case of the classical Kelly formulation, we begin with a bounding set P⊆[0,1] for this probability. Then, we provide a theorem, our main result, which gives a closed-form description of the optimal robust nonlinear controller and a corollary which establishes that it robustly outperforms linear controllers such as those found in the literature. A second, less significant, contribution of this letter bears upon the computability of our solution. For an n-flip game, whereas an admissible controller has 2n−1 parameters, at the optimum only O( n2 ) of them turn out to be distinct. Finally, it is noted that the initial assumptions on payoffs and the use of the uniform distribution on p are made mainly for simplicity of the exposition and compliance with length requirements for a Letter. Accordingly, this letter also includes a new Section with a discussion indicating how these assumptions can be relaxed

Kelly betting with quantum payoff: A continuous variable approach

The main purpose of this study is to introduce a semi-classical model describing betting scenarios in which, at variance with conventional approaches, the payoff of the gambler is encoded into the internal degrees of freedom of a quantum memory element. In our scheme, we assume that the invested capital is explicitly associated with the quantum analog of the free-energy (i.e. ergotropy functional by Allahverdyan, Balian, and Nieuwenhuizen) of a single mode of the electromagnetic radiation which, depending on the outcome of the betting, experiences attenuation or amplification processes which model losses and winning events. The resulting stochastic evolution of the quantum memory resembles the dynamics of random lasing which we characterize within the theoretical setting of Bosonic Gaussian channels. As in the classical Kelly Criterion for optimal betting, we define the asymptotic doubling rate of the model and identify the optimal gambling strategy for fixed odds and probabilities of winning. The performance of the model are hence studied as a function of the input capital state under the assumption that the latter belongs to the set of Gaussian density matrices (i.e. displaced, squeezed thermal Gibbs states) revealing that the best option for the gambler is to devote all their initial resources into coherent state amplitude

Kelly betting with quantum payoff: A continuous variable approach

The main purpose of this study is to introduce a semi-classical model describing betting scenarios in which, at variance with conventional approaches, the payoff of the gambler is encoded into the internal degrees of freedom of a quantum memory element. In our scheme, we assume that the invested capital is explicitly associated with the quantum analog of the free-energy (i.e. ergotropy functional by Allahverdyan, Balian, and Nieuwenhuizen) of a single mode of the electromagnetic radiation which, depending on the outcome of the betting, experiences attenuation or amplification processes which model losses and winning events. The resulting stochastic evolution of the quantum memory resembles the dynamics of random lasing which we characterize within the theoretical setting of Bosonic Gaussian channels. As in the classical Kelly Criterion for optimal betting, we define the asymptotic doubling rate of the model and identify the optimal gambling strategy for fixed odds and probabilities of winning. The performance of the model are hence studied as a function of the input capital state under the assumption that the latter belongs to the set of Gaussian density matrices (i.e. displaced, squeezed thermal Gibbs states) revealing that the best option for the gambler is to devote all their initial resources into coherent state amplitude

The Reality Game

We introduce an evolutionary game with feedback between perception and reality, which we call the reality game. It is a game of chance in which the probabilities for different objective outcomes (e.g., heads or tails in a coin toss) depend on the amount wagered on those outcomes. By varying the `reality map', which relates the amount wagered to the probability of the outcome, it is possible to move continuously from a purely objective game in which probabilities have no dependence on wagers to a purely subjective game in which probabilities equal the amount wagered. We study self-reinforcing games, in which betting more on an outcome increases its odds, and self-defeating games, in which the opposite is true. This is investigated in and out of equilibrium, with and without rational players, and both numerically and analytically. We introduce a method of measuring the inefficiency of the game, similar to measuring the magnitude of the arbitrage opportunities in a financial market. We prove that convergence to equilibrium is is a power law with an extremely slow rate of convergence: The more subjective the game, the slower the convergence.Comment: 21 pages, 5 figure

In-game betting and the Kelly criterion

When a bet with a positive expected return is available, the Kelly crite-rion can be used to determine the fraction of wealth to wager so as to maximizethe expected logarithmic return on investment. Several variants of the Kelly cri-terion have been developed and used by investors and bettors to maximize theirperformance in inefficient markets. This paper addresses a situation that has not,hitherto, been discussed in academic literature: when multiple bets can be placedon the same object and the available odds, true probabilities, or both, vary overtime. Such objects are frequently available in sports betting markets, for example,in the case of in-game betting on outcomes of soccer matches. We adapt the Kellycriterion to support decisions in such live betting scenarios, and provide numericalexamples of how optimal bet sizes can sometimes be counter-intuitive

Convert index trading to option strategies via LSTM architecture

AbstractIn the past, most strategies were mainly designed to focus on stocks or futures as the trading target. However, due to the enormous number of companies in the market, it is not easy to select a set of stocks or futures for investment. By investigating each company's financial situation and the trend of the overall financial market, people can invest precisely in the market and choose to go long or short. Moreover, how to determine the position size of the transaction is also a problematic issue. In the past, many money management theories were based on the Kelly criterion. And they put a certain percentage of their total funds into the market for trading. Nonetheless, three massive problems cannot be overcome. First, futures are leveraged transactions, and extra funds must be deposited as margin. It causes that the position size is hard to be estimated by the Kelly criterion. The second point is that the trading strategy is difficult to determine the winning rate in the financial market and cannot be brought into the Kelly criterion to calculate the optimal fraction. Last, the financial data are always massive. A big data technique should be applied to resolve this issue and enhance the performance of the framework to reveal knowledge in the financial data. Therefore, in this paper, a concept of converting the original futures trading strategy into options trading is proposed. An LSTM (long short-term memory)-based framework is proposed to predict the profit probability of the original futures strategy and convert the corresponding daily take-profit and stop-loss points according to the delta value of the options. Finally, the proposed framework brings the results into the Kelly criterion to get the optimal fraction of options trading. The final research results show that options trading is closer to the optimal fraction calculated by the Kelly criterion than futures trading. If the original futures trading strategy can profit, the benefits after converting to options trading can be further superior

The Value of Information for Populations in Varying Environments

The notion of information pervades informal descriptions of biological systems, but formal treatments face the problem of defining a quantitative measure of information rooted in a concept of fitness, which is itself an elusive notion. Here, we present a model of population dynamics where this problem is amenable to a mathematical analysis. In the limit where any information about future environmental variations is common to the members of the population, our model is equivalent to known models of financial investment. In this case, the population can be interpreted as a portfolio of financial assets and previous analyses have shown that a key quantity of Shannon's communication theory, the mutual information, sets a fundamental limit on the value of information. We show that this bound can be violated when accounting for features that are irrelevant in finance but inherent to biological systems, such as the stochasticity present at the individual level. This leads us to generalize the measures of uncertainty and information usually encountered in information theory