Adaptive Bet-Hedging Revisited: Considerations of Risk and Time Horizon
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Adaptive Bet-Hedging Revisited: Considerations of Risk and Time Horizon Omri Tal1,3
· Tat Dat Tran1,2
Received: 12 June 2019 / Accepted: 14 March 2020 / Published online: 4 April 2020 © The Author(s) 2020
Abstract 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. Keywords Adaptive bet-hedging · Kelly gambling · Growth-optimal portfolio theory · Game theory · Finite time horizon · Extinction risk
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Omri Tal [email protected] Tat Dat Tran [email protected]; [email protected]
1
Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany
2
Institute of Mathematics, Leipzig University, Augustusplatz 10, 04109 Leipzig, Germany
3
Cohn Institute for the History and Philosophy of Science and Ideas, Tel Aviv University, Tel Aviv-Yafo, Israel
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O. Tal,T.D. Tran
1 Introduction ‘‘Adversity has the effect of eliciting talents, which in prosperous circumstances would have lain dormant.’’ -- Horace (65BC-8BC)
Kelly’s work on optimal gambling strategies and the value of side information was arguably the first convincing attempt at applying concepts from information theory for analysis in a different field Kelly (1956). This work was the precursor to growthoptimal portfolio theory which has extended the basic ideas to the realm of capital markets (Cover and Thomas 2006). There has recently been a resurge of interest in employing insights from optimal gambling theory in models of adaptive bet-hedging under fluctuating environments, where close analogies between the economic and biological setting have been convincingly made apparent (Bergstrom 2014; Rivoire and Leibler 2011; Donaldson-Matasci et al. 2010). Biological bet-hedging was originally proposed to explain the observation of ungerminated seeds of annual plants (Cohen 1966). This strategy involves the variable phenotypic expression of a single genotype, rather than a result of genetic polymorphism, although it is difficult to empirically determine whether observed phenotypic diversity in a population arise
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