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.
Highlights
The Standard ModelMost adaptive bet-hedging models are largely based on the classic horse race gambling model associated with Kelly (1956), where the biological counter-part is a lineage apportioning bets on several possible environments
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
Summary
Most adaptive bet-hedging models are largely based on the classic horse race gambling model associated with Kelly (1956), where the biological counter-part is a lineage apportioning bets on several possible environments. Assume that k horses run in a race, and let horse Xi win with probability pi. If horse Xi wins, the odds are oi for 1. A gambler wishes to apportion his bankroll among the horses 0 < fi ≤ 1, such that fi = 1 and participate in indefinitely repeated races n → ∞. How to best apportion the bankroll each time? Wealth is a discrete-time stochastic process over n periods, n Wn = Wi (X ). I =1 where W (X ) = f (X )O(X ) is the random factor by which the gambler’s wealth is multiplied when horse X wins.
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