Abstract
In game theory, Parrondo's paradox describes the possibility of achieving winning outcomes by alternating between losing strategies. The framework had been conceptualized from a physical phenomenon termed flashing Brownian ratchets, but has since been useful in understanding a broad range of phenomena in the physical and life sciences, including the behavior of ecological systems and evolutionary trends. A minimal representation of the paradox is that of a pair of games played in random order; unfortunately, closed‐form solutions general in all parameters remain elusive. Here, we present explicit solutions for capital statistics and outcome conditions for a generalized game pair. The methodology is general and can be applied to the development of analytical methods across ratchet‐type models, and of Parrondo's paradox in general, which have wide‐ranging applications across physical and biological systems.
Highlights
Molecular motors and enzyme transport had been analyzed through Brownwinning outcomes by alternating between losing strategies
In an opposing direction[1,2,3]; underlying the unexpected motion in microorganisms, despite the strategy being likely locally malis an agitation-ratcheting mechanism, formed from alternating adaptive; in ecological systems, environmental fluctuations have diffusive and localizing temporal phases as the potential oscil- been shown to enable the persistence of rare species in invaded lates
We present explicit closed-form solutions for cap
Summary
A generalized capital-dependent Parrondo’s game structure with drawing outcomes is first considered. In game , winning, drawing, and losing outcomes, respectively, occur with probability p, r, and q = 1 − p − r in each round; in game , they respectively occur with probability p1, r1, and q1 = 1 − p1 − r1 when capital c is divisible by M ≥ 3 and p2, r2, and q2 = 1 − p2 − r2 otherwise. Stochastic mixing is controlled by parameter γ, reflecting probabilities γ and 1 − γ of playing game and , respectively, at each round. Each winning and losing outcome results in a unit capital increment (ηp = 1) and decrement (ηq = −1), respectively, and each draw results in unchanged capital (ηr = 0).
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