Second Parrondos Paradox in Scale Free Networks

Parrondo’s paradox occurs in sequences of games in which a winning expectation value of a payoff may be obtained by playing two games in a random order, even though each game in the sequence may be lost when played individually.Several variations of Parrondo’s games apparently with the same paradoxical property have been introduced by G.P. Harmer and D. Abbott; history dependence, one dimensional line, two dimensional lattice and so on. I have shown that Parrondo’s paradox does not occur in scale free networks in the simplest case with the same number of parameters as the original Parrondo’s paradox. It suggests that some technical complexities are needed to present Parrondo’s paradox in scale free networks. In this article, I show that a simple modification with the same number of parameters as the original Parrondo’s paradox creates Parrondo’s paradox in scale free. This paradox is, however, created by a quite different mechanism from the original Parrondo’s paradox and a considerably rare phenomenon, where the discrete property of degree of nodes is crucial. I call it the second Parrondo’s paradox.

💡 Research Summary

Parrondo’s paradox traditionally demonstrates that two losing games, when played in a random or periodic sequence, can yield a winning expectation. This counter‑intuitive result has been reproduced in a variety of simple spatial settings—one‑dimensional lines, two‑dimensional lattices, and history‑dependent variants—by exploiting nonlinear probability updates or state‑dependent rule changes. However, when the same games are transplanted onto scale‑free networks, which are characterized by a power‑law degree distribution and a small set of highly connected hub nodes alongside many low‑degree nodes, the paradox fails to appear. The authors first confirm this failure by implementing the classic Parrondo games (Game A with a fixed losing probability, Game B whose win condition depends on the fraction of winning neighbours) on Barabási‑Albert networks (γ≈3, minimum degree m=3). Across all degree classes the average capital declines, and random mixing of the two games does not reverse the loss.

To overcome this limitation, the paper introduces a minimal modification that retains the original number of parameters: Game B is redefined so that every node, regardless of its degree, uses the same integer threshold k. In each round a node wins if the number of winning neighbours is at least k; otherwise it loses. Because the threshold is independent of degree, low‑degree nodes (d < k) almost always lose, while high‑degree hubs (d ≥ k) enjoy a dramatically higher win probability. This asymmetry leverages the discrete nature of node degrees in a scale‑free graph.

The authors develop a theoretical framework by grouping nodes according to degree d and tracking the mean capital μ_d(t). The dynamics are approximated by a Markov transition equation
μ_d(t+1)=μ_d(t)+p_A·Δ_A+f_d(k)·Δ_B,
where f_d(k) is the probability that a node of degree d meets the threshold k, and Δ_A, Δ_B are the capital changes for Games A and B respectively. Using the degree distribution P(d)∝d^‑γ, they analytically derive a critical interval for k (for the chosen γ and m, roughly k=4–5) in which the overall expected capital E