Quantum probabilities in competing lizard communities

Abstract

Despite predictive success, population dynamics and evolutionary game theory still pose fundamental problems. Violation of the competitive exclusion principle in plankton communities provides an example. A promising solution of this ‘paradox of the plankton’ comes from theories involving cyclic competition, an evolutionary analogue of the classical rock-paper-scissors (RPS) game. However, modeling probabilistic RPS structures, one encounters a fundamental difficulty: the pairs rock-scissors, scissors-paper, and paper-rock possess representations in separate Kolmogorovian probability spaces, but a single global probability space for entire triplets does not exist. Populations that take part in cyclic competition should therefore involve probabilistic incompatibilities, analogous to those occurring in quantum mechanics. Here, using experimental data collected from 1990 to 2011 on the RPS cycles of lizards, we show that the incompatibilities are indeed unavoidable, and the data cannot be reconstructed from a single Kolmogorovian probability space. We then prove that the effect is genuinely quantum probabilistic, i.e. all the probabilities can be formulated in terms of a single density matrix and a set of non-commuting projectors. This formal quantum structure is dormant in games where probabilities of strategies do not entangle with probabilities of payoffs, and thus could be overlooked. In more realistic scenarios, involving games ‘with ace in a sleeve’, the non-Kolmogorovian structure can be activated. Surprisingly, lizards occasionally do play such games. In consequence, the formalism of evolutionary games, similarly to quantum mechanics, should begin with density matrix equations. Implications of our finding extend beyond lizard communities, given that RPS games are common in nature and higher dimensional RPS games may be even more common in ecosystems.