Quantum probabilistuc structures in competing lizard communities

Despite predictive success, population dynamics and evolutionary game theory [1, 2] still pose fun- damental problems. Violation of the competitive exclusion principle in plankton communities pro- vides an example. A promising solution of this “paradox of the plankton” [3] comes from theories involving cyclic competition [4–6], an evolution- ary analogue of the classical rock-paper-scissors (RPS) game. However, modeling probabilistic RPS structures one encounters a fundamental dif- ficulty [7, 8]: 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 prob- abilistic incompatibilities, analogous to those oc- curring in quantum mechanics. Here, using ex- perimental data collected from 1990 to 2011 on the RPS cycles of lizards, we show that the in- compatibilities are indeed unavoidable, and the data cannot be reconstructed from a single Kol- mogorovian probability space. We then prove that the effect is genuinely quantum probabilis- tic, 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 probabili- ties of strategies do not entangle with probabili- ties of payoffs, and thus could be overlooked. In more realistic scenarios, involving games “with ace in a sleeve”, the non-Kolmogorovian struc- ture can be activated. Surprisingly, lizards oc- casionally 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 ex- tend beyond lizard communities, given that RPS games are common in nature [9] and higher di- mensional RPS games may be even more common in ecosystems [6].

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