MARCO PIUMETTI¹*, NIK LYGEROS²
*Corresponding author
1. Department of Applied Science and Technology, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy
2. Université de Lyon, LGPC (UMR 5285), 69616, Villeurbanne, France

Abstract

Synergistic and cooperative phenomena in catalysis are defined here via a formal classification of the catalytic systems, using the combination of game theory and hyperstructure theory. Game theory is based on the idea that players are rational decision-makers; therefore, making a transposition is reported as rational, the chemical choice. 
Three catalytic reactions are considered as characteristic examples: i) the degradation of an azo-dye (Acid Orange 7) with vanadium/hydrogen peroxide/ascorbic acid solutions (homogeneous catalysis); ii) the total oxidation of ethene and iii) the carbon monoxide oxidation over Cu-Ce mixed oxide catalysts (heterogeneous catalysis). According to the game theory, these reactions are paradigms of not-zero sum, asymmetrical and synchronous cooperative-games. The players (i.e. active phases/centres) exhibit a finite number of pure strategies, reflected by their concentrations, and play a strategic game. Moreover, there is the Nash equilibrium and even Pareto equilibrium in the synergistic case. Since the effects of the structural relations among players depend on the catalytic system, a scale of magnitude (in terms of catalytic benefits) is proposed: cooperation < synergetic hyperstructure << strong synergetic hyperstructure.

INTRODUCTION
In catalysis there are several characteristic examples in which the presence of two, or more, components may influence the catalytic activity of a catalyst by changing its solid-state chemistry (1). Many solid catalysts are multicomponent materials (e.g. bismuth molybdates, perovskites, heteropolyanions, vanadium phosphate, and so on) with active phases and promoters mutually-interacting each other over different domains (2-3). Thus, either cooperative or synergistic phenomena may appear and the catalytic behavior typically results in a non-linear combination of complexity structures of active sites (4-6).
For instance, it has been shown through X-ray photoelectron spectroscopy (XPS) and electrical conductivity measurements that multicomponent catalysts, such as bismuth molybdates over Co(Fe)-molybdates, reveal synergistic effects due to the enhanced electrical conductivity of the Co(Fe)-molybdate support, attributable to the presence of both Fe²⁺ and Fe³⁺ in Co²⁺ molybdates, thus favouring the so-called Mars and van Krevelen (MvK) mechanism (7, 8). Similarly, the incorporat ...