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Continuous time, continuous decision space prisoner’s dilemma: A bridge between game theory and economic GCD-models

Glötzl, Erhard (2016): Continuous time, continuous decision space prisoner’s dilemma: A bridge between game theory and economic GCD-models.

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Abstract

General Constrained Dynamic models (GCD – models) in economics are inspired by classical mechanics with constraints. Most macroeconomic models can be understood as special cases of GCD – models. Moreover, in this paper it will be shown that not only macroeconomic models but also game theoretic models are strongly related to GCD – models. GCD models are characterized by a system of differential equations in continuous time while most game theoretical models are set up in discrete time. Therefore it is necessary to build a bridge from game theoretical models denominated in discrete time to game theoretical models using continuous time. This bridge is illustrated in the following using the example of a continuous time, continuous decision space prisoner’s dilemma. Furthermore, it is shown that the differential equations which determine other continuous game theoretic models can be understood to a certain extent as special cases of the GCD – differential equations. Well known types of continuous game theoretic models include for instance “Evolutionary Game Theory” with the replicator equation, “Adaptive Dynamics” with the canonical equation, which is nothing else than a replicator – mutator equation, and the so called “Differential Games”, which are strongly related to optimal control theory with two controls and two different objectives (goals). Most of the GCD – models are characterised by 3 key feature: - mutual influence, - Power-factors - Constraints Nowak (2006b) and Taylor & Nowak (2007) show that there are five mechanisms which, under certain conditions, can lead to the evolution of cooperation in an iterated prisoner’s dilemma. Inspired by this, we apply the 3 key features of GCD – models to the standard prisoner’s dilemma in discrete time which yields 3 additional mechanisms which enable the evolution of cooperation. The assumption or axiom of the free market economy is that an individual optimisation strategy will lead to an overall optimum by virtue of Adam Smith’s invisible hand. Without additional conditions this assumption alone is fundamentally wrong. As in prisoner’s dilemma also in economics cooperation is essential to get an overall optimum. The big question of political economy is to analyse which additional measures could guarantee that the individual optimisation strategy characterising a free market economy leads to cooperation as precondition to get an overall optimum. From this point of view the different economic theories could be characterised in terms of which measures they assume to be sufficient to guarantee an overall optimum without abandoning the principle of individual optimisation.

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