Deduction Theorem: The Problematic Nature of Common Practice in Game Theory

Deduction Theorem: The Problematic Nature of Common Practice in Game Theory Holger I. MEINHARDT † August 2, 2019 We consider the Deduction Theorem that is used in the literature of game theory to run a purported proof by contradiction. In the context of game theory, it is stated that if we have a proof of φ null ϕ, then we also have a proof of φ ϕ. Hence, the proof of φ ϕ is deduced from a previous known statement. However, we argue that one has to manage to prove that the clauses φ and ϕ exist, i.e., they are known true statements in order to establish that φ null ϕ is provable, and that therefore φ ϕ is provable as well. Thus, we are only allowed to reason with known true statements, i.e., we are not allowed to assume that φ or ϕ exist. Doing so, leads immediately to a wrong conclusion. Apart from this, we stress to other facts why the Deduction Theorem is not applicable to run a proof by contradiction. Finally, we present an example from industrial cooperation where the Deduction Theorem is not correctly applied with the consequence that the obtained result contradicts the well-known aggregation issue. MS Classifications 2010: 03B05, 91A12, 91B24 Keywords: Propositional Logic, Deduction Theorem, Herbrand Theorem, Proof by Contradiction, TU Games, Cooperative Oligopoly Games, Partition Function Approach, γ -Belief, Nash Equilibrium, Aggregation across Firms. 1 Introduction We review a common practice in the literature of game theory of applying the Deduction Theorem (Herbrand Theorem, 1930) on a purported proof by contradiction.