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

## TL;DR

This paper critically examines the misuse of the Deduction Theorem in game theory, highlighting logical flaws in common proof practices and providing an industrial example where incorrect application leads to contradictions.

## Contribution

It identifies and explains the problematic assumptions in applying the Deduction Theorem within game theory proofs, emphasizing the need for proper proof validation.

## Key findings

- The Deduction Theorem is often misapplied in game theory proofs.
- Incorrect application can lead to false conclusions and contradictions.
- An industrial example demonstrates the practical impact of these logical errors.

## Abstract

We consider the Deduction Theorem 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 $\phi \vdash \varphi$, then we also have a proof of $\phi \Rightarrow \varphi$. Hence, the proof of $\phi \Rightarrow \varphi$ is deduced from a previously known statement. However, we argue that one has to manage to establish that a proof exists for the clauses $\phi$ and $\varphi$, i.e., they are known true statements in order to show that $\phi \vdash \varphi$ is provable, and that therefore $\phi \Rightarrow \varphi$ is provable as well. Thus, we are not allowed to assume that the clause $\phi$ or $\varphi$ is a true statement. This 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.