Deduction Theorem: The Problematic Nature of Common Practice in Game Theory
This work addresses foundational logical errors in game theory, which could affect theoretical models and applications, but it is incremental as it focuses on correcting existing practices rather than introducing new methods.
The paper critiques the application of the Deduction Theorem in game theory, arguing that it is often misused in proofs by contradiction, leading to incorrect conclusions, as illustrated by an example from industrial cooperation that contradicts known aggregation issues.
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 $φ\vdash \varphi$, then we also have a proof of $φ\Rightarrow \varphi$. Hence, the proof of $φ\Rightarrow \varphi$ is deduced from a previously known statement. However, we argue that one has to manage to establish that a proof exists for the clauses $φ$ and $\varphi$, i.e., they are known true statements in order to show that $φ\vdash \varphi$ is provable, and that therefore $φ\Rightarrow \varphi$ is provable as well. Thus, we are not allowed to assume that the clause $φ$ 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.