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