The Incompleteness of Theories of Games

Axiomatics, the Social Sciences, and the Gödel Phenomenon: A Toolkit

“…Lewis and Inagaki (1991b) pointed out that our results entail the incompleteness of the theory of Hamiltonian models in economics. They also entail the incompleteness of the theory of Arrow-Debreu equilibria and (what is at first sight surprising) the incompleteness of the theory of finite games with Nash equilibria (see da Costa and Doria, 1994a;Tsuji et al, 1998). Those two last questions are discussed below in Section 17.…”

“…Yet the stronger axioms may also be debatable on philosophical grounds, so that the proof of a desired fact from the enriched system eventually turns out to be technically correct but philosophically (and perhaps empirically) doubtful. For details see da Costa and Doria (1994a) and Tsuji et al (1998).…”

Can economic systems be seen as computing devices?

“…We use the Tsuji et al (1998) incompleteness theorem for game theory in the light of the results by Lewis and Velupillai in order to try to understand whether it makes sense to interprete actual economic systems and processes as computing devices.…”

“…Less technically, even if we can describe a game by some recursive function, its equilibria may be noncomputable. This is a remarkable result that inspired the work by Tsuji et al (1998). Tools used in Lewis' argument are taken exclusively from within recursion theory, in contrast to the Tsuji and the da Costa and Doria results, which use a different approach and ask for weaker conditions on the game.…”

“…Velupillai. We then sketch the mathematical background for the Tsuji et al (Tsuji, M., da Costa, N.C.A., Doria, F.A., 1998. The incompleteness of theories of games.…”