# Blog Archives

# Topic Archive: games

# Open determinacy for games on the ordinals is stronger than ZFC

The principle of open determinacy for class games — two-player games of perfect information with plays of length ω, where the moves are chosen from a possibly proper class, such as games on the ordinals — is not provable in Zermelo-Fraenkel set theory ZFC or Gödel-Bernays set theory GBC, if these theories are consistent, because provably in ZFC there is a definable open proper class game with no definable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it implies that there is a satisfaction class for first-order truth, and indeed a transfinite tower of truth predicates TR_α for iterated truth-about-truth, relative to any class parameter. This is perhaps explained, in light of the Tarskian recursive definition of truth, by the more general fact that the principle of clopen determinacy is exactly equivalent over GBC to the principle of elementary transfinite recursion ETR over well-founded class relations. Meanwhile, the principle of open determinacy for class games is provable in the stronger theory GBC + Pi^1_1 comprehension, a proper fragment of Kelley-Morse set theory KM.

This is joint work with Victoria Gitman, with helpful participation of Thomas Johnstone.

See also related article: V. Gitman, J.D. Hamkins, Open determinacy for class games, submitted.

For further information and commentary concerning this talk, please see the related post on my blog.

# Knowledge-Theoretic Aspects of Strategic Voting

It has long been noted that a voter can sometimes achieve a preferred election outcome by misrepresenting his or her actual preferences. In fact, the classic Gibbard-Sattherthwaite Theorem shows that under very mild conditions, every voting method that is not a dictatorship is susceptible to manipulation by a single voter. One standard response to this important theorem is to note that a voter must possess information about the other voters’ preferences in order for the voter to decide to vote strategically. This seems to limit the “applicability” of the theorem. In this talk, I will survey some recent literature that aims at making this observation precise. This includes models of voting under uncertainty (about other voters’ preferences) and models that take into account how voters may response to poll information.

# Causal Decision Theory and Intrapersonal Nash Equilibria

Most philosophers today prefer ‘Causal Decision Theory’ to Bayesian or other non-Causal Decision Theories. What explains this is the fact that in certain Newcomb-like cases, only Causal theories recommend an option on which you would have done better, whatever the state of the world had been. But if so, there are cases of sequential choice in which the same difficulty arises for Causal Decision Theory. Worse: under further light assumptions the Causal Theory faces a money pump in these cases. It may be illuminating to consider rational sequential choice as an intrapersonal game between one’s stages, and if time permits I will do this. In that light the difficulty for Causal Decision Theory appears to be that it allows, but its non-causal rivals do not allow, for Nash equilibria in such games that are Pareto inefficient.