Dear Nisan,
I just found your post via a search engine. I wanted to quickly follow up on your last paragraph, as I have designed and recently published an equilibrium concept that extends superrationality to non-symmetric games (also non-zero-sum). Counterfactuals are at the core of the reasoning (making it non-Nashian in essence), and outcomes are always unique and Pareto-optimal.
I thought that this might be of interest to you? If so, here are the links:
https://www.sciencedirect.com/science/article/abs/pii/S0022249620300183
(public version of the accepted manuscript on https://arxiv.org/abs/1712.05723 )
With colleagues of mine, we also previously published a similar equilibrium concept for games in extensive form (trees). Likewise, it is always unique and Pareto-optimal, but it also always exists. In the extensive form, there is the additional issue of Grandfather’s paradoxes and preemption.
https://arxiv.org/abs/1409.6172
And more recently, I found a way to generalize it to any positions in Minkowski spacetime (subsuming all of the above):
https://arxiv.org/abs/1905.04196
Kind regards and have a nice day,
Ghislain
Thank you for your comment, Vladimir_Nesov.
It is indeed correct that “the result be BE” is a false proposition in the real world. In fact, this is the reason why they are called counterfactuals and why the subjunctive tense (“would have”) is used.
Nashian game theory is based on the indicative tense, for example common knowledge is all based on the indicative tense (A knows that B knows that A knows etc). Semantically, knowledge can be modelled with set inclusion in Kripke semantics: A knows P if the set of accessible worlds (i.e., compatible with A’s actual knowledge) is included in the set of possible worlds in which P is true (and we can canonically identify this set with P, i.e., conflate a logical proposition with the set of worlds in which it is true).
What is important to understand is that counterfactuals can be rigorously captured and anchored into the actual world. This has been researched in particular by Lewis and Stalnaker in the 1960s and 1970s. A statement such as “should the result be BE, then the row player would have known it,” is mathematically modelled as a counterfactual implication P>Q with P=”the result is BE”, Q=”the row player knows that the result is BE” where P and Q are predicates on all possible worlds (including counterfactual worlds) that can be understood as subsets of the set of all possible worlds.
Q=Krow(P) is itself a compound predicate, because it is a knowledge statement: Q is true in a specific world ω if the set of worlds accessible to the row player from ω is included in P.
Counterfactuals imply some sort of distance between possible worlds: the more different they are, the farther apart they are. Given P and Q, the predicate P>Q is defined as true in a world ω if, in the closest world to ω in which P is true, denoted fP(ω), Q is also true.
So all in all, the statement is formally modelled as the predicate P>Krow(P) being true in the actual world and is to be distinguished from the logical implication P⟹Krow(P), which is trivially true in the actual world if P is false. But the counterfactual implication P>Krow(P) on the other hand is not trivial and can potentially be false in the actual world even when P is false.
I hope it helps clarify! For more details, the 1972 book by Lewis “Counterfactuals” is a very interesting read.