Hello,
Intro about standard two player games
Gale-Stewart games are the well-known games played by two Players $I$ and $II$, which in turn play natural numbers for infinitely many ($\omega$) steps. The "outcome" of a play is an element is in $\omega^{\omega}$. Given an objective $A\subseteq \omega^{\omega}$, Player $I$ wins a play, if the outcome in $A$. He loses (and Player $II$ wins) otherwise. A strategy for a player is a function $\sigma : \omega^{ * }\rightarrow \omega$ which, given a history of previously played moves (an element in $\omega^{*}$) tells the player how to choose the following natural number. A game with objective $A$ is determined if one of the two player has a winning strategy. The notion of determinacy, for a given (class of) set(s) $A$, is of course very important in set theory.
Imperfect information two player games
Games recently are also becoming more and more important in theoretical computer science. And partial information games are being studied as well. Let us define, here, a partial information Gale Stewart game, as a standard G.S. game, with the difference that the set of strategies available to the players is a (proper) subset $\Sigma\subseteq \omega^{ * }\rightarrow \omega$.
As an example, we might model that Players do not have memory of the past moves, by restricting the set of strategies to the set
$\Sigma=\{ \sigma : \sigma( s ) = \sigma (t)$ if $last(s)=last(t) \}$,
where $last: \omega^{ * } \rightarrow \omega$ returns the last natural number in the input sequence. These strategies are usually called positional or memoryless strategies.
It is an interesting question to see what classes of sets are determined under positional strategies. But in general I suspect it is interesting to study determinacy under many other sets $\Sigma$ of strategies, in other words, it should be interesting enough to study partial information games.
The case of Blackwell's games
A famous class of imperfect information G.S. games is the class of Blackwell's games. It can be described as follows: the game is played by two players, $I$ and $II$, which at each turn, play independently and at the same time two naturals $a$ and $b$ chosen from a finite set $ K=\{0,\dots,k\}$ for some $k\in\mathbb{N}$. A play takes $\omega$ steps. The result of a play is an element in $(K\times K)^{\omega}$. Given an objective $W\subseteq (K\times K)^{\omega}$, Player $I$ wins if the produced play is in $W$. The set of strategies for the two players in a Blackwell game, is given by the functions $\sigma: (K\times K)^{*} \rightarrow K$. One is often interested in mixed strategies, i.e. randomized strategies; one can model randomized strategies, abstractly, as probability measures over the Borel-sigma algebra on the space of the strategies).
Note that Blackwell's games can be defined as imperfect information G.S. games (in the sense defined above), if desired.
Blackwell's games have been studied by set theorists. For instance D. Martin proved that (in ZFC) all Blackwell's game (with Borel objectives) are determined under mixed strategies. (here determinacy has a slightly different meaning than usual).
Questions
So after this discussion, my questions are:
Q1: Do imperfect informations G.S. games play an important (or any) role in Set theory? I'm not aware of particular use of them. For instance in Jach's Set theory, I think there is no mention at all about Blackwell's games. Could you point to some relevant literature if your answer is positive? Or even any "feeling" about their potential use?
Q2: In particular, do Blackwell's game play any important (or any) role in Set theory? Just the fact that Martin's worked on this, suggests (to me) that they actually might have some role.
An application, of Blackwell's games (or anyway similar concepts) i'm aware of is in Logic, and it can be found in Hintikka's works on Independence friendly logics, where Blackwell's games are used to give "game semantics" to this/these logic(s).
Thank you in advance for any answer!
Matteo Mio
