What is the axiom of quasideterminacy?

Question

This is something mentioned in the "See also" of the wikipedia page for the axiom of determinacy, but when you click on it it takes you to the page for "Determinacy" and the section for quasideterminacy has nothing in it.

Answer 1

It appears to be an extension of determinacy to settings in which the movesets themselves cannot be well-ordered: a quasistrategy is a strategy which maps positions (for the player in question) to a set of moves rather than a single move, and a quasistrategy is said to be winning if every play consistent with it is a win for the player using that strategy. Quasidetermined games are then games in which at least one player has a winning quasistrategy. Note that a quasidetermined game is not necessarily determined; the existence of a winning move set doesn't imply that one can always choose a winning move from it!

(This brief exposition was cribbed from https://groups.google.com/forum/#!msg/sci.math/GxY78XEIGq4/2pgp_6xQWBYJ , which appears to be the first good source I can find for the phrase in a web search.)

Answer 2

A quasistrategy is like a strategy, except that instead of telling what your next move is, it tells you a set of possible next moves. For instance, a quasistrategy for player II for a game on $\omega$ is a map $2^{<\omega, odd}$ to $\mathcal{P}(\omega)\setminus \emptyset$: to each play so far, the quasistrategy assigns a nonempty set of possible next moves.

Two quasistrategies played against each other do not determine a play, of course, but we can speak of a play being consistent with a pair of quasistrategies. For instance, the play $0,1,0,1,...$ is consistent with player I's quasistrategy "play an even number" and player II's quasistrategy "play an odd number."

Now, if we're playing a game on a well-ordered set - such as $\omega$ - there isn't really any difference between quasistrategies and strategies: to each quasistrategy $\Sigma$ we can associate a strategy $\hat{\Sigma}$, which is "Play the least move allowed by $\Sigma$." But, if the set of possible moves is not well-ordered, then this breaks down.

Quasideterminacy for a set $A$ means "Every game on $A$ has a winning quasistrategy:" that is, for every game on $A$, there is a quasistrategy for one player $A$ such that, for any quasistrategy for the other player $B$, every play compatible with those two strategies is a win for $A$. Note that by the previous paragraph, if $A$ is well-ordered then $$QDet(A)\iff Det(A).$$

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Note that in general, quasideterminacy principles are deeply weird! For instance, both players can have winning quasistrategies in the same game!

To see this, let $A$ be an infinite Dedekind-finite set - that is, a set with no countably infinite subset. Then consider the game where players I and II alternate playing elements of $A$, without repetition (neither player can play an element already played by one player or the other). Then "play an uplayed element" is a winning quasistrategy for both players.

This is somewhat vacuous, however: what's really going on is that, in the absence of some choiciness of $A$, there is no way to actually play two quasistrategies on $A$ against each other. In particular, there are no plays compatible with both players playing "play an unplayed element", and so both players win vacuously.