Three-player perfect information games are usually undetermined

Recently, I gave the talk Undeterminacy and choice (Indeterminación y elección), at the XVII Colombian Mathematical Congress, in Cali. Slides can be found at my talks page.

The talk addressed the results of my recent paper on ${\sf AD}^+$ with Richard Ketchersid, mentioned in my previous entry, and some extensions, about which I expect to be posting soon. Afterwards, somebody asked me how much of the theory of determinacy can be extended to three-player (or more) perfect information games. Not much.

The following easy example was suggested by Richard Ketchersid: There is an undetermined one-move game where players I, II, III play 0 or 1, with I playing first, then II, and finally III. To see this, say that $n_I,$ $n_{II},$ and $n_{III}$ are the numbers played, and that:

• Player I wins iff $n_{II} \neq n_{III}.$
• Player II wins iff $n_{II} \neq n_I$ and $n_{II}= n_{III}.$
• Player III wins if $n_I = n_{II}=n_{III}.$

(One may think of this game as a perfect information version of paper-rock-scissors.) I imagine this observation is ancient, and would be grateful for a reference.