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 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 and are the numbers played, and that:

• Player I wins iff
• Player II wins iff and
• Player III wins if

(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.