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.

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2 Responses to Three-player perfect information games are usually undetermined

  1. Daisuke Ikegami says:

    All I know about it is a paper by Benedikt on infinite games with more than two players:

    http://www.illc.uva.nl/Publications/ResearchReports/PP-2003-19.text.pdf

    It will be appear in Journal of Applied Logic. But I suspect that this is not the oldest reference for the undeterminacy.

  2. Thanks! The paper looks interesting, and I didn’t know anything about this area.

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