Re: Four more "oldies"

From	Mike Terry <news.dead.person.stones@darjeeling.plus.com>
Newsgroups	rec.puzzles
Subject	Re: Four more "oldies"
Date	2026-08-27 00:48 +0100
Organization	A noiseless patient Spider
Message-ID	<116ntvu$1226s$1@dont-email.me> (permalink)
References	(1 earlier) <116hnp2$2vnov$1@dont-email.me> <1787591053-4353@newsgrouper.org> <116imdr$3acor$1@dont-email.me> <1787728084-4353@newsgrouper.org> <116non7$10kbv$1@dont-email.me>

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On 26/08/2026 23:18, Mike Terry wrote:
> On 26/08/2026 08:08, James Dow Allen wrote:
>>
>> Mike Terry <news.dead.person.stones@darjeeling.plus.com> posted:
>>
>>> On 24/08/2026 18:04, James Dow Allen wrote:
>>>>
>>>> Mike Terry <news.dead.person.stones@darjeeling.plus.com> posted:
>>>>
>>>>> On 23/08/2026 18:44, James Dow Allen wrote:
>>>>>>
>>>>>> (2)  ...
>>>>>> With optimal strategy, what is the probability that the team wins?
>>>>>> This is tricky. First try it with 3 players.
>>>>>> Or two with 'pass' not an option.
>>>>
>>>> "Tricky" in an almost perverse way.  Do try it first with only 3 players.
>>>
>>> For 3 players I see similar sorts of strategies, except they are a little simpler.
>>>
>>> Specifically, players can effectively bet on a 2-1 split which occurs with probability 3/4, and in
>>> doing so the team wins with that probability.  E.g. the strategy can be:
>>> -   if a player sees team mates with hats RR then choose G
>>> -   if a player sees team mates with hats GG then choose R
>>> -   (otherwise pass)
>>
>> Beautiful!
>>
>>>
>>> But I don't feel that's helped me with the 4-player game.  (I suppose that I could write a program
>>> to enumerate what I consider plausible strategies, but I'm doubting I'm going to get anything better
>>> than 3/4 for the 3-player game...)
>>
>> I assume you DO have a 3/4 solution for the 4-player game?
>> Gur "gevpx" vf gung gurer vf ab jnl gb vzcebir ba gung.
> 
> At the time I previously posted I only had the 1/2 solution I'd previously described.
> 
> Since then I've written a (pretty naive) program to search for the best strategies, and for the 
> 3-player game it confirms the best we can do gives a 3/4 chance for the team to win.
> 
> For the 4-player game I doubt I will be able to run the program to completion - there are just too 
> many potential strategies to enumerate. :(  But as luck would have it, it's found a handful of 3/4 
> strategies near the beginning of its search tree - these strategies have one of the players always 
> passing regardless of what hats that player sees!
> 
> But these results don't really improve my understanding.

I hadn't twigged before, but some of these strategies are just the 3-player strategies in disguise, 
with the 4th player ignored (and 4th player always passing).  Such a strategy can always be used 
with any greater number of players, and yields the same win-rate as the 3-player game strategy.

We can conclude:
a) however many players (>3) there is always a fairly straight-forward strategy
    borrowed from the 3-player game giving a 3/4 win-rate, and
b) as the number of players increases, the best strategy win rate cannot go down...

Mike.

Thread

Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-23 17:44 +0000
  Re: Four more "oldies" David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-08-24 07:40 +0000
    Re: Four more "oldies" David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-08-25 07:27 +0000
      Re: Four more "oldies" ram@zedat.fu-berlin.de (Stefan Ram) - 2026-08-25 12:33 +0000
  Re: Four more "oldies" richard@cogsci.ed.ac.uk (Richard Tobin) - 2026-08-24 09:45 +0000
    Re: Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-24 10:33 +0000
    Re: Four more "oldies" Phil Carmody <pc+usenet@asdf.org> - 2026-08-27 22:45 +0300
  Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-24 16:25 +0100
    Re: Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-24 17:04 +0000
      Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-25 00:41 +0100
      Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-25 01:08 +0100
        Re: Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-26 07:08 +0000
          Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-26 23:18 +0100
            Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-27 00:48 +0100
            Re: Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-27 10:42 +0000
              Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-28 04:55 +0100
  Re: Four more "oldies" Charlie Roberts <croberts@gmail.com> - 2026-08-24 13:27 -0400
  Re: Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-29 00:04 +0000

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