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A287150 Fairest turn sequence for 3 players where the probability of a win for a player on his turn approaches 0.

%I A287150 #30 Jun 05 2017 23:42:21
%S A287150 0,1,2,2,1,0,2,1,0,0,1,2,1,2,0,0,2,1,0,2,1,1,2,0,2,1,0,0,1,2,1,0,2,2,
%T A287150 0,1,0,2,1,1,2,0,2,1,0,0,1,2,1,0,2,2,0,1,0,2,1,1,2,0,1,2,0,0,2,1,0,2,
%U A287150 1,1,2,0,2,1,0,0,1,2,0,1,2,2,1,0,2,1,0,0,1,2,1,0,2,2,0,1,2,0,1,1,0,2,1,0,2,2,0,1,0,2,1,1,2,0,1,2,0,0,2,1
%N A287150 Fairest turn sequence for 3 players where the probability of a win for a player on his turn approaches 0.
%C A287150 0, 1, and 2 are three archers of equal skill, but which have infinitesimally low chances of hitting a target. Still, they try: taking turns until one of them hits: 0 goes first, if 0 misses then 1 will try (these are the first two terms). Subsequent terms are found by giving a turn to the player who has the lowest probability of winning so far.
%C A287150 Joshua Cooper and Aaron Dutle showed that for two players this is the Thue-Morse sequence (A010060).
%C A287150 The following observations are made and verified to be true for the first 5000 terms:
%C A287150 - Every 3 terms completes a new round with each player going once per round.
%C A287150 - Every other round is the same as the last but in reverse order.
%C A287150 - The 48th-101st terms form a block of 54 terms which are repeated from then on.
%H A287150 Joshua Cooper and Aaron Dutle, <a href="https://doi.org/10.4169/amer.math.monthly.120.05.441">Greedy Galois Games</a>, Amer. Math. Monthly, 120 (2013), 441-451, <a href="https://arxiv.org/abs/1110.1137">arXiv:1110.1137 [math.CO]</a>.
%H A287150 Daniel Hug, <a href="http://output.jsbin.com/kilebod/">Generate fairest turn sequence for n players</a> (web app written in JavaScript)
%t A287150 n=3; a=Range[n]; pw=(1-p)^(Range[n]-1)*p;
%t A287150 Do[
%t A287150 next = First[Select[Range[n], And@@NonNegative/@Limit[Sign[pw-pw[[#]]], p->0]&, 1]];
%t A287150 AppendTo[a, next]; pw[[next]]+=p*(1-p)^k
%t A287150 , {k, n, 50}];
%t A287150 a-1
%t A287150 (* _Andrey Zabolotskiy_, Jun 05 2017 *)
%Y A287150 Cf. A010060.
%K A287150 nonn
%O A287150 0,3
%A A287150 _Daniel Hug_, May 20 2017