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dicex.gms : Non-transitive Dice Design - Enhanced

Robert A Bosh, suggested in a recent Optima Newsletter an extension
to the original DICE (SEQ=176) model: Each number on a face can appear
only once. This has been formulated by adding the 0/1 variable fmap
and an appropriate assignment constraint. Note the importance of the
'=e=' type in equation eq1 and the power of eq3.

Probabilistic dice - an example of a non-transitive relation.
We want to design a set of dice with an integer number on each face
such that on average dice1 beats dice2, and dice2 on average beats
dice3 etc, but diceN has to beat dice1.
References:
Large Model of Type: MIP
$title Non-transitive Dice Design - Enhanced (DICEX,SEQ=272) $Ontext Robert A Bosh, suggested in a recent Optima Newsletter an extension to the original DICE (SEQ=176) model: Each number on a face can appear only once. This has been formulated by adding the 0/1 variable fmap and an appropriate assignment constraint. Note the importance of the '=e=' type in equation eq1 and the power of eq3. Probabilistic dice - an example of a non-transitive relation. We want to design a set of dice with an integer number on each face such that on average dice1 beats dice2, and dice2 on average beats dice3 etc, but diceN has to beat dice1. Martin Gardner, The Colossal Book of Mathematics, WW Norton, New York, NY, 2001. Robert A Bosh, Mindsharpener, Optima, MP Society Newsletter, Vol 70, June 2003, page 8-9 Robert A Bosh, Monochromatic Squares, Optima, MP Society Newsletter, Vol 71, March 2004, page 6-7 $Offtext sets f faces on a dice / face1*face6 / d number of dice / dice1*dice3 / ; parameters wn min wins needed fnum(d,f) assigned face values big big m ; wn = floor(0.5 * sqr(card(f))) + 1; fnum(d,f) = card(f)*(ord(d)-1)+ord(f); big = card(d)*card(f) - 1; alias(f,fp),(d,dp); variables wnx number of wins fval(d,f) value of dice - will be integer comp(d,f,fp) one if f beats fp fmap(d,f,dp,fp) assigns values to dice faces binary variable comp, fmap; fval.lo(d,f) = 1; fval.up(d,f) = card(d)*card(f); fval.fx(d,f)$(ord(d)=1 and ord(f)=1) = 1; equation eq1(d) count the wins eq2(d,f,fp) definition of non-transitive relation eq3(d,f) different face values for a single dice eq4(d,f) assign values to faces; eq1(d) .. sum((f,fp), comp(d,f,fp)) =e= wnx; eq2(d,f,fp).. fval(d,f) + big*(1-comp(d,f,fp)) =g= fval(d++1,fp) + 1; eq3(d,f-1) .. fval(d,f-1) + 1 =l= fval(d,f); eq4(d,f) .. sum((dp,fp), fnum(dp,fp)*fmap(d,f,dp,fp)) =e= fval(d,f); model dice1 each dice with unique faces / eq1 ,eq2, eq3 / dice2 all faces are unique / eq1, eq2, eq3, eq4 /; $if set nosolve $exit option iterlim = 1000000, reslim = 50, optcr = 0.0, optca = 0.99; solve dice2 using mip maximizing wnx; option fval:0; display wn,fval.l; parameter rep1 Chance of winning against next; rep1(d,f) = 100*sum(fp, comp.l(d,f,fp)) / card(f); rep1(d,'chance') = sum(f, rep1(d,f))/card(f); option rep1:0; display rep1;