#N.Tideman's statistical election-model for computing Condorcet Winner existence probability. #Maple program by Warren D. Smith, December 2009. #Tideman recommends these parameter values: # a1 := -0.532 +- 0.028; a2 := -0.789 +- 0.055; a3 := -2.486 +- 0.010; a4 := -1.281 +- 0.011; #RegHalfBeta := (alpha, beta) -> # integrate( u^(alpha-1) * (1-u)^(beta-1) , u=0..1/2 ) / # integrate( u^(alpha-1) * (1-u)^(beta-1) , u=0..1 ); #which when you do the integrals becomes the following: #RegHalfBeta1 is one close form from HOMF 26.5.23; #RegHalfBeta2 is a slightly faster close form, according to my timing experiments: # k=1..20 time=17.84 for RegHalfBeta1, time=15.98 for RegHalfBeta2 # k=1..30 time=143.03 for RegHalfBeta1, time=133.85 for RegHalfBeta2 # k=40,50,60,70 takes about 15 minutes run time using RegHalfBeta2. # k=30 value=0.7490373553 for RegHalfBeta1, value=0.7490373550 for RegHalfBeta2 # k=10 value=0.9285237812 for RegHalfBeta1, value=0.9285237742 for RegHalfBeta2 # k=3 value=0.9882900824 for RegHalfBeta1, value=0.9882900822 for RegHalfBeta2 RegHalfBeta1 := (alpha, beta) -> 2^(-alpha)/(alpha*Beta(alpha, beta)) * hypergeom([alpha, 1-beta], [alpha+1], 1/2); RegHalfBeta2 := (alpha, beta) -> 1- 2^(-beta)/(beta*Beta(beta, alpha)) * hypergeom([beta, 1-alpha], [beta+1], 1/2); ProbWin := proc(i, j, k, a1, a2, a3, a4) local x, y, r, n, p, alpha, beta; description "probability candidate i loses to j, where 0