designBer  <-  Vectorize(   function(p, q)  {
    Q  <-  q*(1-q)/(p*(1-p))
    x  <-  1/(sqrt(Q)+1)  # ideal proportion in trial 1,  that is n/N
    x}   )
p  <-  q  <-  seq(from=0.05,  to=0.95,  by=0.05)
pq  <-  expand.grid(p, q)
z  <-   outer(p, q, FUN=designBer)
image(p, q, z, main="Proportions of runs for trial 1")
contour(p, q, z, nlevels=19, xlab="p", ylab="q", add=TRUE)

designRelEff  <-  Vectorize(  function(p, q)   {
     Q  <-  q*(1-q)/(p*(1-p))
     x  <-  1/(sqrt(Q)+1)        # ideal proportion in trial 1,  that is n/N
     eff  <-  (2 * (p*(1-p) + q*(1-q) ) ) / ( ((p*(1-p))/x)  + (q*(1-q))/(1-x) )
     1/eff
}
 )
image(p, q, zz <- outer(p, q, FUN=designRelEff), main="Relative efficyency for equal apportionment" )
contour(p, q, zz,  nlevels=19, xlab="p", ylab="q", add=TRUE)