**A**n almost straightforward riddle from The Riddler involving a triple gambler’s ruin: *Dawn competes against three players Alessandra, Berenike, and Chinue, with probabilities of winning one round ¾, ½, and ¼, respectively, until the cumulated score reaches ±15, ±30, and ±45, for the first, second, and third games. What is Dawn’s optimal sequence of adversaries?*

First, a brute force R simulation shows that the optimal ordering is to play the three adversaries first weakest, third strongest and middle fair:

ord=function(p){
z=2*(runif(1)<p[1])-1
while(abs(z)<15)z=z+2*(runif(1)<p[1])-1
y=2*(runif(1)<p[2])-1
while(abs(z+y)<30)y=y+2*(runif(1)<p[2])-1
x=2*(runif(1)<p[3])-1
while(abs(z+y+x)<45)x=x+2*(runif(1)<p[3])-1
return(x+y+z>0)}
mcord=function(p,T=1e2){
for(t in 1:T)F=F+ord(p)
return(F/T)}
comp=function(T=1e2){
return(c(mcord(c(.5,.55,.45),t),
#mcord(c(.5,.45,.55),t),#1-above
mcord(c(.55,.5,.45),t),
#mcord(c(.45,.5,.55),t),#1-above
mcord(c(.55,.45,.5),t)
#mcord(c(.45,.55,.5),t)))#1-above
))}

where I used probabilities closer to *½* to avoid estimated probabilities equal to one.

> comp(1e3)
[1] 0.051 0.038 0.183

(and I eliminated the three other probabilities by sheer symmetry). Second, checking in Feller’s bible (Vol. 1, XIV.3) for the gambler’s ruin probability, a simple comparison of the six orderings confirms this simulation.

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