The probability of losing half by Kelly's criterion

Question

According to "Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos And Wall Street", the probability of losing half initial money at some point using Kelly's criterion is 50%. This is written without any proof. How to prove this?

If situation is too complex, I want to know the special case, where the winning probability is 50% and return is 200%, the losing probability is 50% and return is 50%. The expected return is 125% > 0. So this gamble offer good odds. According to Kelly's criterion, the optimal strategy is to bet half of total money.

In this case, what's the probability of losing half initial money at some point?

Answer 1

As herb steinberg commented the expected return for a bet of $1$ is clearly $\frac{2+0.5}{2}=1.25$, i.e. an expected profit of $0.25$.

It seems in this case, the probability of losing half or more of your original bankroll is nearer $0.44$ than $0.5$ as suggested by these $100000$ simulations of $1000$ successive Kelly criterion bets

winmultiplier  <- 2
losemultiplier <- 0.5
winprob        <- 0.5
loseprob       <- 1 - winprob
winmultiplier*winprob + losemultiplier*loseprob # expected return for bet 1
kellyproportion <- (1-losemultiplier)/winprob - loseprob/(winmultiplier-1)
kellyproportion # see Kelly criterion value

logwin  <- log(winmultiplier *kellyproportion + 1-kellyproportion)
loglose <- log(losemultiplier*kellyproportion + 1-kellyproportion)

samplerun <- function(n){
  logvalues <- sample(c(logwin,loglose), size=n, prob=c(winprob,loseprob),
                      replace=TRUE)
  values <- exp(cumsum(c(0,logvalues)))
  c(values[n+1], min(values)) # final and worst positions
  }   

set.seed(2021)
sims <- replicate(10^5, samplerun(1000))
median(sims[1,])      # median simulated overall return
# 3.769305e+25
min(sims[1,])         # worst simulated  overall return
# 63854.5
max(sims[1,])         # best simulated   overall return
# 4.450009e+46
mean(sims[2,] == 1)   # proportion never overall loss
# 0.21114
mean(sims[2,] <= 1/2) # proportion worst position half or lower
# 0.43633

It is possible to do very badly early on with the Kelly criterion, even if it is later recovered. In the simulation the worst worst case lost $99.999934\%$ of the initial bankroll, before recovering to many thousands times the initial bankroll. The overall distribution of worst cases is illustrated in this graph.

min(sims[2,])         # worst simulated worst position 
# 6.649331e-07
plot.ecdf(sims[2,])