A Game of Probability

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ironman353
Posts: 257
Joined: Thu Jan 16, 2014 2:29 pm

A Game of Probability

Post by ironman353 »

I need some help to solve this problem.

$A$ and $B$ are two players, each have exactly one turn. $A$ goes first. $A$ keeps on choosing a random number uniformly distributed over $(0,1)$ and add the values. If at one point it exceeds $1$, $A$ loses. If $A$ thinks his cumulative sum is very close to $1$, hence there is a risk of losing, he stops. Then $B$ starts the same process and add the values separately. If at one point $B$ exceeds $A$'s sum and still below $1$, he wins. What is the optimal strategy for $A$ to stop adding and what is the probability of winning in that case ($B$ knows the value $A$ stopped at)?

From simulation It appears that the optimal threshold of $A$'s cumulative sum is approximately $0.5772$, which is very close to the Euler-Mascheroni constant $\gamma$.
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