Mathematics > Probability
[Submitted on 13 Dec 2018 (v1), last revised 12 Jan 2019 (this version, v3)]
Title:Concerning an adversarial version of the Last-Success-Problem
View PDFAbstract:There are $n$ independent Bernoulli random variables with parameters $p_i$ that are observed sequentially. Two players, A and B, act in turns starting with player A. Each player has the possibility on his turn, when $I_k=1$, to choose whether to continue with his turn or to pass his turn on to his opponent for observation of the variable $I_{k+1}$. If $I_k=0$, the player must necessarily to continue with his turn. After observing the last variable, the player whose turn it is wins if $I_n=1$, and loses otherwise. We determine the optimal strategy for the player whose turn it is and establish the necessary and sufficient condition for player A to have a greater probability of winning than player B. We find that, in the case of $n$ Bernoulli random variables with parameters $1/n$, the probability of player A winning is decreasing with $n$ towards its limit $\frac{1}{2} - \frac{1}{2\,e^2}=0.4323323...$. We also study the game when the parameters are the results of uniform random variables, $\mathbf{U}[0,1]$
Submission history
From: Jose Maria Grau [view email][v1] Thu, 13 Dec 2018 12:27:19 UTC (8 KB)
[v2] Wed, 9 Jan 2019 07:01:42 UTC (8 KB)
[v3] Sat, 12 Jan 2019 19:58:13 UTC (8 KB)
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