# Concerning an adversarial version of the Last-Success-Problem

## TL;DR

This paper analyzes an adversarial variation of the Last-Success-Problem involving two players who sequentially observe Bernoulli variables and decide whether to continue or pass, determining optimal strategies and winning probabilities.

## Contribution

It introduces an adversarial version of the Last-Success-Problem, deriving optimal strategies and characterizing winning probabilities under different parameter settings.

## Key findings

- Optimal strategies for both players are established.
- Probability of Player A winning decreases with increasing n for uniform parameters.
- As n grows, Player A's winning probability approaches approximately 0.4323.

## Abstract

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, $\mathbf{U}[0,1]$