This game of two players has public parameters an integer $n\ge2$, and a probability $p$ with $1/n<p\le1$. E.g. $n=4$, $p=1/3$.

In the first phase of a game, a player secretly decides $n$ probabilities $p_i$ with $0\le p_i\le1$ for indexes $i\in[0,n)$, subject to the constraint $n\,p=\sum p_i$. It follows at least two $p_i$ are non-zero.

Then the other player repeatedly chooses an index $i\in[0,n)$ and has a hit for $i$ with probability $p_i$ (thus hits with probability $p$ if choosing $i$ uniformly at random).

A game stops after hits for two distinct $i$. The winner (if any) is the player who needed the least index choices for this, in two games with the roles reversed.

How to play in each of the two phases in order to maximize probability of winning ?

Under optimal strategy, what's the probability (or a tight lower bound for large $n$ or/and small $p$) that a player stops within $c$ choices, as a function of $n$, $p$, $c$ ?

Note: the strategies and probabilities asked are for the first of the two games, and subsidiarily for the second one should that need any adjustment according to the outcome of the first game, which is known by both players. If that matters, add that coin toss decides which player decides who guesses first, and tell how to make that decision.

This game of two players has public parameters an integer $n\ge2$, and a probability $p$ with $1/n<p\le1$. E.g. $n=4$, $p=1/3$.

In the first phase of a game, a player secretly decides $n$ probabilities $p_i$ with $0\le p_i\le1$ for indexes $i\in[0,n)$, subject to the constraint $n\,p=\sum p_i$. It follows at least two $p_i$ are non-zero.

Then the other player repeatedly chooses an index $i\in[0,n)$ and has a hit for $i$ with probability $p_i$ (thus hits with probability $p$ if choosing $i$ uniformly at random).

A game stops after hits for two distinct $i$. The winner (if any) is the player who needed the least index choices for this, in two games with the roles reversed.

How to play in each of the two phases in order to maximize probability of winning ?

Under optimal strategy, what's the probability (or a tight lower bound for large $n$ or/and small $p$) that a player stops within $c$ choices, as a function of $n$, $p$, $c$ ?

Note: the strategies and probabilities asked are for the first of the two games, and subsidiarily for the second one should that need any adjustment according to the outcome of the first game, which is known by both players. If that matters, add that coin toss determines which player decides who guesses first, and tell how to make that decision.

This game of two players has public parameters an integer $n\ge2$, and a probability $p$ with $1/n<p\le1$. E.g. $n=8$, $p=1/3$.

In the first phase of a game, a player secretly decides $n$ probabilities $p_i$ with $0\le p_i\le1$ for indexes $i\in[0,n)$, subject to the constraint $n\,p=\sum p_i$. It follows at least two $p_i$ are non-zero.

Then the other player repeatedly chooses an index $i\in[0,n)$ and has a hit for $i$ with probability $p_i$ (thus hits with probability $p$ if choosing $i$ uniformly at random).

A game stops after hits for two distinct $i$. The winner (if any) is the player who needed the least index choices for this, in two games with the roles reversed.

How to play in each of the two phases in order to maximize probability of winning ?

Under optimal strategy, what's the probability (or a tight lower bound for large $n$ or/and small $p$) that a player stops within $c$ choices, as a function of $n$, $p$, $c$ ?

This game of two players has public parameters an integer $n\ge2$, and a probability $p$ with $1/n<p\le1$. E.g. $n=4$, $p=1/3$.

In the first phase of a game, a player secretly decides $n$ probabilities $p_i$ with $0\le p_i\le1$ for indexes $i\in[0,n)$, subject to the constraint $n\,p=\sum p_i$. It follows at least two $p_i$ are non-zero.

Then the other player repeatedly chooses an index $i\in[0,n)$ and has a hit for $i$ with probability $p_i$ (thus hits with probability $p$ if choosing $i$ uniformly at random).

A game stops after hits for two distinct $i$. The winner (if any) is the player who needed the least index choices for this, in two games with the roles reversed.

How to play in each of the two phases in order to maximize probability of winning ?

Under optimal strategy, what's the probability (or a tight lower bound for large $n$ or/and small $p$) that a player stops within $c$ choices, as a function of $n$, $p$, $c$ ?

Note: the strategies and probabilities asked are for the first of the two games, and subsidiarily for the second one should that need any adjustment according to the outcome of the first game, which is known by both players. If that matters, add that coin toss decides which player decides who guesses first, and tell how to make that decision.