I've been struggling with proving a conjecture concerning order statistics of Brownian motions for a while. The conjecture I'm looking to prove is the following: (I have run Monte Carlo simulations that confirm the conjecture.)

Let $(q_1(t), …, q_n(t))$ be a vector of independent Brownian motions with $q_i(0) = 0$ for all $i$. Let $T > 0$. Each $q_i$ represents an object in a game, and each of $J < n$ players in the game can pick exactly one object $i$ at any time $t \in [0, T]$. Whenever a player picks an object, he picks the one that currently has the highest value amongst those that have not been picked before. A player's payoff is the value of the object at the time he picks it. Let $x_j(t_j; t_1, …, t_{j-1})$ denote the expected payoff (expectation taken at time 0) from picking at time $t_j$, given that players $1, …, j-1$ will pick at times $t_1, …, t_{j-1}$. I want to show that $x_j(t_j; t_1, …, t_{j-1})$ is decreasing in $t_1, …, t_{j-1}$.

I will need this for an economic theory paper I'm writing, and would be very grateful for any help!

Some things that I already thought through:

- I have already shown that $x_j(t_j; t_1, …, t_{j-1})$ is increasing in $t$ and jointly continuous in all arguments.
- The case $j=2$ is quite tractable, and for the case $j = 3$ I have the intuition on how to prove it. That intuition, however, doesn't extend to the general case.