There is a game with $10$ cards(cards are number $1$ to $10$). They are distributed to $2$ players randomly. Player who has the highest sum of card values wins the game.
Now, before the cards are dealt, player A will say a number. This number is the number of cards he will RANDOMLY pull from player B, and then return the cards of his choice to player B.(The cards returned cannot be the cards he has taken from player B).
So, what is the number of cards player A should pull from player B to maximize his chances of winning the game?
If Player A says $0$, his winning probability remains at $0.5$
If Player A says $5$, he takes all of B's card, and give all of his cards to B. His winning probability remains at $0.5$ again
I assume that there is a unimodal winning probability with the peak at the centre.
However, I am facing difficulty in which one of these $2$ will be larger and by how much?
Part $2$ - Can this be generalised to '$n$' cards?
I wrote the following Matlab routine, to find answers for 2 cards up to 20 cards.
The mean of the sum, with 2n cards, and swapped k of them, is $$\frac{n(2n+1)}2+\frac{2n+1}{2n+2}k(n-k)$$ The variance of the sum is $$-\frac{(2n+1)(7n+4)}{12(n+1)^2(n+2)}k^2(n-k)^2 -\frac{(2n+1)(3n+1)}{6(n+1)(n+2)}k(n-k) +\frac{n^2(2n+1)}{12}$$ I then used the Gaussian approximation, assuming the mean and variance with enough cards would look normally distributed. I plotted the Matlab formula $$1/2+(1/2)erf((Mean-250250)./sqrt(2*Var))$$ on the same graph as simulations and got this result. Simulations were 100000 deals of 1000 cards (500 each), and each deal was calculated for all values of k.
The following was the difference, after 100000, between swapping i cards and swapping 500-i cards. Note, the same deals were used for different numbers of swapped cards, so there is a strong correlation between i and i+1. The difference between i and 500-i was less than one percent.
The nearest integers to the Gaussian estimates are given below, and can be compared with exact counts given above: