Question
You and a friend are each given a die. Both of you are allowed to independently determine how many sides each of your individual dice have. You may select $1,2,3,4,5,$ or $6$ sides. The sides are labeled $1-k$ if you select $k$ sides. Each side is equally likely to appear among the number of sides you select. Then, both of you roll your dice. If the sum of the two upfaces is at most $7$, you receive a payout equal to the upface of your die from a third party. Otherwise, neither of you receive anything. Assume both players play optimally. Let $s$ be the number of sides you select and $p$ be the expected payout. Find $sp$.