Monotonic sequence games

Abstract:
From a deck of cards labelled with the integers from 1 through $n$ two players take turns choosing a card and adding it to the right hand end of a row of cards. The game ends when there is a subsequence of $a$ cards in the row whose values form an ascending sequence, or of $d$ cards whose values form a descending sequence.

I’ll discuss the “long run” (large $n$ for fixed $a$ and $d$) behaviour of this game and also some variants (e.g., when the deck of cards is $\mathbb{Q}$.)