It's a two-part problem.
(1) Figure out Player 2's optimal move for every move that Player 1 can make.
(2) Figure out Player 1's optimal move based on what he knows Player 2 will do.
This is all assuming each player is rational and seeks to maximize his expected winnings.
Let \(f(x_1)\) be the value that Player 2 should play given that Player 1 played \(x_1\). It's easy to see that \(f(x_1)=x_1+1\) for \(x_1\in \{1,...,14\}\) and \(f(x_1)=x_1-1\) for \(x_1\in \{15,...,20\}\). (To maximize his expected earnings, Player 2 must pick one greater or one less than Player 1, in the direction in which the sum of all numbers up to the end is greater.)
This shows that under optimal play, Player 1's move determines the expected outcome of the game. If Player 1 plays 14, he knows Player 2 will be forced to play 15 to maximize his earnings, and thus Player 1's expected earnings will be \( (1+\cdots+14)/20=105/20)\). If Player 1 plays 15, then he knows Player 2 will play 14, and his expected earnings are \( (15+\cdots+20)/20=105/20\). Under these circumstances, Player 2's expected earnings are in both cases 105/20.
So the first player should play 14 or 15, and it doesn't seem to matter who plays first.