Unless I'm missing something, I reckon your solution is a lot more complex than it needs to be.

Given that "the other player also picks his best move", then all you need to do is determine which produces the higher score - a pop-first, or a shift-first.

This is my go at it:

#!/usr/bin/perl -wl
use strict;

my @numbers = qw(16 2 10 2 9 17 5 8 15 14 20 19 19 11 10 11 9 13 7 13)
+;
my $shiftscore = checkscore("shift");

@numbers = qw(16 2 10 2 9 17 5 8 15 14 20 19 19 11 10 11 9 13 7 13);
my $popscore = checkscore("pop");

my $bestscore = $popscore > $shiftscore
    ? $popscore
    : $shiftscore;

print "POP:$popscore:SHIFT:$shiftscore";
print "Best Score:$bestscore";

sub checkscore {
    my $first_turn = shift;
    my ($player, $opponent);
    $player = $first_turn eq "pop"
        ? pop(@numbers)
        : shift(@numbers);
    while (@numbers) {
        $opponent += $numbers[0] > $numbers[-1]
            ? shift(@numbers)
            : pop(@numbers);
        last if !@numbers;
        $player += $numbers[0] > $numbers[-1]
            ? shift(@numbers)
            : pop(@numbers);
    }
    return ($player - $opponent);
}
[download]

Which gives:

POP:4:SHIFT:10
Best Score:10
[download]

Update: To test how long it takes with 1000 numbers I added the following lines:.

use List::Util qw(shuffle);

my @numbers = shuffle(1 .. 1000);
print join(" ", @numbers);

my @copy = @numbers;

my $shiftscore = checkscore("shift");

@numbers = @copy;
my $popscore = checkscore("pop");
[download]

And I get something like this:

POP:4354:SHIFT:7452
Best Score:7452

real    0m0.049s
user    0m0.030s
sys     0m0.000s
[download]

PS: Like the OP, I'm not 100% sure that I'm getting correct results - and I'm not sure how to verify it for very large ranges, I guess I'll just wait for some more responses :)

Update 2: - as tilly has pointed out, my solution is wrong. I kindof suspected that it would be - because it seemed too simple. And although I acknowledged further down in the thread that it was wrong, I forgot to add an update here.

Cheers,
Darren :)