I believe that no strategy of that type can possibly work. Here's an example where a number 10 in changes whether it is better to pop or shift first: 
0.631752274168395
0.502325774658843
0.0553136679912676
0.696650395168113
0.904058789590099
0.83366887317402
0.676563261504992
0.620510190454731
0.835401306610937
0.331025798953092
100
0.460845097973213
0.0150072228842113
0.0863915966693014
0.252652201491809
0.0344964741543734
0.277799160386071
0.358041640281542
0.984148718850204
0.846530682637557

[download]
(With the 100 there, you should shift, remove it and you should pop.)

Here is the program that I wrote to find that strategy: 

#! /usr/bin/perl -w
use strict;
my $n = shift || 4;
my $tries = shift || 4**$n;

for (1..$tries) {
  my @num = map rand, 1..$n, 1..$n;
  $num[$n] = 0;
  my $iter = get_move_iter(@num);
  my ($move) = $iter->();
  $num[$n] = 100;
  $iter = get_move_iter(@num);
  my ($move2) = $iter->();
  if ($move eq $move2) {
    print "Try $_ was same\n";
  }
  else {
    print "Got $move vs $move2:\n@num\n";
    last;
  }
}

# Scores is a 2-dim array, $scores[$i][$j] is the best score
# that you can get given a string of $j numbers starting at
# $numbers[$i].
sub get_scores {
  my @numbers = @_;
  my @scores;
  for (0..$#numbers) {
    $scores[$_] = [0, $numbers[$_]];
  }

  for my $j (2..@numbers) {
    for my $i (0..(@numbers - $j)) {
      my $a = $numbers[$i] - $scores[$i+1][$j-1];
      my $b = $numbers[$i+$j-1] - $scores[$i][$j-1];
      $scores[$i][$j] = ($a > $b) ? $a : $b;
    }
  }
  return @scores;
}

sub get_move_iter {
  my @n = @_;
  my @scores = get_scores(@n);
  my $i = 0;
  my $j = $#n;
  return sub {
    if ($j < 0) {
      return;
    }
    elsif ($n[$i] - $scores[$i+1][$j] > $n[$i+$j] - $scores[$i][$j]) {
      $j--;
      return "shift", $n[$i++];
    }
    else {
      return "pop", $n[$i + $j--];
    }
  };
}

[download]

In reply to Re^2: Puzzle: Given an array of integers, find the best sequence of pop / shift... by tilly
in thread Puzzle: Given an array of integers, find the best sequence of pop / shift... by TedPride

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