Thank you for all your interesting ideas and suggestions. The code I have copy with now is

sub already_voted {
  my ($ip_address, $user_agent) = @_;
  my (@lines, $line, $file_ip, $file_ua, %voters, $voted); 
  
  # Open our list of IP address and user agents. 
  # This is in the format:
  #
  # IP_address:::UserAgent
  #
  open (IPADDRESS, "<ips.txt"); 
  @lines = <IPADDRESS>;
  close IPADDRESS; 

  # Parse the array and store IP and User Agent in a hash. 
  foreach $line (@lines) { 
    chomp $line; # Remove \n
    ($file_ip, $file_ua) = split(/:::/, $line); 
    $voters{$file_ip} = $file_ua; 
  }

  # Check to see if the IP of the person currently voting 
  # is in the hash. 
  if (exists($voters{$ip_address})) { 
    
    # It looks like the same person is voting again. Stop the
    # evil person. But wait, let's check the user agents too.

    if ($voters{$ip_address} eq $user_agent) {
      # Same IP, same user-agent. Looks like a duplicate vote.
      $voted = 1;

      sharedcode::logfile("Bad: dup vote $ip_address and $user_agent."
+);.
    }
  }

  # Let's put *this* vote into the file:
  open (IPFILE, ">>ips.txt");
  print IPFILE "$ip_address".':::'."$user_agent\n";
  close IPFILE; 
  
  if ((-M "ips.txt") >= 7) { 
    # File is getting old, let's clear the file. 
    open IPFILE, ">ips.txt"; 
    print IPFILE " "; 
    close IPFILE;
  }
  
  return $voted; 
}
[download]

Obviously, this code isn't going to trap 100% of the cheaters, but that's fine by me. Thanks again for all your suggestions.