This program looks like a good exercise to learn on. I noticed that at the beginning of the program you initialize $dbm{win} and later use $dbm{won}, resulting in an uninitialized value warning.

Another idea for improvement is to put the input in a loop and only exit the loop upon receiving the proper input. This is one way:

my $max;
chomp($max = <STDIN>) while(not defined $max && $max =~ m/^[0-9]+$/);
my $answer = int(rand($max))+1;
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