For what you are doing radio buttons would fit the idea better. It's very easy to set up.
<select name="vote" type="radio">
<option value="Name1">Name 1</option>
<option value="Name2">Name 1</option>
<option value="Name3">Name 1</option>
</select>
then of course the rest of the code to submit, etc.
Then run some sort of sql
$sql = "INSERT INTO voting (candidate) VALUES ('$vote');";
That will ad a new row for each vote. What you could do instead for the sql is
$sql = "UPDATE voting SET count = count + 1 WHERE candidate = '$vote';";
In this case, you would have to have a filed titled count, and candidate.