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I have now modified the script to calculate an attenuated score for each game. This appears in parentheses, and it is used as a tie-breaker in the sort. The code for calculating this value looks like this:

foreach ($votesbyperson as $key => $val) {
    if ($val > 1)
        $votevalue[$key] = min(1, 60/(50+$val));
    else
        $votevalue[$key] = .5;
}

This attenuates the value of single votes and the value of lots of votes while counting 2-10 votes as 1 vote each.