Characterizing Score Distributions in Dice Games


Aaron Isaksen, Christoffer Holmgård, Julian Togelius, and Andy Nealen
NYU Tandon School of Engineering, Game Innovation Lab


We analyze a variety of ways that comparing dice values can be used to simulate battles in games, measuring the ‘win bias’, ‘tie percentage’, and ‘closeness’ of each variant, to provide game designers with quantitative measurements of how small rule changes can significantly affect game balance. Closeness, a metric we introduce, is related to the inverse of the second moment, and measures how close the final scores are expected to be. We vary the number of dice, number of sides, rolling dice sorted or unsorted, biasing win rates by using mixed dice and different number of dice, allowing ties, re-rolling ties, and breaking ties in favour of one player.


[pdf] Isaksen, A., Holmgård, C., Togelius, J. and Nealen, A., ‘Characterising Score Distributions in Dice Games’, Game & Puzzle Design, vol. 2, no. 1, 2016, pp. 24–37. (c) 2016



Figure 1.  We examine games where players roll and compare the individual dice values. Each player’s dice are sorted in decreasing order and then paired up. Whichever player rolled a higher value in a pair wins a point. The points are summed, and whoever has more points wins the battle.



Figure 2. Rolling the dice and then (a) leaving them unsorted before matching or (b) sorting them in numerical order before matching gives significantly different probability distributions of score differences.  Rolling unsorted gives rise to closer games using our closeness metric.



Figure 3. Rolling mixed dice lets the designer control the win bias between the two players.


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