Politely Cheating
by Isabel Harris
There exist many games where an established pattern decides the winner. For example, in rock-paper-scissors, rock always beats scissors, scissors always beats paper, and paper always beats rock. The pattern is circular, with each option beating one and losing to one option. If a player was forced to choose their move first, their opponent could always choose the option that would win.
We can make games with similar patterns out of dice. In these games a die wins by having a larger face showing. Consider a game using a set of two standard dice. Both players roll and have equal likelihoods of winning. There is no strategy and no choice involved. Now consider playing the same game using the modified dice A, B, and C, shown in Figure 1.
Figure 1
In this set A usually beats C, C usually beats B, and B usually beats A. This is a pretty simple game and only allows two people to play. But, there is now some strategy involved. No matter what die Player 1 chooses, Player 2 can choose a die that usually beats Player 1, but Player 2 must insist his opponent chooses first. Oskar van Deventer developed a 3-player game of the same type using seven dice [4]. The first two players can choose any of the seven dice in the set and the third player can find a die that will usually beat the chosen two. The relationships between the dice he created are shown in Figure 2
Figure 2
It is possible to have a set of 19 dice for a four player game, as shown in Figure 3. For clarity, the winning relationships for only one die, the median die, are shown. The red arrows denote that the median die beats the specified dice. The relationships among the remaining dice can be found by rotating the pattern of wins from the median die.
In this set A usually beats C, C usually beats B, and B usually beats A. This is a pretty simple game and only allows two people to play. But, there is now some strategy involved. No matter what die Player 1 chooses, Player 2 can choose a die that usually beats Player 1, but Player 2 must insist his opponent chooses first. Oskar van Deventer developed a 3-player game of the same type using seven dice [4]. The first two players can choose any of the seven dice in the set and the third player can find a die that will usually beat the chosen two. The relationships between the dice he created are shown in Figure 2
The Mathematics of Cheating
The games described above utilize the nontransitive property of probability. Transitivity is the mathematical property that if a relationship exists between A and B and the same relationship exists between B and C, that same relationship exists between A and C. For example, if A > B and B > C, then A > C. Transitivity is very common and is true in many situations. However, transitivity does not hold in probability. The probability an event occurs is calculated by noting the times an event occurs and dividing by the size of the sample space. Take the 2-player game for example. Using Figure 4, it can be observed die A beats die C 5 out of 9 times, a filled square denotes die A wins. Thus, the probability die A wins, 5/9 , is greater than the probability die C wins, 4/9.
The probability an event R occurs is denoted P(R). Thus, in this 2-player game. P(A > C) > P(C > A), and P(C > B) > P(B > C). But P(B > A) > P(A > B), breaking the property of transitivity. This is a simple game of nontransitive dice [3].
In Deventer’s set each die contains one number from each range [1, 7], [8, 14], [15, 21]. Every column in Figure 5 represents a die in Deventer’s set.
Because each row increases by 7(n-1), where n is the row number, we know every number in a certain row automatically beats all numbers in the preceding row or rows. Thus, we can focus on which and how many of the diagonal squares are won by each die. In the example in Figure 6, Die 1 wins the last square on the diagonal and Die 2 wins the other two squares on the diagonal. Therefore, Die 2 beats Die 1. This idea holds as we create larger dice sets for more players and makes for an easy check for winning die.
References
[1] cp4space.wordpress.com/2013/02/15/tournament-dice/
[2] D. Bednay and S. Bozóki, Constructions for nontransitive dice sets, in Proceedings of the 8th Japanese-Hungarian Symposium on Discrete Mathematics and its Applications, Vesprém, Hungary, 2013, 15-23.
[3] en.wikipedia.org/wiki/Nontransitive_dice
[4] grimes.s3-website-eu-west-1.amazonaws.com
[5] L. Angel and M. Davis, A direct construction of nontransitive dice sets, J of Combinatorial Designs 25(11)(2017), 523-529.
[6] S. Bozóki, Nontransitive dice sets relalizing Paley tournaments for solving Schütte’s tournament problem, Miskolc Math Notes 15 (2014), 39-50.
[7] Weisstein, Eric W. “Quadratic Residue,” From MathWorld – A Wolfram Web Resource.
Disclaimer: The views and opinions in this work are those of the author and do not necessarily reflect the views of Auburn University.