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--- fairness [2023/02/28 16:18]
harshec created
+++ fairness [2026/09/19 05:20] (current)
bmenrigh Added info about all-subset place fairness and mentioned how strong perm-fairness is.
@@ Line 3: / Line 3: @@
   * **Go-first-fair.** This is as originally proposed by James Ernest to Eric Harshbarger. Each player has an equal chance of rolling the highest number, and that person would go first.
-  * **Place-fair.** Here, each player has an equal chance of not only rolling highest, but also ending up in any ranking/position. For example, each player would also have an equal chance of rolling the second highest number, the third highest number (if three or more players rolling), and so on.
+  * **Place-fair.** Here, each player has an equal chance of not only rolling highest, but also ending up in any ranking/position. For example, each player would also have an equal chance of rolling the second highest number, the third highest number (if three or more players rolling), and so on. Place fairness comes in an even "stronger" form where all subsets of the dice are also place-fair. We sometimes call this all-subset-place-fair.
   * **Permutation-fair.** This is the strongest type of fairness. Not only does each player have an equal chance of ending up in any position in the ordering, but every possible ordering (permutation) of players has an equal chance of occurring.
-To the casual reader "place-fairness" and "permutation-fairness" might sound like the same thing, but they are different. It is quite possible for a set of dice to be "place-fair" without being "permutation-fair". In such a case, each player would be equally likely to end up in any position of the hierarchy, but some overall orderings of players would be more likely to occur (for example, if the dice are colored Black, Red, Green, and Blue, any color would be equally likely to end up in any of the four sorted positions, but maybe "Green, Red, Black, Blue" ordering comes up more often, than say, "Red, Blue, Black, Green" -- individually the colors are equally likely to end up in positions 1-4, but how their positions relate to the other colors might not be equi-probable). Achieving that equi-probability of the permutations only happens with "permutation-fairness".
+To the casual reader "place-fairness" and "permutation-fairness" might sound like the same thing, but they are different. It is quite possible for a set of dice to be "place-fair" without being "permutation-fair". In such a case, each player would be equally likely to end up in any position of the hierarchy, but some overall orderings of players would be more likely to occur (for example, if the dice are colored Black, Red, Green, and Blue, any color would be equally likely to end up in any of the four sorted positions, but maybe "Green, Red, Black, Blue" ordering comes up more often, than say, "Red, Blue, Black, Green" -- individually the colors are equally likely to end up in positions 1-4, but how their positions relate to the other colors might not be equi-probable). Achieving that equi-probability of the permutations only happens with "permutation-fairness". Permutation fairness is, in some sense, the perfect property, because it guarantees all other fairness, including for all subsets.
 The original problem of looking for a go-first-fair set soon morphed into trying to find a set that was permutation-fair. So, calling this wiki the "Go First Dice Wiki" is actually a misnomer; it should be called "The Permutation-Fair Dice Wiki", but that does not have as nice a ring to it, and the original phrasing has stuck.

Go First Dice Wiki

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