Selecting pairs of non-transitive dice
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Non-transitive dice are sets of dice which break transitivity, namely it is possible for A to have an advantage over B, for B to have an advantage over C, and for C to have an advantage over A. These are well-known and studied for the case of selecting a single die (possibly rolling it multiple times).
We introduce the problem of two players selecting pairs of dice out of a common pool, where they alternate taking turns and the second person selecting dice has an advantage (note there are multiple ways that the dice can be picked up; we seek dice which give an advantage to the second player in all scenarios). We exhibit sets of dice with these properties, found by computational search, and discuss some theoretical aspects by use of Kneser graphs.