Explanation for counter-intuitive discrete probability results

In the second case. If A and B both roll above the mean, then B tends to win. If they both roll below the mean, then A tends to win.
So for three players, A needs both B and C to roll below the mean, or one chance in 4.

The first game is fair by a symmetry argument: there is an involution on the set of possible outcomes that replaces the result $i$ of each ordinary die by $7-i$, and the outcome $k$ of the icosahedral die by $21-k$, and under this involution the gains of the players are interchanged; therefore there expected gains are equal.

However for the three-player game no such symmetry applies, and in fact it is not fair. If every one of the $20\times20\times 6^3 = 86400$ outcomes should come up exactly once (that it one outcome for every second in a day), the gains would be $31446$ for each of the icosahedral players, but only $23508$ for the three-dice fellow. While among each pair of players either has equal change of beating the other, the ones with flatter distributions of their outcome have a markedly better chance of beating both other players at once.