Does Avis or Iago have a better chance of winning a match?
According to Swing Dice theory, Avis has the advantage.

If both players choose the same size X, Iago needs to keep 20 points to win (the 2/3 difference between his 60+X points and Avis' 30+X). So all things being equal, Iago needs to win and hold on to one of his 20-siders. 

The bad news for Iago is that even in the best case (Avis takes a d20, Iago takes a d4), he still needs to hold 9.3 points (which means he still needs that d20, since the d4 won't cut it here). The point difference between the two buttons is wide enough that Iago must always win - there's no case where Avis can end up with a higher point total.

The news gets worse - if Iago ends up with a larger X than Avis, he now needs to hold two dice. And it doesn't help that Avis has five dice vs. Iago's four - given that a normal game of BM involves taking dice each round, he needs Avis to pass at least once.

To sum up - Iago's best action is to take a d4 (regardless of what Avis has), win initiative, force Avis to pass at least once each battle, take all of Avis' dice, and hold one of the d20s.

(Avis, on the other hand, can take a d20 and still force Iago to win and hold a d10 or better to win. There's no bad matchup for him.)

Source: a quick Excel spreadsheet to calculate win conditions for each combination of Xs. :)