[MarkGritter]

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6. We need Gritter to churn out some combinatorics for us! I'm too lazy to do it myself.

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OK, what the heck, I'm feeling bored at work. Random math geekage:

There are 52C4 = 270725 hands in this game. How many are Badugis (4-card hands)? A badugi has one card from each suit and no pairs. So it should be 13*12*11*10 = 17160. (We can reach any possible rainbow hand despite having a fixed suit ordering, so no fancy counting or inclusion-exclusion seems necessary.) Thus badugis form the top 6.3% of hand ranking (from #1 to #715 if I got the math right.)

What hand do you need to be a favorite to win vs any possible one or two draws? (Note that if have a badugi, you are by definition holding one of your opponents' outs.)

Ignoring discards for now, there are going to be 44 unknown cards HU. Say your opponent with A23x and you hold any badugi. Then he has 13 flush cards to hit, but 1 of them is in your hand, and the A,2,and 3 don't give him a badugi either. You are a 35:9 favorite vs. a single draw. If you opponent has already dumped 3 bad cards, you are a 32:9 favorite instead.

What if you have two opponents? Again ignore earlier discards. If they are both drawing for the same suit, then there are again 9 good cards, and 31 bad cards for them. The chance that both will miss is (31/40)*(30/39) = 60%. So it seems likely your badugi holds up vs. any two players drawing one. (Maybe a little more care is needed here because the same pair card may not be bad for both players, so one player could miss and still kill the second one's out. But let's just give them both a23x for now.)

With three opponents old #715 (KQJT) doesn't look so good. 16 known cards, 36 remaining. Again your opponents are better off if they are all drawing for the same suit. Compare schd schh schh schh -- about 9 diamond outs for everybody --- with schd ssch scch schh --- only 6 outs in each suit. Then the probability that your hand is good is only (27/36)*(26/35)*(25/34) = 41%. So you have an edge, but are no longer a favorite.

If we give Hero a rough Q-high badugi instead, that reduces the number of outs by one, and a J by two.

worst Q badugi wins = 28/36*27/35*25/34 = 44%
worst J badugi wins = 29/36*28/35*26/34 = 49%

So, a smooth J-high badugi should usually be the favorite vs. 3 players. (Against the A23x/A23x/A23x example only A23J would be a favorite, but most hands are rougher than that.)

But, your opponents are unlikely to be drawing to the best case, so a Q is probably a favorite in many real-life situations too.