Quote:
Originally Posted by hectorjelly
Browni3141 has hit on the crux of the problem, if you allow C the equities will change. Take this example as an illustration:
Say three players get all in in holdem on the flop
Player A has a flush and $100
Player B has a set and $200
Player C has a straight and $200
Which leaves the main pot of $300 with all contestants, and a side pot of $200 with player B and C
If the pot is run once then the main pot of $300 can either go to player A or B, player C will never win this as a straight is never going to beat a flush. Player C's equity is 34% of $200. If we allow the main pot to be run twice if the shortstack loses the first run out, his equity becomes 34% of $500.
Thanks, great example. I wouldn't have understood without it, and the last sentence is pretty opaque, but the example helped me work it out in my own brain.
Here's how I would phrase it (maybe clearer to some, completely unclear to others). The fairness of the run-twice scheme is based on assuming two events are independent, but alas they are actually not independent:
- The ratio of B's probability : C's heads up is not the same as
- the ratio of B's win probability conditional on A losing the main pot :
C's win probability conditional on A losing the main pot.
So you're choosing to run once or twice based on an event that's not independent to B's or C's winning probability HU. If B is 70-30 vs C for the side conditional on A losing the main, or 50-50* vs C conditional on A winning the main, then this scheme is hurting B. Those times she has the biggest edge on the top board, another board is run for the other half of the pot. Those times she doesn't have an edge (or as big an edge), the first board stands.
The given example is particularly stark because:
p(straight beating the set | the flush winning the main) is 1.0, since the board can't have paired, and
p(straight beating the set | the flush losing the main) is 0.0, since the board pairing is the only way the flush could lose the main
where | is standard notation for "conditional on."
*60-40, 51-49, whatever, could be any edge smaller than 70-30.
Last edited by AKQJ10; 06-24-2018 at 02:35 AM.