Join Date: Jan 2007
Posts: 1,940
I've never tried constructing hypothetical nash betting ranges when chop pots are involved, so this may be incorrect. But from what I can tell, you'd start by betting all your scoops (obviously), then add twice as many of your chop combos. This provides your opponent with the indifference ratio of 2:1 value to bluffs. (If you don't have twice as many chop combos, then you'd include 1 pure bluff combo for every 2 "leftover" scoop combos)
In the situation where you have at least twice as many chops as scoops (which holds true for any equity < 50%), your additional captured share of the pot on the river (by "bluffing" with chop pots) ends up being equal to your % of scoop combos.
Additional share by bluffing = Scoop combos = (raw equity)^2
Total share of pot = Additional share + raw equity = e^2 + e
0.5 = e + e^2
e = 0.5 * sqrt(3) - 0.5 = ~0.3660254
So by changing post-draw rules, his pre-draw strategy changes to checking any draws with equity between ~0.333 and ~0.366 as they no longer capture >50% of the pot on the final betting round as they did before.