I address this issue in one of my posts in my Hold’em Mathology blog on Tumblr.
https://holdemmathology.tumblr.com/p...-and-range-for
I deal with the case of hero facing an all-in bet as last to act. We assume villain makes an all-in bet of X*Pot with a villain shove range of VSR. If hero calls, his EV in pot size units is
EV = eq*(1+X) – (1-eq)*X,
where eq is the probability hero wins against the shove range.
Set EV to 0 and hero’s break-even equity is X/(1+2X).
This says hero’s required equity is always less than 50% and approaches that value as X increases. The post has a table that shows hero’s required equity and associated calling range for various villain shove ranges and for various villain all-in bets in pot size units. I used the ranges provided in Equilab. Other range schemes will give different results but they should not differ by much.
Example: If villain makes a 5xPot shove, hero needs equity of at least 45.5%. If villain’s shove range is 25%, hero can call with a top 17.2% range so that every hand in that range has at least 45.5% equity. This is the 17.2% calling range: {66+,A5s+,K9s+,Q9s+,JTs,ATo+,KTo+,QTo+}.
If villain goes all-in with a bet of 5xPot with any hand, hero can profitably call with a top range of 64.1% or less. For a 75% villain shove range, the calling range should not exceed 48.4%.
Note that this chart differs from the GTO-like Call/Fold charts that have been proposed, which attempt to achieve a Nash equilibrium with both players playing GTO. Here, the values are based on a read of villain’s shove range so that it is an exploitative play approach assuming villain is not playing GTO.
