I have done some work on this using standard EV analysis. The following assumes the only limper is not the small blind.
The EV for BB to just check-call a limp (i.e., no investment) is
EVcheck = 2.5*eq,
where eq is the showdown equity of his hand. Clearly, BB has positive EV.
For a BB raise,
EVraise = fe*2.5 +(1-fe)(eq*(2.5+2*R)-R)
where
fe is fold equity (villain fold probability)
R is the raise-of amount
Then a raise is better than a call if EVraise > EVcheck= 2.5*eq
The following table shows some results for required fold equity given hero equity and raise. Note that since there will be future action unless the raise was all-in or the hand is checked down, this has to be considered a first cut look to be adjusted for factors not explicitly considered such as realized equity, stack sizes, future bets and folds, etc.
Required fe For Raising Limper Rather Than Checking
...........Raise-of in bb Units
eq | 1 | 3 | 5 | 7.5 | 10 |
0.00 | 0.29 | 0.55 | 0.67 | 0.75 | 0.80 |
0.20 | 0.23 | 0.47 | 0.60 | 0.69 | 0.75 |
0.33 | 0.17 | 0.38 | 0.50 | 0.60 | 0.67 |
0.49 | 0.02 | 0.04 | 0.07 | 0.11 | 0.14 |
Example: Assume BB hero has 33% equity and is considering a raise of 3bb. Then villain has to fold 50% of the time for the raise to have better EV than ‘check-calling’ the limp.