Analysing Bluffs with Equity

I did some analysis today looking at bluffing at pots on the flop or turn with a pot sized bet left behind and with a variety of hands which will win some % of the time at showdown. I thought I'd share some of it with you, because it really shows how powerful bluffs with equity (also known as semi-bluffs) really are.

When bluffing with no chance of winning at showdown (a pure bluff) we risk a pot sized bet to win the pot so to break even on our bet our opponent must fold 50% of the time.
[risk/(risk + reward) => 1/(1+1) = 0.5] If he folds more often than that we obviously profit.

When bluffing with equity our break even point comes at x = -1/(3Eq - 2) {Eq = the equity of our hand versus villain's calling range}. To get this I just created the EV equation, set it equal to zero and rearranged for x which is the indifference point. If our hand has 25% equity then x = -1/(3*0.25 - 2) so we break even when our opponent folds just 20% of the time. This is a huge difference considering our hand only wins at showdown a quarter of the time villain calls. If our hand has 33% equity against villain's calling range (think flush or straight draws on the flop) then we should bet all-in every time because any time villain folds we profit, and when he doesn't we own too big of a share of that pot to fold.

To demonstrate with a graph, the following shows our indifference points for a given equity between 0 and 50%.



Any time our indifference point is greater than 1 betting all-in is mandatory.

In conclusion, I hope it's easy to see that in a situation where your opponent will fold some of the time and you have a pot sized bet left semi - bluffs are usually very profitable.