Overbet Math

Poker’s still going pretty meh this month. Rather than moan some more I figured I’d just publish an old article I wrote awhile back to keep my readers interested. Hope you like!

Isildur1 has been tearing it up lately destroying all comers at NLH at the highest possible stakes and utilizing the “NO LIMIT” aspect of the game to the fullest degree with loads of river overbets so I thought I’d do a little bit of math on them as I think it’s a poorly understood concept for a lot of people.

When someone overbets the pot I can’t tell you how many times I’ve heard, “Oh shoot, I better be right A LOT of the time here for calling to be good.” Well how much is “A LOT”. Turns out it’s not that much more than normal.

Intuitively think about it this way. If someone bets you $20 into a $5 pot that he can accurately predict the flip of a coin, would you take his bet? If you said no you fail. Hard. Yet I think a lot of poker players think they have to be right >50% of the time when someone overbets. The fact of the matter is it is impossible for you to have to be good >50% of the time facing a bet on the river into a pot that is already >$0. The reason being of course is that if you bet with someone at even odds then you only have to be good >50% of the time for it to be a good bet. When calling a bet on the river, you are basically taking someone’s bet at even odds, except you have the overlay of what’s already in the pot so even if they bet 100 times the pot you only have to be good <50% of the time for calling to be profitable. Let's start by looking at a hand I observed between Isildur1 and Patrik Antonius. Effective stacks are 127k. Patrik is dealt 8c7c in the small blind. Patrik (sb) 500 Isildur1 (bb) 1000 Patrik (sb) opens to 3000 Isildur1 (bb) 3 bets to 9000 Patrik (sb) calls Pot is 18k Flop Ks Qh 7s Isildur1 cbets 14k and Patrik Calls Pot is 50k Turn 6d Isildur checks, Patrik checks behind River 4d so final board is Ks Qh 7s 6d 4d Isildur overbet jams 102k into the 50k pot. Patrik...calls/folds with his 3rd pair 8 kicker? Before I go through the math, intuitively answer these questions for yourself. From Patrik's perspective: How often does Isildur have to be bluffing for calling to be profitable? From Isildur's perspective: If he's bluffing, how often does Patrik have to fold for it to be profitable? If he instead bluffed 3/4 pot how often does Patrik have to fold for it to be profitable? If he's value betting, what % of the time does he miss value by overbetting? (ie if you think Patrik calls a "normal 3/4 pot bet" 40% of the time, but an overbet only 25% of the time your answer would be 15% (40-25=15) Okay now that you quickly skimmed over my questions and neglected to answer any of them time for me to hand feed you the answers. From Patrik's perspective he has to call 102k into a pot that will be 254k so he has to be good 102/254=40% of the time for calling to be correct. Compare this to the "normal" 3/4 pot bet of 37k into a 50k pot, he's calling 37k into a pot that will be 124k, so 37/124=30% so he only has to be good 30% of the time for calling to be correct. So yes when calling a two times pot bet on the river he has to be correct 33% more of the time (40/30=1.33), but this isn't exactly A LOT or nearly as much as some people think when contemplating a call in this spot. From Isildur's perspective if he's bluffing he either wins 50k or loses 102k. So 50x-102(1-x)=0 where x is the breakeven % of the time Patrik must fold for it to be a profitable shove. Solving for x we get 50x-102+102x=0 152x=102 x=102/152=67% So Patrik must fold at least 67% of the time for shoving to be profitable. Compare this to if Isildur bet 3/4 pot on the river we get 50x-37(1-x)=0 50x-37+37x=0 87x=37 x=37/87=43% Patrik must fold only 43% of the time when betting 3/4 pot for bluffing to be profitable. So if Isildur thinks he gains 24% in fold equity by overbet jamming two times the pot then this is clearly the better play. Let's compare the two scenarios for when Isildur is value betting. He can either win an additional 37k or 102k. Patrik will have to call the 37k nearly 3 times (102/37=2.76) as many times as the 102k bet so suppose Patrik calls a 3/4 pot bet on the river 30% of the time, he only has to call the two times overbet >11% (.3/2.76) of the time for overbetting for value to be better than 3/4 pot betting.

I imagine this is where a lot of Isildur’s edge is coming from. He is just very good at knowing when people are going to fold or call on the river and when they do call he gets three times the value that a normal person in his spot would have gotten. Of course it goes both ways. When he’s bluffing he loses 3 times as much, but like I said he’s probably very very good at gauging when someone’s ready to call or fold on the river and if you always know the answer to that question, there’s really no reason NOT to overbet every river (well if you could, you’d bet less when bluffing and more when value betting but you can’t obviously for balance reasons).

For those wondering about results in the hand, Patrik called and caught Isildur bluffing with AcJh to scoop the 254k pot.