Overbetting

by mersenneary

Imagine a simple heads up bluffing game. Both players start with $1000. After $10 each as an ante, your opponent is dealt one card, and it will be either the duece of clubs, the three of clubs, the four of clubs, the five of clubs, or the one of the two red queens, for six possible holdings. You do not get a card. Your opponent then has two choices: Surrender, and let you take the $20 pot, or bet some amount. If your opponent bets, you have to decide whether to call the bet. If you do, you win the pot if your opponent has a baby club, and lose the pot if your opponent has a red queen. After the hand, a new one starts, and the players do not switch positions.

So, 4/6 (or 2/3) of the time, your opponent will have a loser in the hole. If your opponent makes his betsizing the size of the pot, $20, his bet needs to be a bluff at least one in three times for calling to be profitable. If your opponent uses exactly that frequency, it means you're indifferent between calling and folding.

In this case, your expectation playing this opponent in this game would be 0EV, and you'd break even over time. Half the time, your opponent will bet – say, always when he gets the five of clubs, or, of course, a queen. You'll lose $10 when your opponent bets. The other half of the time, your opponent will surrender, and you'll win $10. Thus, when your opponent bets the size of the pot, it makes it seem very much like this is a breakeven game with no edge.

However, there's nothing magical or limiting about the size of the pot. We can bet anything we want in no limit games, and there are often big opportunities to take advantage of this. Let's say, instead, that your opponent opted to bet $40 – twice the size of the pot, whenever he decides to bet. Now your call needs to be right 40% of the time for you to be indifferent between calling and folding. I'll spare you the number crunching, but now, your opponent will be betting about 56% of the time with an expectation of +$10, and folding 44% of the time with an expectation of -$10. This works out to a solid expected winrate of over a dollar per hand for your opponent – suddenly, he's printing money.

It turns out that the Nash Equilibrium of this game is for the player with the card simply to go all-in for $1000 about 2/3 of the time, say, whenever he gets the five of clubs or the four of clubs, plus, of course, whenever he gets dealt a queen. By using this strategy, your opponent wins over $3 per hand against you, and it's completely unexploitable. Your opponent can sacrifice some EV for lower variance, and only bet $100, and it still results in well over $2 per hand. Overbetting is simply the correct way to play this game against a good player.

Of course, we should never get so lost in theory land that we don't learn anything we can actually put to use in the real world to make as much money playing poker as possible. To bridge that gap, here are some characteristics about this game to look for in HUSNGs when identifiying overbetting situations.

1. The overbettor credibly represents a big hand. (In the example, a queen).

2. The bluffcatcher has a capped range, and very rarely has a hand that doesn't lose to the overbettor's value betting range. (In the example, the bluffcatcher always loses to a queen).

3. The bluffcatcher either won't change the fact that his range is capped because he's not a good thinking player, or he can't, either due to the rules of the game or because it loses too much value in other situations. (In the example, it's the rules of the game).

The first is pretty self explanatory. If you don't rep anything, expect to get hero called by good players when your overbet makes no sense for your line, even if your opponent doesn't have many strong holdings.

The second and the third are where all of the extra thinking comes in with HUSNGs. Here are some common situations where many regs have capped ranges:

a. Check/calling flops they would check/raise with big value hands.

b. Failing to continuation bet on boards where most strong-but-vulnerable hands would always be willing to c-bet and get it in.

c. Checking behind on the turn after c-betting flop on a vulnerable board when strong hands can comfortably bet and call all-in if jammed on.

d. Check/raising flop, barreling turn, and then checking the river when the draws come in.

Having a capped range just means not credibly representing any nutted holdings. Sometimes, people have capped ranges because they play strategies that are excessively straightforward and exploitable. Other times, it's because the downside of being capped is outweighed by the benefit from getting maximum value from big hands, and not playing them deceptively (our OOP flatting range 75bb deep is capped, we never have aces, but this is clearly correct).

On Wednesday, I'll write about one of these examples – overbetting the turn on wet boards after your opponent fails to continuation bet on the flop.

Questions for now?

mers