Poker — minimum defence frequency vs pot odds
Returning to my essays on poker concepts, many players will have come across the ideas of ‘minimum defence frequency’ (MDF) and ‘pot odds’. These two concepts appear very similar and perhaps even contradictory.
I’m coming to the realisation that most poker theory available online has little relevance to the low stakes games I play. Entire books are devoted to big blind vs button heads-up battles, as this is ‘a very common spot’ (apparently). Yet I’ve never been in a big blind vs button heads-up battle in my life. The only situation that would be truly useful to me would be to discuss action that has 7 players see the flop! So how useful are the concepts of MDF and pot odds for the social player?
First, some definitions. Minimum defence frequency (MDF) is the minimum percentage of hands we must not fold to a bet, to prevent the aggressor from profiting by always bluffing us. ‘Pot odds’ is the probability our hand needs to have of winning, in order for us to call a bet (ignoring implied odds; more on that later). Both measures refer to how frequently we should be defending an opponent’s aggression.
Let’s suppose that there are 100 chips in the pot, and the aggressor bets 100. How often should we call? MDF approaches this question from the aggressor’s perspective. They are betting 100 to potentially win the 100 in the pot, giving them 1:1 odds. Even if their hand has zero equity (i.e. no chance of beating our hand) then if we, the hero, fold more than 50% of the time the aggressor comes out ahead. In a heads-up battle the hero needs to call (or raise, but don’t fold!) at least 50% of the time. The hero can spin a coin, or preferably choose the best 50% of hands in their perceived range at that point to continue with, regardless of what the aggressor holds.
Pot odds on the other hand approaches the same scenario from the hero’s perspective. There is now 200 in the pot, and hero needs to put in 100 to call. We are getting 2:1 odds and need to win 33% of the time to make a profit. We therefore need to compare our hand with the perceived average hand strength (‘range’) of the aggressor: will we win 33% of the time? With pot odds we are considering what the villain holds.
In mathematical terms:
MDF = pot size / (pot size + bet size).
Pot odds = call size / (pot size + bet size + call size), where the pot size is before the aggressor’s bet.
If the pot is 100 chips and the aggressor bets 100, the MDF is 100 / (100+100) = 50%, while the pot odds are 100 / (100 + 100 + 100) = 33%.
Both seem to be measures of how often we should call, but the percentages are different. Hmm. How should we interpret these two numbers?
On the one hand there is no contradiction here; the 50% and 33% refer to different things so don’t necessarily need to be the same number. On the other hand, it’s not clear to me that if we set up thousands of scenarios where hero has a calling decision when the aggressor bets 100 into a 100 chip pot, that hero can satisfy both constraints on average — of calling 50% of the time and also only calling when hero has 33% equity.
Game theory optimal (GTO)
In a GTO World the two players (they would have to be computers . . ) will play so that this is true. It’s helpful to add some constraints here that we are on the river and the aggressor has a perfectly polarised range (they either have the nuts or nothing). In this case the aggressor always bets 100 chips with all their strong hands and half as many weak hands. Hero is receiving 2:1 pot odds to call, but there are 2 times as many strong hands as weak hands. Hero can only break even. In GTO parlance, hero is ‘indifferent’ to calling or folding.
To keep the aggressor honest, though, the hero targets aggressor’s bluffs by applying the MDF of calling 50% of the time. The aggressor is getting even money on his bluffs, so the bluffs break even. In GTO parlance, the aggressor is indifferent to bluffing or not. We have a dynamic equilibrium. Hero has 33% equity (pot odds) and should call 50% of the time (MDF). Note that Hero can’t do anything about the aggressor’s value bets, which is why in poker it’s generally advantageous to have a polarised range.
This is all very well, but let’s return to the relevance of all of this for the social player in a loose game, light years away from GTO …
Pros and cons of MDF
Clearly the MDF is a meaningful concept for heads up play and appears to be a consideration for the earlier streets: we need to be a little bit ‘sticky’. The winner of the 2016 World Series of Poker was Qui Nguyen, winning $8m for his victory. He partially ascribed his victory to the fact that he was hugely aggressive, and his opponents simply folded too much. They didn’t follow the minimum defence frequency.
It’s important to note however that the MDF strictly applies to when the villain has zero equity: villain has no chance of winning so hero folding more often than [pot size / (pot size + bet size)] is an immediate gain for the villain. If villain’s hand has some equity however then the calculation is a bit more nuanced. It’s not obvious to me that if villain’s range crushes our range that we still need to continue with x% of our range. To take an extreme example, if the villain turns their hand face up (a straight flush, say) and then bets 100, obviously we don’t need to continue with our best x% of hands! The point of MDF is only to stop the aggressor from bluffing with air.
Another important point is that in multiway pots the task of defending against the aggressor is shared between multiple players. If we are facing a bet and have a weak hand with little chance of improving, then we can fold and rely on the players behind us to ‘keep him honest’.
Pros and cons of ‘pot odds’
Similarly, ‘pot odds’ is clearly a mathematically correct concept, and useful to know for that reason. However, it only strictly applies when we’re on the river and villain goes all-in, betting 100 chips into 100 pot say. In this case we need to be 33% sure of winning to call. How will we come to this determination? Tricky. Generally speaking, weaker player populations don’t bluff enough: ‘they always have it’. Fold!
Within this player population, however, are a few individuals who love to bluff (and who often show up at the river with weak hands so need to bluff). Against them: call! A bit of card reading will inform whether the bet is more likely to be a bluff e.g. a busted draw on the river, or when the betting sequence doesn’t tell a convincing story. (I’m not saying I can do this, mind, but this is the theory!)
Let’s take the situation where we are not all-in and still have 2 streets to go, and villain bets 100 into a 100 pot. Maybe we have an open ended straight draw (OESD): between our hole cards and the board we have 5,6,7,8, say, and we need a 4 or a 9 to make our straight. We have 8 ‘outs’ for a sure win, which has an approximately 32% chance to occur over two streets. We (just about) have the required 33% pot odds to call.
The problem is that if we don’t make your straight on the turn and if villain wasn’t all-in they can just bet again! Now we only have circa 16% chance to make our straight, so we now have to fold. This suggests that we should be folding more often than the pot odds suggest on the previous streets though, interestingly (and for reasons I frankly don’t fully understand), the expert advice is to call any reasonable bet (pot sized or less, say) on the flop when we have an OESD.
The point is that villain doesn’t know what draw we’re on. Maybe we have a flush draw. Maybe we don’t have a draw at all and have a monster hand already! By staying in the pot other good things may happen on the turn or river, even if we don’t make our straight. We must just play confidently and aggressively!
This goes to the concept of implied odds. If we have an OESD on the flop, we only have a 16% chance of getting our straight on the turn, which doesn’t meet our immediate pot odds if villain bets 100% of pot. However, if we do make our straight, hopefully more money will go into the pot, which we will win too. It is difficult to estimate the value of these implied odds. Naturally it is limited to the amount of the villain’s stack, though it would be very optimistic to assume that we will ‘stack’ the villain every time!
My biggest practical objection to the concept of pot odds is how we’re meant to determine whether we have a 33% chance of winning the pot, to that big river bet. Only a computer could analyse all the combinations still remaining in our and villain’s ranges and calculate our equity. The best we can hope for is to determine that we have a ‘high’, ‘medium’ or ‘low’ chance of being ahead. Then you need to overlay whether the villain ‘under bluffs’, ‘over bluffs’ or is balanced. Then we make an educated guess!
In summary
Minimum defence frequency and pot odds are both core concepts in poker. For the social player they are useful conceptually, but there are very few situations where we can apply the mathematical formula and come to a solid conclusion. If we are playing an aggressive player, it is important to be a little bit sticky on the earlier streets, otherwise they will run us over even if they have nothing. The larger the bet, the lower the MDF — sometimes it’s ok to just let them have it.
On the river, when deciding to call a bet, pot odds gives us an idea on how often we need to be good to call. The better the pot odds, the less sure we need to be of being ahead. Good luck at the table on making this decision!