It is suggested in some strategy articles to thin the field when you have the pre-flop advantage – bet so that you are only facing one or two opponents. Presumably, this makes it easier to play and cuts down the likelihood of a bad beat or suck-out.

I thought I would show this with some examples. I looked at four different hands – AA, AKo, 99 and 87s. Against each I assumed five different opponent types with ranges of top 10% (tight), 20% (moderate), 35% (loose), 50% (maniac) and a mix of these ranges. I assumed a hero bet of 3bb with all opponents calling.

Here’s what I found. With the four hands analyzed each against the 5 opponent types, we have results for 20 cases. In 10 of them, EV was highest for five opponents, the maximum value we considered. Facing one opponent had the highest EV in 8 cases. For AKo, facing one opponent had the highest EV for all opponent types. For 99, facing one opponent had the highest EV for the 3 middle types. AA always preferred five opponents while 87s had highest EV at two opponents for the aggressive type and at four opponents for the maniac type.

The results are summarized below:



Admittingly, these results apply to a very specific pre-flop situation and do not account for position, stack sizes, fold equity, etc. and, perhaps, most important, future action where realized equity becomes critical. Nevertheless, they surprised me that one opponent was not the best situation the large majority of the time.

This preliminary analysis led me to think that perhaps the suggestion to minimize the number of opponents deserves further study.

A more detailed summary of this analysis can be found at the following Tumblr blog site:

https://holdemmathology.tumblr.com/post/182712065725/

I think it's interesting. Particularly those combinations that are not preferred:

87s dislikes heads up and 4 way pots(I made a post once titled "four way pots suck")

AKo and 99 dislike(using the term loosely) 3,4,5 way pots(someone's going to bet you out of the pot much more often = unrealized equity).

I think an analysis of the suited connectors in particular would help a lot with preflop range construction. Which suited connector prefers the highest average number of opponents? How can we best manipulate the action with the marginal hands(call/3bet)?

I also think it's considerably valuable information for shorthanded facing a raise in the small blind(fold or 3bet, with very selective calling ranges) because I would probably rather see a 3 handed pot with 87s than get it heads up(note that a high big blind 3 bet frequency could force me into 3bet or fold strategy in the small blind).

Interesting stuff.
Can you tell us which hands were in your "top 10%"? I may have misunderstood the methodology, but I'm struggling to see why (on the blog) 99 apparently only had 39% equity vs the "tight top 10%".
In most of the 10% ranges I've experimented with, 99 is close to flipping.

I'll first check the calculation and then get back to you with a detailed explanation of what I did.
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Oops, you're right. I mistyped in the spreadsheet- instead of 39% it is 49%. That makes the EV 0.68 meaning one opponent is the highest EV for the 99 "tight" cases examined.

I double-checked a number of other cases and they were all OK.

I also love this type of stuff so thanks for doing it and please do more.

Maybe I am totally misunderstanding something here. Do all of your villains call Hero's raise with their entire (respective) ranges? If so, it seems to me, that you aren't really analyzing the total "thinning the field" phenomenon. Doesn't "thinning the field" often relate to calling vs raising (or re-raising)? Pot size can grow not only with number of villains who call but also whether I myself call, raise small, re-raise large, etc.

And is there an order in which the Villains call? That is, in your scenario in which only one villain calls, is it always the tightest villain? That seems to, potentially, be important too.

If this is all explained in your link, I apologize that I have not taken the time to read it yet.

Here is some relevant detail.

I set up 5 opponent types: For each type I assumed there is/are 1, 2, … ,5 opponents

Tight – all players play the top 10% hands. Moderate – all players play the top 20% hands
Loose – top 35% hands. Maniac top 50% hands
Mix – I opponent who is top 10% 2 opponents, 1 top 10% ,1 top 20% … 5 opponents 10%,20%, 35%, 50%, 50%

I assume hero makes a bet of 3bb and all opponents call. I use Equilab to estimate hero's equity for each situation, e.g. hero has AA and faces 4 Moderate opponents who each have a top 20% hand and each call the 3bb pre-flop hero bet.

I then use the standard EV equation for a bet to estimate hero's EV where the amount to be won increases with the number of opponents but the chance of hero winning decreases with more opponents.

As stated, this is a very specific and perhaps not very realistic scenario but I thought it would still confirm that reducing the field would usually be most beneficial and found that is not so, at least for the cases I analyzed. After doing a little search after posting I found a Mike Caro article that had the following statement:

“On average, when holding aces before the flop, you’ll make more profit in hold ’em against four or five opponents. Trying to cut your risk by playing against just one or two opponents is called “thinning the field.” It’s probably the most misunderstood and misused tactic in poker. Listen closely: Usually you shouldn’t try to thin the field”

Our analysis confirmed this for the Aces hand.

As you probably know, Caro's analysis was done in the Limit Hold'Em environment.

Do your results change in any way if there are large post-flop bets (i.e., to mimic NLHE)? Say a pot-size flop bet that is called by a villain if and only if villain wins the hand.

I think that would be a fairly straightforward factor to add to your EV equation.

Thanks again for doing this.

Doing flop bets would be difficult since the flop obviously affects the showdown equity. so choosing which flops is critical.

I actually varied hero's bet size in the pre-flop case and found very similar results for a limp as well as somewhat larger bets, say up to 10bb .

However, for much larger bets there was more variation and it appeared to favor having more rather than less opponents. But, I haven't spent much time on this

I didn't really mean to explicitly model different flops.

I meant to "reverse engineer" the (unknown) flop to mimic the pre-flop equity. Maybe I should have said the entire board rather than the flop.

Take the case of Hero having Aces. Suppose one villain has 35% pre-flop equity vs Aces. Then imagine that 35% of the hands go to showdown with Villain beating Hero and an additional post-flop pot-sized bet is made and called. In the other 65% of the cases, model it exactly as you have done to date.

I think this is easy to incorporate into Hero's EV equation but, admittedly, I have given this very little thought.

The analysis scenario you suggest is more realistic than what I had done but I’m not sure it is a simple extension.

I start with a hero hand and villain range and use Equilab to find hero/villain showdown equities. Then I use these showdown equities to compute EV. As I understand it, your scenario starts with the showdown equities and includes a future bet.

Anyway, the suggestion to look at other scenarios is just what I think should happen - - a further look at the “thin the field” strategy may be worthwhile. What I did generated in my mind that the strategy may not always be a good idea.

Here’s another result from the model. In all the bet size variants I looked at with aces, from limping to very large bets, facing five opponents was always EV better than a lesser number. And the larger the bet the higher the EV. But the larger you bet, the less likely you will get that many callers. So what would be the best bet size? This would require incorporating a fold equity factor into the EV equation where folding frequency depends, at least partially, on the risk/reward ratio. However, this extension still leaves open the question of how future action in the hand might change the results.

I suppose if more complete analysis was easy, it would have already been done.

Forgive me for being a bit facetious, but using your methodology, isn't it the case that almost every hand (not AA, obviously) will maximise its EV by thinning the field down to zero opponents?
e.g. If I'm reading the table on the blog correctly, for 99, you have a EV of 0.68 (big blinds presumably) vs one "random" opponent and 0.9 for 5 randoms. But if 99 raises and steals the blinds, the EV is 1.5.
In short, when I open with 99, I'm not hoping to "thin the field" to one, two, or three players (although facing one player with a VPIP of 50% of hands should be good for me). I'm hoping to get no calls at all.
I'm very sure that 87s is in the same boat. Although it might do reasonably well multiway, I'd much rather play it against no-one and just add 1.5bb to my stack.

No, it’s not true that EV is maximized at zero opponents in almost every case that I examined. The 99 hand EV table in the blog shows that for aggressive (top 35% hands) and maniac opponents, having I opponent has higher EV than no opponent. For other hands and larger bet sizes, 0 opponents was not highest EV. Sometimes it was. I had to make some simplifying assumptions to do the analysis and the one I made was that all opponents call.

Again, I am not advocating for or against thinning the field. I simply wanted to verify if the dictum was a good one and based on the simple analysis I did, I think it warrants further study. You and others commenting are a step in that direction and hopefully further math analysis will be done.

OK, cool. I'll have another read of the blog tomorrow. It's interesting stuff, as I said.

Great stuff!

I think intuitively some of it makes sense. In each test case we are paying a fixed amount so our cost is fixed and our return is only dependent on the size of the pot and our equity.

Since equity doesn't decrease as fast as the pot increases it's not surprising that some hands would have a gap with a few players but then sharply pick up with more players entering the pot.

I know you mentioned fold equity in the article but did youn specifically mean the opener's fold equity or were you also including our loss of equity due to folding in situations where we still have the best hand?

I would imagine our loss of equity due to folding the best hand increases much more quickly than both the pot size increases and our absolute equity decreases.

In the end I would imagine that balances out in favor of more people folding.

It might be easy to model that by slightly adjusting down hero's equity for each additional caller, maybe even a higher negative factor for aggressive opponents and less for tight ones?

Seems like that could lead to some interesting results.

The equities I used were showdown equities, obtained through Equilab. If fold equity were included in the model, the "total" equity would actually increase since villain folding gives you a sure win. However, this is balanced out by a lower win amount with a fold than that for a win with a call.

So my reference to fold equity was not set for any particular street or situation, but was made to emphasize that the model I used is certainly ripe for improvement.