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Debunking Nash
When villain deviates from NASH and we are playing NASH we gain EV
Ok we saw in the wall of text above that whilst Heads Up with both players playing SHOVE / FOLD and adopting equilibrium strategies if one player deviates from NASH then the other player gains in EV. However this is only if we remain within the toy game and are restricted to SHOVING or FOLDING. Mixed strategies can destroy NASH calling ranges as we saw above (and we will look into this more later). But what if we remain in the toy-game of NASH? [this might closely approximate real-life play in some formats].
Let’s look at three-handed play when we are in the BB. Let’s assume we are all playing PUSH / FOLD and all players currently are at the equilibrium. We have a base rate EV. The button then decides to play looser than NASH. What happens to our EV?
Most would say our EV goes up as the button is now playing sub-optimally. This is a myth. Our EV now drops and the EV of the SB goes up.
How can this be? We saw that in HU play the exact opposite: villains moving away from the equilibrium sacrifice EV to the other player. Well the reason is that in 3max+ whilst no player can increase his own EV by moving away from the equilibrium, he can lower another player’s EV by doing so.
In the above example by shoving looser than NASH the button lowers his own EV, and the gives value to the SB’s value range. We in the BB get fewer walks and so have to give up our BB more. So both we and the Button surrender EV which goes to the SB.
When the button plays tighter the opposite happens. Again the Button surrenders EV, but now it goes to us rather than the SB. The SB doesn’t gain as he still has to play PUSH / FOLD v us and we have seen that it is a negative EV proposition for him. Also we get more uncontested walks, increasing our EV.
People deviating from the equilibrium isn’t always a good thing for you if you stick to NASH. In real games, you are nearly always better moving from the pseudo-safety of the equilibrium and to go hunting in the fertile lands of exploitable play.
Comments
Not been able to post on the forums [fora?] for 6 months due to a self-exclusion from the site [playing on a stake elsewhere]. Have still been lurking though, and there’ve been some interesting discussions on NASH equilibriums [equilibria?]
NASH is one of the most intriguing and misapplied aspects of poker theory and there are a number of myths that need debunking. I’ll tackle a few of them in this post. Here are a few erroneous beliefs about NASH I’ve encountered recently:
Playing NASH is always +EV and you won’t lose money playing it [esp. HU]When villain deviates from NASH and we are playing NASH we gain EVShoving a hand that is in the NASH range is un-exploitable and / or always +EVPlaying NASH HU is un-exploitableNASH doesn’t SUCKHave a little think, and if you believe any of the statements above are true, then read on.
For those uninitiated, in poker the NASH equilibrium traditionally refers to a toy poker game where one player has the choice to SHOVE or FOLD and the other players have the option of CALLING or FOLDING. The NASH solutions describe ranges that once adopted by one of the players mean that the other players’ best responses are to also play the corresponding NASH strategies. Once all players are playing NASH no player can increase his EV by deviating from this strategy – indeed if they do they are sure to lower their EV.
Thus the game is in a stable equilibrium where a player’s EV will decrease the further they move away from their NASH equilibrium strategy. It is therefore unexploitable within this toy game [it is also maximally exploitative as players playing NASH are also maximally exploiting each other; NASH and Game Theory Optimal play could be described as strategies that maximally exploit the strategies that maximally exploit their own strategies – a point often missed].
The ranges for HU games are well known and can be found here:
http://www.holdemresources.net/h/poker-theory/hune.html
For those uninitiated the unsuited hands appear below the pocket-pair diagonal in red, and suited hands above in green. The numbers in the box refer to stack sizes. If the effective stack size is equal or smaller than the number then they should shove or call [depending on what table they are using]. Otherwise they should fold.
So let’s debunk those myths:
[this is coming down in multiple tl;dr walls of text, so bear with me lol...]
Playing NASH is always +EV and you won’t lose money playing it [esp. HU]
I’ve seen winning regs post similar statements to this on the forums recently. Its plain wrong. Let’s look at why:
The idea that NASH is always + EV rather depends on your definition of + EV, or reference point.
Certainly playing NASH is not necessarily +EV from start of hand. Consider heads-up play [the format NASH is most readily applicable to]. Playing NASH, the SB can SHOVE or FOLD and the BB can CALL or FOLD. Above 7.8bb eff. this is a losing proposition for the SB. Below 7.8bb eff. it is a losing proposition for the BB.
We can intuitively see why this is. At sufficiently short stacks the SB commits less of his stack pre-flop and so folding cost him less. At sufficiently deep stacks the SB has to risk more to win the BB's blind and so has to open-fold more. The BB's positional advantage then kicks in and he gets to be more selective with his all-in range.
What this is means is that at so called short-stacks such as 12bb [where bad players may incorrectly decide to play NASH], the game value for the SB at the equilibrium is around -8bb/100. At ever increasing stacks it encroaches on -50bb/100 [note it cannot go below -50bb/100 as open folding all of your range would yield -50bb/100 and so the SB will never settle on a strategy worth less than that].
The EV of playing SB NASH v BB NASH HU at different stacks can be seen here:
http://pasteboard.co/dQkdz3f.png
Any 'strategy' that has an EV of negative 8bb/100 from start of hand cannot be described as being +EV in my book. There are 3 caveats to this:
1. This is a guaranteed minimum expectation. Once the SB shoves NASH at 12bb, the BB cannot construct a calling range that lowers the SB's EV below -8bb/100. Any deviation away from the equilibrium by the BB will only increase the SB's EV.
2. -8bb/100 is much better than open folding your entire range. That would yield an EV of -50bb/100.
3. Within the toy games NASH solves the BB would also be forced to play PUSH / FOLD poker, so the overall game value @ 12bb at the equilibrium is 0 EV due to reciprocity: any losses in the SB are reclaimed as winnings in the BB
These caveats are why people erroneously believe adopting NASH is + EV. Let’s look at them in more detail. First from the SB’s view point @ 12bb eff.
Being better than open folding your entire range is a pretty muggy standard to aspire to. And just because a strategy has a guaranteed minimum EV, doesn’t mean we should settle for being negative EV from start of hand - certainly not when losing at a clip of 8bb/100 in position. Indeed it is very easy to construct a SHOVE range that is both better than open folding your entire range and also guarantees a minimum EV.
Let’s fold everything, and shove only AA. Here we do better than folding 100% and the BB’s best response is to also fold everything other than AA. If he deviates from this strategy then he loses EV and that EV goes to us. So we have an un-exploitable strategy that beats open folding and gives us a minimum EV. Boss, or not?
Of course we wouldn’t settle for shoving AA and folding everything else, as we can see it’s a terrible strategy and that being better than open folding 100% and also un-exploitable are not desirable things in themselves when we know we can do better. The same apllies to NASH.
[Note that if villain starts calling only AA we can move back towards the equilibrium strategy and increase our EV by exploiting his new nitty call range; only when both players are maximally exploiting each other and unable to increase their EV by changing strategies are they playing equilibrium strategies].
The reciprocity is the reason that NASH is seen as an EV neutral solution. Any losses in the SB are reclaimed when we have our turn in the BB. However outside of these toy games villains can and do play mixed strategies, and ones that vastly outperform NASH. So adopting NASH when you don’t have that reciprocity means you play a losing strategy in your SB and don’t gain from villain having to return the favour when positions are reversed.
Let us consider the BB for a moment. Let’s look at the SB shove only AA strategy. Well we can still profitably play the BB NASH range @ 12bb. Remember that at the equilibrium in the BB we win @ 8bb/100 v the NASH shove range, and if the SB deviates then we can only gain in EV.
Here the SB is miles from the equilibrium and shoving only 0.45% of hands instead of the 54% NASH recommends. All calls perform terribly except AA, yet the SB is folding so much that the game value for NASH BB is a whopping +48bb/100. [Just think on how bad a heads up strategy shoving AA and folding everything else is @ 12bb. And remember it yields an ugly -48bb/100.]
So despite all calls except AA being horrendously bad we profit even when calling 33% of the time due to the SB folding so much
But what if villain is playing a mixed strategy, does calling NASH still give you that solid minimum return of 8bb/100? NO!
Against my readless 12bb shove range the NASH calling range performs terribly. Indeed my EV is 50bb/100 when I shove into it. The BB doesn’t gain from me folding when not shoving as I play 100% vpip readless at this depth. So calling NASH - far from giving a gold plated 8b/100 in EV - can be as bad as -50bb/100.
For comparison’s sake, calling NASH v my readless shove range @ 12bb does worse in EV terms than the strategy of open folding all of your range except AA [which you shove] does in the SB. [the reason the EV is so low, is that I am not folding my SB the times I don’t shove – NASH is not un-exploitable outside of toy PUSH-FOLD games].
Now the above comparison is slightly disingenuous as my readless shove range is un-exploitably +EV from start of hand. The BB’s best response is still worth 22bb/100 to me. Still calling NASH is 28bb/100 WORSE than the best response, an ugly strategy by any measure. And not one that can be described as optimal.
So we can put the idea that playing NASH is always +EV well and truly to bed.
Shoving a hand that is in the NASH range is un-exploitable
I saw this exchange take place in a thread recently about a hand that took place @ 10bb eff.:
“If we're vs a villain who's calling 90% of the time, shoving the bottom end of the Nash shoving range can no longer be profitable”
“No that’s not true”
Yes it is. Emphatically so.
First let’s look at the value of shoving the NASH push range into the NASH call range @ 10bb. The game value of this scenario for the SB is -4bb/100 from start of hand.
http://pasteboard.co/dVwOY7k.png
We can see hands like 53s are better shoved than folded. This is due to the fold equity they receive coupled with the fact there are precious few 3x and 5x that in the call range
Let’s now see what happens when villain starts to call 90% of hands.
Well first of all the BB has moved from the equilibrium so our EV has necessarily increased. Indeed the game value for us in the SB has moved from -4bb/100 to +36bb/100.
But what of the hands at the bottom of our shove range?
http://pasteboard.co/dWcdoyl.png
We can see that the hands at the bottom of our shove range now perform terribly, and are indeed very unprofitable shoves. We are MUCH better open folding them. Of course our overall EV has increased due to the fact that our value hands gain massively from the extra calls and that dwarfs the losses at the bottom of our range. But that does not mean that all the hands in the NASH range are profitable shoves – many are clearly better open folded.
Our best response to a 90% caller out-performs NASH by a huge margin, and we can increase our game value to +41bb/100 by moving from the equilibrium ourselves.
If you want un-exploitable shoves with individual hands, then consult chubukov charts. They will tell you if folding is too nitty HU as if the hand appears in a chubukov range then you have at least one better option: ie to shove.
But shoving hands that appear in the NASH range can be extremely unprofitable and often much worse than open folding them.
Playing NASH HU is un-exploitable
Outside of the guaranteed reciprocity of toy games, playing NASH heads up is both unprofitable and exploitable.
Now some may bridle at this and state that playing NASH is by definition un-exploitable. Well, you are wrong. Let’s demonstrate why.
Game state:
12bb eff.
You play NASH SHOVE / FOLD in SB
You call NASH CALL / FOLD when shoved on in the BB
You are extremely exploitable and very unlikely to win against anyone remotely competent
Let’s look at why:
When you are in the SB:
At 12bb you will be shoving NASH. The NASH calling range maximally exploits you and indeed I will win at a rate of 8bb/100 by adopting it. You may say that I don’t know you are playing NASH so I will lower my EV by being away from the equilibrium. True, but even my readless call range @ 12bb means I still have a win-rate of 5bb/100 against your shove range, and the fact I am not going to be playing SHOVE / FOLD will hurt you when you are BB. But NASH ranges maximally exploit each other and I can maximally exploit your shove range by calling NASH, and deny you the chance to do the same to me when I am SB.
When you are BB:
When I shove you will call NASH. We saw earlier that my shove range v your NASH call range is worth a whopping +50bb/100 to me. And I am not going to do you the favour of folding when I don’t shove. My all-in adjusted win rate in the SB playing a mixed strategy @ 12bb HU is +4bb/100 the polar opposite of what a NASH SB strategy v me would be.
It worth noting that readless I play 100% of hands in the SB @ 12bb [my actual vpip lifetime is 86% due to playing with reads]. NASH is nitty, playing only 54% of hands. Also I am by no means a great or good player. Plenty of players will be crushing my SB win rate at this depth.
Also many approximate full GTO solutions exist for 12bb eff. No GTO SB strategy remotely resembles NASH PUSH / FOLD. All involve mixed strategies. What this tells us is that GTO mixed strategy outperforms the NASH PUSH / FOLD strategy. If it did not, GTO would adopt it. Therefore even if you play perfectly v a GTO mixed strategy you wouldn’t make up for your losses in when you are SB playing PUSH / FOLD.
So I hope I have by now shown that NASH does indeed suck as a readless strategy. Especially in heads up play and HUSNG’s please forget you have even heard of it above 6bb. It is profitable below 6bb for the SB and you can and probably should adopt it readless below 5bb head up from both positions.
But above that narrow subsection of the game: move away from the equilibrium, play mixed strategies and forget the myths surrounding NASH. It SUCKS.
we always want the max ev, always, 100%!