The equity chops keep the pots (and variance) smaller by lowering the average size. A 200BB pot can only happen if a player is all in on the river, or if he goes all in prior to the river with 0 equity.

The mirroring reduces the variance by ensuring that players are dealt the same whole cards and run outs as the bot. So if the player is dealt aces 60k hands in a row, so too will the bot.

I'm not sure that the use of mirroring will change the efficacy of the equity chops in terms of reducing variance. I'm just an amateur observer of this competition though. Maybe someone more knowledgeable can confirm.

What happened today, where's the streams??

What are you doing Dong stahp

It will lower variance in the situation where one mirror ends up all in before river while the other does not.

Ok so for this one you know if you get it in with 67% equity you should win 67% of the time right? But in real world terms for that 1 hand, you only win 100% or 0% (I guess chop some too), but anyways, the whole point is that you will only converge to 67% wins over a really large sample. By doing equity chops it makes it so you converge directly, you just win 67%, and it's over. That's why it lessens variance.


Ever played somebody and had them whine relentlessly about card distribution? This takes that away. You play the hand from both sides, without anybody getting any extra knowledge, and it doesn't come down to "well they set over setted me 10x more" or "they flopped flush / got AA / whatever way more often"



<3 Dong

Does running it twice reduce variance? Yes? Four times? Now just pretend that all in equity chops means running all board combinations (it's essentially the same thing). There, easy to get GL guys, all five of you (Libratus included) are monsters

If you take the big cooler situations preflop it wont make any difference since you both play the hands the same so if you get AA and the bot KK and a K hits its the same on either side. So its not really a factor for them pots because of the mirrored hand effect.

Its main purpose is for when there's a chance you play the hands differently. If you 3 bet 23ss and get allin with a flushdraw on the flop but the bot just folds 32ss pre in its mirrored hand then that's a chance for a big disparity depending on the results of the allin.

Won't both players remember the hand, at least by the flop?

Donger Kim you are right.

I will explain to al the people above here what (i think) you mean with an other example.

4 libratus bots are going to play each other they get mirrors hands.
so 2 are a team (they get mirrord hands) and other 2 are a team (they get mirror hands)
The EV in ANY hand will be EXACTLY the same as net results if you look to 1 team combined. It does not matter if you run the board with an allin in this case or just split the equity. Ofcourse if you look to the result of 1 libratus bot and not as a team of 2 there will be more variance.

only if they take the same line and both sides get allin. If one side takes a funky line and doesn't get allin then it leaves room for variance with allins.

not true. bot A from team 1 does funky line bot C from team 2 does funkly line.

In this example a team will never get above a results of 0$ win as wel never get as a team above the EV of 0$ does not matter how much hands the teams will play against each other

what i say above is partly true because bots also randomize with the same hand some times.

What i dont get is why they just not run it out. then you have at the end an net result and you can figure out the ev result. the ev result and net win result combine is extremly strong evidence in mirror hand play

Variance is a measure of how far data is from its mean. Here the data is bb won or lost per hand. Because the mean bb/hand for either side is 'close' to 0, having data points of +/-200bb (no equity chop) will lead to higher variance, compared to +/-50bb (equity chops).

Dong is right that all in equity chops reduce the effectiveness of mirroring. When you combine the two they work against each other. But I'm pretty sure the combination still reduces variance more than either one on its own.

Is it really that hard to understand what dong is saying?

example hand dong has AhKh bot has 8d8s

dong opens 200 bot 3bet 900 dong 4bets 2700 bot cals

flop 8h10sjh

bot checks, dong bet 3400 bot goes allin dong cals - split equity

mirrow hand bot AhKh dan polk has 8d8s

bot opens 300 dan 3bets 1100 bot 4bets 3300 dan cals

8h10sjh

dan ckecks bot bet 3700 dan cals

turn js

dan ckecks bot goes allin dan cals


Is it that hard to understand that NOT running the cards instead of split equity INCREASE the variance in this hand with dong and dan together.

I really dont get it that you guys dont understand this

Yeah, running the cards out rather than splitting equity decreases the variance of results in cooler situations. In your example the net result is zero. But that also means you're ignoring a possible skill edge.

In your example the humans get it in with equity and the bot gets it in drawing dead, so the humans played it "better" and deserve to win chips in this situation.

But like I said above, in this case mirroring isn't as effective in evening out the luck in card distributions because you're not always seeing the same number of flop/turn/river cards on both mirrored instances.

To add to my previous post.

The data actually being the sum of the mirrored hands complicates this a bit. But I think on average equity chopping will reduce the variance in the data. I would be very interested to hear the researchers formal argument here.

Yes you are correct its entirely possible that mirroring + equity chop can increase variance when they sum up the results of the mirrored hands.

You could similarly construct an example where the variance is reduced from the equity chop. Take 88 vs AK where one sees a flop in a single raised pot and AK wins, and the other where they get all in pf.

It's not very clear which scenario will happen more frequently and for what magnitude. But it does make it less clear if equity chopping helps.

how does a superior strat wins over an inferior one? it plays the same hand differently if it were in opponents place. so by mirrored hands it converges faster to the true winrate.

hilarious

Wow this variance stuff isn't that hard guys.

All in equity splits reduces variance ON AVERAGE.

Mirror hands reduces variance ON AVERAGE.

Are there one off situations where this isn't true? OBVIOUSLY

Do equity splits and mirror hands reduce variance overall? OF COURSE

Poker is dead. Everyone's understanding of variance is solid.

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Any updates on results?

Btw, Libby's twitter account is gold:

^lmao

Half way-ish results. RIP Humans (and RIP Timex lol)

That Twitter account is amazing