The Case for Mixed Strategies

S Yama suggested that I pull a discussion of mixed strategies out of the thread, so I will.

So, let me start with a reasonably common scenario. The scenario is a UBT final table with UBT rules, and your opponent is a competent player:

You: $137,000
Opponent: $137,500

Both of you have a secret bet remaining and you bet first. What is your best bet?

P.S. Yes, this is meant to be the last hand.

 

2 cents

my 2 cents worth
max bet
assuming this is last hand
really not enough info to go on

 

I would contend that this is one the worst possible options here. To quote David Sklansky, "Do you see why?"

P.S. What information do you feel that you are missing? You are playing against a Bet21 regular, like Zweeky or myself.

 

?

since you are on the button i still say max bet since you both still have secret bets
even tho seasoned players i still believe you have to go for the win hands down

maybe my 1 cents worth

with option to double for less

or take the low and hope for a swing

51500 bet and possible double or split for win

no good plays that i can see

maybe 51500 and hope for surrender and a swing

let me know what i am missing

 

Guess right and hope you win?

If I remember correctly $100,000 is the max bet on the final table.

For me, being down by $500 and having to bet first I guess I'd bet $45,000.

My reason is if player one only bets $500 there is nothing I can do to win. They surrender and keep the low no matter what I do.

Should I get a chance to split and re-split, or split and Double-down.

I figure I need to play a little lower and hopes the other player bets bigger and loses or that I get a swing.

Now this maynot be the best play, but at a quick glance it is how I'd play it.

 

Toonces, I might follow your advice from the other thread and check my watch.

Maxwell, I think what you are missing is the potential for mind games between BR1 and BR2. If the bets were not secret, then whatever BR2 bet, BR1 would just match it in order to correlate and be a big favourite. With BR2 betting in secret, BR1 can either try to predict BR2's bet and match it, or just give up on the whole idea of correlating, and bet the minimum to ensure the low.

So the thrust of what this thread is about is the need to be unpredictable. If you always bet the max in this situation, people may begin to notice, and then it really is no longer a secret bet.

But you will want to limit the randomness to a range of equally good bets, betting enough so that you cannot be overtaken by a min bettor, even if they get the opportunity for a number of splits and DDs, and perhaps also ensuring you hold enough back to be able to split if you get the chance. Betting > 2/3 of a max bet would also be of benefit, if that still leaves enough for a split (which presumablly it doesn't in this case), since a BJ would then beat a max-bet win.

 

Bad situation there, toonces. Good teaser, though!

I rather like the idea of a 98K bet here. Obviously it'll take out any chance of a split, but here's the advantages I can see - and all are assuming a 100K max bet from my opponent since there's no dependable way to determine anything else. I'm also assuming this is an online tournament, where they do not deal you a down card unless you DD (which gives away your move). Since there are no live UBT EBJT's right now, assuming an online event is fair.

1. If we both surrender, I lose 1K less and win.

2. I've got the option of doubling for $1K, still retaining the low and hopefully suckering my opponent into an all-in DD in an attempt to retain the high.

3. I've got enough out there to lock him out if I get a natural and he doesn't (this is the only reason I don't like Rick's 45K bet as much).

4. If the dealer shows an Ace, I can insure for a mere $500, in secret, without losing any of the above options and possibly fooling the other player into a much larger insurance bet. If neither player insures and the dealer does get a natural, I lose 2K less and win.

Thoughts?

 

I think London's point here is what I'm trying to tease out. In a situation like this (which a situation I've been in many a time online), you will have no idea what you opponent bet behind you, so you really won't know after the cards come out if you are going low, going high, going high with a chance to surrender into the low, going low with a chance to double into the high, etc. Now, it is true that you can make certain bets that make one of these more likely, but I think it has limited value.

So, essentially here, you have 100 bets available. And your opponent knows that. And your opponent's goal (as London pointed out) is to get into your head and figure out what you might be thinking and match it. If he does, it is a disaster for you.

So, to start with, the 3 obvious bets to avoid are min, half-bankroll, and max-bet. Especially with the online buttons, these are just too easy to guess. And this is why I responded the way I did to Maxwell.

But I think that if you take the mindset that there is a perfect bet here, like LeftNut's $98,000, then you risk your opponent taking the same thought process as you and making a lucky guess. This is less true for Tex's $45,000 as that bet is more arbitrary. Keep in mind that if you opponent concludes that there are only 5 bet sizes that he can expect from you, and he's right, he has a 20% chance of matching you.

So, I think I would probably avoid min, half and max, and do my best to choose randomly from the rest. I would lean toward the smaller end of the spectrum, since given a choice, you would rather have the low than the high. I think you would like to leave your splitting options open, but I would argue that most of your play scenarios won't be that helpful, since you are only guessing as to his bet size, and you have no idea if you need to resplit to win, surrender to win, etc.

 

Toonces, There is a term for what I believe you are describing, it's called a "Swag". Swag is an acronym for "Scientific Wild Ass Guess".

 

So if you are playing on-line, 4 things to remember:

1) Do not bet min.
2) Do not bet max.
3) Do not split bankroll
4) Do, SWAG on down SWAG on down the road. ​

But I play all my tournaments in B&M Casinos where I don't have to worry about someone picking up a consistent way of playing the last hand. So what is the best bet in that scenario? Or do you just SWAG again?

 

Very interesting conclusion by Monkeysystem:

Sooooooooo, is SWAG on down SWAG on down the road in B&M casinos the answer? I think not. Here are some suggestions for BR2 any one of which I feel is a whole lot better than SWAG. When the amount bet plus the possible playing actions are considered there is enough variety to keep BR1 off-balance.

1) Bet max - double a 10 or 11 with remaining bankroll, never hit a breaking hand
2) Bet min - split anything (even re-split) and double on any non breaking hand, or surrender
3) Bet half bankroll - then double on any non breaking hand or split & don't hit a breaking hand
4) Bet 1/3 bankroll - then double on any non breaking hand or split (and re-split or double a split hand) & don't hit (or double) a breaking hand
5) Bet $51,000 - then double on any non breaking hand
6) Bet $5,000 - split anything (even re-split) and double on any non breaking hand, or surrender

These are suggestions just off the top of my head. My point is that mixing it up entails a lot more than just the amount of the bet. Seems to me that BR2 needs to adapt some poker bluffing skills (his physical actions as well as betting amounts) in order to keep BR1 off-balance. So betting SWAG = :vomit: in my opinion - I like to have a purpose in mind when I bet.

So does anyone have a opinion other than SWAG? Keep in mind that SWAG requires no skill or knowledge whatsoever. Or don't we need any skills when in BR2's predicament?

 

What amounts are available for you to bet? 1000 TO 100000 . I do not play on line, so I am not familiar with the bankrolls. I would like to take a try with this hand anyway.




Bets 1000 too small Br1 wins with surrender of 1000 bet Bets 100000 too large easily correlated, and probably the most frequent bet for br1 and br2.
Bets 50000 good choice for double but is a hand that you will most of

ten need to win in otder to advance
Bets 6000 to 10000 my choice coverss splits , resplits, doubles of min. bet. D0esn't lock you into a must win unless the leader bets small. I think with abet of this size that you will often get the low and less frequently the high. Punch holes in my bet if you will. I haven't done well in my recent tournaments and would welcome any help.

 

half penny

Ok This Is My Half Cent Worth
Since I Have Time To Think A Little Bit
I Think I Need All The Options I Can Muster Out Of This Hand So With That Said I Believe I Would Bet 55k And Hope For My Options To Come Into Play Split,double.surrender Etc And Hope For The Oh Luck Factor
This Question Really Made Me Think About My Game And I Believe I Have Been Iluminated To Another Piece Of The Puzzle In Tournament Play Thanks TOONCES For The Teaser And Feedback From Everybody:d :d

 

Monkey, I think that you have articulated the best explanation for this hand. Using a mixed strategy is not SWAG. Granted, I picked this situation because more than any others it should be obvious that there is no correct answer. If there was a correct answer, than BR1 would match it and you would be screwed.

As for toolman's playing alternatives, I don't like them. The fact is that your bet is secret and your opponents bet is secret. You know not much more than you did before when you see your hand. For example, You won't know if doubling down is enough for the win or if surrendering gets you the low, etc. I see an argument for bets between 51K and 1/2 bankroll, but that also means that you will likely only have the high, not the low.

I think what toolman is saying is that he wants to have either the low with a chance to double into the high, or the high with a chance to surrender into the low (rather than a pure low or pure high). But what you are missing is if you put yourself in that position (which of course you won't know whether you were in that position until the hand is done), you put your opponent in just as good of shape, with position to boot!

For example, you bet $55,000 with expectations of BR1 betting $100,000. You have the low with a chance to double into the high. But your opponent has a high with the chance to surrender into the low. And you can't risk surrendering all stiffs yourself because it could be that your opponent bet $27K or less and your surrender locked the game for him.

 

None of them??? Jeez!!!

Ya, we could go round and round why any alternative may work under some circumstances and every alternative can be countered if circumstances are different and BR1 bets different and/or reacts different to BR2's playing his cards. But nobody has come up with better alternatives because every alternative can be countered if BR1 makes the correct bet. We can go on with that for years without resolving anything.

The conclusion I'm leaning toward now in this thread is, as toonces said, "there is no correct answer". So if this is true, you take a shot and hope it works. Maybe SWAG is the way to go after all.

If someone has a better answer than SWAG, let's hear it!!! Not a hypothetical narrative but specific examples of the play(s) that give BR2 his best mathematical chance.

 

Mixed strategy

After reading posts on the subject of secret bets, here and in the thread “Last hand, secret bet” I decided that there is so much uncertainty that it needs a broader explanation. What I present here may be too much for some but at the same time it is only a very small part of the subject. I am using some of the numbers not for their exact values but to represent the situation, and the whole post is rather informal because: a) I am not a mathematician, b) the subject of game theory (statistics) involves many aspects where neither standard algebraic nor arithmetic manipulations can be applied.

Blackjack seems like a simple game but in conjunction with tournament strategy it produces literally thousand if not millions of possibilities.
In some situation a player have definite best betting and/or playing strategy and for best efficiency (payout), when it should not be changed even if that strategy would be known and the opponent allowed to bet/play in any desired way - then such situation is called strategic dominance.
If from a set of possible strategies a player always chooses the same one (regardless if it is dominance or not) then it will be a pure strategy.
A mixed strategy is when a player randomly chooses between possible moves.
The usage of the word randomness may cause a lot of confusion, as most people understand its meaning just as a chaotic, uncontrolled and unpredictable. However, that “mixed strategy”, we are talking about, is a mathematical term which encompasses a random input, provides no deterministic patterns, correlations, and biases, but follows a probability distribution – a set of chances of success (values) for all assumed/known strategies with attributed frequencies (probabilities) that have to add up to 100%.
Mixed strategy is a complex and a precise process that has almost nothing to do with commonly understood act of: “I throw out a few “bad” plays or pick a bunch of seemingly good plays (or at least better than some others) and randomly (but rather evenly) chose amongst them.”

I personally prefer to look at situations like these (e.g. secret bets) as a complex but fixed and completed, and specifically weighed (by me) that form one specific state of chances and strategies. I use simplification by grouping bets into ranges, and transform the game of “I think that he thinks that I think that...” into weighed chances of each of my opponent bets/plays (once again grouped). Some of the decisions may seem to me very close in effectiveness but then I may have to focus on finding more detailed arguments (and tournament provide them aplenty) to decide between them. After that work there will be one bet/play (or a specific range, from which again I can separate one single bet/play that I find to believe is the most effective).
We play tournaments with people who have and always will have patterns. It is fun to try to detect them, often times it is not a “tell” but recognition of how much the other person may know about the game (and the tournament blackjack depth seems to have no limits). We can always go deeper and deeper and find finer points that may offer us additional game advantage... It is an effort of how much focus I can muster, and how much I can expect from my opponent. Even if we suspect our opponent to be more experienced, we want to play our best game, we need to always strive to outplay her or him –that’s the way to make it really happen.
Even when we encounter a real intransitivity (as opposed to dominance strategy) and there would be usage for mixed strategy it most likely would be impossible to make that complicated optimal play. I would rather wade through my options, to the best of my abilities, finding arguments for better and better plays and then live with the consequences.
Mixed strategy assumes a thorough understanding of the situation and applying optimal play. One can of course try to apply it knowing that it may be not a truly optimal but a close one – and that is absolutely fine.
I think that blackjack tournaments offer so many intricacies that I can analyze them until I reach the time limit (my own set, or official) and then pick one “my best” play. I prefer it over analysis of the situation concluding that it was thorough enough to make a statement that it requires mixed strategy and that I can assign right frequencies and apply randomness to my actions.

Both methods can be fun, I value the process itself as much as the result it causes (unless I would play for a million bucks – then: Just give me the money. Well, we all have our prices, don’t we?)

S. Yama

PS
I haven’t use any numbers, I can’t believe it.
I will use them presenting an example of a full “secret bet” analysis and my “grouped”; my free time allowing doing so.

 

S. Yama,

Good background on mixed strategies as they relate to game theory.

Clearly, at the table, there's no time to do a full mixed strategy analysis, but I still think there is this concept that it can be important at times to be unpredictable, and that requires the acceptance of a mixed strategy.

I heard a similar discussion on the 2+2 pokercast about a month ago. Take the concept of rock-paper-scissors. Let's say that after studying a player, that you think that ROCK has an advantage over the mix of things your opponent throws. You shouldn't always throw rock, or your opponent will adjust. You need to weight your throws toward rock, but not exclusively or your opponent will adjust.

In the same way, if there is a best bet in a scenario where strategic dominence does not apply, you ought to weight your choices towards the better bet, but you cant make it exclusively, in my opinion.