The Case for Mixed Strategies

Won't it very often be the case that you come up with quite a wide range of bets, all of which are entirely equivalent as far as the 'pure' strategic analysis is concerned? In that situation, picking one of these at random provides the needed unpredictability at no cost to you.

For instance, rightly or wrongly, I came up with the range 68K..98K as my preferred bet for this example. That leaves me a lot of scope for varying the actual amount I bet, each time that I'm unfortunate enough to be faced with this situation.

 

It depends on the situation. For example, if BR1 has a 10,000 lead over you, he will be able to correlate if he guesses your bet plus or minus 10,000.

 

weighing opponent's chances of bet/play

I think Colin’s proposed range of bets (68K to 98K) was in reference to toonces’s original post with BR2 and BR1 bankrolls respectively 137K and 137.5K.
We can use this example to make a case for weighing chances of opponent’s bets and decision.
Colin, could you try to come up with a specific numbers for BR1’s secret bet being from 1K to 100K? Sure, some bets could be grouped. You could assume an “imagined” opponent with a specific betting characteristics, but it might be more fun if you assume a particular player with a style of playing very well known to you.
I will pick up the subject after this weekend.

S. Yama

 

In arriving at the range 68K..98K, I really only considered two possible BR1 bets: 1K and 100K. These do seem to be the most likely possibilities, and if you restrict your thinking to just those two, then I don't think it actually matters what weight you give them, you can chose your bet to fully cater for both.

1K and 100K seem likely because -

• a skilled player might bet 1K to take the low, or might anticipate that I bet 100K and match me.
• a novice might bet 100K, as a matter of course.

If BR1 bets 1K, then the most they can conceivably win is 8K, so we could guarantee the high with a bet >= 9K.

If BR1 bets 100K, then 99K takes the low, and 98K allows for both players to surrender.

So that's an initial range of 9K..98K. But to be able to DD or split past a 100K win, we need to raise our lower limit to 51K, and to be able to overtake with a BJ, we need to raise it still further, to 68K.

Toonces, I take your point about the effect of different leads. As BR1's lead increases it has a double impact; the range of equivalent bets shrinks, and also the leeway BR1 has for getting close to your bet grows. If I were to apply the same criteria as above to a 10K deficit, I would end up with a bet range of 68K..78K, and if BR1 knew I was going to bet in that range they could guarantee correlation by betting within that same range.

So I would have to change my criteria. If I drop the BJ and surrender considerations, then the range expands to 51K..89K. Even so, BR1 still has a coin-flip chance of correlating by betting in the middle of that range.


If I was to try to assign a probability to some possible BR1 bets (for 137.5K vs 137K), then I suppose it might be something like this.

A complete novice, who has shown a tendency to bet an amount, X, in previous rounds, with no thought for correlation -
1K: 20%
X: 30%
Split bank: 10%
100K: 40%

A complete novice, with no particular favourite bet, who has been mostly betting big -
1K: 10%
Split bank: 25%
100K: 65%

A complete novice, with no particular favourite bet, who has been mostly betting small -
1K: 40%
Split bank: 20%
100K: 40%

A skilled player -
1K: 60%
Split bank: 5% (That would be to correlate with my own, supposed split bank)
100K: 35%

S. Yama, I'm not sure if that's quite the sort of thing you were looking for, but it was quite fun, picking those numbers out of thin air.

 

Nice to see the level of discussion on Blackjack tournament theory remains at a suitably high level here. This is what I love about the game.

Cheers

Reachy

 

Amen to THAT! Not one insult or personal cheapshot in the bunch. :cheers:

I still very much like my 98K answer to toonces' original teaser question. Trying to put your opponent on a specific bet, or trying to make a bet that he/she will be unlikely to guess, is an interesting concept. But that's all a guessing game in this case. Perhaps 96K would be a good choice as well, in case the opponent happens to guess my logic correctly and matches my secret 98K number. In any event, since I am behind before the deal, I like the idea of having enough ammunition on the table to allow many options, as well as knowing that my unmatched blackjack means Game Over. When I won the online tournament to gain the UBT TV seat, the final hand was somewhat similar to this although I had a decent lead before the deal in that case. My very good but unfortunate opponent got a non-splittable hand, I got J-A on a large enough bet to put me slightly more than 2x max bet ahead, game over.

Of course, if you run into me online in a similar situation, this doesn't mean I'll actually bet either 98K or 96K. :laugh:

 

But in this case, the extra ammunition does not alllow any extra options; in fact it precludes one - splitting.

68K would be the most flexible bet, in terms of how you can use your ammunition. A BJ can win it for you, and you can still spilt a pair. But 68K..68K is not much of a range! Without even realizing it, I had already discarded the splitting criterion, to arrive at a wider range, much as I described widening the range by discarding the BJ and surrender criteria, when BR1 has a bigger lead. An alternative would have been to discard the BJ and keep the split, making the range 51K..68K. Maybe that would be better?

Thinking about it now, I probably should have assigned a greater probability to the 'split bank' option for a skilled BR1. It looks like such a tempting bet for BR2, that BR1 might be more inclined to correlate with it.

Any bet in the 68K..98K range is just as good as 98K. So rather than say to yourself - "I should bet 98K, but maybe he'll guess that, so I'll knock one or two thousand off" - the best approach would surely be to generate a random number in that range.

 

Good points, Colin, particularly about losing the option of splitting. But I would not prefer 68K for two reasons:

1. It's a perfect 1/2 bankroll bet and easier for the opponent to guess.

2. A BJ would not win it. It would force the opponent to get more chips in play but he/she could still win with a successful split/double.

I am enjoying this discussion quite thoroughly!

 

Indeed.

Oops. You're right, of course. I overlooked that. Although, it's ameliorated to some extent by the fact that BR1 won't know how much you bet. In a live game, having not bet enough to lock out BR1, perhaps some serious bluffing would be in order? A loud cry of "Yippee, I can't lose!" when you see your BJ, might induce them not to bother with the 'pointless' DD.

So, for completeness, the range of bets that does all that you want would be 92K..98K. All you need is a seven-sided die to roll, and then you can make your choice!

 

It doesn't have to be a guessing game

"Trying to put your opponent on a specific bet, or trying to make a bet that he/she will be unlikely to guess, is an interesting concept. But that's all a guessing game in this case."

LeftNut, you are basically right: it is a guessing – though, something has to be added to that statement.
It is a guessing game when one chooses not to pay attention, or disregards, or doesn’t look close enough, for lots of influencing factors. Some clues are obvious, some have only trace evidence and are very subtle -but they all can be translated, with better or worse approximation, into solid numbers (precluding some clairvoyance, which I don’t condemn but don’t consider worthy of discussing it here).
Blackjack tournaments are (at least to me) “games of numbers” that include a well known facts and statistics, but also transform psychology and game-time adjusted expectation.
The more precise the numbers we use are the more evident the scale and relation to anything else one would want refer it to is. For many people numbers don’t have to be exact. Rounding can be just fine, for some others -seeking for a directional indication (bigger or smaller, meaning better or worse) is satisfying, and for quite a few -the social and other aspects of tournament participation outweigh any mathematical analysis. And this is all fine, as long it works for the interested parties.
We all come to play bj tournaments with different backgrounds and experiences and we utilize unlike processes and we are having different expectation.
However, for people inclined to know more, there is almost always an extra step that can increase their accomplishments. It effectiveness is sometimes very small, but those small improvements can be made almost every single time you act. The minimally better situations can be continuously boosted up by minimally better bets/play, so the final product of many very small better plays can make a significant difference in the end.
Is the effort worth the work – the answer to it will be different for different players.

S. Yama

 

Mixed Strategies:

Profound, Yama.

I'd agree and state it is evident in the number of wins of particular people who pay very close attention to small significant differences. You and two others who post, or run this site, or used to post on this site are examlpes.

Unfortunately for me, I tend to be a rounder and a dismisser of small significants. I've kind of learned it tells. But the energy and committement for the two percent reward, well, utility has many ramifications. Oh my head hurts!

 

yeah, yeah, too kind

Noman, the other description of my posts would be stuffy, and affectedly wise, or perhaps as Reachy and Colin would say, too clever by half;
But thank you very much for your kinder version.

S. Y.

 

Theories

Just a few words before I go into specifics of the discussed case.
Weighing chances, in many, rather differentiated forms (including game-time related weighing) serves as a clasp and the core of the optimal play in my tournament strategy system. Applied as overall and ever-present component, it is probably the least understood subject by general players, though all (or majority) individual uses are well known.

In our situation we have to attribute a chance of our opponent different bets, which at first may seem as picking it out a thin air. But after a few trials, with understanding of what to look for, it hopefully would become a useful tool.
The “guessing” has to be as much scientific as possible. It has to based on observable and repeatable actions, set in various but specific fields, producing data that allows us, in conjunction with known laws, to form hypothesis being a part of a theory, which can be implemented and produce desired and measured results.

We started with hypothesis of usefulness for “mixed strategy”, countered by hypothesis of “one play -the best we can come up with” (both are parts of broader tournament strategy theory used by authors/players), the second one needs to have chances for opponent bets assigned/guessed. Once we have this we will need to know/find the effect of correlation between two players and their bets combinations. Then we will quarry which one, or what group of bets is better and how much better that the others.
Upward and Onward,

S. Yama

 

Thank you, S. Yama, I'll be looking forward to your further discussion!

 

overcoming your opponent

Before we get to Colin’s great response, and his version of weighing chances of our opponent bets, we should spend some time looking into the consequences of having particular two bets (my numbers disclaimer from the post #19 applies here).
Generally speaking, it is a very complex function of two bets and their relation to each other and both of them to the gap between the two bankrolls. In toonces’s case, since BR1 leads by less than a minimum bet, at least the gap issues are reduced -as any difference in betting is at least twice the magnitude of the gap, or more.

Let’s look at a few characteristic groups:
First group can be form by BR1 betting so small that even winning a three-bet does not protect her against BR2 winning a single bet. Simplifying, we can call it BR1 bet < 1/3 of BR2 bet. In those cases all that BR2 has to do is to win his bet and that chance is about 44%.
If BR1 bets more than 1/3 and less than ½ of BR2’s bet she can protect against BR2’s winning a single bet by winning a triple bet. This does not happen too often but it slightly reduces chances of BR2, who in response to possibility of BR1’s triple bet can double his bet, and so on, the total BR2 chances to overcome BR1 by winning should be a bit more than 43%.
But wait a minute. If BR1 bets more than the gap (in our case that would be any bet) most of the pushes by BR2 will have BR1 losing, bringing up total chance of BR2 surpassing BR1 to about 48%. There are a few more other specific situations in those ranges, but on average it should be around 45%.

When BR1 bets less than ½ of BR2’s bet, we can round up BR2’s chances (betting and playing first) to about 45%.

When BR1 bets ½ or more, but less than BR2's bet, then we have the most interesting situation and often times blackjack basic strategy should not be used by either of the players. Based on the tournament strategy skills of both players BR2’s chances may vary from under 30% to over 40%.

When BR1 bets at least ½ of BR2’s bet but less than exactly BR2’s bet, we can round up BR2’s chances to about 35%.

(We can use the same chance and apply it to one more specific situation, where BR1 bets exactly 1K more than BR2. Main reason for it would be cases when best play for BR2 is to surrender and BR1 following the suite, causing a play-off)

When BR1 happens to bet exactly what BR2 bets, then on top of any swing and winning bj vs. BR1’s regular win, BR2 may slightly increase his chances by winning a double bet to a single win, or worse, by BR1.

When BR1 bets exactly the same amount as BR2, we can round up BR2’s chances to about 25%.

And finally, when BR1 bets more than BR2 by more than a double gap, we deal with two basic situations. One where BR2’s bet is less than half of BR1’s, and then BR2’s rate of success is about 50%, and the second when BR2’s bet is more than half of BR1's and then his chance to overcome BR1 is a few percent better than 50%. We can sum it up by saying:

When BR1 bets more than BR2, we can round up BR2’s chances to about 50%.

It turns out we can approximate BR2 chances to be contained in only three categories: 25%, 35%, 45%, and 50%.
For most of the possible bets, BR2 chances, starting with his smaller bets and going to bigger bets, will look like that: 45%, 35%, one bet of 25%, one bet of 35%, and 50%.

Knowledge of these numbers can be used to identify a few ranges of preferred bets based on how we weight our opponent chances of different bets. There are some key identifying numbers that could be seen with a helpful graph or using some mathematical functions. But also, we can calculate BR2’s total chances for all of his bets once we agree on BR1 bets.

So onward to BR1 bets...

S. Yama

PS
I made correction to situation when BR1 bets more than BR2, at first I tried to push it into the category of 45%, while it is slightly better than 50%.

 

opponent's bets

As in Colin’s post, we can guess opponents’ secret bets (or when they bet after us) based on a category of players we put them in.
We can assign the chances for particular bets and ranges, and in spite of the fact that often time we will misjudge, the more often we practice it the better our results become. We can practice this “guessing” on almost every bet our competitors make, through out the rounds we play, and whenever we are the spectators.
The clues could include player behavior at the table, specific plays, way of making bets, plying style of the particular group, or even a prevailing style of locals/unknown players, etc..

However, the most important clue is the opponent skill level. We always have to ask ourselves if and to what degree the opponent would play less skillfully, given a chance to make a mistake by our intentional play that is less than perfect against perfect response. This is important because we encounter those type of questions (and opportunities) all the time, unlike rather uncommon secret bet situations where other observation are helpful.

When we have no idea what the opponent will bet (after asking this ourselves), then it means that all-possible bets are equally likely. This means that for allowed bets from 1K to 100K, each and every bet should be assign chance of 1.0%.
Almost always though, some bets are much more likely, and some very unlikely. When I have time to make a thourough study of somebody’s bet, I like to take 5% to 20% and spread it evenly along the whole spectrum- this base covers good bets and some incidental bets. The 20% bet base would apply for a novice, more unpredictable player, and it would mean that each bet out of 100 possible would have a chance of 0.2%. This leaves us with 80% of bets ---the rest - that will be assigned to specific bets.

Let’s take three very different types of players, suggested by Colin, make it a bit more precise, and see how our different bets would perform against them.

Even a complete novice would bet on the minimum bet, more on “half the bankroll”, and most on the maximum bet. We could use the 80% of bets (left after assigning 20% for the incidental base) and assign 10% for the min bet., 20% for half brl, and 40% for the max. This leaves us with 10%, and I would tend to assign it to three other possibilities: bets very close to the minimum bet (2K, 3K), half the max bet (50K), and just under the max bet (99K).

Our weighing could look like this:
1K..........2K..........3K..........4K...to...49K..........50K..........68K..........69K...to...98K..........99K..............100K
10.2%....5.2%......1.2%.........0.2% each............2.2%........20.2%..........0.2% each............2.2%.............40.2%

For a different type of a player, let’s use about even chances for min., mid., and max bets:
1K..........2K..........3K..........4K...to...49K..........50K..........68K..........69K...to...98K..........99K..............100K
23.2%....5.2%......1.2%.........0.2% each............3.2%........23.2%..........0.2% each............2.2%.............23.2%

And let’s assign greater deal of minimum and smaller bets to more experienced type, and the incidental base of 10%:
1K..........2K..........3K..........4K...to...49K..........50K..........68K..........69K...to...98K..........99K..............100K
60.1%....5.1%......3.1%.........0.1% each............1.2%........10.1%..........0.1% each............1.2%.............10.2%

What are your feelings about what are the best bets for BR2 against each type, and how big will be the differences?

S. Yama

 

Pretty even results

Instead of showing effectiveness of all BR2’s possible bets, let’s look at the ones at critical intervals, right before them, and right after them. The numbers represent chances of BR2 overcoming BR1, playing expertly.

The top line represents BR2’s secret bets.
The second line is for BR1 being a novice, betting heavy on max bet.
The third line is for BR1 being a novice, betting about evenly on min., max, and half bankroll.
The fourth line is for BR1 being more experienced, betting mostly minimum bet.

1K..........2K........3K.......10K........48K.......50K.........66K.........68K.......70K.......98K........99K........100K
46.7......47.0......48.4.....48.9.......49.9.......49.4.......49.2.......45.4.......47.3......43.6.......37.4.........33.6
43.4......45.0......47.7.....48.4.......49.5.......48.7.......49.5.......42.8.......45.0......43.7.......38.7.........36.6
34.2......39.2......45.4.....46.5.......47.1.......46.7.......47.0.......44.1.......45.0......44.4.......42.2.........41.3


Guys, feel free to leave any comments. I will be back in a week.
S. Yama

 

My real difficulty would be to try to find a way of applying this kind of logic at the tables, working within the time constraints, without causing my brain to implode.

On more than one occasion, I've attempted to think too deeply about a situation and ended up making a worse bet than if I had gone with my initial, more simplistic thoughts.

I think the hope has to be that by analysing specific examples in great depth, after the fact, some insights may emerge that can be applied more generally. Like the chess player who glances at a board and recognises a familiar situation, we hope to be able to spot recurring patterns and not have to think everything through from first principles, with the clock ticking.

Alas, I am a long way from such grand-mastery.