You're BR1, with two hands...

let's find out

Nice discussion, though somewhat peculiar because more than two answers were offered without the final consensus. Nice contributions by Archie, fgk, Reachy, Monkeysyste, BlueLight, and toolman.
I was hesitant to make any posts but since I decided to make them, this thread is a good one to put in my two cents.

There are many approaches to a problems like this: calculate all possibilities with weighted approach (including anticipated % likelihood of different bets and plays for all players), assume optimal bets and plays by all, pick one most likely play, pick just one variant that seems most significant, etc.
The first approach is the best, but rarely we would have time and “coolness” to do it at the real tournament.
Here –we can, and should, and perhaps we will. It is not going be that difficult, and we don’t have to be exact. It may be more important (it usually is to me) to use the right methodology and the final numbers can only be an approximation.
Before we roll up our sleeves and start working let’s agree on the set up and clear some thesis.
It is almost the “winner-take-all” tourney (prizes: 50K first, 5K second). There are five players but the last two’s bankrolls are so small that practically it is three players game. Two hands to play.
BR1 (both times bets first) has 17,000,
BR3 (bets always second) has 6,200, and
BR3 (bets last) has 6,450
Minimum bet 25, no max bet, no surrender.

A clarification. It was said in the thread that the value of a ½ max bet lead is of little importance compared to other type of tournaments. Just the opposite is true, as betting ½ max by the leader guarantees first place barring a close opponent getting a blackjack, which still can be protected by doubling, also losing this ½ max bet makes BR1 a winner if nobody else with a close bankroll wins their bets. In limited maximum bet tournament other players winning their max double bets can overcome the same ½ max bet.
Also, significance of other players winning by getting a blackjack is one of the last aspects of the game I would worry, and pay attention to it only when most other aspect of the game got analyzed. The chance of uncontested blackjack for two opponents in two hands (at least one of them will get a bj) is 16.9%, and this can be protected by BR1 winning “a right” bet, but when you exclude players who lose on the first trial it is “only” 13.1%.

My 30 seconds analysis at the table would most likely look like this:
There is a good chance that BR2 and BR3 will bet big or max.
The most (without bj) BR2 can get to is (6,450 x 2 x 2) = 25,800.
The most (without bj) BR3 can get to is (6,200 x 2 x 2) = 24,800
I have 17,000 – I need a net win (at the most) of 8,900 (or 7,900 to BR3) to cover it.
Bet 9,000? Bet 4,500 (4,000 to BR3)- chance for lock with a good double? Bet 6,000 – bj gets me there?
A two step bet, so two wins would get me to my goal – that would be betting at least 3,000 on the first of the two hands remaining?
How to get there without giving too much if they swing me on the first hand?
BR2’s all-in gets her to 12,900 – I can bet up to 4,000 and still be the chip leader even if I lose and they win.
Conclusion: bet 4,000

Let’s try to go through more exact process to figure out the chances and values.
Toolman, would you be interested in working some real numbers and correlation we may think as the right ones for the three players involved? Approximations will be okay, we are trying to find out just what are/may be better bets and plays.
First we need to look at frequencies of our opponents’ results. Since we are dealing with all-in bets for BR2 and BR3 we can use Wong’s published chances and correlation. Normally we would use just the three outcomes: win/push/lose for two players, but since it was specified in the thread that blackjacks may be important – lets make blackjack as the separate category. We can distinguish four outcomes: bj – 4.5%, no bj win – 39.5%, push 8%, and loss – 48%. Now let’s look of outcomes for both players, it is not difficult to work in blackjacks as they can be treated as independent results and happen proportionally to all other outcomes.
Let’s write down in rows (spreadsheets would be a convenient medium, but are not necessary) what are (approximately) the chances for BR2 and BR3 for all sixteen possibilities (both bj, bj/win, bj/push, bj/loss, win/bj, win/win, etc.).
Done?
We don’t need the exact numbers for each category right now, when BR1 bets minimum, but will need it for analyzing BR1 making a bigger bet.
Some results will present identical consequences and will be grouped together. For BR1 betting minimum we will have only three groups. Once all participants (if any) of this exercise are ready we will proceed.
In the next steps we will have to check what bet and play is best for BR1 based on the lead size and the number of opponent left (one or two). And finally we will calculate chances for BR1 betting 4,000 (or other bets) to see how they perform against the minimum bet.

S. Yama

 

Betting minimum

Assuming that BR1 bets minimum (25) he will have about 17,000 regardless of his hand result.
BR2 will have 12.900 if wins, 16,175 if catches bj, 6,450 if pushes, and zero if loses, assuming betting all-in.
BR3 will have 12.400 if wins, 15,500 if catches bj, 6,200 if pushes, and zero if loses, assuming betting all-in.

It is obvious that if both opponents push or lose and BR1 has about 17,000, there is no chance for them to catch BR1 even with a blackjack on the last hand. In those cases BR1 bets minimum and has lock.

If both opponents win, BR1 has to rely on luck and bet enough to cover winning an all-in bet by either one of opponents. It is almost certain that in response to BR1’s big bet at least one player, BR2 or BR3, would take low. In effect BR1’s chances are about 47% and comprise of winning the last hand when neither BR1 nor BR2 receives a blackjack and when BR1 pushes and neither opponent win. The play for BR1 is similar to basic strategy but it would be interesting to check if in some cases “push as good as win” would not change the optimal strategy (for example hitting BR1 twelve vs. dealer’s four and the opponent seventeen, and standing on twelve vs. dealer’s three and the opponent twenty).

When only one opponent wins his or her big bet and the other loses or pushes, then BR1 best option is to take the low. Since BR1’s lead is less than one-third of the contender bankroll, betting anything from minimum to the lead minus a chip is correct, but if the second opponent have pushed protecting against her blackjack should be considered. Without BR2 making a mistake on the last hand, BR1 takes first place whenever BR2 doesn’t win, about 56% of the times.

Though we recognized only three categories that effect BR1 bets on the last hand, there are sixteen possibilities of different results for two players:


We should have something like this:
..........BR2......... BR3......... frequency %
1......... bj.......... bj............. 0.21
2......... bj.......... w............. 1.79
3......... bj.......... p.............. 0.36
4......... bj.......... l............... 2.18
5......... w.......... bj............. 1.79
6......... w.......... w............ 26.22
7......... w.......... p............. 16.4
8......... w.......... l............... 9.82
9......... p.......... bj.............. 0.36
10....... p.......... w............... 1.64
11....... p.......... p................ 1.0
12....... p.......... l................. 5.0
13....... l........... bj............... 2.18
14....... l........... w............... 9.82
15....... l........... p................ 5.0
16....... l........... l................ 31.0

For our purpose we grouped 11, 12, 15, and 16 as lock. Configurations 1,2, 5, and 6 -taking high – having 47% chance. The rest of possibilities - low - having 56% chance.
When we multiply frequencies by the chances the altogether chance for BR1 to take first place by betting minimum on the next-to-last hand and then playing optimally is almost 72 %.

S. Yama

 

Thanks for the analysis S. Yama. I am a little pressed for time from now until the end of June so I didn't have the time to respond. Hopefully the members endorsing other plays will have time to respond with their probabilities of succeeding. I would be interested in the results.

 

bogged

Excellent post Mr Yama. I started to analyse this situation in exactly the same way that you have just done (i.e. looking at every possible outcome and the likelihood that it would happen) but it ended up getting far to complicated and I had trouble getting the numbers to work. I started on every possible outcome inc. BJs for the penultimate hand and then the final hand so the total number of final hand outcomes just became massive and unmanageable. Also, because I looked at 3 players outcomes, Wongs numbers from Table 4 don't have enough detail, so I tried to extrapolate and it didn't really work; I lost confidence in the accuracy of the numbers I was generating so I dropped it. Nice to see I was sort of on the right track though...

Looking forward to your other workings so that we can finally decide which would be the better bet.

Cheers

Reachy

 

Just one more comment about S. YAMA's work. Like Reachy, I too am lacking the specifics of the probabilities for the outcomes that S. YAMA quoted. Glad to see he jumped in with these.

 

Divagations

Reachy and others,
I am glad you were on the right track. Don’t get discouraged by not getting to the final numbers. My belief is that by going through the process you learn and benefit more than by knowing any one specific answer.
It is especially true if you have time to notice and observe conditional relationships that occur in tournament situations. Once you found and examined them you may be surprised how often you will encounter them again and again - that’s because of your familiarity with the subject.
Let yourself explore the peripheral subjects while you working on any specific project.

Here are just a few notes one could make about many “associated” topics while trying to follow up our “main” subject of what to bet on next-to-last hand against two opponents. They may seem like digressions and rumbling but really they are part of knowledge necessary to develop better intuition and help in becoming an expert in the field.

We are trying to see how betting more than minimum, let’s say 4K, would perform. We are set to find out correlation of BR1’s hand to sixteen possible combinations for BR2 and BR3 results. We can use Wong’s number for just two players and try to analyze and deduct the rest, we can use his numbers for three players to cross reference for accuracy.
Let’s start with the first combination when both opponents get blackjacks. Since blackjacks are almost independent events you could say that BR1's hand has chance of a win 44%, push 8%, and loss 48%. Now you could say that among the wins there will be 4.75% blackjacks, but if you make adjustments for card removals and the fact that dealer could not have blackjack then this number becomes only 4.2%. So we would have: BR1-bj, BR2- bj, BR3 - bj - 4.2%, win - 39.8%, push – 8%, and loss - 48%.
With bet of 4K, losing it puts BR1 at a huge disadvantage by dropping him to third place going into the last hand and having to bet first. A push preserves BR1 lead and practically is as good as winning. BR1 should modify his play, which should be reflected in his hand outcomes. He could nudge pushes slightly up at the “expense” of wins and loses. The total EV drops but the total number of wins and pushes increases. We could separate “new” results into bj/win/push/loss if we are doing calculations for all possible combinations but we could simplify it and say that total wins and pushes would be about 53% and loses 47%. The exact numbers are not as important as realizing possibilities of changes and their direction. These relations once observed should be easier to identify later in other tournament blackjack situations.

Another observation flowing from the first case, when both opponents get blackjacks, and we lose our hand is realization that in rare cases we can drop from quite favorable to very disadvantaged position. As BR3 and betting first we need to swing (at least gain to) both players. What are the chances of it? Not high. It has to be much lower then chances of gaining on one player but it has to be higher that simply getting a blackjack while no one else gets the snapper. Setting it at slightly more than ten percent seems about right. But that is low, can we do anything else? Well, what are the chances that both players would give us low? If we think of it, such a chance exists, but only when BR2 is a good player and she bets second, after us, but in front of the new BR1. If we bet just a chip more that the difference to BR1, BR2 can’t take the low to us to protect herself from the possibility of us winning a double bet. Since BR2 goes for high she may bet close to half her bankroll or high enough so her blackjack beats BR1’s double (all-in). In this scenario it becomes quite possible that BR1 would make almost equally big bet, thus giving us the low and about 33% total chances of winning the tournament. But betting the difference to BR1 plus a chip gives up almost 2.5% when we get blackjack and at least one opponent wins his/her hand. That, in turn, would mean that we would need to assess chances for BR1 going high in the situation described above at about 10% or more. Etc., etc.

S. Yama

 

Win-Push-Loss three players

Thanks Monkeysystem for finding that in my previous post in combination 7 (win-push) “the dot” was moved one spot to the right. That frequency should be 1.64% instead of originally posted 16.4%.

Below is the table of win-push-loss for three players, wich for clarity I rounded to one percent. Wins include blackjacks so there are twenty-seven combinations in nine rows.
With blackjack as the separate result the table would consist of sixty-four combination. For interested in the subject it should not be difficult to construct the full table. The table below is not result of a simulation, it is just a logical conclusion of numbers published by Wong in his Table 4 from the CTS book.
It was fun to work on it, a bit like Sudoku game. Due to rounding, a couple positions are within one percent of the other previously posted numbers.

Is anybody interested in attempting to calculate chances for BR1 betting 4K from our original post?

........P1....P2.....................P3
...........................W..........P..........L
1......W.....W........23..........01.........06
2......W......P........01..........00.........01
3......W......L........05..........01.........06
4.......P......W.......01..........00.........01
5.......P.......P.......00..........00.........01
6.......P......L........01..........01.........03
7.......L.....W........05..........01.........06
8.......L......P........01..........01.........03
9.......L......L........07..........02.........22

Total:..................44..........08.........48

S. Yama

 

Wrap-up

Time to wrap-up this boisterous exchange of ideas.
I checked all sixty-four possibilities, frequency adjusted, for other bets and the results are:
Bet of minimum on the next-to-last hand makes BR1 the winner of the tournament 71.5% of the time.
Bet of 4,000 on the next-to-last hand makes BR1 the winner of the tournament 69.5% of the time.

I was pretty sure that 4K bet would perform better by about 5 percent. Oh, well, calculate and learn.
Because of a somewhat surprising result I checked bet of 10,400 – winning covers BR2 and BR3 winning their all-in bets twice, and losing it still leaves more than half of what the opponents would have by winning. And...
Bet of 10,400 on the next-to-last hand makes BR1 the winner of the tournament 69.6% of the time.

Since we were dealing with possibilities of slightly different plays and counteractions and I used approximations, those numbers should have margin of error of a few percent points. All three bets basically show an equal efficiency.

This analysis, once again, shows that different bets and plays may end up with the identical outcomes, and this would not be very uncommon occurrence. It also stresses importance of “secondary” aspects of the game, like counting and profiling.

S. Yama

 

Thank you Yama

for this final analysis.

I meet BR1 regularly in tournaments in my area. I now know that he is an advantage player at regular BJ. In that particular tournament, he won the drawing and was the wild card. He must have known that he would be on the button on the last hand because he started very agressively up to the official count (hand 16) to build his big lead and he then started betting the minimum the rest of the way.

I can never forget this play because it was only my third tournament and my first final table. I was BR2 and was lucky enough to win the tournament. I remember playing the minimum from the start, waiting to see the situation develop. I finally made a move on hand 13 (I was BR5 at the time) with the Wong progression. I won my first big bet and decided to sit for a few more hands and wait for the count. BR1 was still betting big and had a huge lead over everybody else.

On hand 16, after the count, I was BR4. I decided to bet the minimum one more time before making a big move. Good idea because BR3 and BR2 made their move on hand 17 and lost their bets. BR1 bet the minimum for the first time to my surprise!!! Don't forget I'm a newbie at the time.

Suddenly, I found myself BR2 and decided to wait for the second to last hand to make an all-in move, knowing that I would be betting last against my main opponents and that two consecutive all-ins would give me my only chance to win if BR1 kept betting the minimum.

BR1 told me later that he thought I was playing for second place. No way.

I now realize how so lucky I was! I now played 11 tournaments altogether, made two final tables (on first, one second), but did not even qualify for the semis in the other 9 tournaments.

I love it. I'm hooked.

 

Missed

I hadn't noticed that S. Yama had given us his final result until Ken just posted. I not sure whether the final result was calculated manually (but using excel of course!) and was therefore fairly time consuming to compute, or whether some other software was used to come up with the answer fairly quickly without too much labour. The reason I ask is because, as usual, there were several different answers and I think it'd be interesting to check all of them. In fact it should be possible to produce some sort of graph/chart to show how the probabilities change. I'd be really interested to know what my bet of $2325 would have come out as since this the nearest suggestion to the minimum.

Cheers

Reachy