The following situation came up in my most recent tournament: Last hand. One will advance to finals. Min bet 10, max bet 2,000 in increments of 5. No surrender, insurance, or even money allowed. Blackjacks pay 2:1. This is a face down pitch game. BR2 has been measuring his bets and keeping a watchful eye on the chip stacks. BR3 is first to act. BR3...…………... stack 6,175 bet 2,000 Monkeysystem stack 8,205 bet ? Thinking BR2 stack 7,400 BR4...…………... stack 6,000 As I saw it I had two choices for my bet. Choice #1. Bet 800. This forces BR2 to bet the max and win his bet. However, I will not be able to cover the double downs by BR3 and BR4. Choice #2. Bet 2,000. BR2 can respond by betting 1,190 to take the low on me. However, he will not be able to double down aggressively to retake the high if I tuck my hand. My bet covers the double downs by BR3 and BR4. This seems to me to be a pretty close decision. If it weren't for the double down threat from BR3 and BR4, this would be an easy decision to bet 800. If BR3 and/or BR4 were within a max bet I would probably have to bet 2,000. Can anybody generate some numbers to analyze this situation? Would my decision change if BR4 didn't exist? What if we add a BR5 with a stack of 5,995?
I appreciate Monkeysystem thoroughness (as usual) in setting up the situation. Gronbog, can you add to your sim his bet of 1,000 and BR2 betting both low and high, thanks, S. Yama
Running this situation across Stanford Wong's Tournament Blackjack Software I got the results below. All the results have a standard error of 0.016, except as stated in the three-player scenarios. Bet of 800: Winning probability = 0.378 Bet of 2,000: 0.503 Bet of 1,000, BR2 bet 190: Winning probability = 0.327 Bet of 1,000, BR2 bet 2,000: 0.422 BR2's results: BR2 bet of 190: Winning probability = 0.290 Bet of 2,000: 0.340 If we have no idea how BR2 might respond to our bet of 1,000, we can weight the two results evenly and get a winning probability of 0.375. If we think there is a strong but uncertain possibility that BR2 will bet 2,000, we can give that possibility a 2/3 weight. This makes our winning probability 0.395. If we remove BR4 from the equation, we should still bet 2,000. The numbers with a standard error of 0.012: Bet of 800: Winning probability of 0.458 Bet of 2,000: 0.530. Bet of 1,000, BR2 bet 190: Winning probability = 0.418 Bet of 1,000, BR2 bet 2,000: 0.463. BR2's results. Standard error of 0.01: BR2 bet of 190: Winning probability = 0.329 BR2 bet of 2,000: 0.382 I tried the three-player situation but with our lead over BR2 less than 1/5 of the max bet. This allows BR2 to take the low and then double aggressively to take the high over our single bet. Bet of 200: Winning probability = 0.442 Standard error of 0.01 Bet of 2,000: 0.510 The numbers are probably not as precise as a more sophisticated simulation would yield. But the results are pretty consistent. So the principle that you should bet the max when you are BR1 and have two or more opponents applies, even when you are a little more than a max bet ahead of BR3. As the old adage says, "Go big or go home."
My apologies. I completely forgot about this. I started to run the situations above this morning, but the resources required for the four player scenario quickly overwhelmed the computer I tried it on. I will try it on another computer I have which has more memory some time today. My software first computes the optimal strategy for each player given their bets and then simulates the situation using those strategies. By the time we get to player 4 responding to every possible situation that the first 3 players can end up in, the tree of possibilities gets quite large. I can save time and resources by assuming that some of the other players will play basic strategy, but that would probably not be useful for this situation. I have no idea what assumptions Wong's software makes about how the other players will play.
According to the booklet that comes with the software, you can program the opponents as an opponent called, "Toughie." Toughie usually uses the optimal strategy available to him but mixes it up at times. They wrote Toughie this way to make him unpredictable. I would have to assume that Toughie's usual play in a situation is what Wong recommends in "Casino Tournament Strategy." In my opinion a bigger problem than the AI's in this software is that it's so slow. You can't run millions of tournaments in a simulation. When I was running this one-handed scenario it was playing through less than a thousand trials an hour. It doesn't run any faster in my Asus gamer than it did twenty years ago in the glacial computers of the 20th century. Because of the small sample sizes you have to use the statistics it generates carefully. You are more interested in confidence intervals than in hard numbers. The simulator gives you the standard error of its results. You should run the simulations until your results are at least three standard errors apart. Then you should use the results to evaluate the decisions you take from them, rather than on the hard numbers. If a decision is between two results that are less than a percentage point apart, you may have to run the simulation for days to get a difference of three standard errors. I'm running S_Yama's scenario as we speak, and plan to let it go for days, because the difference in results for BR2's low and high bet are so close. So far BR1 has a winning probability of 0.406 if BR2 takes the low and 0.488 if BR2 takes the high, with a standard error of 0.011. At least it uses so little memory that there's no risk of melting your computer.
I was able to coax my software through the strategy generation phase by stopping it just as it began to run out of memory. This allowed for approximately 380,000 simulated iterations for generating the strategies. Once generated, playing the strategies can be easily simulated as many times as needed, since no further resources are needed at that point. So far I have done this for the case of a suggested bet of 800 and assuming that BR2 and BR4 both bet 2000. All players play the generated optimal strategy, which covers every case encountered during the 380,000 iterations used to to generate it. For cases not analyzed, the players revert to basic strategy. 100 million iterations resulted in reasonable standard errors. Here are my results: Player 1: Finishes 1: 11,992,491/100,000,000 = 11.9925% Standard Error: 0.003249% Finishes 2: 25,199,577/100,000,000 = 25.1996% Standard Error: 0.004342% Finishes 3: 44,234,502/100,000,000 = 44.2345% Standard Error: 0.004967% Finishes 4: 18,573,430/100,000,000 = 18.5734% Standard Error: 0.003889% Reaches goal: 11,992,491/100,000,000 = 11.9925% Standard Error: 0.003249% Player 2: Finishes 1: 42,733,893/100,000,000 = 42.7339% Standard Error: 0.004947% Finishes 2: 37,673,688/100,000,000 = 37.6737% Standard Error: 0.004846% Finishes 3: 15,803,548/100,000,000 = 15.8035% Standard Error: 0.003648% Finishes 4: 3,788,871/100,000,000 = 3.7889% Standard Error: 0.001909% Reaches goal: 42,733,893/100,000,000 = 42.7339% Standard Error: 0.004947% Player 3: Finishes 1: 28,758,669/100,000,000 = 28.7587% Standard Error: 0.004526% Finishes 2: 26,725,807/100,000,000 = 26.7258% Standard Error: 0.004425% Finishes 3: 22,794,980/100,000,000 = 22.7950% Standard Error: 0.004195% Finishes 4: 21,720,544/100,000,000 = 21.7205% Standard Error: 0.004123% Reaches goal: 28,758,669/100,000,000 = 28.7587% Standard Error: 0.004526% Player 4: Finishes 1: 16,514,947/100,000,000 = 16.5149% Standard Error: 0.003713% Finishes 2: 10,400,928/100,000,000 = 10.4009% Standard Error: 0.003053% Finishes 3: 17,166,970/100,000,000 = 17.1670% Standard Error: 0.003771% Finishes 4: 55,917,155/100,000,000 = 55.9172% Standard Error: 0.004965% Reaches goal: 16,514,947/100,000,000 = 16.5149% Standard Error: 0.003713% I'll look at the case of a bet of 2000 next.
If you are running it under Wind0ws 98 in VMWare, then that may be down to a configuration issue (in VMWare and/or the Windows 98 'guest' OS). There's also a 'hardware virtualization' BIOS setting on the 'host' PC that needs to be enabled (though I'm not sure if failing to enable this would mean VMWare simply refused to work at all, rather than run with reduced performance). https://www.vmware.com/support/ws3/doc/ws32_performance4.html https://www.howtogeek.com/213795/how-to-enable-intel-vt-x-in-your-computers-bios-or-uefi-firmware/ You could try a different guest OS, such as Linux+Wine, or Windows NT/XP. Whatever OS you use to run Wong's software, it might be better to avoid virtualization altogether, and either create a bootable USB stick or a dual-booting system.
Here are my results for a bet of 2000 and BR2 responding with 1190. BR4 still bets 2000: Player 1: Finishes 1: 14,531,711/100,000,000 = 14.5317% Standard Error: 0.003524% Finishes 2: 9,541,409/100,000,000 = 9.5414% Standard Error: 0.002938% Finishes 3: 49,872,355/100,000,000 = 49.8724% Standard Error: 0.005000% Finishes 4: 26,054,525/100,000,000 = 26.0545% Standard Error: 0.004389% Reaches goal: 14,531,711/100,000,000 = 14.5317% Standard Error: 0.003524% Player 2: Finishes 1: 49,399,024/100,000,000 = 49.3990% Standard Error: 0.005000% Finishes 2: 36,896,033/100,000,000 = 36.8960% Standard Error: 0.004825% Finishes 3: 10,807,388/100,000,000 = 10.8074% Standard Error: 0.003105% Finishes 4: 2,897,555/100,000,000 = 2.8976% Standard Error: 0.001677% Reaches goal: 49,399,024/100,000,000 = 49.3990% Standard Error: 0.005000% Player 3: Finishes 1: 13,617,679/100,000,000 = 13.6177% Standard Error: 0.003430% Finishes 2: 49,298,345/100,000,000 = 49.2983% Standard Error: 0.005000% Finishes 3: 29,613,215/100,000,000 = 29.6132% Standard Error: 0.004565% Finishes 4: 7,470,761/100,000,000 = 7.4708% Standard Error: 0.002629% Reaches goal: 13,617,679/100,000,000 = 13.6177% Standard Error: 0.003430% Player 4: Finishes 1: 22,451,586/100,000,000 = 22.4516% Standard Error: 0.004173% Finishes 2: 4,264,213/100,000,000 = 4.2642% Standard Error: 0.002020% Finishes 3: 9,707,042/100,000,000 = 9.7070% Standard Error: 0.002961% Finishes 4: 63,577,159/100,000,000 = 63.5772% Standard Error: 0.004812% Reaches goal: 22,451,586/100,000,000 = 22.4516% Standard Error: 0.004173%
Here are my results for a bet of 1000 with BR responding with 190 and BR4 still betting 2000: Player 1: Finishes 1: 15,354,877/100,000,000 = 15.3549% Standard Error: 0.003605% Finishes 2: 18,970,526/100,000,000 = 18.9705% Standard Error: 0.003921% Finishes 3: 46,428,636/100,000,000 = 46.4286% Standard Error: 0.004987% Finishes 4: 19,245,961/100,000,000 = 19.2460% Standard Error: 0.003942% Reaches goal: 15,354,877/100,000,000 = 15.3549% Standard Error: 0.003605% Player 2: Finishes 1: 34,565,332/100,000,000 = 34.5653% Standard Error: 0.004756% Finishes 2: 45,551,398/100,000,000 = 45.5514% Standard Error: 0.004980% Finishes 3: 15,406,856/100,000,000 = 15.4069% Standard Error: 0.003610% Finishes 4: 4,476,414/100,000,000 = 4.4764% Standard Error: 0.002068% Reaches goal: 34,565,332/100,000,000 = 34.5653% Standard Error: 0.004756% Player 3: Finishes 1: 29,765,053/100,000,000 = 29.7651% Standard Error: 0.004572% Finishes 2: 28,952,057/100,000,000 = 28.9521% Standard Error: 0.004535% Finishes 3: 20,282,435/100,000,000 = 20.2824% Standard Error: 0.004021% Finishes 4: 21,000,455/100,000,000 = 21.0005% Standard Error: 0.004073% Reaches goal: 29,765,053/100,000,000 = 29.7651% Standard Error: 0.004572% Player 4: Finishes 1: 20,314,738/100,000,000 = 20.3147% Standard Error: 0.004023% Finishes 2: 6,526,019/100,000,000 = 6.5260% Standard Error: 0.002470% Finishes 3: 17,882,073/100,000,000 = 17.8821% Standard Error: 0.003832% Finishes 4: 55,277,170/100,000,000 = 55.2772% Standard Error: 0.004972% Reaches goal: 20,314,738/100,000,000 = 20.3147% Standard Error: 0.004023%
Here are my results for a bet of 1000 with BR2 responding with 2000 and BR4 still betting 2000: Player 1: Finishes 1: 10,833,904/100,000,000 = 10.8339% Standard Error: 0.003108% Finishes 2: 24,118,223/100,000,000 = 24.1182% Standard Error: 0.004278% Finishes 3: 45,992,444/100,000,000 = 45.9924% Standard Error: 0.004984% Finishes 4: 19,055,429/100,000,000 = 19.0554% Standard Error: 0.003927% Reaches goal: 10,833,904/100,000,000 = 10.8339% Standard Error: 0.003108% Player 2: Finishes 1: 43,392,220/100,000,000 = 43.3922% Standard Error: 0.004956% Finishes 2: 36,332,291/100,000,000 = 36.3323% Standard Error: 0.004810% Finishes 3: 16,527,517/100,000,000 = 16.5275% Standard Error: 0.003714% Finishes 4: 3,747,972/100,000,000 = 3.7480% Standard Error: 0.001899% Reaches goal: 43,392,220/100,000,000 = 43.3922% Standard Error: 0.004956% Player 3: Finishes 1: 30,110,711/100,000,000 = 30.1107% Standard Error: 0.004587% Finishes 2: 29,532,460/100,000,000 = 29.5325% Standard Error: 0.004562% Finishes 3: 21,369,949/100,000,000 = 21.3699% Standard Error: 0.004099% Finishes 4: 18,986,880/100,000,000 = 18.9869% Standard Error: 0.003922% Reaches goal: 30,110,711/100,000,000 = 30.1107% Standard Error: 0.004587% Player 4: Finishes 1: 15,663,165/100,000,000 = 15.6632% Standard Error: 0.003635% Finishes 2: 10,017,026/100,000,000 = 10.0170% Standard Error: 0.003002% Finishes 3: 16,110,090/100,000,000 = 16.1101% Standard Error: 0.003676% Finishes 4: 58,209,719/100,000,000 = 58.2097% Standard Error: 0.004932% Reaches goal: 15,663,165/100,000,000 = 15.6632% Standard Error: 0.003635%
