Self-Styled Tournament Experts Take a Bath in Reno
by Peter A. Griffin
(From Blackjack Forum Volume VI #4, December 1986)
© Blackjack Forum 1986
“Nice guys finish last!”
– Leo Durocher
The highlight of the 1986 Festival Reno was a city-wide blackjack tournament pitting 12 different casinos against each other. Each casino fielded a 12 man (correction: person) team. Every day, for three days, each player would play 52 hands at one of the casinos. Each casino would host 12 players for about one hour each day. Players were given $500 in non-negotiable chips at the beginning of each day’s 52-hand round, regardless of what they finished the previous round with. Play took place with two rounds dealt per shuffle to four players at each of three tables.
Among the entrants was Karl’s (Sparks) Hotel & Casino’s dirty dozen – a collection of lockmasters, first-basers, rodmen, theoreticians, card counters, spooks, advisors, hole-carders, basic strategists, couponomists – you name it, every kind of degeneracy was represented. The theme of this assemblage was “the cream will rise to the top.” Karl’s finished last.
The prize structure of the tournament was such that all the entry fees would be returned to the players in prizes: $48,000 to the team finishing first and $24,000 to the runners-up. The winning team was the one with the greatest total cash out from the three days of play. The rules required a $10 minimum bet per hand with a $500 maximum. A team whose members all broke even at every session would have cashed out with 3 x 12 x $500 = $18,000 as a final total.
Karl’s strategy going into the first round of play was to wager the table minimum on neutral or negative decks, 10% of bankroll on moderate advantages and 20% of bankroll on strong advantages. And with the choice of this strategy their fate was probably sealed!
The results of the first day’s play were depressing indeed: Karl’s had been unlucky, posting a total of $4600, well below the daily par of $6000. But more disturbing than this was the leading total: the Nugget had come in with almost $11,000! At Karl’s strategy session the next morning their consulting theoretician advised, “Gentlemen, the Theory of Blackjack no longer applies.” Little did they and their Strangeloveian hireling (three Heinekens an hour) realize it never had. It was decided not to panic, but to raise the previous day’s 10% and 20% to 20% and 40%.
On Friday, Karl scored another $4600, while the Nugget hit par, maintaining the lead with a second day total of $16,900, far in excess of Karl’s $9300. The final day was marked by the tapping out of all 12 Karl’s players, having committed themselves to Kamikaze tactics of each playing for a $10,000 table win. On the twelfth and final round, Karl’s representative mercifully and fittingly went out on the first hand – 16 vs. 5 succumbing to the dealer’s five card 17.
Surprisingly, the Nugget was overtaken on the final day by Western Village, the ultimate winner by $24,000 to $22,000, leaving Karl’s worthies with nothing to show for their weekend but three monogrammed t-shirts and a hat. Had these “experts” been the victims of bad luck?
Well, yes and no. “Yes” in the sense that considering the cards dealt to them they wouldn’t have won regardless of betting strategy. “No” in that the strategy adopted on the first day probably only had about a 2% chance of beating the average best score of the 11 other teams.
The crucial observation came from a bystander watching the carnage of Karl’s suicidal charge halfway through the final day: “Betting the way you were, you didn’t rate to win anyway.” Thus spoke Robert Etter, up in Reno for a weekend bridge tournament and some desert stargazing with his telescope. Etter, not really from Georgia, although his undergraduate degree is from one of their diploma mills, observed intuitively that if there were many competing teams betting big, then Karl’s team would have virtually no chance by betting small, regardless of their level of skill.
Post-tournament analysis confirms Etter’s observation by showing that the winning final total of $24,000 was not at all improbable and, in fact, was just about the expected result. It is difficult to mathematize precisely the 11 competing teams’ styles of play, but the assumption of a flat $150 bet with a 2% negative expectation per hand probably mimics well the drift (expectation) and volatility (standard deviation) of the typical opponent’s play.
Under these assumptions, the expected final result for a team would be to lose 2% of their 1872 bets overall, namely to finish 37.4 big bets below par. The standard deviation for a team’s play would be approximately 1.1 x Ö1872, or about 47.5 bets. Note that expectation is constant (.02) times n, the number of hands, while standard deviation is another constant (1.1) times the square root of n. In the very long run .02 x n will dominate 1.1 x Ön, but in the relatively short space of n = 1872 hands the standard deviation is the controlling factor.
To continue the analysis we dip into the theory or order statistics from a standard normal distribution, it being well known that the result of 1872 blackjack hands subject to a bounded bet is very nearly a Gaussian variable. Obscure sources reveal that the expected value of the largest of a sample of 11 normal variables (the competing, “inferior,” teams) is to be 1.58 standard deviations above the mean. For our blackjack scenario, this translates into a final score of –37.4 + 1.58 (47.5) = +38 big bets. In terms of the tournament, this would be 18,000 + 38 (150) = $23,700 as the typical result for the best of 11 teams.
Now, the questionable part of this analysis is the convenient assumption of a flat $150 bet. If, in fact, the average bet was $150 (experience at the table suggests this), then any deviation from flat betting of $150, but still averaging $150, would increase the fluctuation (standard deviation) of the team’s bankroll). So, if the expected loss of 37.4 x $150 is correct, then the presumed standard deviation of 1.58 x 47.5 x $150 is too small; if the standard deviation is correct, then the expected loss is less than assumed. Conclusion: as long as the average bet is close to $150, the expected winning total for 11 teams would be about $24,000, as it was.
With a bet spread of $10 to $100 what chance would Karl’s have to top $24,000? An overly optimistic analysis of their chances would restrict the discussion to the assumption of 468 large bets of $100, one for every other deck, on the average, with a presumed advantage of 2% on these wagers. Assume the remaining 1404 small bets have no effect. Then Karl’’ would have an expected final total of 18,000 + .02 (46800) = $18,936 with a standard deviation not much in excess of 1.1 x Ö468 x 100 = $2370. Their chance of topping a typical best of $24,000 would be the area under the normal curve to the right of (24000 – 18936)/2370 = 2.14 standard deviations, which is less than 2%.
Moral? The casino blackjack player counts on very large n with a small constant of proportionality, the blackjack tournament player on the square root of relatively small n with a large multiplier. ♠
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