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Two books take place simultaneously within the slim confines of Stewart O'Nan's new novel The Odds. One tells the story of a failed marriage, its two unhappy principals not filled with hate for each other, but worn down by their own and their partner's mistakes and weaknesses. As the Double or Nothing term implies, you either double your buyin or walk away with nothing. In other words, if 10 players buyin with 5$ each, the top 5 finishes will win 10$ each. The payout structure of Double or Nothing SNGs is therefore very different from standard 10 player SNGs where 1st place usually wins 50% of the total prize pool. Double or Nothing, Fifty50, Double Thru. The names may differ slightly but the sit-and-go format stays the same: Be in the top half of the 6 or 10 players to survive and you'll double your stake. Double or Nothing poker tournaments are among the most popular sit-and-go tournaments on most online poker sites. We predict All Elite Wrestling's Double Or Nothing, from outside the MGM Grand in Las Vegas.For more awesome content, check out: F. AEW Double or Nothing 2020 takes place this Saturday night. It’s a pretty stacked card and I know I am looking forward to it. I will be providing AEW Double or Nothing 2020 Betting Tips of course. I will be writing them up here on Friday, and recording them in podcast form on Thursday.

A martingale is any of a class of betting strategies that originated from and were popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins the stake if a coin comes up heads and loses if it comes up tails. The strategy had the gambler double the bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake.

Since a gambler will almost surely eventually flip heads, the martingale betting strategy is certain to make money for the gambler provided they have infinite wealth and there is no limit on money earned in a single bet. However, no gambler possess infinite wealth, and the exponential growth of the bets can bankrupt unlucky gamblers who chose to use the martingale, causing a catastrophic loss. Despite the fact that the gambler usually wins a small net reward, thus appearing to have a sound strategy, the gambler's expected value remains zero because the small probability that the gambler will suffer a catastrophic loss exactly balances with the expected gain. In a casino, the expected value is negative, due to the house's edge. Additionally, as the likelihood of a string of consecutive losses occurs more often than common intuition suggests, martingale strategies can bankrupt a gambler quickly.

The martingale strategy has also been applied to roulette, as the probability of hitting either red or black is close to 50%.

Intuitive analysis[edit]

The fundamental reason why all martingale-type betting systems fail is that no amount of information about the results of past bets can be used to predict the results of a future bet with accuracy better than chance. In mathematical terminology, this corresponds to the assumption that the win-loss outcomes of each bet are independent and identically distributed random variables, an assumption which is valid in many realistic situations. It follows from this assumption that the expected value of a series of bets is equal to the sum, over all bets that could potentially occur in the series, of the expected value of a potential bet times the probability that the player will make that bet. In most casino games, the expected value of any individual bet is negative, so the sum of many negative numbers will also always be negative.

The martingale strategy fails even with unbounded stopping time, as long as there is a limit on earnings or on the bets (which is also true in practice).[1] It is only with unbounded wealth, bets and time that it could be argued that the martingale becomes a winning strategy.

Mathematical analysis[edit]

The impossibility of winning over the long run, given a limit of the size of bets or a limit in the size of one's bankroll or line of credit, is proven by the optional stopping theorem.[1]

However, without these limits, the martingale betting strategy is certain to make money for the gambler because the chance of at least one coin flip coming up heads approaches one as the number of coin flips approaches infinity.

Mathematical analysis of a single round[edit]

Let one round be defined as a sequence of consecutive losses followed by either a win, or bankruptcy of the gambler. After a win, the gambler 'resets' and is considered to have started a new round. A continuous sequence of martingale bets can thus be partitioned into a sequence of independent rounds. Following is an analysis of the expected value of one round.

Let q be the probability of losing (e.g. for American double-zero roulette, it is 20/38 for a bet on black or red). Let B be the amount of the initial bet. Let n be the finite number of bets the gambler can afford to lose.

The probability that the gambler will lose all n bets is qn. When all bets lose, the total loss is

∑i=1nB⋅2i−1=B(2n−1){displaystyle sum _{i=1}^{n}Bcdot 2^{i-1}=B(2^{n}-1)}

The probability the gambler does not lose all n bets is 1 − qn. In all other cases, the gambler wins the initial bet (B.) Thus, the expected profit per round is

(1−qn)⋅B−qn⋅B(2n−1)=B(1−(2q)n){displaystyle (1-q^{n})cdot B-q^{n}cdot B(2^{n}-1)=B(1-(2q)^{n})}
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Whenever q > 1/2, the expression 1 − (2q)n < 0 for all n > 0. Thus, for all games where a gambler is more likely to lose than to win any given bet, that gambler is expected to lose money, on average, each round. Increasing the size of wager for each round per the martingale system only serves to increase the average loss.

Suppose a gambler has a 63 unit gambling bankroll. The gambler might bet 1 unit on the first spin. On each loss, the bet is doubled. Thus, taking k as the number of preceding consecutive losses, the player will always bet 2k units.

With a win on any given spin, the gambler will net 1 unit over the total amount wagered to that point. Once this win is achieved, the gambler restarts the system with a 1 unit bet.

With losses on all of the first six spins, the gambler loses a total of 63 units. This exhausts the bankroll and the martingale cannot be continued.

In this example, the probability of losing the entire bankroll and being unable to continue the martingale is equal to the probability of 6 consecutive losses: (10/19)6 = 2.1256%. The probability of winning is equal to 1 minus the probability of losing 6 times: 1 − (10/19)6 = 97.8744%.

The expected amount won is (1 × 0.978744) = 0.978744.
The expected amount lost is (63 × 0.021256)= 1.339118.
Thus, the total expected value for each application of the betting system is (0.978744 − 1.339118) = −0.360374 .

In a unique circumstance, this strategy can make sense. Suppose the gambler possesses exactly 63 units but desperately needs a total of 64. Assuming q > 1/2 (it is a real casino) and he may only place bets at even odds, his best strategy is bold play: at each spin, he should bet the smallest amount such that if he wins he reaches his target immediately, and if he doesn't have enough for this, he should simply bet everything. Eventually he either goes bust or reaches his target. This strategy gives him a probability of 97.8744% of achieving the goal of winning one unit vs. a 2.1256% chance of losing all 63 units, and that is the best probability possible in this circumstance.[2] However, bold play is not always the optimal strategy for having the biggest possible chance to increase an initial capital to some desired higher amount. If the gambler can bet arbitrarily small amounts at arbitrarily long odds (but still with the same expected loss of 10/19 of the stake at each bet), and can only place one bet at each spin, then there are strategies with above 98% chance of attaining his goal, and these use very timid play unless the gambler is close to losing all his capital, in which case he does switch to extremely bold play.[3]

Alternative mathematical analysis[edit]

The previous analysis calculates expected value, but we can ask another question: what is the chance that one can play a casino game using the martingale strategy, and avoid the losing streak long enough to double one's bankroll.

As before, this depends on the likelihood of losing 6 roulette spins in a row assuming we are betting red/black or even/odd. Many gamblers believe that the chances of losing 6 in a row are remote, and that with a patient adherence to the strategy they will slowly increase their bankroll.

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In reality, the odds of a streak of 6 losses in a row are much higher than many people intuitively believe. Psychological studies have shown that since people know that the odds of losing 6 times in a row out of 6 plays are low, they incorrectly assume that in a longer string of plays the odds are also very low. When people are asked to invent data representing 200 coin tosses, they often do not add streaks of more than 5 because they believe that these streaks are very unlikely.[4] This intuitive belief is sometimes referred to as the representativeness heuristic.

Anti-martingale[edit]

In a classic martingale betting style, gamblers increase bets after each loss in hopes that an eventual win will recover all previous losses. The anti-martingale approach, also known as the reverse martingale, instead increases bets after wins, while reducing them after a loss. The perception is that the gambler will benefit from a winning streak or a 'hot hand', while reducing losses while 'cold' or otherwise having a losing streak. As the single bets are independent from each other (and from the gambler's expectations), the concept of winning 'streaks' is merely an example of gambler's fallacy, and the anti-martingale strategy fails to make any money. If on the other hand, real-life stock returns are serially correlated (for instance due to economic cycles and delayed reaction to news of larger market participants), 'streaks' of wins or losses do happen more often and are longer than those under a purely random process, the anti-martingale strategy could theoretically apply and can be used in trading systems (as trend-following or 'doubling up'). (But see also dollar cost averaging.)

See also[edit]

References[edit]

  1. ^ abMichael Mitzenmacher; Eli Upfal (2005), Probability and computing: randomized algorithms and probabilistic analysis, Cambridge University Press, p. 298, ISBN978-0-521-83540-4, archived from the original on October 13, 2015
  2. ^Lester E. Dubins; Leonard J. Savage (1965), How to gamble if you must: inequalities for stochastic processes, McGraw Hill
  3. ^Larry Shepp (2006), Bold play and the optimal policy for Vardi's casino, pp 150–156 in: Random Walk, Sequential Analysis and Related Topics, World Scientific
  4. ^Martin, Frank A. (February 2009). 'What were the Odds of Having Such a Terrible Streak at the Casino?'(PDF). WizardOfOdds.com. Retrieved 31 March 2012.
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Two books take place simultaneously within the slim confines of Stewart O’Nan’s new novel The Odds. One tells the story of a failed marriage, its two unhappy principals not filled with hate for each other, but worn down by their own and their partner’s mistakes and weaknesses. The other explores how desperation can drive people to take enormous chances, discovering in themselves the possibility of courage and the capacity to reject what fate seems determined to do to them.

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Art and Marion Fowler are running away, fleeing, we are told in the book’s opening lines, toCanada. Art has lost his job, their house is being foreclosed upon, and the denial of their dire circumstances in which they have lived will no longer support the weight of reality. But they aren’t heading north to disappear. Instead, Art has converted most of their remaining assets to cash in order to gamble it all on the roulette tables of a casino atNiagara Fallsin a desperate attempt to win back what they’ve lost.

But they’ve lost a lot more than money.Marionis still deeply wounded by Art’s infidelity, an affair that took place decades before and for which Art has been doing penance ever since. O’Nan deftly weaves the present and the past together, slowly revealing Art’s motivations and bottomless guilt, and over the chapters that move back and forward in time he paints a pointillist picture of nor just Art’s indiscretion, but of their entire married life. Art loves his wife deeply, yet he was capable of betraying her, and O’Nan doesn’t try to resolve this apparent paradox because he knows there is no resolution beyond the fact that Art is flawed, complex and not completely self-aware or totally in control. In other words, he’s human.

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Whether this means he ought to be forgiven isMarion’s decision, and O’Nan sees her as the most tragic and sympathetic character in the text. She is still angry, and probably always will be, but is hardly guiltless herself. The divorce they are planning is ostensibly for financial reasons, butMarionsees it also as a final cutting of the cord, a way to get on with her life. Yet despite everything she still loves her husband, and she is tired of her anger and her suspicion even as she sees forgiving him and letting go of her pain as a personal failure. This pain is not all there is toMarion, but it is so deeply entrenched in her psyche that she runs the risk of allowing it to define her. The relationship between these two very real people, their intimate knowledge of each other’s quirks, preferences, weaknesses and strengths is shown skillfully in the fluid shifts in tone and mood that O’Nan effects through the slightest turn of a word or phrase in what anyone watching from the outside would see as a totally innocent conversation.

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All this is set against the backdrop ofNiagara Fallsand Art’s crazy plan. A love letter to the Falls this book certainly is not, and O’Nan (I cannot help but suspect for comic relief) seems almost gleeful as he skewers the kitschy tourist traps and even kitschier tourists Niagara has fattened itself on for a hundred years. BringingMarionback to the site of their honeymoon, Art hopes to rekindle the dying embers of their marriage. He perseveres despite the thousand little inconveniences and small failures that tend to plague budget tourists: the lost reservation, the bad dinner that results in stomach upset, the too-long line for the much-anticipated attraction. A heart concert at the casino, complete with drugs, sex and the spectacle of middle-aged baby boomers rocking arthritically to the soundtrack of their lost youth is the saddest and funniest episode of the text, and it shows vividly one of O’Nan’s minor but complementary points: traveling is fun, but it is also a good course in dealing with disappointment.

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The disappointments Art andMarionexperience on this trip, while always minor, reflect metaphorically the slow, small, disappointing dissolution of their life together, a life Art and perhaps unconsciously evenMarionwishes they could resurrect. The odds are against them, and they know it. But they try and, win or lose, in trying they achieve a kind of success. So does The Odds.