Why What You Got Ain't What You Expected:

Edge, Volatility, and Risk of Ruin in Gambling

(A Work in Process)

by

Alan Krigman

ICON Inc

211 S 45^{th} St, Philadelphia PA 19104

email: alan@icon-info.com

Abstract:Short term risk of ruin can serve as a means of characterizing gambles in a manner meaningful to the vast majority of players -- those concerned with their performance during sessions or casino visits of reasonable duration. Risk of ruin is examined from two viewpoints.1)

The earnings criterion:The probability that players will reach target earnings levels before depleting their bankrolls under specified conditions, regardless of the number of decisions required to reach one point or the other. An exact solution to the earnings-based risk of ruin problem does not appear to be tractable. Predictions obtained using equations incorporating certain simplifying assumptions, however, correlate reliably with results of simulations for common classes of gambling situations. Three more-or-less standard formulations, representing successive levels of approximation, are presented. They are shown to exhibit differences which are not consequential for the purposes to which risk of ruin analyses are properly applied.2)

The survival criterion:The probability that players will not deplete their bankrolls during sessions of stated duration under specified conditions, regardless of intermediate profit levels achieved. An expression is presented incorporating the "barrier effect" first known to be described in a gambling context by Don Schlesinger. This recognizes that the points where players are likely to be after completing some number of trials are not the same as the levels they are likely to reach during the action.

The relevant equations are presented in a uniform format based on edge and variance or standard deviation of individual rounds or decisions, as well as on the amounts specified as bankroll, win bet size, and win goal or desired duration of play. The arithmetic is computationally obtuse, but is readily performed using modern spreadsheet software. A downloadable Excel97 spreadsheet is accordingly provided.

Risk of ruin applications are discussed with respect to sizing bets, establishing earnings targets, and understanding the trade-off opportunities inherent in use of playing and betting strategies to control short-term performance. Results of simulations are shown to indicate how actual and predicted performance might differ when distributions are decidedly non-Gaussian, and possible sources of the discrepancies in several instances are suggested.

This paper is offered as a work in process. In particular, the objective is to summarize the current state of the art, and stimulate refinements to the equations which might be appropriate for certain common gambling situations such as those with skewness so high that a normal distribution is not a valid assumption for relevant numbers of trials.

Expectation and volatilityTraditional analyses of casino and other gambling propositions begin and end by evaluating and comparing the

expected valueof various alternatives. Usually expressed as "edge" or "expectation," the average fraction of the amount wagered players win or lose on the decision, this is the first moment of the statistical distribution associated with the.

Casino games generally have negative edge. The payoffs are offset from the chances of winning such that the laws of probability enable the house to make a profit in the long run.

A few bets, specifically odds at craps and doubles in some video poker configurations, have zero expectation; however, these tend to be auxiliary and not independent wagers. Likewise, proper doubles and some splits at blackjack have positive expectation at the moment the money is wagered, although the house may have had the edge when the initial bet was made at the start of the round.

Under rare circumstances, players have an edge. Blackjack, with card counting and appropriate bet and strategy variations is a prime example. Progressive payout games, usually but not exclusively slot machines, may go positive when the jackpot gets high enough.

Expected value, however, is meaningful only as a long-term phenomenon. Not only does it usually represent a small fraction of each transaction, which requires many decisions to accumulates into money worth mentioning, but it gets buried in the nominal amounts won or lost on individual decisions. That is, players betting $10 on the five at craps are subject to an edge of 4%. The house earns a theoretical $0.40 on the action. But the players either win $14 or loses $10, and rarely think about the fact that in a "fair" game, the house had an edge because the bet could win four ways and lose six, so the $10 at risk should actually have paid $15.

Volatility, on the other hand, characterizes the up- and down-swings likely to be encountered during the course of the action. It also affords a means of quantifying the probability of deviating from the expected value at any juncture. Statisticians traditionally measure volatility by thevariance-- the second moment of the distribution -- or by its more useful square root, thestandard deviation. In the course of the action, the cumulative expectation increases linearly with the number of decisions while the overall standard deviation rises with the square root of this quantity. This is the effect that accounts for volatility dominating performance over the short haul, while edge becomes overriding in the long term.

Casino management appear to be only vaguely aware of the significance of volatility. Normally, ignoring this factor does them little harm. In general, casinos book enough bets over a relatively narrow range of values during most accounting periods that they can safely ignore volatility and project their performance using edge alone. When small numbers of trials are involved, though, this can lead to unanticipated consequences and poor judgements. Such situations include high rollers during short periods, early performance of newly-implemented games or rules, establishing reset and growth levels of progressive jackpots, and unwarranted concern about players enjoying short-term successes -- especially if they are or are suspected to be counting cards at blackjack..

Players understand that they will experience upswings and downswings, but think of the phenomenon as a manifestation of luck (or mystical forces) rather than a characteristic of the game that can be described analytically. However, unlike casinos, players are seriously impacted by the effects of volatility during normal sessions -- much more so than by edge, even on the most outrageous "sucker bets." Problems which accrue to players because of failure to account for volatility generally involve depleting bankrolls during what should be considered the normal downswings of a game, unrealistic aspirations for earnings, and misconceptions about the effects of betting progressions.

Ascendancy of edge as a metric for characterizing games follows from this consideration, and from the fact that -- over the years -- they have performed or sponsored most of the analyses because it has been in their business interests or in their legal requirements to do so. On the other hand, few players make enough decisions in their gambling lifetimes, let alone single sessions or even extended casino vacations, to approach the point where the effect of edge dominates their performance.

Drawing inferences from expected values and variances of distributions is standard practice in broad areas of commerce and politics. The subject is well-covered in the literature of many fields. A simple example will show the implications for gambling and highlight the dichotomy between short- and long-term phenomena.

At a six-deck blackjack game with reasonable rules, the edge and standard deviation for each hand following basic strategy are about -0.5% and 1.12, respectively.

A) After 100 hands at $10 each, the player's expectation is to lose 0.5% of the $1,000 handle or $5. The standard deviation for the session is the standard deviation per unit bet, times the size of the bet, times the square root of 100 hands or 1.12 x $10 x 10 = $112. The so-called "empirical rule" of statistics says that for wide ranges of situations like this, approximately 68% of all players will be within one standard deviation of the mean. In this case, after 100 hands, the range is therefore from a loss of $117 to a win of $107. The $5 represented by edge is buried in the $112 standard deviation.

B) After 10,000 hands at $10, the player's expectation is to lose 0.5% of the $100,000 handle or $500. The standard deviation is $1,120. About 68% of all players can now expect to be from $1,620 behind to $620 ahead. The $500 biases the range of anticipated results, but volatility is still dominant.

C) After 1,000,000 hands, the expected loss would be $50,000 and the standard deviation is $11,200. Roughly 68% of all players can expect to be from $61,200 to $38,800 in the hole. Edge has surpassed volatility in effect.

Risk of ruinUnfortunately, characterizing gambling in terms of expected value and standard deviation does not relate well to the way either casinos or players understand the activity. Risk of ruin offers a more viable alternative. It not only relates directly to quantities players and casinos alike find meaningful, but affords opportunities to make rational decisions about parameters such as bankroll, bet sizes, and objectives.

Risk of ruin is the probability of depleting a bankroll and being unable to continue. This quantity, when qualified with bankroll, bet size, and either a profit target or projected duration of play, characterizes a gamble in a manner that incorporates the effects of expectation and volatility while being directly meaningful to players and casinos alike. The concept has two interpretations, both of which are relevant to short-term gambling performance.

The earnings criterionShort term risk of ruin determined by the earnings criterion is the probability that players will achieve stated profit levels prior to depleting their bankrolls, without regard to the number of decisions involved before one or the other is achieved. An illustration might be a player at a single-zero roulette table with a $500 bankroll betting $25 per spin on

redand planning to play until he or she earns $1,000 -- or goes bust trying. Risk of ruin in this situation -- 91.5%, and the complementary 8.5% chance of success, show that this player has set unrealistic goals.

Earnings-based risk of ruin is of primary importance to the large class of players who simply won't quit unless they reach some profit level. An understanding of the concept can help these players in several ways, including setting more pragmatic goals, satisfying themselves to quit at levels below their goals which actually represent good profits for their mode of operation, changing their betting strategies or their games, recognizing the trade-offs between low probabilities of large wins and high probabilities of small wins, and re-evaluating what they consider appropriate gambling stakes for a session.

Earnings-based risk of ruin is a function of bankroll, earnings target, bet size, and the probabilities and payoffs associated with various possible results. The relationship, however, is not straightforward and the equations used to predict risk of ruin under various actual gambling conditions incorporate a variety of simplifying assumptions.

The survival criterionShort term risk of ruin determined by the survival criterion is the probability that players will complete a stated number of decisions, typically rounds, without depleting their bankrolls, regardless of any intermediate earnings level achieved. As an example, the single-zero roulette player with the $500 bankroll betting $25 per spin on

redmight anticipate remaining at the table for three hours hoping to catch a lucky streak and strike it rich. This player faces only 7.5% risk of ruin within three hours -- a reasonable prospect for survival, notwithstanding the scant likelihood of reaching that $1,000 target.

For a great many players, including quite a few who don't consciously realize it, survival-based risk of ruin is the primary consideration. They do not want to deplete the stake they considered sensible when they were still at home and their budgeting was not influenced by the aura of the casino. Bettors may nurture fantasies about converting their $100 stakes into $10,000 returns, and be willing to sink the $100 into the effort; however, they realize that if they run out of money prematurely, they either have to admit defeat or replenish their stakes so they can continue trying. Even players who dream of the limo ride to Easy Street, and are willing to go down in a blaze of glory trying, tend to have different perspectives on sessions in which they lost their stakes but spent an entire day doing it, and in which they tapped-out prematurely, then wandered around woefully for the remainder of the time they allocated to gambling, or went to one or another of the all-too-readily available wells to replenish their bankrolls for another try.

Survival-based risk of ruin is a function of bankroll, bet size, number of decisions, and the probabilities and payoffs associated with various possible results. The relationship involves straightforward statistical principles, although the traditional calculations have recently been shown to be conceptually erroneous. The fallacy lay in considering only the points at which player could expect to be at the end of their sessions, rather than the downswings which might put them out of business before the specified number of decisions could be completed and from which they might recover if they had additional capital.

Earnings-based risk of ruin analysisThree alternate equations for earnings-based risk of ruin will be presented. These have all been implemented on an Excel97 spreadsheet which accompanies this paper, and is available for downloading. The formulations are essentially variations of one another, derived using increasing degrees of simplifying assumptions.

Even-Money equivalentsRichard Epstein (

The Theory of Gambling and Statistical Logic-- page 59 of Revised Edition) presents a relationship of the form given below in Equation 1, for the case when all bets are paid at even money. Richard Reid developed the same equation from a different perspective (Goal-Oriented Risk of Ruin --www.bjmath.com/bjmath/ror/goror.htm).

Where: p(B,g) = probability of reaching g before losing B

B = initial bankroll

g = target win goal

p = probability of winning even-money wager

w = amount of wager

Equation 1, alone, is of limited value because of the restriction to even-money bets. Peter Griffin (

The Theory of Blackjack, Extra Stuff: Gambling Ramblings), however, suggested that arbitrary bets can be reduced to even-money equivalents -- amounts and probabilities of winning -- with the same expectations and variances as the prototype wagers. A method for calculating the even-money equivalents has been described by Alan Krigman (50-50 and Even-Money Equivalent Bets-- www.bjmath.com/bjmath/Betsize/equmb.htm). This shows that the even-money equivalent bet size and probability of winning can be found from Equations 2 and 3, respectively. Calculate the even-money p and w from the edge and variance of the prototype bet using Equations 2 and 3, then substitute the results into Equation 1 to get risk of ruin.

Where: E = edge or expectation

V = variance

Power-law risk of ruinEquation 3 can be simplified by assuming that the square of the edge is negligible with respect to the variance. With this assumption, Patrick Sileo (

The Evaluation of Blackjack Games Using a Combined Expectation and Risk Measure-- www.bjmath.com/bjmath/sileo/sileo.pdf), has shown that Equations 1 through 3 can be combined into Equation 4. This expression is widely used for earnings-based risk of ruin -- as, for instance, by Bryce Carlson (Blackjack for Blood; Risk, Ruin, and Trip-Stake Wipeout-- www.bjmath.com/bjmath/ror/tripror.htm).

Where: p(B, g) = probability of reaching g before losing B

E = edge (per decision)

S = standard deviation (per decision)

B = bankroll

g = target win

w = amount of wager

Exponential risk of ruinMaking the further assumptions that target earnings are much greater than bankroll and that standard deviation is much greater than edge, Patrick Sileo (ibid) reduces Equation 4 to the form of Equation 5. This form is not only computationally simpler, but offers advantages in solving for win goal or bankroll as a function of predetermined risk of ruin.

Where p(B, g) = probability of reaching g before losing Be = exponential function (base of natural logarithm)

B = bankroll

g = target win goal

E = edge (per decision)

V = variance (per decision)

w = amount of wager

ComparisonThe simplifying assumptions leading from Equations 1 through 3 to 4 and 5 turn out to be quite robust for situations of practical interest. A few comparisons will illustrate that the differences in results obtained using the alternate techniques are minor.

In double-zero roulette, betting on individual numbers, edge is -5.26% and variance is 33.2. For a player having a $250 bankroll and betting $5 per spin, the probability of reaching a $100 profit before losing $250 is predicted as 69.1314% with the even-money equivalent method, 69.1312% with the power law approach, and 69.1313 with the exponential technique. Lowering the variance with respect to edge by betting $5 per round on an outside proposition such as

even-- assuming a 100% loss on 0 and 00 -- the predicted probabilities of success are 12.1877%, 12.1178%, and 12.1411%, respectively.

At craps, a place bet on the six has an edge of -1.515% and a variance of 1.1639. A place bet on the four has an edge of -6.667% and a variance of 1.7422. Assume a player has a bankroll of $100 and bets either $6 on the six or $5 on the four. Chances of earning $100 for the former, calculated using each of the methods, are 39.3212%, 31.3192%, and 39.3199%. Chances of earning $100 with the latter, again calculated using the alternate approaches, are 17.8170%, 17.7599%, and 17.7789%

In these illustrations, the discrepancies increase as the edge becomes less insignificant relative to the variance -- as the simplifying assumptions would indicate. However, in any conceivable applications of risk of ruin, the final results would be considered equivalent. One reason is that the implied presumption of a normal distribution with zero skewness, which underlies all three equations, undoubtedly introduces greater uncertainty than any of the later assumptions. Another is that the probabilistic nature of gambling ascribes little meaning to a few percentage points difference in risk of ruin from one situation to the next.

The survival criterion for risk of ruinSurvival-based risk of ruin is a function of bankroll, bet size, number of decisions, and the probabilities and payoffs associated with various possible results. The calculation is performed by assuming that after a reasonable number of trials, the

Central Limit Theoremof statistics would ensure that the distribution of players' fortunes (bankroll plus win or minus loss) would beNormalorGaussian.

Traditionally, however, risk of ruin was taken to be the probability that the player would be below the cut-off point after the prescribed number of decisions had been completed. In what must be recognized as a brilliant insight, however, Don Schlesinger (

Blackjack Attack) pointed out that survival-based risk of ruin is a barrier and not an end-point problem. That is, the concept should be interpreted literally as players exhausting their stakes and therefore being unable to continue. They may accordingly hit the barrier before the prescribed number of decisions has been completed, even though they might recover were they able to continue -- for instance, by selling their souls to the nearest available devil as a means of raising additional capital.

Schlesinger's expression for survival-oriented risk of ruin as a barrier problem is given in Equation 6. The first term in this expression gives the traditional probability of having exhausted a bankroll at the end of a session. The second term, which includes the exponential formulation for long term risk of ruin, modifies the result for the barrier effect. Schlesinger has noted that for many practical conditions, the barrier risk of ruin is roughly twice the end-point risk of ruin.

Where: p(n) = probability of losing B within n decisions

f(x) = cumulative normal distribution function of argument x

n = target number of decisions

E = edge (per decision)

V = variance (per decision)

B = bankroll

w = amount of wager

ApplicationsThe most straightforward applications of risk of ruin are to estimate players' chances of surviving sessions of specified duration, or of achieving target win goals, playing particular games with certain initial bankrolls and intended bet sizes. These values can be obtained by solving the equations presented. While the arithmetic may seem tedious, it is readily implemented on most contemporary spreadheets -- especially those with integral

cumulative normal distribution functions. For convenience, an Excel97 spreadsheet can be downloaded from this web site and used to obtain results directly.

Risk of ruin, alone, can be a revelation -- if not a rude awakening -- for large numbers of recreational gamblers. And more than a few who consider themselves in the more experienced category, as well. Most such players seriously overbet their bankrolls and wonder why they keep encountering cold streaks that knock them off the box in reasonably short order. At the same time, they greatly overestimate the amount they can reasonably anticipate winning with their starting stakes and the size bets they make.

This situation presents itself in games with which players are familiar, and whose volatility shouldn't be -- but often comes as -- a surprise. It becomes even more pronounced when players try different games, assuming that a bankroll and bet sizing strategy that affords them good playing time and certain levels of gain in one situation will do the same in another.

The inverse applications can also be useful. Namely, assuming players are comfortable with certain gambling budgets, how much should they bet to have a reasonable level of confidence they'll survive the number of hours they intend to gamble. And, with this established, what level of earnings should they consider adequate. In principal, the equations can be solved to yield bankroll or bet size, given the other parameters and a target risk of ruin. In practice, this is generally infeasible. Iterative programs could be written to obtain the answers by repetitive evaluation of the equations in the form presented. However, owing to the ease with which answers can be obtained using spreadsheets, it is relatively easy to plug a reasonable set of starting values into the present equations, then raise or lower bets or bankrolls until the risk of ruin gets close enough to the target value.

The trade-offs between the two risk of ruin criteria can also be illuminating. In particular, by changing values manually, it quickly becomes clear that bets small relative to bankroll improve the probability of survival, while decreasing the likelihood of achieving any stated win goal before depleting a stake. Players may wish to achieve a balance between these factors, sacrificing the one to enhance the other -- or at least recognizing that by emphasizing one factor, the other will necessarily suffer.

Example I -- Blackjack with basic strategyIn a six deck game with reasonable rules for resplitting and doubling, edge per hand for the player is -0.5% and standard deviation per hand is 1.225. Assume a player has a bankroll of $1,000 and wishes to bet flat at $25 per hand. What is the likelihood that the player will still be in the game after 500 rounds?

Equation 6 gives the risk of ruin for this situation as 12.97%. The player can therefore be roughly 87% sure of surviving the 500 rounds at this level.

If the player also has the goal of doubling his or her money by betting in this manner, regardless of the number of rounds required, and is willing to sacrifice the original stake trying, what is the likelihood of success?

Equations 1, 4, and 5 all give the risk of ruin as 57.87% under these conditions. The chance of success is the complement, or approximately 42%.

Example II -- Blackjack with card countingSchlesinger (

Blackjack Attack) uses an example in which a card counter achieves an edge of 1.79% with a betting spread such that the standard deviation is 3.35. A player starting at $25 per round, with a bankroll of $1,000, would have a risk of ruin equal to 9.7% under these conditions within 500 rounds, using Equation 6. Chance of survival is therefore 90.3%.Equations 1, 4, and 5 indicate that this player has 64.4% chance of doubling his money before going broke under these conditions.

Example III -- Switching from blackjack to Caribbean StudA recreational player is accustomed to a $500 buy-in at $10 blackjack, but decides to try Caribbean Stud because of the opportunities for big payouts. Assume that in blackjack, the player has been betting $10 flat and following a strategy that gives the house an edge of 1%. At Caribbean Stud, the player also starts with $500 and bets $10 on the ante.

In terms of survival, this player has only 6.8% risk of ruin before the completion of 500 rounds of blackjack. The foray into at Caribbean Stud turns out to be a disaster. With an edge of -5.27% and a standard deviation of about 2.25, risk of ruin within 500 rounds is as approximately 50%.

As an added point of interest, suppose the player has been content with a profit of $150 at blackjack. His or her chance of reaching this level before exhausting the $500 is the complement of the risk of ruin -- about 67%. The player will somewhat less readily achieve this level of profit at Caribbean Stud -- the chance of success being roughly 64%. Moreover, betting $5 instead of $10 to enhance survival (to 88% in 500 rounds) reduces the chances of reaching the $150 profit level on a $500 bankroll to just over 50%. Of course, a big hit sometime during the session will change the picture radically for an individual. However, the estimates based on the equations appear to be reliable when the performance of many players is taken into account.

Example IV -- Buying the four and betting "no four" at crapsA "buy" bet on the four at craps is a wager with three ways to win and six to lose, and a payoff of 2-to-1, except that the house collects a

vigorishequal to 5% of the wager regardless of outcome. Therefore, to buy the four for $20, players must drop $21 on the layout. If they win, the house returns the $20 nominal bet along with a $40 payoff. For all practical purposes, therefore, this is a wager of $21 to win $39, with a probability of success equal to 0.333%. At this level, on a per unit basis, the edge on the bet is -0.04761%, the standard deviation is 1.346, and the skewness is +0.707.

A "no four" bet at craps, also known as a lay bet against the four, is a wager with six ways to win and three to lose, and a payoff of 1-to-2, except that the house collects a

vigorishequal to 5% of the expected payoff regardless of outcome. Therefore, to bet a nominal $40 against the four, players must drop $41 on the layout. If they win, the house returns the $40 along with a $20 payoff. In reality, therefore, this is a wager of $41 to win $19 with a probability of success equal to 0.667%. At this level, on a per unit basis, the edge on this bet is -0.02430, the standard deviation is 0.6898, and the skewness is -0.707.

These bets are essentially the inverses of one another, and exhibit small skewnesses of equal magnitude but opposite direction. Note that the differences in edge and standard deviation results because of the inverse probability-payout relationship between the bets. The theoretical casino win is the

vigorish, equal to $1 for nominal $20 buy or $40 lay bets.

Earnings-based risk of ruin, calculated for these bets using Equations 1, 4, and 5 at various levels of play, all yielded essentially the same results. Simulations were also performed for these wagers, to verify the analytical predictions. The criteria for the simulations differed from those in the analysis in one dimension that could be expected to impact results. Namely, the end points in the simulation were levels when players' fortunes equalled or were less than zero, or equalled or exceeded the established target. The end points in the analysis are assumed to be exactly zero or the target earnings.

The following tables, A and B, compare risk of ruin determined analytically and by simulation for the two propositions, with bets and win goals of various amounts, and bankrolls of $1,000.

Table A

Risk of Ruin for Buy Bets

on the Four at Crapsbet win

goalrisk of ruin

simulatedrisk of ruin

analytical21 100 26% 23% 63 100 17% 13% 105 100 20% 12% 21 500 73% 73% 63 500 48% 48% 105 500 46% 42% 21 1000 93% 92% 63 1000 71% 70% 105 1000 63% 62%

Table B

Risk of Ruin for Buy Bets

against the Four at Crapsbet win

goalrisk of ruin

simulatedrisk of ruin

analytical41 100 26% 23% 123 100 15% 13% 205 100 18% 12% 41 500 74% 73% 123 500 48% 47% 205 500 43% 42% 41 1000 93% 92% 123 1000 70% 69% 205 1000 62% 62% The tables show the greatest discrepancies between the analytically predicted and simulated results when bets are large relative to the win goal. The effect is ascribed to the win levels at which play is terminated overshooting the nominal value by an amount averaging half the payout, such that a greater risk of ruin would be predicted were the actual quitting points used in the equations. The tables also suggest that the differences in skewness do not affect risk of ruin calculations. Results are essentially the same for the analytically derived values on the buy and lay bets, as would be expected because skewness is ignored. However, the results of the simulations are also similar for the alternate bets.

Example V -- Betting a Single Spot at 00 RouletteIndividual bets on single spots at double-zero roulette have edge equal to -5.26%, standard deviation of 5.762, and skewness of +5.918. This statistical distribution not only has large edge and variance, but is significantly skewed from the normal with a moderately long narrow tail on the positive side showing low-probability large returns.

Values of earnings-oriented risk of ruin, calculated using Equations 1, 4, and 5, all yield essentially equivalent results. Representative values found in this manner are compared with risk figures obtained from a simulation in Table C, below, for representative conditions, for a player with a starting bankroll of $100.

Table C

Risk of Ruin for Bets

on Single Spots at 00 Roulettebet win

goalrisk of ruin

simulatedrisk of ruin

analytical1 50 44% 39% 5 50 60% 34% 9 50 73% 34% 1 110 63% 60% 5 110 63% 58% 9 110 73% 53% 1 190 76% 75% 5 190 74% 68% 9 190 72% 67% The analytical predictions for risk of ruin tend to be smaller than those determined by simulation, with the most pronounced effects evident at win goals and large bets. This is probably due to the same phenomenon noted for buy and lay bets on the four at craps, namely that the simulated sessions were considered to have failed or succeeded when players' fortunes went below or above the limits, while the analytical predictions assume that the limits are reached exactly. Owing particularly to the size of the payout, successes can be expected to exceed the nominal limit in most cases, often by large amounts. If actual termination points rather than thresholds could be used, the analytical predictions for probability of earnings-oriented ruin would undoubtedly be much greater.

On the other hand, as values of win goal increased relative to the sizes of both the wager and the payoff, the figures converged. This suggests that the simplifications underlying the equations are reliable for these situations.

Example VI -- skewed betting strategiesPlayers can introduce skewness into the probability distributions for their sessions using various progressive betting schemes. In negative progressions or

Martingalesystems, bets are raised after losses and returned to the base value after wins; these tend to yield sessions with high probabilities of small wins. In positive progressions, bets are raised after wins and returned to the base value after losses; these produce sessions with small probabilities of high wins. As a practical matter, the offsetting factor in either case is bankroll depletion.

To evaluate the reliability of the earnings-oriented risk of ruin equations with skewed betting strategies, a game was hypothesized with chances of winning and losing each equal to 50%, and a 2% commission charged on all winning bets. This gamble was simulated with the following betting strategies:

a) Negative -- bets start at 1 unit, and progress to 2, 4, 8, ... 8 following successive losses. That is, they remain at 8 units regardless of the number of successive losses greater than three. Bets return to 1 unit after a win.

b) Positive -- bets start at 1 unit, and progress to 2, 4, 8, ... 8 following successive wins. That is, they remain at 8 units regardless of the number of successive wins greater than three. Bets return to 1 unit after a loss.

The amounts and frequencies of the bets are the same for both strategies. Neglecting the initial bet effect, these are 0.5 @ 1 unit, 0.25 @ 2 units, 0.125 @ 3 units, and 0.125 @ 8 units. The mean and variance are therefore equal -- average bet is 2.5 units and bet variance is 5.25. For purposes of analysis, the edge is -1% and the unit variance is (5.25/2.5)*(0.5)*(0.98

^{2}+ 1^{2}) = 2.05842.

Table D shows the risks of ruin determined by simulation for the two betting strategies, along with the values predicted analytically using the even-money equivalent calculation. The table indicates that the analytical results were extremely close to the risks of ruin determined by simulation with the negative progression, and were near enough for most practical purposes to those for the positive scheme.

Table D

Risk of Ruin for Proposed Gamble

with Positive and Negative Progressionsbank win

goalpositive

progressionnegative

progressionanalytical 50 50 57.2% 54.7% 54.8% 100 100 62.5% 59.8% 59.6% 100 50 41.9% 39.8% 40.0% 50 100 74.9% 73.7% 72.9%

Areas for future investigationThe analytical approaches to risk of ruin appear to be reliable for the broad classes of applications in which bankrolls and win goals are large relative to bet sizes and payoffs. Further investigation seems warranted in a number of areas.

A) Means to allow for effects of overshooting win goals caused by high-ratio payoffs.

B) Means to allow for effects of being unable to make the next bet, even though bankroll is not depleted, because of high minimum wager requirements. This would be of most interest in lay bet situations or in instances where players bet on combinations of propositions such as multiple place bets at craps.

C) Effect of the extremely high positive skewness characterizing jackpot oriented games which pay according to schedules rather than fixed amounts per bet.

D) The impact of a single large return, early in a session, on final probabilities of survival and of reaching target earnings.