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  • This post continues Part 3 of a series on the Kelly criterion and its application to sports betting. Part 1 provides an introduction to the Kelly criterion and a worked example. Part 2 provides a simple derivation. This post discusses fractional Kelly betting, where a fraction of the amount recommended by the Kelly criterion is placed on each bet.

    Why consider fractional Kelly betting?

    There are two popular criticisms of the Kelly criterion. The first concerns Kelly betting in general, and the second concerns the application of the Kelly criterion to sports betting.

    The first criticism is the roller-coaster ride that your account balance takes when you implement a Kelly strategy. The Kelly criterion can provide optimal bet amounts that exceed 50% of the account balance, which creates a high degree of volatility.

    The second criticism is that, when applied to sports betting, the Kelly criterion does not account for the uncertainty in your perceived probability of occurrence. Kelly betting can be applied to Blackjack with a high degree of certainty in the calculated probabilities. With sports betting, however, you may feel the probability of an outcome is 50%, but you do not know that with certainty. The true probability may lie between 40% and 60%, or even 20% and 80%. Applying the Kelly criterion without acknowledging the uncertainty in the probability can lead to ruin.

    These criticisms have led many practitioners to adopt fractional Kelly betting, also known as partial Kelly betting. It involves taking the bet amount recommended by the Kelly criterion and multiplying it by a certain fraction. This results in less volatile returns and a lower chance of the account balance hitting zero.

    The Kelly criterion formula revisited

    The Kelly criterion calculates the fraction, f, of the account balance that should be placed on a bet, given the available odds and your perceived probability of winning. The formula depends on how you express the betting odds, so two versions are presented below. Version A uses the decimal odds system that is popular in Australia. Decimal odds of 2.50 mean that if you win, a $10 bet would result in a $25 payout and a $15 profit. Version B uses the “b to 1″ odds system, where odds of “3 to 1″ mean that if you win, a $10 bet would result in a payout of $40 and a profit of $30.

    Version:

                         A                      B
    Formula:

               
    Variables:

          f = fraction of your bankroll to bet
    d = decimal betting odds

    p = probability of bet winning
    q = 1 – p = probability of bet losing

          f = fraction of your bankroll to bet
    b = “b to 1″ betting odds
    p = probability of bet winning
    q = 1 – p = probability of bet losing

    Fractional Kelly betting

    With fractional Kelly betting, you multiply f by a number between 0 and 1. Hence, if we use h as our multiplier, where 0 < h < 1, we now bet h*f of our account balance instead of f. The lower the value of h, the more conservative the strategy, because you are betting a smaller percentage of your account balance. For example, a quarter Kelly strategy, where h = 1/4, is more conservative than a half Kelly strategy, where h = 1/2. The graph below provides a stylized comparison of a full Kelly versus a half Kelly and a quarter Kelly. The graph has been drawn by simulating 100 bets with odds of 2.10 and a perceived probability of occurrence (which is modeled as being true) of 50%. Observe that the full Kelly criterion results in the most volatility, but is also expected to provide the highest long-term growth.

    How to implement fractional Kelly betting

    With full Kelly betting, the only ambiguous variable is your perceived probability, p, of an outcome occurring. The betting odds are known with certainty, and q is simply 1 – p. With fractional Kelly betting you now have the additional decision of how much to reduce f by. An aggressive strategy is to multiply f by a number close to 1, and a conservative strategy is to multiply f by a number close to zero.

    So how do you choose a value for h?

    A non-technical approach is to set an arbitrary value for h, say h = 0.25, and commence Kelly betting. If after a few hundred bets your strategy has enjoyed positive results, you could increase h to 0.30 for the next set of bets, and proceed in this manner until you find a value for h that you feel comfortable with.

    A more technical approach is to record for a few hundred bets over time:

    1. The betting odds
    2. Your perceive probability of occurrence
    3. The Kelly criterion value for f
    4. The result of that bet (win/lose)

    For an arbitrary value of h, simulate your portfolio performance had you bet h*f on these fixtures. Excel solver can be used to find the value of h that would have maximised your account balance growth over time. Alternatively, you could calculate the value of h that, in retrospect, would have provided a level of volatility that you feel comfortable with. This value of h could then be applied, and tweaked over time, as more betting data are recorded.

    Variable fractional Kelly betting

    While some practitioners use the same value of h for all bets, one alternative is to vary h depending on your confidence in the chosen value of p. Recall that the Kelly criterion calculates the optimal bet amount based on the disparity between p and the probability that is implied by the betting odds (1/d). By applying variable fractional Kelly betting, your bet amount will depend both on the disparity between p and 1/d, and your confidence in the chosen value for p.

    Let h equal the maximum fraction you would ever consider to multiply f by. If you set h = 1, then in some circumstances you are willing to implement full Kelly betting. Alternatively you may only wish to bet at most 0.5*f, hence h = 0.5.

    Now let the variable g represent your confidence in the chosen value of p, where 0 < g < 1. g = 0 equates to no confidence, and g = 1 represents full confidence. For each bet, calculate f and g, and bet the fraction f*h*g of your account balance. If you are absolutely certain in p, then g = 1, and you would bet your maximum fraction h*f. The weakness in this approach is it provides no guidance on how to set g. How do you measure the level of certainty in your perceived probability? You may be more confident in your knowledge of some sports than others, but how to you quantify that? If you use a betting tracker spreadsheet you could compare your historical performance across different sports. Stronger performance indicates that you are better able to assess the probabilities in that sport. One approach is to set g based on your performance percentile for that sport compared to the rest. For example, if you regularly bet on eleven sporting markets, then you could set g as follows.

    Sport Return    Rank    g  
    Super Rugby 3% 7 0.4
    Test Match Cricket 2% 8 0.3
    Premier League Football   8% 3 0.8
    Men’s Tennis 6% 4 0.7
    NBA Basketball 5% 5 0.6
    Women’s Tennis -3% 10 0.1
    Formula 1 -4% 11 0.0
    NCAA Basketball -1% 9 0.2
    NRL Line Betting 12% 1 1.0
    NRL Head to Head 11% 2 0.9
    AFL Line Betting 4% 6 0.4

     

    g has been calculated above as (11 – rank)/(11 – 1). Hence, for h = 0.5, you would bet 0.4*f on Premier League fixtures and 0.3*f on NBA fixtures. The value of h could be chosen using the methods outlined in the previous section.

    Coming up in Part 4

    The next post will provide further discussion and a critique of the Kelly criterion.

  • This post continues Part 3 of a series on the Kelly criterion and its application to sports betting. Part 1 provides an introduction to the Kelly criterion and a worked example. Part 2 provides a simple derivation. This post extends the Kelly criterion to incorporate the possibility of a refund. It is recommended that you read Part 1 and Part 2 before proceeding.

    Introduction

    In the Kelly betting framework, recall from Part 2 the variable p (the probability that an outcome will occur) and the variable q = 1 – p (the probability that the outcome will not occur). If you are betting on this outcome, you believe you will win with probability p, and lose with probability q.

    Note that q is defined as 1 – p to ensure that p + q = 1. This implies that no other outcome is possible: you either win or lose. There are, however, bets where refunds are a distinct possibility. In this case there are three possible outcomes and p + q < 1. This post will show that only a slight tweak to the Kelly criterion is required to incorporate the possibility of a refund.

    When are refunds a possibility?

    Refunds are a distinct possibility for sports such as:

    • Football (soccer)
      bet365 refunds all losing pre-match Correct Score, Half-Time/Full-Time and Scorecast bets in the event of a 0-0 draw.
    • Tennis
      Each bookmaker has it’s own policy, but Sportsbet, among others, refunds all bets if a player retires injured. Note that Betfair does not do this, which is something to keep in mind when comparing odds.
    • Boxing
      For example, Sportsbet refunded losing bets on the Green v Briggs bout.
    • Promotions
      For popular events bookmakers often run promotions where they refund certain losing bets in the event of a particular outcome. Some of these refunds are announced without warning. For the 2010 FIFA World Cup winner, Sportsbet refunded all losing bets on Australia because they felt the red cards Australia received were harsh.

    Notation

    This post continues with the notation used in Part 1 and Part 2. Let:

    • W0 = your account balance before you make a bet
    • Wn = your account balance after making n bets
    • f = the fraction of your bankroll (account balance) to bet on a particular outcome
    • d = the decimal betting odds for that outcome (2.50 means a winning $10 back bet would have a payoff of $25 and a profit of $15)
    • p = your perceived probability of that outcome occurring (where 0 < p < 1)
    • q = your perceived probability of that outcome not occurring (where 0 < q < 1 and p + q < 1)

    A new variable is introduced in this post. Let:

    • j = your perceived probability of receiving a refund (where 0 < j < 1 and p + q + j = 1)

    Incorporating the possibility of a refund

    In the Kelly betting framework we now have three possible outcomes:
    – win (d-1)f with probability p
    – lose f with probability q
    – have the bet refunded with probability j

    The account balance after making one bet can be one of three possible values:
    – W1 = W0(1 + f*(d-1)) with probability p
    – W1 = W0(1 – f) with probability q
    – W1 = W0 with probability j

    Recall that in Part 2 the exponential rate of growth of the gambler’s capital was expressed as:

    (1/n)*log(Wn/W0) = log[(1 + f*(d-1))k(1 – f)(n-k)]1/n

    The blue text represents the gross return for a win and the red text represents the gross return for a loss. Note that in the event of a refund, the account balance is unchanged, so W1 = W0 and W1/W0 = 1. If, after n bets, you had k wins, m refunds, and n-k-m losses, then this expression can be modified to be:

    (1/n)*log(Wn/W0) = log[(1 + f*(d-1))k(1 – f)(n-k-m)(1)m]1/n

    The number 1, raised to any power, is simply 1 so this expression simplifies to:

    (1/n)*log(Wn/W0) = log[(1 + f*(d-1))k(1 – f)(n-k-m)]1/n

    Note that for a sufficiently large n, (n-k-m)/n = q. If you plug the expression above into the workings in Part 2 and solve for f, you get:

    Note that when q + p = 1 this expression simplifies to the standard Kelly criterion discussed in Part 1 and Part 2. Also note that the denominator term (p + q) could be rewritten (1 – j), where j is the probability of a refund.

    The formula above uses the decimal odds system that is popular in Australia. For those who are more familiar with the “b to 1″ betting odds used in other literature, the corresponding formula is:

    Interpretation

    The impact of a potential refund depends on whether it reduces the possibility of winning, reduces the possibility of losing, or both. With football betting on bet365, the 0-0 draw refund has no impact on the possibility of winning, but it does reduce the possibility of losing, because the refund only applies to losing bets.

    In tennis betting, if the bookmaker voids and refunds all bets in the event of injury, this reduces the probabilities of winning and losing because both players have the potential to retire injured. Having said that, it’s not uncommon for players to pick up a an injury during a tournament, but continue playing. You may feel that the injured player is more likely to withdraw hurt than their opponent. This would have an asymmetric impact on your perceived probabilities of winning and losing. Another point to consider is, for someone playing with an injury, there may be a higher chance they will retire hurt if they are losing the match, than if they are winning.

    The Arsenal vs Chelsea fixture revisited

    Recall the hypothetical Arsenal vs Chelsea fixture used in Part 1. To recap, the odds are as follows, with your perceived probabilities of occurrence shown in brackets:

    Full-time result:

    Arsenal: 2.60 (p = 20% chance of occurring)
    Chelsea: 2.65 (p = 50% chance of occurring)
    Draw: 3.25 (p = 30% chance of occurring)

    Based on the standard Kelly criterion, you should bet
    f = [0.5(2.65 – 1) – 0.50]/(2.65 – 1) = 0.197 = 19.7% of your account balance on Chelsea.

    Now pretend the bookmaker refunds all losing full-time bets in the event of a 0-0 draw. Suppose you believe there is a 5% chance of a 0-0 draw, and you believe the outcome probabilities are now:

    • Arsenal: 2.60
      p = 20% chance of winning, q = 75% chance of losing, j = 5% chance of a refund
    • Chelsea: 2.65
      p = 50% chance of winning, q = 45% chance of losing, j = 5% chance of a refund
    • Draw: 3.25
      p = 30% chance of winning, q = 70% chance of losing, j = 0% chance of a refund

    Based on the Kelly criterion, after adjusting for the possibility of a refund, you should bet
    f = [0.5(2.65 – 1) – 0.45]/[(2.65 – 1)(0.5 + 0.45)] = 0.239 = 23.9% of your account balance on Chelsea.

    The optimal bet size has increased in this example because the perceived probability of losing has dropped.

    Coming up in Part 3d

    Part 3d will discuss fractional Kelly betting.

  • This post continues Part 3 of a series on the Kelly criterion and its application to sports betting. Part 1 provides an introduction to the Kelly criterion along with a worked example. Part 2 provides a simple derivation of the Kelly criterion. Part 3a extends the Kelly criterion to incorporate backing and laying bets using an exchange such as Betfair. This post extends the Kelly criterion to account for the time value of money. It is recommended that you read Part 1 and Part 2 before proceeding, if you have not done so already.

    What is the time value of money?

    The time value of money is the concept that a dollar today is worth more than a dollar in the future. If someone offered you a choice between receiving $1,000 now and receiving $1,000 in a year’s time, you would prefer to receive the money today. This is because even if you don’t need the money now, you can earn interest on the $1,000 between now and next year.

    The time value of money depends on two variables: time and interest rate. The greater the time until you receive a payment, the less that payment is worth. This is because you have to forgo additional time that the money could be sitting in an interest bearing account. Also, the greater the interest rate, the less the payment in the future is worth. This is because money in a high interest account would have grown by a larger amount in the meantime had you received it now.

    In essence, the time value of money relates to the opportunity cost of the funds. By receiving $1,000 in a year’s time instead of today, you have lost the opportunity to earn interest on that money in the meantime.

    Visit Wikipedia to learn more about the time value of money.

    The implication of time in sports betting

    With sports betting, punters have access to a wide range of betting options for events around the globe. These bets can vary drastically in terms of the time it takes for settlement to occur. Settlement is the process of determining whether your bet has won or lost. Obviously, settlement can only occur after the conclusion of the sporting fixture. Typically, most bookmakers settle your bets within minutes of the end of the sporting event. For some events, settlement occurs within a few seconds of the conclusion of the fixture.

    Below is a selected list of betting options and their associated settlement dates. Note that this post was written at 2pm on July 21, 2010.

    Event Settlement date Approximate time (years)
    Forex Market Movement 5 minutes or less 9.51E-06
    ATP Atlanta Tennis Fixture 6:00 AM July 22nd 2010      0.0018
    Tour de France winner July 25th 2010 0.011
    Tri Nations Winner September 11th 2010 0.14
    Rugby World Cup 2011 Winner      Octboer 23rd 2011 1.26
    FIFA World Cup 2014 Winner July 13th 2014 3.98

     

    For fixtures that settle within the next week or so, the time value of money can be safely ignored. However for tournaments that are held in upcoming years, the time value of money is an important consideration.

    This post will illustrate that for fixtures that settle in the distant future, the optimal bet amount drops. The conclusion is that you should bias your betting towards fixtures that conclude sooner rather than later.

    Notation

    This post continues with the notation used in Part 1 and Part 2. Let:

    • W0 = your account balance before you make a bet
    • Wn = your account balance after making n bets
    • f = the fraction of your bankroll (account balance) to bet on a particular outcome of a sporting event
    • d = decimal betting odds for that outcome (2.50 means a winning $10 back bet would have a payoff of $25 and a profit of $15)
    • p = your perceived probability of an outcome occurring (where 0 < p < 1)
    • q = 1 – p = probability of that outcome not occurring

    We will introduce some new notation for this post. Let:

    • r = the annual interest rate that you could get if you invest your funds elsewhere
    • t = the time, measured in years, between now and the settlement of the bet

    Calculating the time value of money

    If you were given the choice of receiving $1,000 today or $x,xxx in three year’s time, what future figure would make you indifferent between your two options? Most people would turn down $1,001, and most would accept $3,000. If you could invest $1,000 at an interest rate of 5%, then you would have $1,000(1.05) = $1,050 in a year’s time. If you reinvested that figure for the second year you would have $1,050(1.05) = $1,102.50 in two years time. If you reinvested that figure for the third year you would have $1,102.50(1.05) = $1,157.63 in three years time. This means you would have to be offered at least $1,157.63 to consider taking the future amount rather than the $1,000 today.

    We can recalculate this figure as $1,000(1.05)3 = $1,157.63. More generally, let PV (present value) denote a payment today, and let FV (future value) denote the future payment that is of equivalent value to the PV. We can calculate the FV as FV = PV(1+r)t. If we knew the FV and wanted to know the PV, we could calculate PV = FV/(1+r)t. Note that this is identical to the mathematical statement PV = FV(1+r)-t.

    For those who have a stronger mathematics background, you may prefer to use an exponential interest rate to be consistent with the Kelly criterion’s objective of maximizing the exponential growth rate of your sporting account balance. In this case FV = PV*ert and PV = FV*e-rt. Note that the value for r in this formula will be slightly lower than that used in the previous paragraph, because it is the continuously compounded rate.

    To make the workings understandable for a greater audience, PV = FV/(1+r) will be used henceforth. Simply substitute this expression with PV = FV*e-rt if you wish instead to use a continuously compounded rate.

    The Kelly criterion revisited

    Recall that in Part 2 the exponential rate of growth of the gambler’s capital is expressed as:

    (1/n)*log(Wn/W0) = log[(1 + f*(d-1))k(1 – f)(n-k)]1/n

    The blue text represents the gross return for a win and the red text represents the gross return for a loss. Let $x denote the dollar value of a bet that you make. In terms of cash flow timings, a losing bet can be treated as occuring immediately. You make your bet of $x, which is taken from your account balance today and never returned. If you make a winning bet, you have $x taken from your account balance today and $x(d-1) returned to your account balance at some point in the future, where d are the decimal odds of your bet. You can think of this as an investment. You deposit $x now and receive $x(d-1) in the future. This payment in the future (FV) must be discounted to get our PV using the formula PV = FV/(1+r).

    When we take into account the time value of money, the gross return for winning becomes (1 + f*(d-1)/(1+r)t)k, while the gross return when your bet loses remains unchanged. If you plug this expression into the workings in Part 2 and solve for f, you get:

    The formula above is for the decimal odds system that is popular in Australia. For those who are more familiar with the “b to 1″ betting odds used in other literature, the corresponding formula is:

    Observe that as r and t increase, the value of f decreases.

    The Arsenal vs Chelsea fixture revisited

    Recall the hypothetical Arsenal vs Chelsea fixture that was used in Part 1. To recap, the odds are as follows, with your perceived probabilities of occurrence shown in brackets:

    Full-time result:

    Arsenal: 2.60 (pA = 20% chance of occurring)
    Chelsea: 2.65 (pC = 50% chance of occurring)
    Draw: 3.25 (pD = 30% chance of occurring)

    Based on the Kelly criterion, you should bet
    f = [0.5(2.65 – 1) – 0.50]/(2.65 – 1) = 0.197 = 19.7% of your account balance on Chelsea.

    We will now recalculate the optimal bet amount assuming that the fixture will take place: in (A): 3 day’s time, (B): 30 day’s time, (C): 1 year’s time, and (D): 3 year’s time. Assume you can get 5% per annum in an interest bearing account. Using the formula above we get:

    (A): t = 3/365 = 0.008219
    (B): t = 30/365 = 0.08219
    (C): t = 1
    (D): t = 3

    f(A) = 0.5 – 0.5(1.050.008219)/1.65 = 0.197
    f(B) = 0.5 – 0.5(1.050.08219)/1.65 = 0.196
    f(C) = 0.5 – 0.5(1.051)/1.65 = 0.182
    f(D) = 0.5 – 0.5(1.050.008219)/1.65 = 0.149

    Hence, for fixtures within the next month or so, you can pretty much ignore the time impact of money unless you are making substantially large bets. However, for events that conclude more than a year from now, the time value of money becomes an important consideration.

    You may find that your optimal bet size drops below zero, meaning you shouldn’t make a bet at all. Consider the 2014 FIFA World Cup. You can get odds of 4.50 with bet365 for Brazil to win the tournament. Suppose you believe that Brazil has a 1 in 4 chance of winning. Ignoring the time value of money, you would calculate your optimal bet as:

    f = [(0.25)(4.50 – 1) – 0.75]/(4.50 – 1) = 0.036

    This means you would bet 3.6% of your account balance on Brazil to win the tournament.

    However, if you consider the time value of money, where t = 3.98 and r = 5%, your optimal bet size becomes:

    f = 0.25 – 0.75*1.053.98/3.5 = -0.01

    Hence, if the time value of money is taken into account, you would chose not to place a bet on Brazil.

    Coming up in Part 3c

    The next post will provide an extension to the Kelly criterion when refunds are a distinct possibility. For example, bet365 refunds losing football (soccer) bets in the event of a 0-0 draw.

Bet with the Best!
  • This post is Part 3 of a series on the Kelly criterion and its application to sports betting. Part 1 provides an introduction to the Kelly criterion along with a worked example. Part 2 provides a simple derivation of the Kelly criterion. Part 3 in this series provides some extensions to the Kelly criterion. This post extends the Kelly criterion to incorporate backing and laying bets using an exchange such as Betfair. It is recommended that you read Part 1 and Part 2 before proceeding, if you have not done so already.

    What is backing and laying?

    With standard bookmakers you can only bet for an outcome to occur. Because Betfair is an exchange, rather than a bookmaker, you can bet both for and against sporting outcomes.

    If you ‘back’ a bet on an outcome, you are betting that the outcome will happen. For example you could back a bet on Federer to win Wimbledon. If the odds are 3.50, a $10 bet would result in a $25 profit if he wins, and a $10 loss if he doesn’t.

    If you ‘lay’ a bet against an outcome, you are betting that the outcome will not happen. For example you could lay a bet on Federer to win Wimbledon. If the odds are 3.50, a $10 bet would result in a $25 loss if he wins, and a $10 profit if Federer doesn’t win. Essentially, when you lay a bet, you act as the bookmaker. You get to keep the other punter’s money if they lose, but have to pay them at the agreed odds if they win.

    The Kelly criterion is designed for making ‘back’ bets. According to the formula, you would not make a bet if f < 0. This post adjusts the Kelly criterion to account for Betfair commissions. It also provides the corresponding formula for a lay bet.

    Betfair commissions

    Unlike standard bookmakers, who incorporate their fees directly into the odds themselves, Betfair charges a commission fee to the winner of each bet. The commission fee is calculated as a percentage of your net winnings (profit) as follows:

    Commission = (market base rate) * (1 – discount rate)

    The market base rate is typically 5%. The discount rate ranges between 0% and 60%, depending the level of your account activity. The more bets you make, the greater the discount. A discount rate of 60% corresponds to a commission of 2%, hence the commission fee charged by Betfair ranges between 2% and 5% of your net winnings.

    Henceforth we will denote the commission fee (as a percentage of your net winnings) as c.

    Notation

    This post continues with the notation used in Part 1 and Part 2. Let:

    • W0 = your account balance before you make a bet
    • Wn = your account balance after making n bets
    • f = the fraction of your bankroll (account balance) to bet on a particular outcome of a sporting event
    • d = decimal betting odds for that outcome (2.50 means a winning $10 back bet would have a payoff of $25 and a profit of $15)
    • p = your perceived probability of an outcome occurring (where 0 < p < 1)
    • q = 1 – p = probability of an outcome not occurring

    This post will refer to the hypothetical Arsenal vs Chelsea fixture that was used in Part 1. To recap, the odds are as follows, with your perceived probabilities of occurrence shown in brackets:

    Full-time result:

    Arsenal: 2.60 (pA = 20% chance of occurring)
    Chelsea: 2.65 (pC = 50% chance of occurring)
    Draw: 3.25 (pD = 30% chance of occurring)

    For simplicity’s sake, it is assumed that due to the high liquidity of the betting market, the lay odds equal the back odds for each outcome. In reality, the lay odds are always slightly above the back odds. The more popular (and liquid) the market, the closer the back and lay odds are to each other.

    Example odds in a popular market:

    Example odds in an unpopular market:

    The Kelly criterion for backing a bet

    Recall that in Part 2 the exponential rate of growth of the gambler’s capital is expressed as:

    (1/n)*log(Wn/W0) = log[(1 + f*(d-1))k(1 – f)(n-k)]1/n

    The blue text represents the gross return for a win and the red text represents the gross return for a loss. In Betfair with a commission fee of c, the gross return for winning becomes (1 + f*(d-1)(1-c))k, while the gross return when your bet loses remains unchanged (no commissions are charged to the loser). If you plug this expression into the workings in Part 2 and solve for f, you get:

    The formula above is for the decimal odds system that is popular in Australia. For those who are more familiar with the “b to 1” betting odds used in other literature, the corresponding formula is:

    Note that for c = 0, the above formulas collapse to the original Kelly criterion used in Part 1 and Part 2 of this series. By taking the derivative of the expression for f with respect to c you will find that as c increases, f decreases. This is because the expected payoff for the bet drops as the commission rises, which makes the bet less worthwhile. Recall the example from Part 1:

    Arsenal: fA = [0.2(2.6 – 1) – 0.80]/(2.6 – 1) = –0.300
    Chelsea: fC = [0.5(2.65 – 1) – 0.50]/(2.65 – 1) = 0.197
    Draw: fD = [0.3(3.25 – 1) – 0.70]/(3.25 – 1) = –0.011

    If these odds were offered by Betfair, we need to adjust them for the commission fee using the formula presented above. For a commission fee of 5% the figures become:

    Arsenal: fA = [0.2(2.6 – 1)(1 – 0.05) – 0.80]/((2.6 – 1)(1 – 0.05)) = –0.326
    Chelsea: fC = [0.5(2.65 – 1)(1 – 0.05) – 0.50]/((2.65 – 1)(1 – 0.05) = 0.181
    Draw: fD = [0.3(3.25 – 1)(1 – 0.05) – 0.70]/((3.25 – 1)(1 – 0.05) = –0.027

    Now you would bet 18.1% of your account balance on Chelsea instead of 19.7%.

    The Kelly criterion for laying a bet

    When you lay a bet in Betfair you receive the other punter’s bet as a payout if the outcome does not occur, but you have to pay the punter at the agreed odds if the outcome does occur. As before, let p denote your perceived probability of the outcome occurring. In this case, you will lose the bet if the outcome does occur, and win the bet if it doesn’t.

    Your account balance if you win will be W0(1 + f*(1-c)), while your account balance if you lose will be W0(1 – f*(d-1)).

    Note that we must place the following restriction on f: 0 < f*(d-1) < 1, or equivalently, 0 < f < 1/(d-1). This prevents losing an amount that is greater than your account balance. You can think of this as your budget constraint. This contrasts with the budget constraint for backing an outcome: 0 < f < 1. Note that for a low odds bet of 1.01 and an account balance of $100, you could lay a bet amount up to $10,000, which is more than your account balance. This is because your total potential liability is ($10,000)(1.01 - 1) = $100. In contrast, for high odds of 101.00, the maximum you could lay for this bet is $1, which is far lower than your account balance of $100. We use < rather than < for the upper limit to avoid the possibility of your account dropping to zero.

    After n bets, k of which result in the outcome occurring (a loss for you), and (n-k) of which result in the outcome not occurring (a win for you), your account balance will be:

    Wn = W0(1 – f*(d-1))k(1 + f*(1-c))(n-k)

    We can express the exponential rate of growth of the gambler’s capital as:

    G = (1/n)*log(Wn/W0) = log[(1 – f*(d-1))k(1 + f*(1-c))(n-k)]1/n

    Setting the derivative of dG/df equal to zero and solving for f gives:

    The formula above is for the decimal odds system. For those who are more familiar with the “b to 1” betting odds used in other literature, the corresponding formula is:

    If the above workings are difficult to follow, I recommend re-reading Part 2 of this series.

    Referring back to the Arsenal vs Chelsea example, in the absence of commission fees you would calculate f as :

    Arsenal: fA = [0.8 – 0.2*(2.6 – 1)]/(2.6 – 1) = 0.300
    Chelsea: fC = [0.5 – 0.5*(2.65 – 1)]/(2.65 – 1) = –0.197
    Draw: fD = [0.7 – 0.3*(3.25 – 1)]/(3.25 – 1) = 0.011

    Note that these values are the exact opposites in sign of the corresponding figures for back bets in the absence of commission fees. For a commission fee of 5%, you would calculate f as:

    Arsenal: fA = [0.8(1 – 0.05) 0.2*(2.6 – 1)]/[(2.6 – 1)(1 – 0.05)] = 0.289
    Chelsea: fC = [0.5(1 – 0.05) – 0.5*(2.65 – 1)]/[(2.65 – 1)(1 – 0.05)] = –0.223
    Draw: fD = [0.7(1 – 0.05) – 0.3*(3.25 – 1)]/[(3.25 – 1)(1 – 0.05)] = –0.005

    Based on these figures you would choose to lay a bet against Arsenal using 28.9% of your account balance. It should be immediately noted that in doing so, it isn’t 28.9% of your balance that is at risk. If Arsenal fails to win, you would receive a profit equal to 28.9%(1 – 0.05) = 27.7455% of your account balance. If Arsenal does win, you will lose (2.60 – 1)(.289) = 46.24% of your account balance. Hence, when making a lay a bet, the fraction (d-1)f of your account balance is at risk and unavailable to bet elsewhere.

    Should I back a bet or lay a bet?

    In the example fixture between Arsenal and Chelsea, based on the Kelly criterion, you could back Chelsea using 18.1% of your account balance, or you could lay a bet against Arsenal using 28.9% of your account balance. So which action should you take? Should you do both?

    Recall that the objective in Kelly betting is to maximise the exponential growth rate of your account balance. Let’s compare the expected exponential growth rate for:

    A: Making a back bet on Chelsea
    B: Making a lay bet against Arsenal
    C: Making a back bet on Chelsea and making a lay bet against Arsenal

    In formula form, your account balance for each strategy after each possible outcome is provided below.

    Outcome Arsenal Win Chelsea Win Draw
    Probability pA pC pD
    Option A Payoff W0(1 – fBC) W0[1 + (1-c)(dC-1)fBC] W0(1 – fBC)
    Option B Payoff W0[1 – (dA-1)fLA] W0[1 + (1-c)fLA] W0[1 + (1-c)fLA]
    Option C Payoff W0[1 – fBC – (dA-1)fLA] W0[1 + (1-c)(dC-1)fBC + (1-c)fLA] W0[1 – fBC + (1-c)fLA]

     

    Plugging in the values for our Arsenal vs Chelsea example gives:

    Outcome Arsenal Win Chelsea Win Draw
    Probability 0.2 0.5 0.3
    Option A Payoff W0(1 – 0.181) W0[1 + (0.95)(1.65)(0.181)] W0(1 – 0.181)
    Option B Payoff W0[1 – (1.6)(0.289)] W0[1 + (0.95)(0.289)] W0[1 + (0.95)(0.289)]
    Option C Payoff W0[1 – 0.181 – (1.6)(0.289)] W0[1 + (0.95)(1.65)(0.181) + (0.95)(0.289)] W0[1 – 0.181 + (0.95)(0.289)]

     

    Note that the numbers below will differ slightly to the values calculated using the rounded figures above. Solving gives:

    Outcome Arsenal Win    Chelsea Win   Draw
    Probability 0.2 0.5 0.3
    Option A Payoff W0(0.8190) W0(1.2838) W0(0.8190)
    Option B Payoff W0(0.5368) W0(1.2750) W0(1.2750)
    Option C Payoff W0(0.3558) W0(1.5588) W0(1.0940)

     

    Let vi denote the profit/loss for outcome i. A win will be associated with vi > 0 and a loss will be associated with vi < 0. We can calculate the expected exponential growth rate for each strategy using the formula:

    W1/W0 = pA*log(1 + vA) + pC*log(1 + vC) + pD*log(1 + vD)

    Option A: W1/W0 = 0.2*log(0.8190) + 0.5*log(1.2838) + 0.3*log(0.8190) = 0.011
    Option B: W1/W0 = 0.2*log(0.5368) + 0.5*log(1.2750) + 0.3*log(1.2750) = 0.030
    Option C: W1/W0 = 0.2*log(0.3558) + 0.5*log(1.5588) + 0.3*log(1.0940) = 0.018

    In this example, option B of laying a bet against Arsenal provides the greatest expected exponential growth rate of the account balance. Hence based on the Kelly criterion and the above perceived probabilities of occurrence, you should make the single bet of laying 28.9% of your account balance against Arsenal to win.

    Coming up in Part 3b

    The next post will provide an extension to the Kelly criterion to account for the time value of money. Time is an important consideration when the bet settlement is in the distant future.

PinnacleSports.com Online Sports Betting
  • This post is Part 2 of a series on the Kelly criterion and its application to sports betting. Part 1 of this series provides an introduction to the Kelly criterion along with a worked example. This post provides a simple derivation of the Kelly criterion, which will hopefully provide additional insight into the model. Please read Part 1 before proceeding, if you have not done so already.

    The proof below uses the decimal betting odds system. To use the “b to 1” odds system, simply replace d with b + 1 in the workings below. Please note that this is not a rigorous proof. Readers should consult Kelly’s original paper for a formal derivation of the Kelly criterion.

    Notation

    We will continue with the notation used in Part 1. Let:

    f = the fraction of your bankroll (account balance) to bet on a particular outcome of a sporting event (where 0 < f < 1) d = decimal betting odds for that outcome (2.50 means a winning $10 bet would have a payoff of $25 and a profit of $15) p = your perceived probability of the bet winning (where 0 < p < 1)
    q = 1 – p = probability of the bet losing

    Furthermore, let W0 denote your account balance today, and let Wn denote your account balance after the settlement of n bets. Both W0 and Wn are in monetary units.

    Derivation

    Suppose you have an account balance, W0 and have the option of betting a on a sporting outcome that pays odds of d. You believe that the probability of this outcome occurring is p. What fraction, f, of your account balance should you place on this bet?

    If you win, your bet will result in a payout of f*W0*d. This corresponds of a profit of f*W0*(d-1). If you lose, then your bet results in a payout and profit of –f*W0, which is a loss.

    At the end of your bet, your new bankroll will be:

    W1 = W0 + f*W0*(d-1) = W0(1 + f*(d-1)) if you win, and:
    W1 = W0 – f*W0 = W0(1 – f) if you lose

    Now suppose you make the same bet two times. If you win both bets, you end up with:

    W2 = W0(1 + f*(d-1))(1 + f*(d-1)) = W0(1 + f*(d-1))2

    This is because your initial bankroll after the first bet is W1(1 + f*(d-1)) instead of W0.

    If you lose both bets, you end up with:

    W2 = W0(1 – f)(1 – f) = W0(1 – f)2

    This is because your initial bankroll after the first best is W1(1 – f) instead of W0.

    If you win the first bet and lose the second, you end up with:

    W2 = W0(1 + f*(d-1))(1-f)

    Note that this is the same value as if you had lost the first bet and won the second. This result holds true for any series of bets in this setup. If you win x times and lose y times, the order of the wins and losses is irrelevant to the final outcome.

    Observe that (1 + f*(d-1)) is the multiplier whenever a win occurs, and (1-f) is the multiplier whenever a loss occurs. So, after 8 games, if you won 5 times and lost three times you would end up with:

    W8 = W0(1 + f*(d-1))5(1 – f)3

    If we generalise this to playing the game n times and winning k times, we have:

    Wn = W0(1 + f*(d-1))k(1 – f)(n-k)

    When we divide both sides of the equation by W0, we have an expression for the growth of our initial bankroll.

    Wn/W0 =(1 + f*(d-1))k(1 – f)(n-k)

    This equation gives us the growth or our bankroll after n games. To calculate the growth in the bankroll per game, we need to raise both sides of the equation by a power of 1/n. To illustrate why we do this, suppose you invested $100 that returned $144 after two periods. If you wanted to know the growth rate per period, you would calculate it as follows:

    144 = 100*(1+growth)2
    144/100= (1+growth)2
    1.44 = (1 + growth)2

    We now raise both sides of the equation by 1/n, which is 1/2 in this case. This enables us to solve for the growth rate.

    1.441/2 = (1 + growth)2*1/2
    1.2 = (1 + growth)
    0.2 = growth
    growth = 20%

    Hence a growth rate of 20% results in a balance of $144 after two periods.

    Returning to our example:

    Wn/W0 =(1 + f*(d-1))k(1 – f)(n-k)
    (Wn/W0)1/n = [(1 + f*(d-1))k(1 – f)(n-k)]1/n

    The Kelly criterion seeks to maximise the exponential rate of growth per game, hence we seek to maximise the log of (Wn/W0)1/n. We achieve this by choosing the optimal fraction of our wealth to bet: f.

    In Kelly’s original paper he expressed “the exponential rate of growth of the gambler’s capital” as (pg. 919):

    Note that this is consistent with our workings because a known property of logs is log(Wn/W0)1/n = (1/n)*log(Wn/W0).

    (1/n)*log(Wn/W0) = log[(1 + f*(d-1))k(1 – f)(n-k)]1/n

    Let G = (1/n)*log(Wn/W0). Using known properties of logs we get:

    G = (1/n)log[(1 + f*(d-1))k(1 – f)(n-k)]

    G = (1/n)[k*log(1 + f*(d-1)) + (n-k)*log(1 – f)]

    Anyone who remembers their calculus will recall that if we seek to maximise a function by choosing the optimal value for f, then we need to take the first derivative of the function with respect to f, and set it equal to zero.

    Note that if our perceived probability p of success is true, then in the limit as n approaches infinity, we have p = k/n and q =(n-k)/n. Recall that k is the number of successes out of n games, and (n-k) is the number of failures out of n games.

    Noting that p + q = 1 and solving for f gives:

    Recall that this solution is for the decimal odds system that is popular in Australia. If you use the "b to 1" odds system then the corresponding formula is.

    You can take my word for it that this value for f corresponds to a unique maximum for G(f) on [0,1). A formal proof of this can be found here.

    Note that if f < 0, then you would anticipate a loss by placing a bet at the available odds. Hence in this circumstance you would bet nothing, and the optimal fraction, f, equals 0. However if you had a Betfair account you may choose to lay a bet against the outcome, although you would have to recalculate f using a model that pays odds of 2.00 if the outcome does not occur, and costs you the lay odds if the outcome does occur. This extension of the Kelly criterion will be discussed in Part 3 of this series.

    Coming up in Part 3

    Part 3 will provide some extensions to the Kelly criterion. At this stage I may actually break it down into a number of posts. The extensions will be:
    3a – the Kelly criterion for lay bets in Betfair
    3b – the Kelly criterion that accounts for the time value of money
    3c – the Kelly criterion when refunds are a distinct possibility
    3d – fractional Kelly betting

    Sources

  • In an earlier post I showed how to determine whether to place a bet on a particular sporting outcome. While the post discussed whether to bet, it didn’t determine how much to bet.

    This post will serve as the first of a series to discuss the Kelly criterion. They Kelly criterion is a formula used to determine how much of your money to place on a particular gamble. The formula was derived by J.L. Kelly, Jr in 1956. The formula has a number of applications, one of which is sports betting.

    This post provides an introduction to the Kelly criterion. At this stage, my intention for the rest of this series is as follows:
    Part 2 will provide a simple derivation of the formula
    Part 3 will discuss some extensions to the model
    – Part 4 will provide further discussion and a critique of the model

    The formula

    The formula calculates the fraction, f, of your account balance that you should place on a bet, given the available odds and your perceived probability of winning. The formula depends on how you express the betting odds, so I have provided two versions below. Version A uses the decimal odds system that is popular in Australia. Decimal odds of 2.50 mean that if you win, a $10 bet would result in a $25 payout and a $15 profit. Version B uses the “b to 1” odds system, where odds of “3 to 1” mean that if you win, a $10 bet would result in a payout of $40 and a profit of $30.

    Version:

                         A                      B
    Formula:

               
    Variables:

          f = fraction of your bankroll to bet
    d = decimal betting odds
    p = probability of bet winning
    q = 1 – p = probability of bet losing
          f = fraction of your bankroll to bet
    b = “b to 1” betting odds
    p = probability of bet winning
    q = 1 – p = probability of bet losing

     

    The betting odds are observable and easy to obtain. The key variable in this formula is the probability, p, of the bet winning. Observe that q is simply 1 – p. So if the probability of winning is 60%, then the probability of losing must be 40%. Note that this ignores the chance of a refunded bet. If you play around with the formula you will find that the greater your perceived odds of winning, relative to those implied by the bookmaker, the larger the percentage of your account you should bet. The probability implied by the bookmaker for a particular outcome is the inverse of the decimal odds. For example, 2.50 odds imply a 1/2.5 = 40% chance of winning.

    A worked example

    The example below uses the decimal odds system. To obtain b, if that’s what your most comfortable using, simply subtract 1 from d.

    Suppose you observe the following odds for the football game between Arsenal and Chelsea.

    Full-time result:
    Arsenal: 2.60
    Chelsea: 2.65
    Draw: 3.25

    You take a look at the current form of the two sides, and believe the probabilities of the respective outcomes are as follows:

          p     q
    Arsenal     20%   80%
    Chelsea   50%   50%
    Draw   30%   70%
        ——-    
    Total   100%    

     

    Hence, in your opinion, there is a 20% chance of Arsenal winning, a 50% chance of Chelsea winning, and a 30% chance of a draw.

    Note that the sum of p an q for each outcome is 100%. Also note that the sum of the values for the p’s also equals 100%. Using the Kelly criterion, you calculate the optimal fraction of your wealth to bet on each outcome as follows:

    Arsenal: fA = [0.2(2.6 – 1) – 0.80]/(2.6 – 1) = -0.300
    Chelsea: fC = [0.5(2.65 – 1) – 0.50]/(2.65 – 1) = 0.197
    Draw: fD = [0.3(3.25 – 1) – 0.70]/(3.25 – 1) = -0.011

    Because the fractions fA for Arsenal and fD for the Draw are below zero, you would not bet on these outcomes. The fraction, fC, for Chelsea is a positive value of 0.197. This means you should bet 19.7% of your betting account balance on Chelsea to win. If your account balance is currently $200, then you should place a $39.40 bet on Chelsea.

    Note that if your probabilities for Arsenal, Chelsea and a draw were 35%, 35% and 30%, respectively. Then you would not make any bet on this event. Plug the corresponding probabilities into the formula to verify this result for yourself.

    Coming up in Part 2

    This post simply provides the formula and illustrates how to use it. The derivation of the formula will be discussed in Part 2. To give you some idea of the methodology, the Kelly criterion aims to maximise the log of the expected growth rate of your account balance.

    Sources:

Centrebet Aus Freebet
  • I managed to “green up” for the Woman’s title in the 2010 French Open. By this I mean I managed to lock in a profit regardless of the outcome. I’m now on a streak for woman’s tennis. I prefer betting on the woman than the men because I find that woman’s tennis is more unpredictable, so there’s more value in backing the the underdog.

    Like most people though, I never predicted Schiavone would win. Her backer’s would have made a fortune, because even halfway through the tournament you could have got 90-1 on her. I had bet heavily on Stosur to win the tournament using Betfair and Luxbet, so in the tournament final I hedged by bets by backing Schiavone at 5.10 odds. All up I made $150.13 on $76.79 of bets, making a $73.34 profit.

    To elaborate on “greening up”, you can do this by backing an outcome at X odds, and then betting against the outcome for Y odds later, where Y < X. During the tournament I bet on Venus Williams, Stosur, Hantuchova, and Rezai. I was able to make counter bets against everyone except Rezai later on in the tournament to hedge my positions. Below is a screen shot of my position on May 30th. Basically you try to green up by picking people who you think are undervalued and who will progress further in the tournament. For example, I was able to back Hantuchova for 540-1 odds on May 26th, and by May 30th, I could bet against her at 180-1 odds. I plan to see if I can do the same for the 2010 FIFA World Cup. I have enjoyed sustained success with tennis, but have yet to try it with football.

  • As a follow up to its previous promotion, Luxbet is now offering new members a 2010 FIFA World Cup jersey of your choosing from the 2010 FIFA World Cup.

    Sign up before July 18 2010 and deposit $150 or more to receive your jersey. I’ve just looked around, and it appears a Socceroos jersey would normally set you back $140. Be sure to use the link provided in this post and follow the instructions on the Luxbet landing page. Note that Queensland residents are unfortunately excluded from this promotion.

    Click here to get your free jersey

    You can read my review of Luxbet here.

    As I’ve said before, I think this is an excellent promotion because in the worst case scenario you’ve paid $10 for a football jersey! Unlike other account deposit freebies at least you won’t be tempted to gamble your bonus funds away.

  • Most punters know that in deciding whether to bet, you should first estimate the probability of a particular outcome, and then compare that probability to the inverse of the betting odds. But how many of us factor the time value of money into our betting decisions? I believe this is an important consideration, because any money you lock into a bet can’t be used for other purposes until the event has finished and your winnings are paid out. There is an opportunity cost to this money, the calculation of which is discussed here.

    The main result from this article is that the further away the payout for your winning bet, the higher the odds need to be to compensate you for the delay in receiving your funds. Bookmakers typically don’t provide any compensation for this, which means you should bias your betting towards bets with more immediate payouts.

    Example without time value of money

    Consider a betting decision in the absence of time. Here you compare the inverse of the betting odds to your perceived probability of occurrence. Suppose the odds for the NHL fixture ‘Los Angeles Kings at Vancouver Canucks’ are as follows.
         LA Kings 2.60
         Vancouver Canucks 1.52

    The bookmaker’s odds translate to an implied probability of 1/1.52 = 65.8% for Vancouver to win, and 1/2.60 = 38.5% for LA to win. Note that the probability of LA winning equates to a 1 – 38.5% = 61.5% probability of Vancouver winning. At this point, or preferably before you’ve looked at the odds, you determine your expected probability of Vancouver winning, which I will denote P. Your probability can fall within one of three ranges:
         P < 61.5%      61.5% < P < 65.8%
         P > 65.8%

    If you believe the probability of Vancouver winning is less than 61.5%, you should bet on Los Angeles. If you personally believe the probability is greater than 65.8%, you should bet on Vancouver. Finally, if you believe the true probability lies between 61.5% and 65.8%, then you should not make a bet on this fixture.

    Example with time value of money

    In the previous example, the outcome will be determined within a few days of placing a bet, so timing considerations aren’t important. The funds you place on the bet will unavailable for use for only a short period of time. But what if you make a bet on, say, the 2013 Rugby League World Cup? Yes, I know I’ve chosen an extreme example here, but it will illustrate the concept well. Sportsbet’s odds for the three favourites to win the tournament are as follows:
         Australia 1.20
         New Zealand 6.00
         England 8.00

    These odd correspond to the implied probabilities below:
         Australia 83.3%
         New Zealand 16.7%
         England 12.5%

    You should bet on Australia if your perceived odds of them winning are more than 83.3%, but this ignores the fact that you have made your money unavailable for other uses until after the tournament final in 2013. We clearly need to take this into account before making a betting decision.

    How to determine the time value of money

    The time value of money equates to the opportunity cost of your betting funds. It is the amount you could earn elsewhere had you not made your bet. This takes into account two things: time and interest rate.

    The interest rate is what could get for your money had you chosen not to bet it. I looked at ING Direct this morning, and Australian residents can currently get 4.65% p.a. The time is the date of the expected payout (if your bet wins) minus the date on which you place your bet. Today is the 24th of April, 2010, and let’s assume the 2013 final will take place on November 24th, 2013, so that the payout will be in three years and 7 months time. Mathematically, this is 3 + 7/12 = 3.583 years from now.

    We now have the two components with which to calculate the time value of money:
    Interest rate = r = 0.0465
    Time = t = 3.583

    Using the equation 1/(1+r)^t we can calculate how much to discount any future payout to arrive at the equivalent value had we won the bet today. Note, to keep things simple I have only compounded the interest once per year. You can learn more about compound interest here.

    In our case we get 1/(1.0465)3.583 = 0.8497 when applying the above formula.

    Now that we have determined our discount for the time value of money, we need to multiply the bookmaker odds by this value, and then recompute the implied probabilities:

         Australia 1.20 x 0.8497 = 1.0196
         New Zealand 6.00 x 0.8497 = 5.0982
         England 8.00 x 0.8497 = 6.7976

    The new implied probabilities are:

         Australia 98.1%
         New Zealand 19.6%
         England 14.7%

    This means that instead of betting on Australia if you believe the odds are over 83.3%, you should only consider betting if your perceived odds are over 98.1%. These required probabilities are drastically higher because if you put your money into a savings account you could get the equivalent of 1.04653.583 = 1.1769 odds without taking any risks! Note that 1/0.8497 = 1.1719.

    General formula

    Below is a general formula for comparing adjusted betting odds to your perceived probability of occurrence. Note that I have used the property (1/x)/(1/y) = y/x. Play around with the formulas to verify the result for yourself.

    PX = your perceived probability of outcome X occurring.
    BX = the bookmaker odds for outcome X
    BXC = bookmaker odds for outcome X to not occur (where available)
    L denotes “low”
    H denotes “high”
    NB denotes “no bet”

    When you can bet either for or against outcome X, like in a hockey game, your decision is based on the comparisons below:

    If PX < PL then you should bet AGAINST outcome X
    If PX > PH then you should bet on outcome X
    If PX = PNB where PL < PNB < PH then you should not bet on the event

    When you can’t easily bet against outcome X, like for the Rugby League World Cup winner, your decision is based on the equation below:

    If PX > PH then you should bet on outcome X
    If PX = PNB where PNB < PH then you should not bet on the event.

    Closing remarks

    While the Rugby League example I have used is extreme, I honestly believe that you should take into account the time value of money when making bets that are to take place more than three months away. For three months your time is 3/12 = 0.25, so you should multiply the available odds by 1/1.04650.25 = 0.9887 before comparing them to your perceived probability of occurrence. Looking again at our LA Kings vs Vancouver Canucks example, if the fixture was to take place three months from now, the range for which you would not bet would expand from 61.5% < P < 65.8% to 61.1% < P < 66.5%. Note that the range of perceived probabilities for which you would not bet always expands when you factor in the time value of money. At this stage I must point out some caveats: First, the above analysis assumes that you are 100% confident that your estimated probability is correct. In reality, there is uncertainty in your probability estimate. For example, you may guess that probability of an outcome is 60%, but believe that the true probability could lie anywhere between 50% and 70%. It turns out that the decision of whether to bet is independent of your confidence in the probability estimate. The decision of how much to bet will take into account this uncertainty. I will discuss the issue of how much to bet, which directly relates to your confidence in the estimated probability, on a later date. Second, most sports punters are pure gamblers. Few consider withdrawing money from a sports betting account and putting into a bank account. This makes the use of a savings account interest rate as an opportunity cost irrelevant. I suppose I'm reaching out to a more rational sports betting audience here. Most likely, any decision not to bet simply frees up money to bet elsewhere. In this case your opportunity cost is actually a comparison between two possible bets. If you are constrained for funds, you could bet on event E1 that takes place t1 days time, or you could bet on event E2 that takes place in t2 days time. You would need to discount one using the expected payoff from the other to determine whether to bet on it. Another option is to look at your historical betting performance, and determine an expected return on your betting account each year. If it is 20%, then a bet on an event that pays out one year from now should have the odds discounted by 1/1.2 first. Of course, the downfall of this method is that many punters have negative earnings history, which would lead to erroneous results.

    The analysis here is intended to provide food for thought. The main concept is that the further away the payout for your winning bet, the higher the odds need to be to compensate you for the delay in receiving your funds. Bookmakers typically don’t provide any compensation for this, which means you should bias your betting towards bets with more immediate payouts. This result draws from the fact that the PNB region expands as t increases, making you less inclined to make a wager on later fixtures.

  • As most Australian punters know, live in-play betting is typically only available by telephone. There are a few exceptions to this, however, like financial betting and now, the Australian Open tournament winner betting through Betfair. Recently I have been using opposing bets in tournament betting to effectively bet in-play for particular Australian Open tennis fixtures This strategy is based on the principal that under certain circumstances, the odds on a player to win the tournament drop if they win their next round, especially if they weren’t expected to beat their opponent.

    Live in-play betting enables punters to place bets on the outcome of a match or tournament while it’s still in progress. For example you can bet in-play during a football match, with the odds changing depending on the outcome of the game so far. Unfortunately, for Australian punters, this feature is only available by telephone, not online. I find this frustrating because you can react to game outcomes much faster with the click of a mouse than through a telephone call.

    Betfair is typically no exception to this, however for the 2010 Australian Open punters can make bets on the tournament winner even when there are games in progress. This is in contrast to bookmakers like Sportsbet and Sportingbet, who close down tournament winner betting during each day’s play. They do this because typical bookmakers need time to determine updated odds based on who’s left in the tournament. Betfair is a marketplace, so this information is interpreted immediately by market participants.

    The reason I’m excited about this feature is because online tournament betting enables you to effectively bet in-play online for individual fixtures. If a player looks like they may be eliminated from the tournament, the odds on them winning the tournament increase. If a player is on the verge of losing a match their odds to win the tournament typically rise to 1000-1. Conversely, if an underdog makes a good start to the match, the betting odds on them winning the tournament drop drastically. In many circumstances you can make money by correctly predicting the result to a game in progress by taking advantage of odds shifts in the tournament winner betting as a result of that match.

    The following example illustrates this principle. It involves betting on an underdog who is expected to be beaten in the upcoming round. Say at the beginning of the tournament you bet $10 on Maria Kirilenko to win the Australian Open at odds of 500-1. In her first round match against Maria Sharapova she wins the first set 7-6. As a result the odds of her winning the tournament drop to 300-1. At this point, if you can hedge your bets by laying a bet of $16 against her at the 300-1 odds. This means you stand to win $200 if she wins the tournament and $6 if she doesn’t. Now let’s suppose you don’t hedge your bets and Kirilenko wins the match. As of the time of writing this article, you can now bet against Kirilenko winning the tournament at odds of 180-1. If you bet $27 against Kirilenko at these odds you would net $140 if she wins the tournament and $17 if she doesn’t, a profit either way.

    As a second example, you can bet on a top seed after they have made a poor start to a match. Prior to her game against Kleybanova, the odds on Henin to win the tournament were around 4.5-1. Henin lost the first set 3-6 and in the middle on the second set the odds on her to win the tournament were 7-1. I believed Henin would win the match but didn’t want to make a head-to-head bet over the phone so I placed a $5 bet online for her to win the tournament at the 7-1 odds. After Henin won the match I then placed a $5 bet against her to win the tournament at 5.6-1 odds. This means I will net (7)(5) – (5.6)(5) = $7 if she wins the tournament, with no loss if she doesn’t.

    In essence, the changes in the odds for tournament winner betting can act as a proxy for individual match betting. There are a few caveats to this method, however. First, this strategy relies on adequate liquidity in the Betfair marketplace. There’s nothing more infuriating than taking up odds of 200-1 and then seeing the ‘Back’ odds drop as you had predicted to 150-1 only to find that the ‘Lay’ odds are still above 200-1. I have typically found that more liquidity is available in the men’s draw than the women’s, however there is typically sufficient liquidity for the top eight or so female seeds.

    A second caveat to this strategy is you don’t know the odds you will receive if you are correct. In head-to-head betting, you know the exact payoff if you have a winning bet. However, when using the tournament winner for your bets, you are trying to profit based on odds movements, which you simply can’t know in advance. With my Henin bet I felt the odds against her would drop if she won the match, but I didn’t know if they would drop to 4-1, 5-1, 6-1, etc.

    A third caveat is that the odds won’t move substantially in certain circumstances, so this strategy doesn’t work in all scenarios. Typically a top seed’s odds to win the tournament won’t drop drastically after an early round victory. The strategy outlined here works best in two scenarios: (1) when the player you wish to back is an underdog in their fixture and (2) when a top seed makes a poor start to the match, but you still expect them to win.

    Overall, if you can bet by phone and the odds look reasonable, I would bet in-play using head-to-head bets, but if you want to make an immediate online bet at the available odds, I find tournament winner betting to be an effective way to bet in-play on certain fixtures by proxy.

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