This report describes a parimutuel betting system for managing bets on sporting events and similar competitions.
Caution. Nearly all nations have laws that restrict gambling, and in many places legislative action will be required before this system can be lawfully implemented. Also, some jurisdictions will impose taxes. However, if government regulation includes frequent audits of the gambling operator's finances, patrons are more likely to have confidence that payouts are calculated correctly, and this perception can increase business.
A Greatly Simplified Example of the Basic Idea.
The Springfield Athletic Federation has teams in eight towns, each represented by a two-letter symbol:
| teams of the Springfield Athletic Federation | |
|---|---|
| town | symbol |
| Springfield, Florida | FL |
| Springfield, Georgia | GA |
| Springfield, Illinois | IL |
| Springfield, Kentucky | KY |
| Springfield, Missouri | MO |
| Springfield, Ohio | OH |
| Springfield, Tennessee | TN |
| Springfield, Virginia | VA |
A conventional single-elimination tournament is scheduled:
| round one | game #1: FL vs GA |
|---|---|
| game #2: IL vs KY | |
| game #3: MO vs OH | |
| game #4: TN vs VA | |
| round two | game #5: winner of game #1 vs winner of game #2 |
| game #6: winner of game #3 vs winner of game #4 | |
| round three | game #7: winner of game #5 vs winner of game #6 |
Before the tournament begins, patrons place bets on which teams they think will win. The house (in other words, the business that organizes the gambling) sells shares; each share is a $10 bet that a particular team will win. For instance:
After round three, when the champion is selected, the total of all bets, winning and losing, is paid out to the winning bettors in proportion to how many shares they hold. This is the fundamental mechanism of parimutuel gambling.
Over the long run, a bettor who is considerably more knowledgeable than average can make a profit at parimutuel betting; but this is not easy, as it requires overcoming the house's take, as well as taxes and miscellaneous expenses. By contrast, at most casino games a loss is guaranteed for all patrons in the long run, as the patron is betting against the house, rather than other patrons, and the house builds into each game an advantage for itself.
Because many patrons are "scientific", relying on extensive study, methodical analysis, and advanced mathematics in placing their bets, any implementation of the Springfield system should make all fee and payout calculations as clear and simple as possible. As presented here, the basic bet (to buy a share from the house and retain it) is as elementary as a parimutuel system can be. Some patrons may choose to pursue the resales of shares (more complicated), the buying and selling of options (quite intricate), or the transferring of options (for experts only).
The Springfield system does not have the full range of financial derivatives found in stock markets, because of the Springfield requirement that when undertaking an obligation, a patron must already have, and must retain, the resources to satisfy that obligation. This prevents failure to deliver. Along the same lines, the house never extends credit of any sort.
Much More Detail.
The Springfield system can be feasibly implemented on an internet site, and this report assumes such. Patrons can communicate via public or private messages. The house never reveals a patron's real name to the public, but each patron chooses a pseudonym (here called a moniker) which is public.
The house maintains an account for each patron. The patron deposits money into the account from time to time, and withdraws money as desired. When the patron places a bet, the money comes from his account, and any winnings go back into that account.
Although these examples use dollars, other currencies will work just as well. When necessary, conversion to and from other nations' currencies will take place only when funds are deposited or withdrawn, which will be relatively infrequently; thus no conversion will be required in the more numerous instances when shares are bought, or winning shares redeemed. Because exchange rates fluctuate, a patron might end up with a profit or loss of a different amount than expected.
In the Springfield system, the house charges a fee when each bet is placed; this is to cover the expenses of the operation, and hopefully to make a profit. In order to make the financial calculations abundantly clear to patrons, the fee is in addition to the amount of any bets. While there are many ways that fees might be calculated, the following examples assume for simplicity and concreteness a fee of four percent, which works out to $0.40 on a ten-dollar share. Thus Abe pays $156.00, Gus $62.40, Joe $239.20, and Val $291.20.
Immediately before completing any transaction, the house furnishes to the patron a comprehensive statement of exactly what is to be done; this gives the patron a final chance to cancel the transaction if unsatisfactory.
Although bets are made in multiples of $10, payouts are to the nearest $0.0001, and account balances are maintained to this four-place precision. This virtually eliminates rounding error, also known as breakage. In this report, dollar amounts are written with no decimal places ($35), two places ($35.62) or four ($35.6256).
Reducing the house's expenses is that it need not hire sports experts to set odds or point spreads; the parimutuel system makes that unnecessary. Aside from tiny amounts of rounding error, and the occasional minus pool (explained below), the house makes the same profit no matter who wins; thus it has little incentive to "fix" the tournament.
There need not be any affiliation between the athletic organization and the gambling organization, but it would not be surprising if the athletic required royalty payments from the gambling.
The house makes public the total number of shares sold on each team as the quantity changes during the pre-tournament selling phase, and later when sales are complete.
The house needs to establish a policy outlining the extent to which one patron can see the transaction history of another patron (identified, in any case, only by moniker). Of course each patron has access to full information, past and present, about his own account.
The following is a comprehensive example of betting in the tournament introduced above:
| schedule 1 | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| team | bettors | shares | fees | if this team wins the tournament … | game | |||||
| FL | Ann | 4 | $1.60 | each share is paid $10 × 1133 ÷ 122 = $92.8689 | Ann is paid $92.8689 × 4 = $371.4756 | total house payout is $92.8689 × 122 = $11330.0058 | total breakage is −$0.0058 | #1 | #5 | #7 |
| 6 others | 118 | $47.20 | ||||||||
| sub totals | 122 | $48.80 | ||||||||
| GA | Bob | 6 | $2.40 | each share is paid $10 × 1133 ÷ 85 = $133.2941 | Bob is paid $133.2941 × 6 = $799.7646 | total house payout is $133.2941 × 85 = $11329.9985 | total breakage is +$0.0015 | |||
| 4 others | 79 | $31.60 | ||||||||
| sub totals | 85 | $34.00 | ||||||||
| IL | Dan | 10 | $4.00 | each share is paid $10 × 1133 ÷ 91 = $124.5055 | Dan is paid $124.5055 × 10 = $1245.0550 | total house payout is $124.5055 × 91 = $11330.0005 | total breakage is −$0.0005 | #2 | ||
| 2 others | 81 | $32.40 | ||||||||
| sub totals | 91 | $36.40 | ||||||||
| KY | Flo | 8 | $3.20 | each share is paid $10 × 1133 ÷ 105 = $107.9048 | Flo is paid $107.9048 × 8 = $863.2384 | total house payout is $107.9048 × 105 = $11330.0040 | total breakage is −$0.0040 | |||
| 5 others | 97 | $38.80 | ||||||||
| sub totals | 105 | $42.00 | ||||||||
| MO | Len | 11 | $4.40 | each share is paid $10 × 1133 ÷ 232 = $48.8362 | Len is paid $48.8362 × 11 = $537.1982 | total house payout is $48.8362 × 232 = $11329.9984 | total breakage is +$0.0016 | #3 | #6 | |
| 3 others | 221 | $88.40 | ||||||||
| sub totals | 232 | $92.80 | ||||||||
| OH | Mac | 7 | $2.80 | each share is paid $10 × 1133 ÷ 110 = $103.0000 | Mac is paid $103.0000 × 7 = $721.0000 | total house payout is $103.0000 × 110 = $11330.0000 | total breakage is $0.0000 | |||
| 7 others | 103 | $41.20 | ||||||||
| sub totals | 110 | $44.00 | ||||||||
| TN | Peg | 12 | $4.80 | each share is paid $10 × 1133 ÷ 187 = $60.5882 | Peg is paid $60.5882 × 12 = $727.0584 | total house payout is $60.5882 × 187 = $11329.9934 | total breakage is +$0.0066 | #4 | ||
| 8 others | 175 | $70.00 | ||||||||
| sub totals | 187 | $74.80 | ||||||||
| VA | Sam | 15 | $6.00 | each share is paid $10 × 1133 ÷ 201 = $56.3682 | Sam is paid $56.3682 × 15 = $845.5230 | total house payout is $56.3682 × 201 = $11330.0082 | total breakage is −$0.0082 | |||
| 9 others | 186 | $74.40 | ||||||||
| sub totals | 201 | $80.40 | ||||||||
| totals | 1133 | $453.20 | ||||||||
In the case of GA for instance, the payout per share is $133.2941, but the bettor's profit per share is $122.8941, reflecting the $10 price and $0.40 fee for each share.
No matter the outcome in this tournament, the total breakage is less than one cent.
If there is a low level of betting interest in a tournament, there might be a team on which no patron has purchased any shares. What if such a team wins the tournament? The procedure for handling this can be illustrated with an example. Suppose 187 shares were sold on TN, but none on VA, and yet VA defeats TN in game #4. Then the TN shares are converted to VA shares. If VA goes on to win the tournament, the TN bettors receive a payout calculated as though TN had won.
Early betting discount.
In the hours leading up to a competition, many patrons prefer to buy their shares as late as possible, in order to have gathered the maximum information about the teams and players, and to have obtained the best estimate of what the ultimate odds will be. However, if all bettors act at the last minute, it is likely to overload the house's computer, leaving some patrons disappointed because they are unable to buy shares.
The house can mitigate this problem by offering a fee discount for patrons who buy shares early. For example, the first game of some tournament begins at 7:30pm on a Wednesday, and the house sets the betting deadline at 30 minutes before the competition starts:
A different way to spread out the betting is to have separate pools, one after the other, on the same tournament. For instance, with a tournament that starts on 14 May, pool 'A' might accept bets from 5 May to 9 May, and pool 'B' from 10 May to 14 May. The two pools are completely separate for monetary calculations, and as a result will usually offer different payouts. If desired, each pool could have its own schedule of early betting discounts.
The early betting discount is one reason that fees are not included in the price of a share.
Resales.
Once the tournament begins, patrons can buy and sell shares among one another at a price agreeable to the patrons involved — the house does not set resale prices. The total number of shares issued on each team remains constant. All resales are managed by the house, guaranteeing that the seller gets his money and the buyer gets his shares. Activity in this secondary market arises because patrons make different predictions about how a team will perform in the remaining portion of the tournament: a pessimist sells, and an optimist buys. In most cases, resales of shares on a team will occur after that team has won at least one game, although earlier resales are permitted.
Patrons publicly advertise offers to buy or sell shares; a patron may have several offers outstanding at any one time. Each offer is for shares on only one team in only one tournament. There is a locking mechanism to prevent unsupported offers:
For simplicity, the resale price of a share must be a multiple of $0.10. The examples below assume a fee of 2% to post an offer, and 2% to be paid by a patron who accepts an offer. Thus if a resale takes place, the combined fee works out to 4%, which equals the fee for buying shares from the house.
Whether or not they have yet purchased shares, patrons may informally discuss potential resales and unofficially suggest transactions. To allow patrons to better assess potential profits, the house makes public the total number of shares sold on each team in the tournament, perhaps in a listing similar to schedule 1X. Among other things, it reveals that nobody buying FL shares should pay more than $92.80.
| schedule 1X
excerpt from schedule 1 | |||||
| team | shares | payout | game | ||
|---|---|---|---|---|---|
| FL | 122 | $92.8689 | #1 | #5 | #7 |
| GA | 85 | $133.2941 | |||
| IL | 91 | $124.5055 | #2 | ||
| KY | 105 | $107.9048 | |||
| MO | 232 | $48.8362 | #3 | #6 | |
| OH | 110 | $103.0000 | |||
| TN | 187 | $60.5882 | #4 | ||
| VA | 201 | $56.3682 | |||
| total | 1133 | ||||
The mechanics of offers to sell and offers to buy are similar, as follows:
| offers to sell | offers to buy |
|---|---|
| A patron wants to sell a proposed number of shares on a particular team at a certain price (the ask price). To advertise this he pays a fee which is 2% of the ask price multiplied by the proposed number of shares. Any of the following events can subsequently happen:
A. Some other patron may accept the offer, buying any part of the proposed number of shares at the ask price. The buyer pays a fee of 2% of the ask price multiplied by the number of shares that actually change hands. If that is fewer shares than proposed, the offer remains in effect for the remainder. B. These can occur multiple times:
C. If this team loses a game or wins the tournament, the offer is automatically terminated. | A patron wants to buy a proposed number of shares on a particular team at a certain price (the bid price). To advertise this he pays a fee which is 2% of the bid price multiplied by the proposed number of shares. Any of the following events can subsequently happen:
A. Some other patron may accept the offer, selling any part of the proposed number of shares at the bid price. The seller pays a fee of 2% of the bid price multiplied by the number of shares that actually change hands. If that is fewer shares than proposed, the offer remains in effect for the remainder. B. These can occur multiple times:
C. If this team loses a game or wins the tournament, the offer is automatically terminated. |
| Here is an example sequence:
1. Wes offers to sell 24 shares on VA, advertising an ask price of $14.70 each.
2. After a period of no response, Wes lowers the ask price to $14.20.
3. Zak buys 10 shares of VA from Wes.
4. After another period of no response, Wes lowers the ask price on the 14 remaining shares to $13.60.
5. Eli buys 8 shares of VA from Wes.
6. Wes adds 9 shares to the offer.
| Here is an example sequence:
1. Lou offers to buy 35 shares of OH, advertising a bid price of $16.50 each.
2. After a period of no response, Lou raises the bid price to $16.90.
3. Gil sells 12 shares of OH to Lou.
4. After another period of no response, Lou raises the bid price on the 23 remaining shares to $17.30.
5. Cal sells 9 shares of OH to Lou.
6. Lou terminates the offer on the 14 remaining shares.
|
Note the asymmetry of fees: a patron pays a fee to increase an ask or bid price, but receives no refund on a decrease; similarly for an increase and decrease in the number of shares. This helps deter a patron who would, repeatly, increase and decrease the price or number of shares in order to manipulate the market. For instance, consider a patron who makes an offer (and pays the fee) for 17 shares, later decreases the number of shares to 5 (no fee), and after that increases the number of shares back to 17. The patron must pay a fee for this 12-share increase, even though the offer only ends up at the original number of shares.
A sell offer and a buy offer are complementary if they are for the same team in the same tournament, and if the bid price is no less than the ask price. In a busy trading session, two patrons may not realize that they have advertised complementary offers, so the house will send each of them a message. However, the house does not automatically complete the transaction, for two reasons:
A sell offer and a buy offer are almost complementary if they are for the same team in the same tournament, but the bid price is slightly less than the ask price. The house can inform the respective patrons when this situation occurs, but no dealing happens automatically. Instead, the offerer-to-buy might choose to raise the bid price, or the offerer-to-sell reduce the ask price.
An offer is exhausted if every share was either bought or sold, as applicable.
A share has no "memory" of:
It is a good idea for the house to suspend resales of a team's shares while that team is playing a game, and fifteen minutes before and after. This is because some patrons might get information about the game's progress sooner than others, creating an unfair information distribution. Also, the interval after the game allows the house to cleanly perform automatic termination of any offers involving the losing team's shares.
Specifically, this suspension applies to the acceptance of offers. There is little harm in allowing a patron to post an advertisement or to modify an offer during play. Certainly, a patron should be permitted to terminate an offer at any time.
Highly skewed betting.
Consider a tournament held by this league:
| teams of the Summerfield Athletic Federation | |
|---|---|
| town | symbol |
| Summerfield, Connecticut | CT |
| Summerfield, Kansas | KS |
| Summerfield, Montana | MT |
| Summerfield, Wisconsin | WI |
Assume that nearly all bets are placed on MT, and then MT goes on to win. Then the per-share payout might be less than $10.40 ($10 share price plus $0.40 fee), meaning that correct bettors would suffer a loss. Here are some numbers as an example:
| schedule 2 | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| team | bettors | shares | fees | if this team wins the tournament … | game | ||||
| CT | Dora | 3 | $1.20 | each share is paid $10 × 640 ÷ 5 = $1280.0000 | Dora is paid $1280.0000 × 3 = $3840.0000 | total house payout is $1280.0000 × 5 = $6400.0000 | total breakage is $0.0000 | #1 | #3 |
| 1 others | 2 | $0.80 | |||||||
| sub totals | 5 | $2.00 | |||||||
| KS | Ivan | 4 | $1.60 | each share is paid $10 × 640 ÷ 7 = $914.2857 | Ivan is paid $914.2857 × 4 = $3657.1428 | total house payout is $914.2857 × 7 = $6399.9999 | total breakage is +$0.0001 | ||
| 2 others | 3 | $1.20 | |||||||
| sub totals | 7 | $2.80 | |||||||
| MT | Stan | 8 | $3.20 | each share is paid $10 × 640 ÷ 622 = $10.2894 | Stan is paid $10.2894 × 8 = $82.3152 | total house payout is $10.2894 × 622 = $6400.0068 | total breakage is −$0.0068 | #2 | |
| 27 others | 614 | $245.60 | |||||||
| sub totals | 622 | $248.80 | |||||||
| WI | Vern | 2 | $0.80 | each share is paid $10 × 640 ÷ 6 = $1066.6667 | Vern is paid $1066.6667 × 2 = $2133.3334 | total house payout is $1066.6667 × 6 = $6400.0002 | total breakage is −$0.0002 | ||
| 2 others | 4 | $1.60 | |||||||
| sub totals | 6 | $2.40 | |||||||
| totals | 640 | $256.00 | |||||||
Each share of MT yields a $0.1106 = $10.4000 − $10.2894 loss. This problem can happen in any parimutuel system where the house takes a cut, which is, practically speaking, every parimutuel system. A typical response is that the house will guarantee a minimum payout to each correct bettor. If the house takes a loss, the situation is known as a minus pool.
A reasonable payout floor would be $10.80 per share, guaranteeing the patron at least a $0.40 profit per share. With the betting of schedule 2, the house takes in $6656.00 = 640 × $10.40, but pays out $6717.60 = 622 × $10.80, suffering a nominal loss of $61.60 = $6717.60 − $6656.00, instead of the expected profit of $256.00. Breakage, which in this case is negative, makes the house's actual loss slightly greater.
Under a less traditional but more rigorous approach, the house simply pays $10.2894 per share, and patrons take their losses. This policy might be chosen by a house that has a very low fee structure.
Cancellation values.
In extraordinary circumstances, the remainder of a tournament might have to be canceled after some games are completed, and the Springfield system has a way to handle this. The principle is: because this is a single-elimination tournament, shares on any team that played a complete game and lost it are worth zero. Here is an example:
Suppose after games #1 through #5 have finished, an exceptional event occurs, and games #6 and #7 must be canceled. Then:
The payouts are calculated as if these are three unrelated competitions.
| schedule 3 | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| sub-tournament "A" | |||||||||
| team | bettors | shares | fees | if this team wins game #5 … | game | ||||
| FL | Ann | 4 | $1.60 | each share is paid $10 × 403 ÷ 122 = $33.0328 | Ann is paid $33.0328 × 4 = $132.1312 | total house payout is $33.0328 × 122 = $4030.0016 | total breakage is −$0.0016 | #1 | #5 |
| 6 others | 118 | $47.20 | |||||||
| sub totals | 122 | $48.80 | |||||||
| GA | Bob | 6 | $2.40 | each share is paid $10 × 403 ÷ 85 = $47.4118 | Bob is paid $47.4118 × 6 = $284.4708 | total house payout is $47.4118 × 85 = $4030.0030 | total breakage is −$0.0030 | ||
| 4 others | 79 | $31.60 | |||||||
| sub totals | 85 | $34.00 | |||||||
| IL | Dan | 10 | $4.00 | each share is paid $10 × 403 ÷ 91 = $44.2857 | Dan is paid $44.2857 × 10 = $442.8570 | total house payout is $44.2857 × 91 = $4029.9987 | total breakage is +$0.0013 | #2 | |
| 2 others | 81 | $32.40 | |||||||
| sub totals | 91 | $36.40 | |||||||
| KY | Flo | 8 | $3.20 | each share is paid $10 × 403 ÷ 105 = $38.3810 | Flo is paid $38.3810 × 8 = $307.0480 | total house payout is $38.3810 × 105 = $4030.0050 | total breakage is −$0.0050 | ||
| 5 others | 97 | $38.80 | |||||||
| sub totals | 105 | $42.00 | |||||||
| totals | 403 | $161.20 | |||||||
| sub-tournament "B" | |||||||||
| team | bettors | shares | fees | if this team wins game #3 … | game | ||||
| MO | Len | 11 | $4.40 | each share is paid $10 × 342 ÷ 232 = $14.7414 | Len is paid $14.7414 × 11 = $162.1554 | total house payout is $14.7414 × 232 = $3420.0048 | total breakage is −$0.0048 | #3 | |
| 3 others | 221 | $88.40 | |||||||
| sub totals | 232 | $92.80 | |||||||
| OH | Mac | 7 | $2.80 | each share is paid $10 × 342 ÷ 110 = $31.0909 | Mac is paid $31.0909 × 7 = $217.6363 | total house payout is $31.0909 × 110 = $3419.9990 | total breakage is +$0.0010 | ||
| 7 others | 103 | $41.20 | |||||||
| sub totals | 110 | $44.00 | |||||||
| totals | 342 | $136.80 | |||||||
| sub-tournament "C" | |||||||||
| team | bettors | shares | fees | if this team wins game #4 … | game | ||||
| TN | Peg | 12 | $4.80 | each share is paid $10 × 388 ÷ 187 = $20.7487 | Peg is paid $20.7487 × 12 = $248.9844 | total house payout is $20.7487 × 187 = $3880.0069 | total breakage is −$0.0069 | #4 | |
| 8 others | 175 | $70.00 | |||||||
| sub totals | 187 | $74.80 | |||||||
| VA | Sam | 15 | $6.00 | each share is paid $10 × 388 ÷ 201 = $19.3035 | Sam is paid $19.3035 × 15 = $289.5525 | total house payout is $19.3035 × 201 = $3880.0035 | total breakage is −$0.0035 | ||
| 9 others | 186 | $74.40 | |||||||
| sub totals | 201 | $80.40 | |||||||
| totals | 388 | $155.20 | |||||||
If a tournament is canceled at such an early stage that some teams play no games, any bets on those teams are paid out at $10 per share. Aside from breakage, the house's total payout on an incomplete tournament is the same as on a complete one.
What about refunding fees if part or all of a tournament is cancelled? Fees pertinent to a team fall into three categories:
A patron does not receive a refund of:
When a cancellation takes place, the house might have to refund so many fees that, taking its expenses into consideration, it loses money on the event. However, cancellations are rare.
Cancellation values also seve as a rough guide to how share resale prices might vary as competition progresses. Suppose the tournament of schedule 1 is completed normally. If MO not only defeats OH in game #3, but plays much better than the average patron anticipated, resale prices for MO will likely be higher than the $14.7414 of schedule 3. On the other hand, if MO plays worse than expected but still manages to win, resale prices will probably drop below $14.7414.
Brackets.
A popular kind of betting is based on predicting the winners of all the games within a single-elimination tournament such as that of schedule 1. The bettor makes his selections (in the aggregate, termed his bracket), buys shares before the first game of the tournament, paying the usual price of $10 plus $0.40. A bettor cannot change his picks after his bet is placed.
With 8 teams, there will be 7 games of which to forecast the winners, thus 128 = 27 different combinations of selections. With a larger tournament of 16 teams and 15 games, there are 32,768 = 215 different combinations. For a 32- or 64-team tournament, the numbers become so huge that, even with plenty of bettors, it is almost certain that nobody will get them all correct.
How, then, to distribute the payout? One way is to award points to bettors for correct picks. After the tournament, all the bettors tied for the greatest number of points receive payouts in proportion to shares held. Here is an example:
| round | game | points |
|---|---|---|
| one | #1: FL vs GA | 1 |
| #2: IL vs KY | 1 | |
| #3: MO vs OH | 1 | |
| #4: TN vs VA | 1 | |
| two | #5: winner of #1 vs winner of #2 | 2 |
| #6: winner of #3 vs winner of #4 | 2 | |
| three | #7: winner of #5 vs winner of #6 | 4 |
With these numbers, a perfect bracket earns 12 points.
A bettor's selections must be consistent, meaning that if he chooses some team to win in a later round, he must also choose that team to win in all the earlier rounds. For instance, if the bettor selects KY to win game #7, he must also select KY to win games #5 and #2.
A game in the second round or later is in perdition if the bettor incorrectly picked the winners of both games feeding into it: he has no hope of having correctly picked the winner of this later game. For example, if he chooses MO to win game #3 and VA to win #4, then to be consistent he must choose either MO or VA to win #6. Yet if MO and VA lose games #3 and #4, then game #6 is in perdition for this bettor.
A bettor might be mathematically eliminated before the tournament is over. For instance, if bettor Sam picks all of the first-round games wrong, the consistency rule forces his final score to be zero. If bettor Tom gets at least one of the first-round games right, earning a point, Sam becomes an assured loser before the second round even begins.
There is nothing special about 1, 2, and 4 as numbers of points; other values may be preferred. Still, it makes sense that all games within the same round be worth the same number of points, and that the number of points increase from one round to the next.
The largeness of the number of possible combinations affects the resale market. Although offers to sell are quite feasible, offers to buy might sometimes be unmatched by shares that any bettor actually holds. A public list of every bracket that at least one patron has specified could be prepared, but it risks being unwieldily long.
Here is an extended example of brackets under the Springfield system. In a conventional single-elimination tournament with 16 teams, each represented by a two-letter abbreviation, bettor Joe has selected the team that he believes will win each of the 15 games numbered #1 through #15. He earns 1 point for each correct pick in round one, 2 in round two, 4 in round three, and 8 in round four. His choices are summarized here in tabular form:
| Joe's bracket — before game #1 | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| round | AL | CO | DE | FL | GA | HI | KS | LA | MN | OK | PA | SD | TX | UT | VA | WV | points per | points total |
| one | #1: CO | #2: DE | #3: GA | #4: KS | #5: MN | #6: PA | #7: UT | #8: WV | 1 | at most 8 | ||||||||
| two | #9: CO | #10: GA | #11: PA | #12: UT | 2 | at most 8 | ||||||||||||
| three | #13: GA | #14: PA | 4 | at most 8 | ||||||||||||||
| four | #15: PA | 8 | at most 8 | |||||||||||||||
| grand total | at most 32 | |||||||||||||||||
The winners of games #1 through #4 are CO, DE, HI and KS. Joe's bracket is updated as below, with point counts replacing team names for those games where the correctness of Joe's prediction is known. Because GA lost game #3, it cannot win game #10 or #13.
| Joe's bracket — after game #4 | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| round | AL | CO | DE | FL | GA | HI | KS | LA | MN | OK | PA | SD | TX | UT | VA | WV | points per | points total |
| one | #1 = 1 | #2 = 1 | #3 = 0 | #4 = 1 | #5: MN | #6: PA | #7: UT | #8: WV | 1 | at least 3 at most 7 | ||||||||
| two | #9: CO | #10 = 0 | #11: PA | #12: UT | 2 | at most 6 | ||||||||||||
| three | #13 = 0 | #14: PA | 4 | at most 4 | ||||||||||||||
| four | #15: PA | 8 | at most 8 | |||||||||||||||
| grand total | at least 3 at most 25 | |||||||||||||||||
The winners of games #5 through #8 are MN, PA, UT and VA. Update:
| Joe's bracket — after game #8 | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| round | AL | CO | DE | FL | GA | HI | KS | LA | MN | OK | PA | SD | TX | UT | VA | WV | points per | points total |
| one | #1 = 1 | #2 = 1 | #3 = 0 | #4 = 1 | #5 = 1 | #6 = 1 | #7 = 1 | #8 = 0 | 1 | 6 | ||||||||
| two | #9: CO | #10 = 0 | #11: PA | #12: UT | 2 | at most 6 | ||||||||||||
| three | #13 = 0 | #14: PA | 4 | at most 4 | ||||||||||||||
| four | #15: PA | 8 | at most 8 | |||||||||||||||
| grand total | at least 6 at most 24 | |||||||||||||||||
CO wins game #9, PA wins #11, but UT loses #12. For Joe, #10 is immaterial. Update:
| Joe's bracket — after game #12 | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| round | AL | CO | DE | FL | GA | HI | KS | LA | MN | OK | PA | SD | TX | UT | VA | WV | points per | points total |
| one | #1 = 1 | #2 = 1 | #3 = 0 | #4 = 1 | #5 = 1 | #6 = 1 | #7 = 1 | #8 = 0 | 1 | 6 | ||||||||
| two | #9 = 2 | #10 = 0 | #11 = 2 | #12 = 0 | 2 | 4 | ||||||||||||
| three | #13 = 0 | #14: PA | 4 | at most 4 | ||||||||||||||
| four | #15: PA | 8 | at most 8 | |||||||||||||||
| grand total | at least 10 at most 22 | |||||||||||||||||
PA wins game #14 but loses #15. Update:
| Joe's bracket — final | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| round | AL | CO | DE | FL | GA | HI | KS | LA | MN | OK | PA | SD | TX | UT | VA | WV | points per | points total |
| one | #1 = 1 | #2 = 1 | #3 = 0 | #4 = 1 | #5 = 1 | #6 = 1 | #7 = 1 | #8 = 0 | 1 | 6 | ||||||||
| two | #9 = 2 | #10 = 0 | #11 = 2 | #12 = 0 | 2 | 4 | ||||||||||||
| three | #13 = 0 | #14 = 4 | 4 | 4 | ||||||||||||||
| four | #15 = 0 | 8 | 0 | |||||||||||||||
| grand total | 14 | |||||||||||||||||
Joe collects a payout if no bettor earned more than 14 points.
Best-of-several tournament betting.
The Springfield system readily extends to a best-of-five (or some other number) series between two teams. In the following example, Winterfield, North Dakota (ND) is facing Winterfield, South Carolina (SC). The two teams will play three, four, or five games, and the tournament will end as soon as one team has won three games. Bettors thus have six choices:
| schedule 4 | |||||||
|---|---|---|---|---|---|---|---|
| outcome | bettors | shares | fees | if this outcome is correct … | |||
| ND wins in 3 games | Hal | 5 | $2.00 | each share is paid $10 × 731 ÷ 112 = $65.2679 | Hal is paid $65.2679 × 5 = $326.3395 | total house payout is $65.2679 × 112 = $7310.0048 | total breakage is −$0.0048 |
| 5 others | 107 | $42.80 | |||||
| sub totals | 112 | $44.80 | |||||
| ND wins in 4 games | Jan | 9 | $3.60 | each share is paid $10 × 731 ÷ 72 = $101.5278 | Jan is paid $101.5278 × 9 = $913.7502 | total house payout is $101.5278 × 72 = $7310.0016 | total breakage is −$0.0016 |
| 6 others | 63 | $25.20 | |||||
| sub totals | 72 | $28.80 | |||||
| ND wins in 5 games | Kay | 8 | $3.20 | each share is paid $10 × 731 ÷ 95 = $76.9474 | Kay is paid $76.9474 × 8 = $615.5792 | total house payout is $76.9474 × 95 = $7310.0030 | total breakage is −$0.0030 |
| 3 others | 87 | $34.80 | |||||
| sub totals | 95 | $38.00 | |||||
| SC wins in 5 games | Rob | 12 | $4.80 | each share is paid $10 × 731 ÷ 113 = $64.6903 | Rob is paid $64.6903 × 12 = $776.2836 | total house payout is $64.6903 × 113 = $7310.0039 | total breakage is −$0.0039 |
| 7 others | 101 | $40.40 | |||||
| sub totals | 113 | $45.20 | |||||
| SC wins in 4 games | Ted | 6 | $2.40 | each share is paid $10 × 731 ÷ 191 = $38.2723 | Ted is paid $38.2723 × 6 = $229.6338 | total house payout is $38.2723 × 191 = $7310.0093 | total breakage is −$0.0093 |
| 6 others | 185 | $74.00 | |||||
| sub totals | 191 | $76.40 | |||||
| SC wins in 3 games | Vic | 10 | $4.00 | each share is paid $10 × 731 ÷ 148 = $49.3919 | Vic is paid $49.3919 × 10 = $493.9190 | total house payout is $49.3919 × 148 = $7310.0012 | total breakage is −$0.0012 |
| 2 others | 138 | $55.20 | |||||
| sub totals | 148 | $59.20 | |||||
| totals | 731 | $292.40 | |||||
Next are examples of payout calculations if a cancellation occurs. Suppose the remainder of the tournament is canceled after ND has won two games and SC one game. Then:
Schedule 5 below shows two different ways to calculate the payouts; the house needs to select one before betting begins.
| schedule 5 | ||||
|---|---|---|---|---|
| outcome | live shares | dead shares | proportional payout plan | equal payout plan |
| ND in 3 | 112 | zero | zero | |
| ND in 4 | 72 | each share is paid $10 × 731 ÷ 3 ÷ 72 = $33.8426 | each share is paid $10 × 731 ÷ 280 = $26.1071 | |
| ND in 5 | 95 | each share is paid $10 × 731 ÷ 3 ÷ 95 = $25.6491 | each share is paid $10 × 731 ÷ 280 = $26.1071 | |
| SC in 5 | 113 | each share is paid $10 × 731 ÷ 3 ÷ 113 = $21.5634 | each share is paid $10 × 731 ÷ 280 = $26.1071 | |
| SC in 4 | 191 | zero | zero | |
| SC in 3 | 148 | zero | zero | |
| total | 280 | 451 | ||
Single-game betting.
For betting on a single game, the house can provide point-spread choices. For example, in a low-scoring game like baseball, bettors might choose from these six outcomes in a game between ND and SC:
In basketball, there might be a four-way list:
In those sports where ties are frequent, a tie score will be available as an ordinary betting choice.
Due to the brief duration of a single game, resales are not feasible.
Regular-season betting.
The Autumnfield Athletic Conference has ten teams arranged in two divisions:
| teams of the Autumnfield Athletic Conference | |||
|---|---|---|---|
| Western division | Eastern division | ||
| town | symbol | town | symbol |
| Autumnfield, Colorado | CO | Autumnfield, Alabama | AL |
| Autumnfield, Oregon | OR | Autumnfield, Louisiana | LA |
| Autumnfield, Nebraska | NE | Autumnfield, Massachusetts | MA |
| Autumnfield, South Dakota | SD | Autumnfield, Vermont | VT |
| Autumnfield, Texas | TX | Autumnfield, West Virginia | WV |
In the regular season, each team plays 26 games: 4 games against each opponent within its division, and 2 games against each opponent in the other division. The winner of each division is simply the team with the best overall won-loss record, and after the regular season the two division champions meet each other in a playoff.
The house can offer betting on which of the five teams will win each division. Over the course of a full season, there will be plenty of opportunity for resale action. If before the season ends …
The house might also offer combination bets, for instance where the bettor must correctly pick both division winners out of the 25 pairings.
Options.
For patrons who seek a more advanced betting experience, the house might enable the selling and buying of options. These are most useful when there have been both a great deal of betting action and so much resale activity that bid and ask prices on each team's shares are nearly equal at any time. Options are not likely to work well on a single game or a short tournament, but are quite suited to betting on full-season competition, as there will be enough time to establish trends in resale price movements.
Note the terminology:
Consider these two examples with the bettors Adam, Beth, Carl, and Doug:
| Adam has acquired an option to buy from Beth as many as 17 shares of MO that Beth currently owns. Adam would pay a price of $21.20 per share.
Adam acquired this privilege by paying Beth a nonrefundable premium of $18.70 = 17 × $1.10 per share. Adam is under no obligation. On the other hand, Beth must sell if Adam chooses to exercise the option; Beth has no cancellation privilege. As long as this option is outstanding, Beth's 17 shares of MO are locked, and cannot be used for any other purpose. If Adam when exercising buys fewer than the total number of shares, the option remains active for the remainder. | Carl has acquired an option to sell to Doug as many as 21 shares of KY that Carl currently owns. Doug would pay a price of $19.80 per share.
Carl acquired this privilege by paying Doug a nonrefundable premium of $31.50 = 21 × $1.50 per share. Carl is under no obligation. On the other hand, Doug must buy if Carl chooses to exercise the option; Doug has no cancellation privilege. As long as this option is outstanding, $415.80 = 21 × $19.80 of Doug's funds are locked, and cannot be used for any other purpose. If Carl when exercising sells fewer than the total number of shares, the option remains active for the remainder. |
In investment parlance, Adam has bought a long call, Beth has sold a short call, Carl has bought a long put, and Doug has sold a short put.
Typically, the premium for an option will be much less than the price of the underlying share. A possible fee plan:
Note that 0.8% (to offer) plus 0.8% (to accept) plus 2.4% (to exercise) equals 4.0%, the usual fee for buying shares. That portion of fees based on the premium is on top of this.
The price and premium should each be a multiple of $0.10.
| four kinds of option offers | ||
|---|---|---|
| to buy shares | to sell shares | |
| to buy options | The offer to buy an option to buy shares specifies:
After the offer is established, these can occur multiple times:
The offerer is who would choose to exercise the option, but is never required to have sufficient funds to actually do so. The accepter must have enough of the team's shares, and they will be locked. | The offer to buy an option to sell shares specifies:
After the offer is established, these can occur multiple times:
The offerer is who would choose to exercise the option, but is never required to have sufficient shares to actually do so. The accepter must have enough funds to buy the shares, and the funds will be locked. |
| to sell options | The offer to sell an option to buy shares specifies:
After the offer is established, these can occur multiple times:
The offerer must have enough of the team's shares, and they will be locked when this offer is placed. The accepter is who would choose to exercise the option, but is never required to have sufficient funds to actually do so. | The offer to sell an option to sell shares specifies:
After the offer is established, these can occur multiple times:
The offerer must have enough funds to buy the shares, and the funds will be locked when this offer is placed. The accepter is who would choose to exercise the option, but is never required to have sufficient shares to actually do so. |
If a team is eliminated or it clinches the championship, any outstanding option offers on that team that are automatically terminated. A patron who advertises an option offer may omit the termination time, in which case this automatic termination prevails.
Here is a detailed example of an option:
Are option fees complicated? Yes. A mitigation is that, as always, the house's computer will calculate the exact fee of any contemplated action and display it to the patron, who can cancel if desired.
In a highly sophisticated trading environment, there can be a means for transferring (from one patron to another) either the choice or the obligation under an option. Continuing the examples above:
As with resales, options relating to a team should not be bought, sold, or exercised while that team is playing a game, and fifteen minutes before and after.
If the two-way options of the next section are instituted, the ordinary options of this section can for distinction be termed one-way.
Two-way options.
Two patrons might agree to a two-way option. Consider this example:
Patrons Tony and Fran have agreed to a two-way option, with Tony on the buy side and Fran on the sell side:
Neither Tony nor Fran paid a premium to establish this option. Fran must sell at $38.20 if Tony exercises his option, while Tony must buy at $31.60 if Fran exercises her option. Neither has a cancellation privilege. As long as this option is outstanding, Fran must keep 22 shares of MO, which are locked. Similarly, Tony must keep $695.20 = 22 × $31.60, which is locked. If either patron exercises the option for fewer than the total number of shares, the option remains active for both patrons for the remaining shares. |
A possible fee plan is derived from that for one-way options:
Note that 0.4% (to offer, bid) plus 0.4% (to offer, ask) plus 0.4% (to accept, bid) plus 0.4% (to accept, ask) plus 2.4% (to exercise) equals 4.0%, the usual fee for buying shares.
The bid and ask prices should each be a multiple of $0.10.
Details of the offers:
The buy-side offer for a two-way option specifies:
After the offer is established, these can occur multiple times:
| The sell-side offer for a two-way option specifies:
After the offer is established, these can occur multiple times:
|
In a two-way option the bid price will typically be higher than the ask price. This a because, in any market, a patron who directs a purchase generally sees a higher price, and a patron who directs a sale generally sees a lower price.
In general there will be no particular relationship between the two expiration times. If they are different, then when one expiration time has passed, the two-way option devolves into an ordinary one-way option.
Much as one-way options, two-way options can be transferred. Depending the attractiveness of the option, the patron might be able to sell it and receive a premium, or the patron might have to pay a premium to unload it. Of course, the patron who ultimately holds the option must have suitable funds or shares to lock. Also possible are discharges, where one patron pays the other a premium to abandon the agreement. Procedures for these actions can be established if desired.
Across-the-board betting.
For betting on a regular-season division winner, many patrons will find the regular Springfield system to their liking, particularly if resales are implemented. However, experienced patrons may prefer an alternate betting system called across-the-board (a-t-b). The house can provide both systems, but funds of the two must be kept separate because they have substantially different payouts. Shares from regular betting and a-t-b betting, even if on the same team in the same competition, are not equivalent or interchangeable in any way.
Recall from above:
| Autumnfield Athletic Conference Eastern division |
|---|
| Autumnfield, AL |
| Autumnfield, LA |
| Autumnfield, MA |
| Autumnfield, VT |
| Autumnfield, WV |
Under a-t-b, when patrons are buying shares from the house, they do not choose a team; instead they buy a basket, which is a package containing one share on each team. For example, in betting on the regular-season winner of Autumnfield Eastern, a basket would cost $52.00 = 5 × $10.00 + 5 × $0.40. Each patron can buy as many baskets as desired. Because the same number of shares are sold on all teams, the per-share payout for the winning team is inevitably $50.00.
A bettor who buys a basket and merely holds it suffers an overall loss of $2.00, which equals the betting fee — this fact highlights that the whole point of a-t-b is to transact with other bettors, buying and selling shares on individual teams. Resales, options, and other patron-to-patron transactions are handled without regard to any basket.
The mechanisms of a-t-b resales and options are exactly the same as in regular Springfield betting; the only difference with a-t-b is when patrons are buying shares from the house. The house might even allow basket sales to continue after competition begins, because the payout value of shares will not be affected. Similarly, the house might be willing to redeem baskets at $10.00 per share. Patrons who buy a-t-b shares from other patrons need not themselves have purchased a basket at any time.
Here is the scenario of a sale under the a-t-b system: