WO2005102477A1

WO2005102477A1 - Method of generating a result of a racing game

Info

Publication number: WO2005102477A1
Application number: PCT/AU2005/000594
Authority: WO
Inventors: Clyde Wilson Torkington; Howard Silver
Original assignee: Tabcorp International Pty Ltd
Priority date: 2004-04-27
Filing date: 2005-04-27
Publication date: 2005-11-03

Abstract

A method of generating a result of a racing game having a plurality of participants to enable fixed odds betting on the racing game, the result being generated from at least part of the result of an identifier selection game. The method comprises allocating a unique result label to each result in a set of possible results obtained from at least part of a subset of identifiers selected from a set of identifiers in an identifier selection game so that there is a set of result labels, determining the permutations of finishing order for the participants in the race required to produce a result of the race and the odds of each permutation, dividing the set of result labels to produce subsets of result labels for each permutation which correspond in size to the relative odds of each permutation, running said identifier selection game, determining an outcome result label from said at least part of the result of said identifier draw game, and determining the result of the racing game by determining the permutation to which the outcome result label corresponds.

Description

METHOD OF GENERATING A RESULT OF A RACING GAME

Field of the Present Invention

The present invention relates to a method and apparatus for generating a result of a racing game which enables fixed odds betting on the game. The method and apparatus use the result of an identifier draw such as a number draw to generate the racing game result.

Background to the Invention

Keno is an ancient Chinese numbers game based on the drawing of 20 balls from a cage containing 80 balls numbered 1,2, .... ,80. In the last twenty or so years the game has become fully computerised and this has lead to a very fast game with a draw every 3 - 5 minutes . Many jurisdictions allow the use of an internal software based result generation technique using an approved pseudo Random Number Generator or an external "Black Box" result generator based on a software pseudo-Random Number Generator or a hardware white noise sampler.

Keno results are typically graphically presented as an 8 x 10 grid (matrix) as shown in Figure 1.

Examples of common and exotic Keno bet types are as follows: In the bet type "Spots & Ways", players bet on N numbers (1 <= N <= 15) . The result is determined by finding out how many, say C, of the N numbers were drawn ("caught") and consulting a Prize Table that specifies the prize for catching C from N. In the bet type "Heads or Tails", players bet on more numbers being drawn from the lower 40 numbers - "Heads" OR vice versa - "Tails" OR on an exact split (10:10) between the top & bottom half - "Evens". The result is determined by counting how many of the drawn numbers are less than 41. If this exceeds 10 all "Heads" bettors win a published prize and all "Tails" & "Evens" bettors lose. If this equals 10 all "Evens" bettors win a different prize whilst all "Heads" & "Tails" bettors lose. If it is less than 10 all "Tails" bettors win the same prize as the "Head" prize whilst all "Evens" & "Tails" bettors lose. In the bet type "Lucky Last", players bet on the last number drawn. A fixed prize is won if the selected number is the last one drawn. In the game "Keno Racing", players bet on eight groups of ten numbers (1..10), (11..20), .... ,(71..80) The groups are represented as a race between animated horses . As each number is drawn, the horse whose group the number falls within is moved forward a fixed amount. After the last number is drawn and applied to "horse" movement, the most advanced "horse" wins followed by next most advanced second & so on. Dead heats are decided in favour of the first horse to arrive at the final position - i.e. the first horse whose last number was drawn. A fixed prize, independent of the group number or "horse" is offered for bets on the winner. In the game "Keno Roulette", players bet on what part of the matrix of Figure 1 the first drawn number lies in. A variety of pattern propositions are available to bet upon including: "Straight Up" = nominate the exact first number; "Quarters" = given the results matrix is divided into four quarters, nominate which of the quarters the first number resides within; and "Rows" , "Corners" , etc .

Trackside is a game developed by the present applicant that provides an animated race between a number of "participants" . Players are offered fixed odds on a sub-set of standard horse racing bet types. The win odds are nominated by the game operator and the system derives the place, quinella and trifecta odds from these using published algorithms. Each game result is generated by either an approved internal software algorithm or an external mechanical ball draw taking into account the unequal chance of winning of each "participants". For betting purposes, a Trackside result comprises the first three "participants" . These are called the race "placings" . After the winner has been determined from 12 participants, second is determined by the same algorithm as used to determine the winner excepting the trial ("race") is between 11 participants and their respective chance of winning the trial for second has been adjusted to take account of removal of the winner of the trial for first. Third placing is similarly determined by a further trial between the remaining 10 participants.

In a Win bet, players bet on which participant wins the race. Odds are offered dependent on the participant number, for coming first past the post (winning) .

In a Place bet, players bet on a participant being placed. Odds are offered dependent on the participant number, for being placed.

In an Each Way bet, players bet on a participant winning and/or being placed. Odds are offered dependent on the participant number, for winning and/or being placed

In a Quinella bet, players bet on two participants coming first and second in either order. Odds are offered based on each permutation of the quinella.

In a Trifecta bet, players bet on three contestants coming first, second and third in exact nominated order. Odds are offered based on each permutation of the trifecta. Accordingly, it would be desirable to generate a result for a Trackside type race game from a Keno type identifier draw game as this will allow Trackside type games to be run concurrently with a Keno type game without the need to run a separate random number generator. Providing such a game may also provide an additional enjoyable game to improve player enjoyment in conjunction with a Keno type game.

Summary of the Invention

Accordingly, the invention provides a method of generating a result of a racing game having a plurality of participants to enable fixed odds betting on the racing game, the result being generated from at least part of the result of an identifier selection game, the method comprising: allocating a unique result label to each result in a set of possible results obtained from at least part of a subset of identifiers selected from a set of identifiers in an identifier selection game so that there is a set of result labels; determining the permutations of finishing order for the participants in the race required to produce a result of the race and the odds of each permutation; dividing the set of result labels to produce subsets of result labels for each permutation which correspond in size to the relative odds of each permutation; running said identifier selection game; determining an outcome result label from said at least part of the result of said identifier draw game; and determining the result of the racing game by determining the permutation to which the outcome result label corresponds . The size of the part of the subset of identifiers used is typically determined by the size of the set and the desired error in the return to player. The larger part of the subset, the less the error in the return to player.

In one embodiment, the result of the race is the positions of the first three place getters. In one embodiment, there are twelve participants and hence 1320 possible permutations. The odds of each permutation are determined from the odds offered for each participant.

In one embodiment, the step of allocating a unique result label to each result comprises providing an algorithm which maps each possible result to a unique result identity. In this embodiment, the result labels and the identifiers are numbers. For example, as in the game of Keno where a subset of twenty numbers are drawn from a set of eighty numbers .

Thus, the step of determining a result label can be performed by evaluating an algorithm for a given subset of numbers .

In a preferred embodiment where the identifiers are numbers and the result labels are index numbers, the algorithm is:

Index number = [ [[ (nl-1) * (K-l) + (in2-l) ] * (K-2) + (in3-

1) ]*(K-3)+ ... ]*(K-N+1) + (xnN-1)

where: K = number of numbers in the draw; N = number of numbers drawn; nj = the jth number drawn; i j = the index of the jth drawn number, determined by how many numbers into a sequence of numbers consisting of numbers which were not drawn before the jth number the jth number is. The invention also provides apparatus for generating a result of a racing game having a plurality of participants to enable fixed odds betting on the racing game, the apparatus comprising: a processor configured to: process at least part of a subset of identifiers selected in an identifier selection game to determine an outcome result label from a set of unique result labels corresponding to respective ones a set of possible results derivable from said at least part of a subset; and determine a result of said racing game by determining to which one of a plurality of subsets of the set result labels said outcome result label belongs, each subset corresponding to a permutation of finishing order for participants in the racing game required to produce a result and being formed by dividing the set of result labels into subsets for each permutation which corresponding in size to the relative odds of each permutation, thereby determining to which permutation the outcome result label belongs .

In a preferred embodiment, the apparatus further comprises display means for displaying said race in the form of an animation of the race.

Brief Description of the Drawings Figure 1 is a matrix of Keno results; and Figure 2 is a schematic diagram of an apparatus for determining the result of a game.

Description of the Preferred Embodiment

Figure 2 is a schematic diagram of an apparatus for determining the result of an electronic game of the preferred embodiment. In the preferred embodiment, the electronic game is a horse racing game called Trackside™ where players bet or wager on the result of the electronic game. The individual horses in the electronic game provide a plurality of participants which can be bet upon. Players of the game can place fixed odds bets on the horses in the same manner as fixed odds bets can be placed on conventional horse races . To stimulate customer interest in such a game, it is necessary to allow players to bet at a variety of horses at different odds.

In the preferred embodiment, the electronic game is controlled by software run by a processor of a host computer 101. Once the result of the game has been determined, the host computer 101 instructs the graphics engine 102 to create the horse race simulation. The graphics engine 102 then causes the simulated horse race to be displayed on displays 105.

For a 12 participant horse race, one suitable distribution of odds is shown in Table 1.

Table 1

In Table 1, the house return is intended to be 20%. Accordingly, a betting market is set at 125% and distributed amongst a number of horses .

Persons skilled in the art will appreciate that using the actual win odds will not always be practical because they will require payouts of very small denomination coins . Accordingly, slight variations on these odds are used. For example, the set of odds shown in the right hand column of Table 1 presume that 5 cent pieces will be available as a reasonably convenient form of payment. Accordingly, the actual payment odds give slight variations compared to the absolute odds which lead to slight variations in the house return. The variations in the house return are sufficiently small as to have a barely noticeable effect on the outcome of betting on individual horses .

The preferred embodiment provides a technique for generating a result of the horse racing game from a Keno draw. That is, where twenty numerical identifiers are drawn from a set of eighty numerical identifiers, the set being 1, 2, 3 ... 80.

The method is performed by providing a mapping of Keno results to Trackside results by allocating a plurality of Keno results to each Trackside result in accordance with the odds of particular Trackside trifectas. This is achievable because a large number of permutations of Keno results are available. Indeed, it is only necessary to use part of the subset of numbers drawn in a Keno game for the purpose of generating a Trackside result. This is done by allocating a unique result label to each result in a set of possible results which can be derived from at least part of the subset. In the preferred embodiment, the result labels are a set of unique index numbers. (However, other unique labels, such as symbols or characters could be used.) For example, from the first five numbers of the twenty number subset. The set of unique index numbers is then divided amongst the different Trackside trifectas to form subsets which correspond in size to the relative odds of each trifecta permutation. Accordingly, each possible trifecta will correspond to a set of possible index numbers . As there are a large number of permutations, it is not convenient or computationally easy to do this by way of look up table. That is by storing all of the index numbers because there are a very very large number of results. Accordingly, we have developed an algorithm which enables the generation of the index number which corresponds to the Keno result. Therefore it is only necessary to determine what range the outcome result identifier falls in and it is only necessary to store the bounds of each range to determine the result of the race.

If we let:

K = Number of Keno numbers (typically K=80) N = Number of drawn number that will be used to generate the Trackside result (so N <= 20) .

ND(K,N) = Number of ordered draws without replacement of N from K numbers with order

T = number of Trackside numbers (or participate in the racing game) (currently T = 12)

{di, d₂ , .. , d_τ = Trackside $1 dividends}

{Pi P2 • • PT = chance of 1, 2 , ..., T winning Trackside "race"} i - e ∑ Pi = 1 =!

{p_ijk: l<=i<=T, l<=j <=T, l<=k <= T = probability Trackside result i 1st, j 2nd, k 3rd i ! =j ! =k} . So p_ijk = probability of Trackside trifecta i/j/k

Now:

ND(K,N) = K! / (K - N) ! (1)

so the number of Keno draws with order important,

ND(80,20) = 80! / 60! = 80*79*78*...*61 ~= 10³⁶

& the number of Trackside results (1st, 2nd & 3rd in order) ,

ND(12,3) = 12! / 9! = 12*11*10 = 1320

Pi = 1.0 / (M * di) (2)

Pijk = Pi . ( Pj / (1 - Pi) ) * (Pk / ⁽1 - Pi - Pj) ) (3)

For the Trackside odds sets typically offered, the least probable trifecta lies in the range:

1/25,000 <= Pι_2/ιι/ιo <= 1/5,000

So if we were to use all the 20 numbers drawn, the smallest trifecta range would be of length 4 x 10³² (i.e. 10³⁶ / 10⁴) . This is so large that the worst case rounding error when breaking the line up to reflect the chance of the 1320 trifectas is infinitesimal ( O(10^"30) ).

In order to simplify the expression of the algorithm we have chosen a standard twenty from eighty Keno draw for which the first four numbers drawn are used in draw order - i.e. this is the at least part of the subset which is used.

Given the first four numbers of a Keno draw four - {nl,n2,n3,n4}, the task is to produce a Trackside three from twelve result, i.e. a trifecta permutation using an algorithm that always produces the same output for a given input .

The number of possible combinations of the first four Keno numbers is

L = ND(K,N) = 80*79*78*77 = 37,957,920

We break the range of integers from 0 to L-1 into a set of ND(12,3) = 1320 sub- ranges:

Rl = [0..Ri] , R2 = [Rι+1..R₂] ,

R [Rii+1..L-1] where each range has a length commensurate with the chance of the trifecta winning, so

Length (range for trifecta i/j/k) = Round ( pi_jk * L)

The number of Keno drawn numbers (i.e N) used when determining the Trackside result determines the amount of required computation & the rounding error on the trifecta ranges, in particular that for the least probable trifecta (12/11/10) . For the Trackside odds sets typically offered, the least probable trifecta probability lies in the range :

1/25,000 <= P12_/11/10 <= 1/5,000

We have determined that using just the first three Keno drawn numbers (N=3) along with a standard Trackside "odds set" leads to a range of length 492,960 and for the worked example has a rounding error of 1.4 % for the trifecta 12/11/10. Because this trifecta has such a small chance of occurring this means there is an insignificant impact on the "Return To Player" (RTP) .

The number of Keno drawn numbers to use will depend on the required RTP and individual outcome tolerances allowed by a Jurisdiction however we would expect that the first four numbers should satisfy most Jurisdictions as this provides a range of 38 million integers.

The ranges Rl, R2,... remain the same whilst the Trackside win dividends are unchanged.

If Keno results were sorted into an order, the ordering would be:

1/2/3/4, 1/2/3/5, ... , 1/2/3/80, 1/2/4/3, 1/2/4/5, ... , 1/2/4/80, 1/3/2/4,1/3/2/5, ... ,1/3/2/80, ....80/79/78/77

Observations :

• in the sorted list there are (nl-1) *79*78*77 results before the first result starting with the number nl;

• within the list of results starting with a given number (nl) , there are (i2 - 1)*78*77 results before the first result starting with nl/n2, where i2 is the index into the list of numbers left after removing nl from the list {1, 2,3,..., 80};

• within the list of results starting with given first two numbers (nl/n2) , there are (i3 - 1) *77 results before the first result starting with nl/n2/n3, where i3 is the index into the list of numbers left after removing nl & n2 from the list {l, 2, 3,..., 80};

• there are 77 results that start with nl/n2/n3

So the formula is :

Index Number ( result nl/n2/n3/n4) = (nl - 1) *79*78*77 + ( i2 - 1) *78*77 + ( i3 - 1) *77 + ( i4 - 1)

for example, consider the result 70/5/1/20. The index numbers are :

i2 = 5, i3 = 1, i4 = 18

so Index Number (result 70/5/1/20) = (70 - 1)*79*78*77 + (5 - 1)*78*77 + (1 - 1)*77 + (18 - 1) = 32,762,747

This can be generalised to an algorithm of the form:

Index Number = [ [[ (nl-1) * (K-l) + (in2-l) ] * (K-2) + (in3-

1) ]*(K-3)+ ... ]*(K-N+1) + (inN-1)

where: K = number of numbers in the draw; N = number of numbers drawn; nj = the jth number drawn; inj = the index of the jth dawn number, determined by how many numbers into a sequence of numbers consisting of numbers which were not drawn before the jth number the jth number is.

Accordingly, it will be appreciated that using the algorithm it is possible to obtain a unique result identifier (the Index Number) without storing the individual results. Thus, the step of allocating a result identifier to each result is achieved by providing an algorithm which maps each possible result to a unique identifier. This provides substantial computational savings .

Persons skilled in art will appreciate that there may be other algorithms which can be used to produce a similar mapping. All that is important is the mapping of Keno results is unique so that each identifier is unique and there are no duplication of results.

The size of the part of the subset which is used will be determined by the size of set of numbers, the size of the subset which is drawn, and the desired error in the return to player. The larger the part of the subset which is used, the less the error in the return to the player.

Example

To enable further understanding of the algorithm a working example of a simple case is provided that, without loss of generality, illustrates the approach and the requirement for a very large number of Keno outcomes in order to acceptably generate a Trackside result.

Suppose we use a "Keno like" draw of 3 from 5 numbers to determine the result of a "Trackside like" race between three "participants". There are 5x4x3 = 60 possible results for this draw when the order of drawing a number is significant. Each of these results is equally likely.

Suppose the win prizes (dividends) being offered for 1 dollar win bets on a race between three Trackside

"contestants" are {$3, $3, $4} respectively (odds of 2/1, 2/1, 3/1) . The relative chance (the probability) of each contestant winning can be easily seen to be {4/11, 4/11, 3/11} - these add up to 1. Suppose bets are offered on first & second in order (the reduction of a trifecta bet to an "exacta") & therefore require an algorithm to generate first and second. (Given the field size, there is a one-one correspondence between all the possible exacta & trifecta bets.) There are 3x2 = 6 possible Trackside results, corresponding to every possible Trackside exacta. A Trackside result is normally generated by a trial (race) for first followed by a race for second between the remaining contestants . The chance of each contestant winning the trial for second is obtained by simply re-scaling the probabilities as below:

The probability of "contestant" one winning (henceforth stated as "1 winning") is 4/11. If contestant one wins, the probabilities of contestants two & three winning the trial for second are simply (4/11) / (1 - 4/11) and (3/11) / (1 - 4/11), i.e. {4/7, 3/7}. The probability of the exacta result 1/2 is simply calculated as follows:

Probability{l/2} = Probability{l wins trial for first} x Probability{2 wins trial for second} = (4/11) x (4/7) = 16 / 77 similarly

Probability{1/3} = Probability{l wins trial for first} x Probability{3 wins trial for second} = (4/11) x (3/7) = 12 / 77

proceeding in this way we simply calculate the probabilities of all the exactas - Table 2.

Table 2 - Probability Of Trackside Exactas

Now we divide the range of integers [0,1,2, ... , 59] as accurately as possible into as many sub-ranges as there are exactas where the "length" of each sub-range - the number of integers it comprises - is proportionate to the probability of the exacta, with the last range correcting for rounding of all the other ranges. This is achieved by multiplying the number of "Keno results" (i.e. 60) by each probability in turn and rounding the product to the nearest integer, leading to the sub-ranges in Table 3.

This illustrates a major inaccuracy in this worked example. Whilst the two least probable exactas 3/1 & 3/2 have equal chance, the exacta 3/2 has been given a range of 9 integers versus 8 for 3/1. Thus, there are insufficient "Keno like results" in this simplistic example to be able to accurately generate a "Trackside result" .

Table 3 - Breaking The Keno Result Range According To Trackside Exacta Probabilities

Nevertheless having broken the integer range [0,1, ... ,59] into six sub-ranges, given that each Keno result is equally likely, the Trackside result is generated as follows :

Find a unique mapping of the Keno result to an integer in the range [0,59] . The Trackside result is simply the subrange (Table 2) that this integer falls into.

The problem remains to find such a mapping. This is achieved by ordering the Keno results (these are permutations of 3 from 5 integers) in ascending order as illustrated in Table 4.

Table 4 - Ascending Ordering Of Keno Results

Observations :

• in the sorted list there are (nl-l)*4*3 results before the first result starting with the number nl;

• within the list of results starting with a given number (nl) , there are (i2 - 1) *3 results before the first result starting with nl/n2, where i2 is the index into the list of numbers left after removing nl;

• there are 3 results that start with nl/n2

So we have a simple enough formula:

Index Number ( result nl/n2/n3) = (nl - 1) *4*3 + (i2 1)*3 + (i3 - 1)

Where : i2 = index into list of 4 numbers remaining to n2 when nl is cast out (so 1 <= i2 <= 4) i3 = index into list of 3 numbers remaining to n3 when nl & n2 are cast out (so 1 <=i3 <= 3)

for example, consider the result 4/5/1. Casting out the first number (4) from the original list {1,2,3,4,5} produces the list {1,2,3,5}. 5 is the fourth number in this list so i2 = 4. Casting out 5 from this list produces the list {1,2,3}. 1 is the first number in this list so i3 = 1. So we calculate:

Index Number ( result 4/5/1) = (4 - 1) * 12 + (4 - 1) * 3 + (1 - 1) = 45

& this agrees with the tabulated index (Table 4) . The integer 45 lies within the 5th sub-range in Table 3 and that corresponds to the Trackside result 3/1.

Persons skilled in the art will appreciate that a number of variations may be made to the invention without departing from the scope of the invention. For example, the game can also use Bingo or Lotto type identifier draws. Further, depending on the embodiment, a unique result of the identifier draws may or may not take into account the order in which balls are drawn. These and other variations should be considered as falling within the invention described herein.

Claims

CLAIMS :

1. A method of generating a result of a racing game having a plurality of participants to enable fixed odds betting on the racing game, the result being generated from at least part of the result of an identifier selection game, the method comprising: allocating a unique result label to each result in a set of possible results obtained from at least part of a subset of identifiers selected from a set of identifiers in an identifier selection game so that there is a set of result labels; determining the permutations of finishing order for the participants in the race required to produce a result of the race and the odds of each permutation; dividing the set of result labels to produce subsets of result labels for each permutation which correspond in size to the relative odds of each permutation; running said identifier selection game; determining an outcome result label from said at least part of the result of said identifier draw game; and determining the result of the racing game by determining the permutation to which the outcome result label corresponds.

2. A method as claimed in claim 1 comprising selecting the size of the part of the subset of identifiers on the basis of the size of the set of identifiers and the desired error in the return to player.

3. A method as claimed in claim 1 wherein the result of the race is the positions of the first three place getters .

4. A method as claimed in claim 1 wherein there are twelve participants .

5. A method as claimed in claim 1 wherein the step of allocating a unique result label to each result comprises providing an algorithm which maps each possible result to a unique result identity and the identifiers are numbers and the result labels are index numbers such that each participant is allocated a subset of index numbers, whereby determining a result label is performed by evaluating an algorithm for a given subset of numbers to determine an index number and the index number determines the result.

6. A method as claimed in claim 1 wherein in the identifier selection game a subset of twenty numbers are drawn from a set of eighty numbers.

7. A method as claimed in claim 5 wherein the algorithm is:

Index number = [....[[ (nl-1) * (K-l) + (in2-l) ] * (K-2) + (in3- 1) ]*(K-3)+ ... ]*(K-N+1) + (inN-1)

where: K = number of numbers in the draw; N = number of numbers drawn; nj = the jth number drawn; inj = the index of the jth drawn number, determined by how many numbers into a sequence of numbers consisting of numbers which were not drawn before the jth number the jth number is.

8. Apparatus for generating a result of a racing game having a plurality of participants to enable fixed odds betting on the racing game, the apparatus comprising: a processor configured to: process at least part of a subset of identifiers selected in an identifier selection game to determine an outcome result label from a set of unique result labels corresponding to respective ones a set of possible results derivable from said at least part of a subset; and determine a result of said racing game by determining to which one of a plurality of subsets of the set result labels said outcome result label belongs, each subset corresponding to a permutation of finishing order for participants in the racing game required to produce a result and being formed by dividing the set of result labels into subsets for each permutation which corresponding in size to the relative odds of each permutation, thereby determining to which permutation the outcome result label belongs .

9. Apparatus as claimed in claim 8 wherein the subset of identifiers used is typically determined by the size of the set and the desired error in the return to player.

10. Apparatus as claimed in claim 8 wherein the result of the race is the positions of the first three place getters.

11. Apparatus as claimed in claim 10 wherein there are twelve participants and 1320 possible permutations.

12. Apparatus as claimed in claim 8 wherein the step of allocating a unique result label to each result comprises providing an algorithm which maps each possible result to a unique result identity the result label can be determined by evaluating an algorithm for a given subset of numbers .

13. Apparatus as claimed in claim 1 wherein in the identifier selection game a subset of twenty numbers are drawn from a set of eighty numbers.

14. Apparatus as claimed in claim 1 wherein the identifiers are numbers and the result labels are index numbers, the algorithm is:

Index number = [ [[ (nl-1) * (K-l) + (in2-l) ] * (K-2) + (in3- 1) ]*(K-3)+ ... ]*(K-N+1) + (xnN-1)