Ticket generation for approximating an unknown reference ticket

Abstract

Disclosed is a method for generating tickets, such as lottery tickets, to approximate an unknown reference ticket, e.g., winning numbers. After receiving a request for tickets, seed tickets are acquired that bear random numbers from a set of numbers. The seed tickets are generated using a mixed-integer programming model that ensures a near-identical pairwise similarity measure among the tickets while minimising total similarity measures for all pairs by uniformly distributing the numbers. The tickets are generated from the seed ticket using a one-to-one number mapping module, by replacing numbers from respective seed tickets with other ones of the numbers, to maintain identical structure between generated tickets and seed tickets, and avoid repetition between the tickets and further tickets that have previously been issued in response to respective further requests for tickets.

Description

[0001] TICKET GENERATION FOR APPROXIMATING AN UNKNOWN REFERENCE TICKET

[0002] Technical Field

[0003] The present invention relates, in general terms, to a method for generating tickets to approximate an unknown reference ticket, and a system for implementing the method. The present invention relates to, but is not limited to, generating numbers for tickets in lottery games.

[0004] Background

[0005] The Lottery Quick-Pick System offers a convenient option for players to purchase lottery tickets without the need for manual number selection. Utilizing a random selection process, quick-pick systems generate distinctive number combinations for each ticket, streamlining the process and guaranteeing a fair outcome. An illustrative example is the Singapore Toto Quick-Pick, which adeptly produces the maximum number of bets based on the player's selected ticket type or number of tickets.

[0006] Nevertheless, the current Lottery Quick-Pick System faces various challenges, with the primary concern being the algorithm effectiveness. In general terms, effectiveness is the likelihood that a payout will be received. If the algorithm proves to be less effective, thereby leading to fewer payouts, the Quick- Pick System will not meet player expectations and could lead to player dissatisfaction. This is because players typically have certain expectations regarding the frequency of winning, especially with smaller prize categories.

[0007] There are various algorithms which can be employed by quick-pick ticket generation systems. These include random selection and finite-geometry based methods. Random selection is employed in the current Lottery Quick-Pick System to randomly generate a unique combination of numbers for each ticket. This process ensures that the selection is truly random and not influenced by any patterns or biases. The finite-geometry based method, introduced by Cushing and Stewart (2023), uses finite geometry, specifically Fano planes, to pick numbers for lottery ticket selection. More importantly, under the assumption that the potential selectable numbers are less than 70, their method provides configurations for minimal lottery designs, ensuring a consistent match with two winning numbers.

[0008] It would be desirable to overcome or otherwise address at least one of the above-described problems with existing lottery ticket generation systems, or at least to provide a useful alternative.

[0009] Summary

[0010] Embodiments of the present invention provide distinct differences over the previously known methods. In cases where random-selection methods result in multiple tickets having very similar numbers, embodiments of the present invention ensure a comprehensive consideration of number distribution and overlap management when generating multiple lottery tickets, in cases where finite-geometry promises a match with two winning numbers, embodiments of the present invention prioritize achieving a greater number of matches, such as accurately predicting three or four numbers.

[0011] Disclosed is a ticket generation system for generating tickets to approximate an unknown reference ticket, comprising: an interface for receiving requests from a user and transmitting tickets to the user; a database; memory; and a processor, the memory storing instructions that, when executed by the processor, cause the system to: receive a request for m tickets via the interface; acquire m seed tickets, each comprising / < random numbers from a set of n numbers, generated by employing a mixed-integer programming model, to:

[0012] - ensure a near-identical pairwise similarity measure among the m tickets; and

[0013] - minimise total similarity measures for all pairs by uniformly distributing the numbers; generate each ticket of the m tickets from a respective said seed ticket, utilizing a one-to-one number mapping module that replaces the random numbers of the respective seed ticket with other numbers from the set of n numbers, to:

[0014] - ensure that the generated tickets maintain an identical structure to a structure of the corresponding seed tickets; and

[0015] - avoid repetition between the tickets and further tickets previously generated for one or more further requests, each further request being a request for m tickets, associate a unique identifier with the request or with each said ticket, each unique identifier being for verifying numbers of the m tickets against a reference set of numbers, the reference set of numbers corresponding to the unknown reference ticket; and output the m tickets to the interface and database, each unique identifier to the database.

[0016] Also disclosed is a method for generating tickets to approximate an unknown reference ticket, comprising: receiving a request for m tickets; acquiring m seed tickets each comprising k random numbers from a set of n numbers, generated by employing a mixed-integer programming model, to:

[0017] - ensure a near-identical pairwise similarity measure among the m tickets; and - minimise total similarity measures for all pairs by uniformly distributing the numbers; generating each ticket of the m tickets from a respective said seed ticket, utilizing a one-to-one number mapping module that replaces the random numbers of the respective seed ticket with other numbers from the set of n numbers, to:

[0018] - ensure that the generated tickets maintain an identical structure to the structure of the corresponding seed tickets; and

[0019] - avoid repetition between the tickets and further tickets previously generated for one or more further requests, each further request being a request for m tickets, associating a unique identifier to the request or to each said ticket, each unique identifier being for verifying numbers of the m tickets against a reference set of numbers, the reference set of numbers corresponding to the unknown reference ticket; and outputting the m tickets to the interface and database, each unique identifier to the database.

[0020] Advantageously, embodiments of the invention afford a Smart Quick-Pick that takes into account both the distribution of numbers across tickets, and overlap in numbers between tickets. This ensures numbers are distributed with respect to other numbers that have already been selected, rather than randomly selecting numbers. Moreover, pairwise similarity is targeted to increase the likelihood of winning with three or four matches, the number depending on the parameters of the game.

[0021] Advantageously, embodiments of the invention aim to maximise the probability of breaking even with a given prize structure. The methodologies disclosed herein focus on attaining more, smaller prizes rather than focussing on a significant, but probabilistically very unlikely, larger prize. Brief description of the drawings

[0022] Embodiments of the present invention will now be described, by way of nonlimiting example, with reference to the drawings in which:

[0023] Figure 1 : a computer network for implementing the method described herein.

[0024] Figure 2: Enumerating the winning probabilities of 27 tickets constructed using (1) the finite-geometry based solution {Cushing), (2) random selection {Random), (3) Greedy Based SDP approach in Liu et al. (2023) {Moment), and (4) The MIP based approach in this paper {MIP).

[0025] Figure 3: Break-Even Probability versus Number of Tickets Purchased.

[0026] Detailed description

[0027] The system described herein is used to generate tickets, such as lottery tickets, that have a higher chance of matching a reference set of numbers, such as a lottery draw. The methodology endeavours to ensure a near-identical pairwise similarity between tickets, where the pairwise similarity is defined as the quantity of common numbers between any two tickets. This is characterized by maintaining a discrepancy within a range of 1 between the maximum and minimum pairwise similarity. Simultaneously, the method aims to maintain a relatively uniform distribution of numbers across all ticket purchased by a particular user. For this reason, the following description will be made with reference to lottery tickets, though the invention will not be limited to such an example.

[0028] Some previous ticket generation models have involved random selection. This relies solely on chance and lacks strategic analysis. The present methodology incorporates algorithms and strategic analysis, involving number distribution and overlap management, with the aim of optimizing the selection process to enhance the probability of winning smaller prizes.

[0029] Other, more recent, ticket generation models have involved finite geometrybased methods to consistently guarantee a match with two winning numbers. The present methodologies have a greater likelihood of matching numbers on a ticket with at least three or four winning numbers. In effect, the winning numbers are a reference set of numbers of an unknown ticket, against which the numbers of each of a user's tickets can be compared to determine if the user has won.

[0030] Figure 1 illustrates a network 100 for generating tickets. The network 100 includes a system 102 that receives requests for tickets and generates them according to the present methodology. The system 102 comprises an interface 108 for receiving requests from a user device 160 and transmitting tickets to the user device 160. The user device 160 may be any suitable device such as a smartphone, laptop or PC, and may be one of a plurality of user devices, each for a respective user.

[0031] The system 102 further comprises a database 120, presently shown as being within the same physical system as the memory 104 and processor 103, though it could equally reside remotely therefrom. The database stores historical data such as previous winning numbers, also referred to as historical reference sets of numbers. Some people believe there are "hot numbers", being numbers that have recently been winning numbers. The database 120, or individual user devices 160, may also store historical user sets of numbers, being numbers previously selected by a user - where those numbers appear frequently, they can indicate a preference by the user for selecting those numbers.

[0032] The system 102 also comprises memory 104 and processor 103. The memory 104 storing instructions (program code 106) that, when executed by the processor 103, cause the system 102 to implement the methods described herein. The system 102 communicates with the user devices 160, and extracts data from database 120 (if separate from the processor 103 and memory 104) over network 130.

[0033] The program code 106 implements a seed ticket generator 107, for generating a predetermined number of seed tickets. The predetermined number may be fixed based on the maximum number of tickets that can be sold for a particular draw or game, an exhaustive list of all possible combinations of numbers for a particular draw or game, and other approaches. For example, in the UK Lotto, the maximum limit for the sale of lottery tickets is set at 70. Mixed-integer programming (MIP) model in module 113 is employed in this seed ticket generator 107 to generate all types of seed tickets, covering quantities from 1 to Mmax. Here, Mmaxsignifies the maximum permissible number of tickets that are allowed to be compared to the reference set and also the maximum number of tickets allowed to be sold in each specific draw. Thus, when the system receives a request for m tickets via the interface 108 (m being a number specified by the user), m seed tickets are acquired. Each seed ticket comprises k random numbers from a set of n numbers, where k is the number of numbers required to fill a ticket and n is the maximum number available in a draw. For example, a lottery ticket may show six numbers, with k thus being six, and each number on the ticket may be a number from 1 to 59, with n thus being 59. Note that, in terms of computational time, generating a set of seed tickets, each comprising k = 6 random numbers selected from a set of n = 49 numbers with quantities less than 50, takes less than an hour. However, addressing the set of seed tickets with quantities exceeding 50 demands an exceptionally extended timeframe - as the quantity m increases, the computational demand increases significantly.

[0034] The program code 106 implements a one-to-one number mapping module 115 to generate a unique set of m (1 < m < Mmax) tickets for each user, where m is determined by the user. Utilizing the type of seed tickets provided by the seed ticket generator 107, with corresponding quantities of tickets m, the one-to-one number mapping module 115 performs number substitution by replacing the numbers on this type of seed tickets with alternative feasible numbers. For instance, if there is a number 3 in the seed tickets, and a draw involves numbers from 1 to 59, the replacement number must be within the range of 1 to 59, excluding 3. Through this mapping module, for the same request to generate m quick-pick tickets, a diverse set of m quick-pick tickets can be generated without repetition.

[0035] The program code 106 also implements a unique ID generator 111 that generates a unique identifier for each ticket, or for all m tickets requested by the user. Each unique identifier enables a ticket, or the m tickets, to be verified during matching of numbers on the ticket against the reference set of numbers. In some examples, the tickets are lottery tickets and the reference set of numbers is the set of winning numbers drawn in a particular draw after purchase of lottery tickets for that draw closes. Since the number of numbers in the set of winning numbers is k (i.e. is that same as the number of numbers in a ticket purchased for the corresponding draw), the set of winning numbers can be said to form an "unknown reference ticket".

[0036] The m tickets are then outputted to the interface 108, thereby to be shared with the user device 160. The m tickets are also outputting to a database along with each unique identifier. The unique identifier or unique identifiers may also be sent to the user so that the user can, e.g., after the corresponding lottery numbers are drawn at, check with the system 102 to determined how many matched numbers they have on individual ones of the m tickets they have purchased. In some instances, it is not necessary to transmit the unique identifier to the user if, for example, the user has an account with a lottery company and all tickets they purchase are associated with that account - the account number corresponding to the account can be used as the unique identifier. The program code 106 also implements a comparator 109 that determines, for each ticket that is verified using a corresponding unique identifier, which numbers on a ticket match those in a reference set. Based on the number of matches between numbers in an individual said ticket and numbers in the reference set of numbers, a prize will be awarded for each ticket.

[0037] The system 102 implements a method for for generating tickets to approximate an unknown reference ticket. The "unknown reference ticket" can refer to the set of winning numbers of a lottery draw, or some other unknown set of numbers.

[0038] In some embodiments, the MIP module 113 employed by seed ticket generator 107 initially generates all possible types of seed tickets. Each type of seed ticket is, or corresponds to, a quantity of tickets from 1 to Mmax, where Mmaxdenotes the maximum number of tickets (e.g. the maximum number of quick-pick lottery tickets, or tickets in general, allowed by the lottery firms). The requested m can thus similarly be selected by the user between 1 and Mmax. Thus, the m seed tickets are acquired by selecting the type of seed tickets with a corresponding quantity of m. Each seed ticket may comprise k random numbers, k being the number of numbers that a ticket carries for a particular lottery game or other application. Each of the numbers on a seed ticket is unique with respect to that ticket, although it may appear on more than one ticket. Such a diverse set of optimized lottery tickets serves as the seed lottery tickets, ensuring a varied range of tickets to meet the distinctive requirements for the number of tickets for each individual player.

[0039] Upon receiving a request denoted as i, from an individual player (i.e. user i) specifying the desired number of lottery tickets (m,), this request is fulfilled by employing a two-step process. The lottery tickets can be referred to as quickpick tickets. Initially, the type of seed tickets with a quantity of jnj is chosen to serve as the seed generator. Then, one-to-one number mapping module 115 generates m, unique lottery tickets for each request i. It is important to note that even if multiple customers have requests for the same quantity of lottery tickets, the generated quick-pick tickets will have no duplicates. Furthermore, the generated tickets maintain the same structure as their corresponding seed tickets, involving only the replacement of ticket numbers - e.g. maintaining or changing as little as possible the pairwise similarity measure and distribution of numbers.

[0040] There are two rules governing the generation of tickets when employing the one-to-one number mapping module based on the seed. The first rule provides for homogeneous ticket number replacement, where each number in the original seed can be substituted with any ticket number. This is because every number holds an equal probability of matching the winning number from a theoretical level. This rule is usually applied in scenarios where there are no specific requirements from individual players regarding their requests. The second rule is termed heterogeneous ticket number replacement, indicating that there are specific hot or preferred numbers designated for replacement in the original seed. This rule comes into play when an individual player specifies requirements for the selected numbers, such as preferences or hot numbers they favour.

[0041] With homogeneous ticket number replacement, one-to-one number mapping module 115 applies an equal probability replacement rule, where any feasible number has an equal chance of being selected to replace the numbers in the seed tickets. This can be achieved by simply re-arranging the numbers generated randomly using a random permutation. To enhance the system, heterogeneous ticket number replacement is used to retrieve numbers from the historical winning numbers (derived from historical reference sets), employing unequal probability replacement - i.e., particular numbers, namely those retrieved from the historical winning numbers, will have a higher likelihood of being selected to replace a number on a seed ticket - in contrast to the equal probability replacement used in homogeneous replacement. More specifically, by employing the advanced machine learning methods, the probability of each number becoming a winning number can be predicted based on past draws. This prediction can be helpful in identifying hot numbers and selecting the hot numbers during the quick-pick ticket generation, to replace numbers on seed tickets. Thus, the processor 103 can replace one or more random numbers on each seed ticket with a respective one or more numbers from the one or more historical reference sets, thus enabling hot numbers to be selected more frequently than if they were to appear purely by a probabilistic model applied equally to all numbers. Moreover, the processor may identify the historical reference sets with reference to a predetermined timeframe from current time. This enables more recent hot numbers to be preferentially picked over older, hot numbers. The processor 103 may then select the numbers or numbers to replace the seed ticket random numbers from the historical reference sets that were generated within the predetermined timeframe. The timeframe can be determined by reference to time itself - e.g. draws taking place over the last six months - or by reference to the number of draws - e.g. the last ten lottery draws. A similar process can be applied to historical user sets of numbers or tickets. Where a user preferences particular numbers, e.g., numbers corresponding to the dates of family birthdays, they may wish those numbers to appear in their tickets more frequently than other numbers. Thus, the processor may identify repeated numbers across multiple draws and select those for replacing the random numbers in the seed tickets for that user, and the same timeframe-based selection may be used.

[0042] In general, the MIP module 115 implemented by seed ticket generator 107 is structured as follows:

[0043] • Random Binary Variable uy. For each number j in the set of n numbers (presently n = 59), is a binary variable indicating if it is a winning number in a draw - i.e. if it is in the reference set of numbers. Specifically, Uj = 1 if j is a winning number, and uj = 0 otherwise. The vector u = [u1:...,un]Trepresents the winning numbers.

[0044] • Decision Variable x, : This binary vector represents the number selection for ticket i (where i e [m]). Here, = 1 indicates the selection of the jth number for ticket l. The constraint for each ticket is to select exactly k numbers.

[0045] • Payoff Calculation z^xt) The payoff for each ticket is calculated as the sum of the payoffs of the selected numbers, where denotes the number of winning numbers matched using the ticket xf.

[0046] • Vector Representation x and z(x) : The vector x = vec( a binary vector representing the number selections for all m tickets. The vector represents the payoffs for all tickets.

[0047] • Objective Function / (x,u): The objective is to maximize the performance of the highest-performing ticket among the set of m tickets. This is represented by

[0048] The objective is to design x to maximize the expected performance of the random function ereby increasing the chances of winning a prize. This model provides a structured approach to optimizing ticket selection, such as lottery ticket selection, increasing the probability of winning by strategically choosing numbers across multiple tickets. This strategy can be extended by modelling the problem of ticket number selection for increasing the probability of matching a reference set of numbers, such as matching lottery winning numbers, using a class of moment-based distributionally robust models. These distributionally robust models translate probabilistic challenges into a more manageable moments-based optimization problem.

[0049] The distributionally robust problem can be equivalently reformulated into a completely positive program, which is subsequently approximated as a mixed 0-1 semidefinite program (SDP). Nevertheless, the computational complexity stemming from the high-dimensional completely positive cone presents challenges in solving this problem. A more compact formulation can be used, which optimizes the lottery ticket selection directly within the ambiguity set of z(x) rather than within the ambiguity set of the random binary indicator u. This refinement significantly reduces the problem's complexity, reducing computational requirements in exchange for efficacy in selecting probabilistically robust set of numbers for tickets. However, it would be useful to avoid relying on such a trade-off. Accordingly, the present methodology provide a better and more efficient approach here by reformulating the momentbased distributionally robust optimization model into a mixed integer model (MIP) model, as set out below.

[0050] Let the (pairwise) similarity measure denote the overlap (quantities of common numbers) between the ithand t tickets, in m tickets purchased by a player. More precisely,

[0051] By exploiting symmetry in the Lotto game, it can be shown that the distributionally robust based optimization model formulation can be reformulated into the following optimization problem :

[0052] For the n-choose-k game, and with m tickets, an optimal solution to the continuous relaxation of the above model can be solved in the following closed form : These observations provide (approximate) structural properties of the optimal configuration in the numbers selected for each ticket. More precisely, a configuration can be found with the following properties:

[0053] 1) The tickets are selected to ensure that the similarity measures rtl> are almost identical (differ by at most 1) for all pairs of tickets, and

[0054] 2) The total similarity measure foral 1Pairs areminimum. This ensures, within the bounds of the numbers to be selected, that there is a substantially uniform distribution of numbers from the n numbers, across all m tickets.

[0055] The search for the optimal configuration can now be executed using a MIP model, where commercial solvers are readily available.

[0056] With the above analysis, the optimal configuration, which can be executed using a MIP model, requires that the discrepancy between the maximum and minimum pairwise similarity measures among these lottery tickets does not surpass 1 for near-identical pairwise similarity. Moreover, to ensure a reasonably uniform distribution of numbers across the m tickets, the total occurrences of each number from the potential n number set can be limited, thus uniformly distributing the numbers for the minimum total similarity. To this end, a MIP model can be constructed to minimize the maximum pairwise similarity between any two tickets, all while adhering to constraints that necessitate achieving the most uniform distribution of numbers possible, as outlined below.

[0057]

[0058] In the above MIP model, (la)-(lb) aligns with the goal of minimizing the maximum pairwise similarity between any two tickets, ensuring a near-identical pairwise similarity measure. Inequalities (lc)-(lf) defines the pair-wise similarity measure using a linearization technique, (1g) imposes the constraint that each ticket must contain k numbers, and the last constraint (lh) is a consequence of the solution with the minimum overall similarity.

[0059] The present MIP model demonstrates superior performance in terms of the probability of matching at least 3 or 4 numbers, between tickets in the m tickets and the winning numbers forming the reference set of numbers. When employed in a quick-pick lottery system, this improvement not only boosts consumer appeal but also aligns with customer expectations regarding the frequency of winning as much as possible.

[0060] The present methodology can be extended to determine the optimal number of tickets, based on the performance of the tickets generated by the present smart quick-pick system (i.e. automated ticket generation in response to requests for tickets), to achieve a more attainable objective: breaking even. This insight challenges the common belief that buying more tickets always increases the likelihood of winning, highlighting the importance of a strategic approach in ticket selection. Thus, the present method may further involve, and the processor 103 of system 102 may further be configured to, recommend purchasing pmaxtickets, where pmaxis the number of tickets with probabilistically highest likelihood of breaking even. For example, the processor may determine a wager quantum corresponding to each of a plurality of candidate requests. The wager quantum is an amount for purchase of the number of tickets in a candidate request - e.g. £2 per ticket, being £6 for three tickets and so on. Thus, the wager quantum is determined by calculating or obtaining a total amount the player has paid or would pay to have their request fulfilled. Each candidate request is a simulated request for a particular number of tickets and is thus, during simulation, a dummy request for p tickets, where p differs between some, and potentially all, candidate requests. During the simulation, we initially generate a substantial number of simulated requests. Each simulated request is, in fact, derived from the corresponding seed tickets based on its quantity p, and then match each request to a simulated reference set numbers to determine the number of matches - the reference numbers may also be obtained from historical data, obtained by enumerating all potential draw numbers, or acquired through any other suitable method. The process then calculates, for each candidate request, a likelihood of matching a predetermined number of numbers of respective ones of the p tickets with numbers of the reference set, and a corresponding payout quantum. The processor then identifies the highest probability candidate request, the highest probability candidate request being the candidate request for which the probability for the corresponding payout quantum being equal to or greater than the wager quantum is the highest - i.e. a candidate request for pmaxtickets, where pmaxtickets has the greatest likelihood of the payout equalling, or exceeding, the wager quantum. The payout quantum is the amount paid out, against the ticket, based on numbers in the ticket matching those in winning numbers of the corresponding simulated draw. A recommendation can then be displayed, by pushing the recommendation to the user device 160 over network 130 via interface 108, that m be selected to correspond to the p tickets of the highest probability candidate request.

[0061] Taking the UK Lotto game as an example, it is demonstrated below that it is optimal to purchase 15 lotto tickets. It is further demonstrated that increasing the size of bets beyond 15 tickets leads to a diminishing return in terms of the break-even probability. This highlights a strategic balance between investment and potential return from a portfolio of Lotto numbers.

[0062] In addition, the experimentation below validates the efficacy of the present MIP number selection model by comparing it to other existing strategies. This comparison is carried out regardless of the specific outcome, enumerated across the vast array of 45,057,474 possible draws.

[0063] In terms of metrics, the proposed methods are compared with the benchmarks using the "Probability of Matching a Minimum Number of Numbers" as the evaluation metric. This aligns with the objective outlined in Cushing and Stewart (2023), where the focus was on achieving a minimum of 2 matching numbers. Present experiments extend this to demonstrate the probabilities of matching at least 2, 3, and 4 numbers, as shown in Figure 2.

[0064] When the objective is to maximize the probability of matching at least 3 or 4 numbers, the MIP model demonstrates superior performance. For instance, in comparison to both the random selection method, the finite geometry-based strategy and the moment-based model, the MIP model exhibits an average improvement of 2.02%, 2.78%, and 0.60%, respectively, in the probability of matching at least 3 numbers. While these improvements may appear modest, they carry significant value when considering the overall odds in lottery games. These findings underscore the substantial advantage of the MIP model over the other three benchmarks in enhancing the likelihood of achieving a moderate win, specifically matching at least 3 or 4 numbers - the numbers selected for a 27- ticket strategy, using finite geometry and MIP, is shown in Table 1. Table 1 : numbers selected for the 27 ticket strategy, using finite geometry and MIP

[0065] In the realm of lottery participation, a common goal is to win the jackpot. However, a more pragmatic approach is to focus on maximizing the probability of recovering the money spent on lottery tickets. This perspective shifts the emphasis from chasing a highly improbable grand prize to achieving a more attainable objective: breaking even.

[0066] By aiming to recover the money spent on lottery tickets, participants can engage in a more financially responsible practice. This approach encourages the purchase of fewer tickets or participating in lotteries with better odds for smaller prizes, which aligns more closely with the principle of risk management.

[0067] Figure 3 presents the outcomes for the likelihood of breaking even, based on the solutions constructed using the MIP Model. This analysis uses the prize structure in the UK Lotto game, with each ticket priced at £2. The findings indicate that the most effective strategy for maximizing the chances of breaking even involves purchasing 15 tickets. This approach yields a break-even probability of approximately 15.88%. Intriguingly, the total expenditure for 15 tickets is exactly equivalent to the prize awarded for matching three numbers, highlighting a strategic balance between investment and potential return.

[0068] In conclusion, the advancements presented herein represent a significant milestone in the field of lottery number selection. The present teachings provide a computationally efficient method for empowering players to select Lotto numbers with an enhanced probability of winning. This method is not just a theoretical construct, but represents a practical tool that can be readily applied in real-world scenarios.

[0069] The proposed method stands out against known ticket number selection strategies for several reasons. Firstly, it surpasses the popular random picks method, which is widely used in many markets. This traditional approach, while simple, does not leverage the potential of mathematical and statistical insights in number selection. Secondly, it outperforms newer finite-geometry based methods.

[0070] The present methodology enables creation of a Smart Lottery Quick-Pick System that fully considers number distribution and manages overlaps for generating ticket numbers, contributing to enhance the probability of winning smaller prizes. By adopting such a Smart Lottery Quick-Pick System, the superior performance can offer more innovative and appealing gaming experiences, thereby strengthening the competitiveness of lottery companies in the market. The result of maximizing the break-even probability, and thus the likelihood that a player will leave the game happy, indicates that implementing the proposed model in a Smart Lottery Quick-Pick System can transform the lottery into a low-cost entertainment option, maintaining the excitement of participation without jeopardizing financial well-being.

[0071] It will be appreciated that many further modifications and permutations of various aspects of the described embodiments are possible. Accordingly, the described aspects are intended to embrace all such alterations, modifications, and variations that fall within the spirit and scope of the appended claims.

[0072] Throughout this specification and the claims which follow, unless the context requires otherwise, the word "comprise", and variations such as "comprises" and "comprising", will be understood to imply the inclusion of a stated integer or step or group of integers or steps but not the exclusion of any other integer or step or group of integers or steps.

[0073] The reference in this specification to any prior publication (or information derived from it), or to any matter which is known, is not, and should not be taken as an acknowledgment or admission or any form of suggestion that that prior publication (or information derived from it) or known matter forms part of the common general knowledge in the field of endeavour to which this specification relates.

Claims

Claims1. A ticket generation system for generating tickets to approximate an unknown reference ticket, comprising: an interface for receiving requests from a user and transmitting tickets to the user; a database; memory; and a processor, the memory storing instructions that, when executed by the processor, cause the system to: receive a request for m tickets via the interface; acquire m seed tickets, each comprising k random numbers from a set of n numbers, generated by employing a mixed-integer programming model, to:- ensure a near-identical pairwise similarity measure among the m tickets; and- minimise total similarity measures for all pairs by uniformly distributing the numbers; generate each ticket of the m tickets from a respective said seed ticket, utilizing a one-to-one number mapping module that replaces the random numbers of the respective seed ticket with other numbers from the set of n numbers, to:- ensure that the generated tickets maintain an identical structure to a structure of the corresponding seed tickets; and- avoid repetition between the tickets and further tickets previously generated for one or more further requests, each further request being a request for m tickets, associate a unique identifier with the request or with each said ticket, each unique identifier being for verifying numbers of the m tickets against a reference set of numbers, the reference set of numbers corresponding to the unknown reference ticket; andoutput the m tickets to the interface and database, each unique identifier to the database.

2. For the ticket generation system of claim 1, wherein m is a number selected by a user from 1 to Mmax, the reference set of numbers is one of a plurality of reference sets of numbers, the ticket generation system being configured to generate all types of seed tickets, wherein each type corresponds to a quantity of seed tickets from 1 to Mmax, where Mmaxis a maximum number of tickets permitted to be generated for comparison to each said reference set of numbers, wherein the request for m tickets is a request for m tickets for comparison to a specific said reference set of numbers and wherein the m seed tickets are acquired by selecting the type of seed tickets with a corresponding quantity of m.

3. The ticket generation system of claim 1 or 2, wherein the database comprises one or more historical reference sets, the processor being configured to replace one or more said random numbers with a respective one or more numbers from the one or more historical reference sets.

4. The ticket generation system of claim 3, wherein the processor is configured to identify the historical reference sets with reference to a predetermined timeframe from current time, and select the respective one or more numbers from the said historical reference sets that were generated within the predetermined timeframe.

5. The ticket generation system of claim 1, wherein the database comprises one or more historical user sets, the processor being configured to replace one or more said random numbers with a respective one or more numbers from the one or more historical user sets.

6. The ticket generation system of claim 5, wherein the processor is configured to select the respective one or more numbers from the one ormore historical user sets, by selecting k numbers that appear most frequently in the historical user sets.

7. The ticket generation system of claim 5 or 6, wherein the processor is configured to identify the historical user sets with reference to a predetermined timeframe from current time, and select the respective one or more numbers from the said historical user sets that were generated with the predetermined timeframe.

8. The ticket generation system of any one of claims 5 to 8, wherein at least one said historical user set comprises manually selected numbers.

9. The ticket generation system of any preceding claim, wherein the mixed integer programming model ensures near-identical pairwise similarity and minimises total similarity for all pairs as follows:where n is a highest possible number in the reference set of numbers, x is a binary vector representing numbers selections for all m tickets, TUIdefines a pairwise similarity measure between an itflticket and an l'thticket.

10. The ticket generation system of any preceding claim, wherein the processor is configured to: determine a wager quantum corresponding to each of a plurality of candidate requests, each candidate request comprising a dummy request for p tickets, where p is different for each candidate request; derive each candidate request from the corresponding seed tickets with quantities p; calculate, for each candidate request, a likelihood of matching a predetermined number of numbers of respective ones of the p tickets with numbers of the reference set, and a corresponding payout quantum; identify a highest probability candidate request, the highest probability candidate request being the candidate request for which a probability of the corresponding payout quantum being equal to or greater than the wager quantum is highest; and display a recommendation that m be selected to correspond to the p tickets of the highest probability candidate request.

11. A method for generating tickets to approximate an unknown reference ticket, comprising: receiving a request for m tickets; acquiring m seed tickets each comprising k random numbers from a set of n numbers, generated by employing a mixed-integer programming model, to:- ensure a near-identical pairwise similarity measure among the m tickets; and- minimise total similarity measures for all pairs by uniformly distributing the numbers; generating each ticket of the m tickets from a respective said seed ticket, utilizing a one-to-one number mapping module that replaces the random numbers of the respective seed ticket with other numbers from the set of n numbers, to:- ensure that the generated tickets maintain an identical structure to the structure of the corresponding seed tickets; and- avoid repetition between the tickets and further tickets previously generated for one or more further requests, each further request being a request for m tickets, associating a unique identifier to the request or to each said ticket, each unique identifier being for verifying numbers of the m tickets against a reference set of numbers, the reference set of numbers corresponding to the unknown reference ticket; and outputting the m tickets to the interface and database, each unique identifier to the database.

12. The method of claim 11, wherein m is a number selected by a user from 1 to Mmax, the reference set of numbers is one of a plurality of reference sets of numbers, the ticket generation system being configured to generate all types of seed tickets, wherein each type corresponds to a quantity of seed tickets from 1 to Mmax, where Mmaxis a maximum number of tickets permitted to be generated for comparison to each said reference set of numbers, wherein the request for m tickets is a request for m tickets for comparison to a specific said reference set of numbers and wherein the m seed tickets are acquired by selecting the type of seed tickets with a corresponding quantity of m .

13. The method of claim 11 or 12, further comprising replacing one or more said random numbers with a respective one or more numbers from one or more historical reference sets.

14. The method of claim 13, wherein further comprising identifying the historical reference sets with reference to a predetermined timeframe from current time, and selecting the respective one or more numbersfrom the said historical reference sets that were generated within the predetermined timeframe.

15. The method of claim 11, further comprising replacing one or more said random numbers with a respective one or more numbers from one or more historical user sets.

16. The method of claim 15, further comprising selecting the respective one or more numbers from the one or more historical user sets, by selecting k numbers that appear most frequently in the historical user sets.

17. The method of claim 15 or 16, further comprising identifying the historical user sets with reference to a predetermined timeframe from current time, and selecting the respective one or more numbers from the said historical user sets that were generated with the predetermined timeframe.

18. The method of any one of claims 15 to 17, wherein the at least one said historical user comprises manually selected numbers.

19. The method of any one of claims 11 to 18, wherein the mixed-integer programming model ensures near-identical pairwise similarity and minimises total similarity for all pairs as follows:where n is a highest possible number in the reference set of numbers, x is a binary vector representing numbers selections for all m tickets, TU, defines a pairwise similarity measure between an ithticket and an i'thticket.

20. The method of any one of claims 11 to 19, further comprising: determining a wager quantum corresponding to each of a plurality of candidate requests, each candidate request comprising a dummy request for p tickets, where p is different for each candidate request; calculating, for each candidate request, a likelihood of matching a predetermined number of numbers of respective ones of the p tickets with numbers of the reference set, and a corresponding payout quantum; deriving each candidate request from the corresponding seed tickets with quantities p; identifying a highest probability candidate request, the highest probability candidate request being the candidate request for which a probability of the corresponding payout quantum being equal to or greater than the wager quantum is highest; anddisplaying a recommendation that m be selected to correspond to the p tickets of the highest probability candidate request.

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