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The end of Lotto 6/49 as we know it

Sun 4 Sep 2022 by mskala Tags used: , ,

The rules of Lotto 6/49 are about to change in a way that I think is pretty significant. It's drastic enough that I think we might well say Lotto 6/49 is coming to an end, being replaced by a new game that is not even properly a lotto game anymore and only happens to reuse the name.

Here's a cartoon I found in Wikimedia Commons of "The Chance Seller of the Exchequer putting an Extinguisher on Lotteries," which although referring to a different historical event is kind of like what's happening now.

I've never really been a Lotto 6/49 player; I think I've bought one or two tickets to it in my life. My family growing up (i.e. my parents, sale of tickets to children being illegal) didn't buy lotto tickets as far as I know, and as an adult I seldom do. One exception was when COVID-19 vaccines first became available and there was a lot of talk about blood clot risk from the earliest ones. I decided to buy lottery tickets such that I would have a greater appropriately discounted risk of winning a major prize than of experiencing a serious reaction to the shot.

I did the math and figured achieving that goal would necessitate spending about $60 on lottery tickets - which was interesting in itself, and I think a worthwhile exercise in understanding risk even if one didn't actually buy the tickets. I was surprised that the price tag turned out to be so high, indicating that the vaccine risk was actually higher than I'd imagined going into the calculation. My $60 number was based on playing the Ontario-only "Lottario" game; Lotto 6/49 has less favourable odds, and larger therefore less likely prizes, so it might've cost more like $100, both numbers being approximate and depending on assumptions about the utility curve. In fact, I did buy the tickets and get the vaccine and I had neither a major win nor a serious reaction.

But even for a non-player, Lotto 6/49 was part of the culture in which I grew up. When I heard adults talk about playing "the lotto" it was always understood that 6/49 was the one they meant. Lottery corporations across Canada offer an elaborate menu of products that I'd basically never heard of until I started looking at their Web sites; Lotto Max, introduced 2009, has started to become popular too; but 6/49, introduced in 1982 with a few other significant rule changes between then and now, is still the one all Canadians know about and very many routinely play.

I think there was a survey indicating that some surprisingly large percentage of people in Canada can correctly quote the probability of winning the 6/49 jackpot under the obvious assumptions as "about one in 14 million," despite very few having the necessary combinatorics knowledge to compute it exactly as one in 49 choose 6, that is 13,983,816. When there've been widely publicized situations of "lottery fever" in Canada, with large numbers of people across the country suddenly buying many tickets to chase an unusually large jackpot, they've usually centred on Lotto 6/49. A major change to something so deeply embedded in my culture, especially a change that feels like actually killing it and re-using the name for something else, then feels like a big deal.

As I've written before, there ought to be a moratorium on making unnecessary big changes during the pandemic years. We have a serious global emergency to deal with at the moment. We should not be adding extra bullshit on top of the existing emergency by making breaking changes to things that we don't need to change right now - especially not changes that really shouldn't be made at all, but also not even changes that maybe should be made eventually, but have no urgent reason to be changed right now. There have been far too many renamings, redesigns, cancellations, launches, and so on during the last couple of years. We all should give it a rest.

When the house is on fire is the last time to start retiling the bathroom. Even if you are sure you want to retile the bathroom some day pretty soon, fighting the fire should entail hitting "pause" on non-emergency projects, and especially it should mean not starting new ones.

But I appear to be alone in thinking that. People and institutions in my world seem to think 2020 to 2022 is the perfect time to make as many poorly thought-out major breaking changes to things that didn't need to change at all as they possibly can. New rules for Lotto 6/49 look like part of that trend and there may not be any deeper significance beyond the general itch to break things and create extra problems for ourselves during the pandemic. The desire to fix what ain't broke right now seems to be like a contagious and airborne disease in itself.

Nonetheless, I'm sure that plans for the new 6/49 rules were already being made before the pandemic (because they would've taken a long time to prepare), and I'd love to know what the reasoning behind the change actually was. We will never get a straight answer to that from the Interprovincial Lottery Corporation. Their public communication tries to spin the changes as better for players, but of course they would say that in any case and on general principles it should be presumed to be a lie. They're certainly changing the rules to better achieve their own objectives first and foremost - but what are those objectives?

Maybe we can learn about those objectives by looking closely at the changes to 6/49, especially in context with other major Canadian lotteries. The basic change is that whereas Old 6/49 was fundamentally a lotto game with a small raffle add-on, New 6/49 will be fundamentally a raffle game with a small lotto add-on, inverting the structure.

By Old 6/49 I mean the rules currently in force; there were some significant changes (including the addition of the $1 million raffle prize) in 2013, and a switch to a computerized random number generator from ball-drawing machines in 2019, but it has always been primarily a lotto game since the name Lotto 6/49 was introduced in 1982, so at that level the switch away from being primarily a lotto game at all with what I call New 6/49 starting with the September 14 draw, is the biggest rule change in Lotto 6/49 history. I think it's a big enough change to describe this as ending Lotto 6/49 entirely and starting a new game that just happens to re-use the name and is not even actually a lotto game.

What do I mean by lotto and raffle games? This difference, which I'm making a technical definition but I think accurately reflects the casual usage of these words, has to do with how the draw is conducted. In what I consider a true lotto game, there is a draw of one element uniformly at random from a largish but fixed-size set of possibilities. Usually, not always, in math terms they are selecting a fixed-size subset of size k of the the integers 1 to n for some fixed k and n - like the "six numbers out of 49" main draw that gives 6/49 its name. Players bet on one or more of those combinations. Depending on the game, they may or may not get to choose which numbers to bet on.

There may be some variation in the draw structure. For instance, 6/49 also has a "bonus number" drawn from the 43 remaining after the main draw, and some other lotteries, like the well-known Powerball sold in the USA, are actually drawing from what mathematicians might call the cross product of more than one set of combinations. But the critical and defining feature of a lotto game is that the draw is uniform from an abstractly defined fixed-size set of possibilities.

In a lotto game it's entirely possible that two or more players could end up betting the same numbers. If those numbers win, then there's a question of who gets the prize. For the top jackpot prize in a lotto game it's pretty much universal that when there are two or more winners they split the prize. Smaller prizes, where it is expected that there are usually many winners per draw, might be fixed-size for every winner without sharing, or a fixed amount or a percentage of a pool shared among all winners; with 6/49 in particular, this aspect has changed over the years. It's also possible, and in most games it's a frequent occurrance for the top prize, that there can be no winner at all in a given draw. That's what gives rise to growing jackpots in lotto games: traditionally, prizes not won in a given draw go into a fund that is carried over to the next draw, until eventually there's a winner. The idea of there being a number of top winners other than exactly one, is an essential property of lotto games, a consequence of the fact that the draw is a random selection over numbers rather than over tickets.

Not essential, but a frequent feature of the regulatory landscape, is that there will be a fixed percentage of ticket sales that goes into a prize fund. The prizes are paid out of the prize fund and the prize fund can only be used for paying prizes. I think that's a legal requirement in Canada; it certainly is in many places, and it may be done voluntarily even where not required. The point is to remove any financial interest the lottery operators might have in people not winning; if a prize isn't won or isn't claimed, then the operators cannot pocket it but have to award it to players in some other way (such as by offering extra prizes, or just throwing it back into a growing jackpot). Since the prize fund cannot be used for anything except prizes, the operators have no financial reason to care who wins, or whether anyone wins. That is supposed to increase public confidence, and the real likelihood, that the operators will run the game honestly.

What I call a raffle game differs from a lotto game in that the draw is from the variable number of tickets actually sold, instead of from an abstract distribution. (Minor variations, not relevant here, can be applied for special multi-chance tickets.) There will be exactly one ticket drawn for each prize. That guarantees exactly one ticket will win - eliminating unawarded prizes to carry over (the case of zero winning tickets - though issues like a prize being won and not claimed may still occur) and split prizes (two or more winning tickets).

The lottery in the famous short story The Lottery was not a lotto draw but a raffle draw - or properly a series of three raffle draws, giving uniform distributions over families, households within a family, and individuals within a household, rather than uniform over individuals. An individual in a large household or large family would have a smaller risk of winning.

In a lotto game, the probability of winning is fixed by the abstract distribution of possible draws. Although the amount won may vary because of split prizes, and the risk of that depends on ticket sales and concievably on the popularity of specific numbers and combinations among those players who choose their own numbers, players can primarily see themselves as playing against the draw with relatively little connection to other players' behaviour. In a raffle game, the probability of winning is determined by ticket sales and will vary from one draw to the next; players are more clearly playing against each other's behaviour. When there are many tickets sold, the chance of winning is smaller. On the other hand, the risk of splitting a prize (and the trouble that may cause for both players and game operators) is eliminated. In a sense the split-prize issue is still significant with a raffle but it appears as a smaller chance of winning rather than a possibly-smaller prize given winning.

Old 6/49 was primarily a lotto game: most of the prize money, including the growing jackpot, was awarded through matches and partial matches to the six out of 49 lotto draw. It did also have a raffle component since 2013, the so-called "guaranteed" prize of one million dollars awarded to one ticket every draw; but the large majority of the prize money would go out through the lotto draw, either in the main jackpot or smaller prizes. The top prize in Old 6/49 was for matching all six numbers of the lotto draw, and it was a conventional lotto-style jackpot starting at $5 million, split if won by more than one ticket, and carried over to the next draw if not won by any ticket, with the amount uniquely determined by the rules and potentially growing without limit in time or amount.

New 6/49 still has the six out of 49 traditional lotto draw, but that no longer awards the main prize. Instead, the prize for matching all six numbers of the lotto draw is fixed at $5 million every draw - still subject to splitting if won by more than one ticket. The "guaranteed," that is, raffle, draw is now the sole means of winning the largest and growing (i.e. "jackpot") prize, and the rules for the amount of that prize are a bit complicated and differ from those of a traditional lottery jackpot.

The operators are going to conduct a separate draw of what is thought of as one ball in a pool that starts with 30 balls, though of course they are not really physical balls but RNG possibilities. One of those "balls" corresponds to the growing jackpot; if they draw that ball than the raffle winner gets the big prize. If not, then the raffle winner gets $1 million and they remove the drawn ball from the prize-size pool for the next draw, increasing the chance of drawing the jackpot ball in the future. They remove a ball each time the jackpot is not won. In theory, the prize-size ball pool could eventually shrink to just one ball (the jackpot ball), in which case the jackpot is certain to be awarded in that draw. That would be statistically expected to happen about once every nine years. Whenever the jackpot is awarded, they reset the prize amount and the pool size to 30 balls for the next draw.

Wikipedia says that the New 6/49 jackpot prize will start at $10 million and grow by an additional $2 million each time it is not won; but their source for that claim is a newpaper article that attributes it vaguely to the lottery corporation's Web site, and I haven't been able to find a description of it that seems to be legally binding. What the official rules of the game actually say is only that the jackpot must start at some amount and grow by some amount, with those amounts being at the discretion of the Interprovincial Lottery Corporation. If we believe the $10/$2 million formula, then the maximum possible New 6/49 jackpot is $68 million. The new rules guarantee that a New 6/49 jackpot will be awarded at least once every 30 draws (about 3.5 months at two draws per week) and on average once every 15 draws.

If the starting and growth amounts of the jackpot prize are fixed by the $10/$2 million formula, then the amount of money paid out in major prizes basically does not depend on ticket sales anymore in New 6/49 at all. It is determined solely by the random number generator. Some smaller prizes are still tied to percentages of ticket sales through a complicated formula described in the rules. The input to what the rules call the "Draw Fund," from which the large prizes are paid, comes from a percentage of sales and is therefore variable. So we can assume that the authors of the new rules reasonably expect ticket sales will be enough to comfortably exceed the fixed payments committed to be made from the Draw Fund. Then extra money will tend to build up in the Draw Fund, and it will necessarily be awarded to players through additional draws at the discretion of the Interprovincial Lottery Corporation, because it is not allowed to be used any other way. Those will almost certainly be additional raffle-type draws, not harder-to-control additions to the lotto-type draw.

So, what are the consequences of these changes for the players and operator of New 6/49? As I've said, it represents a shift of a large fraction of the money away from lotto prizes and into raffle prizes. To the extent that there is any strategic element to playing lotto, that is very much reduced in the new game; there is almost no strategy involved on the raffle side. Some of what might be seen as undesirable features of lotto from the operator's point of view are reduced by shifting money away from the lotto side of the game. The new rules invoke "discretion" very much more than the old rules did; a whole lot of the new game is just left up to the Interprovincial Lottery Corporation to decide instead of being spelled out in the rules. And the new rules seem especially to be aimed at eliminating unlimited-size jackpots.

There isn't much strategy in playing lotto, but there is some. First of all, because the prize fund comes from a fixed percentage of ticket sales, on average over a large enough sample, players get back in winnings exactly that percentage of what they spend on their tickets. Under important assumptions it can be said that it doesn't matter which numbers you play. But if you believe in "lucky" numbers, or see problems with the assumptions that lead to considering large-scale average behaviour, then the opportunity to benefit from choosing which numbers to play is significantly decreased under the new rules, where very much of the total prize money, and especially the largest single prize, are moved into the raffle draw. That's a big disappointment to anyone who wanted to apply strategy to Lotto 6/49.

Even if you don't believe in lucky numbers you may believe in unpopular numbers: given that every combination is equally likely, but people are human and Lotto 6/49 players are allowed to choose which numbers to play, you would be foolish to choose for instance {1,2,3,4,5,6} because you can reasonably guess that many people will have chosen that particular combination and if it were drawn, the jackpot would be split many ways. Such things have happened in some famous incidents with other lotteries, for instance when a number combination printed in fortune cookies won the Powerball second prize in 2005, causing 110 winners to split a prize that usually would only have gone to about four or five winners. Such incidents can be a problem both for players and the game operators, and moving money to a raffle draw removes it from the risk of being paid out under such circumstances. Players no longer have the risk of splitting jackpots, but they also lose the opportunity to feel like they are getting an edge through their efforts to avoid splitting jackpots.

If you are buying multiple tickets, there can be strategy involved in the number choices despite all numbers being equally likely, and despite that the average winnings are a fixed percentage. Suppose you want to buy seven 6/49 tickets. The chance of winning some prize per ticket, usually the very smallest prize of a free ticket, is a little better than one in seven. So a very common outcome from buying seven tickets is that exactly one of them will win a small prize. Seven tickets cost $21. If you win one of the smallest prizes, just once, you're likely getting a single free ticket, or if you win money at all, it'll probably be $5. Winning $5 on a ticket combination that cost $21 is probably not going to make you very happy - it feels like losing $16. What if you want to reduce the possibility of small win? You cannot increase your average winnings for a fixed amount spent on tickets, but what if you could trade off a reduced chance of winning at all, in favour of getting a larger payoff in case of winning at all?

You can actually do it! If you buy seven tickets with numbers selected uniformly at random, it is likely they will not have many numbers in common with each other. If you win money at all, it will probably happen because just one of your tickets matched two of the six numbers plus the bonus number in the lotto draw, and then you get $5. But what if you bought seven tickets with exactly the same numbers? Then they all win, or none of them win. The average winnings are the same, but the smallest possible cash prize is $35 for the case of all tickets winning; increasing the minimum prize is balanced by smaller overall chance of winning at all, compared to non-overlapping tickets.

You would not want to really buy seven identical tickets because then you would split with yourself in case of winning the largest lotto prize. But you could buy a set of seven tickets that maximized the overlap between tickets subject to not actually having any two identical. Such a set of tickets might look like {1,2,3,4,5,6}, {1,2,3,4,5,7}, {1,2,3,4,6,7}, and so on. You choose seven numbers instead of six, and then you leave one of those numbers off of each ticket, taking the other six, to form your set of seven tickets with six numbers each and a lot of overlap between them. With that set of tickets, which costs $21, the smallest possible cash payout is $20 plus a free play (equivalent to another $3 ticket), which feels much less like an overall loss than the "winning just $5 for $21" scenario.

This kind of strategy, designed to stack up small prizes to trade off smaller overall chance of winning in favour of larger total payouts at the low end of spectrum, is called a "lottery wheel." It is popular enough that Lotto 6/49 actually offers ready-made multi-play tickets that automatically form wheels without the player needing to manually figure out the overlapping ticket arrangement, for both the seven-ticket case I described and some larger cases.

You can still play a lottery wheel in New 6/49 but the benefit of doing so is reduced because less of the prize money is available through the lotto draw. With the main prize no longer subject to splitting (which is the biggest reason for not buying identical tickets), then when the raffle jackpot is large it might actually be better to just buy however many identical New 6/49 tickets.

Something else that attracts lottery players is the possibility of very large jackpots, and I think it's noteworthy that Old 6/49 was the last Canadian game to potentially offer very large jackpots, and the last nationally-offered game with jackpot growth specified firmly by the rules. New 6/49 officially has its jackpot set "at discretion" of the operator, as a change from Old 6/49. Lotto Max, usually the biggest-money game in Canada, has always had its jackpot at the operator's discretion, but with a fixed maximum specified in the rules. That maximum is currently $70 million (less in the past). It is suggestive that the $10/$2 million formula for New 6/49 jackpot growth results in a maximum of $68 million; that is slightly more than the previous record jackpot for 6/49 (roughly $64 million), but it's also just slightly less than the Lotto Max maximum. It seems pretty clear to me that for whatever reason, $70 million is the magic number. Someone wants to make sure that Canadians can't be allowed to win more than about $70 million in a lottery. They introduced Lotto Max with the then-new maximum jackpot limit to draw large-prize attention away from Lotto 6/49, and now they're nerfing Lotto 6/49 as well with a rule change that creates a maximum jackpot limit.

The powers that be also don't want more lotto-fever outbreaks, especially not those in which which the jackpot grows for an unknown arbitrarily long period. The new rules with the shrinking prize-size ball pool ensure that there will always be a firm deadline for when the pool will run out and the jackpot will necessarily be awarded.

For completeness I should mention that Daily Grand is a nationally-offered game with its top prize fully specified by the rules and not "at discretion"; but its top prize is fixed at $7 million and does not grow at all. Some provincial games, at least including Lottario but probably also others, have top prize amounts specified by the rules instead of "at discretion," and able to grow without theoretical limit. But although I don't know the largest jackpot ever awarded by Lottario, it starts at just $250,000, in practice it maxes out at about $2 million, and it cannot realistically be expected to ever approach the Lotto 6/49 and Lotto Max scales of jackpot size. Other provincial games fall into similar categories: fixed top prizes that do not grow; top prize "at discretion"; or numbers small enough that a theoretically unlimited jackpot will never really be expected to compete with the national games.

In summary, I don't know what the goal of the 6/49 rules change really is, but I'm sure it does not make the game better for players. I'm sure it is meant to make the game better for the purposes of the operators and whoever gives them their orders. Having a limit of $70 million on Canadian lottery prizes in general seems to be a matter of somebody's policy. It's also notable that Canadian lotteries are moving much more toward being subject to "discretion" rather than having rules that firmly constrain the operators' actions.

From the point of view of the player, anyone who wants to play lotto as such and wants to apply some kind of strategy, will find the new Lotto 6/49 rules less appealing. The shift from being primarily a lotto game to primarily a raffle game looks kind of like nerfing 6/49. It is a big enough shift that I'm dubious about even naming the new game "Lotto" anymore. It is the biggest rule change to Lotto 6/49 since its 1982 inception, and given the deep embedding of Lotto 6/49 in Canada's culture, it feels like a cultural shift. I'm not happy about cultural shifts in general, especially not during the pandemic years.

Although I was never really a fan of Lotto 6/49 or lotteries in general, I probably will buy a ticket for the last draw under the current rules - which will be made on September 10, 2022.


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