It took almost a year of pestering, but I finally got my best friend to take me to the neighborhood bar in the town he grew up in. As any good dive bar should, it had a bunch of Keno screens, and I am not surprisingly intrigued by animated numbers, so he asked me if I wanted to play. Like any good economist, I had a suspicion that it was a bad bet, but I was willing to spend some money for the entertainment value. To make a long story short, I didn’t win anything (shocking), but I obviously had to come home and figure out exactly how bad a bet I had made and how I could have done better (read, less badly).
For those of you not familiar with Keno, it’s basically a lottery run every 3 minutes. However, there are two extra features. First, 20 (out of numbers 1-80) numbers are drawn, but you have the option of picking from 1-12 numbers to try to match. Second, there is a bonus option to potentially multiply your payoff by doubling your initial bet from $1 to $2. The Keno card looks like this: (I took the liberty of showing you the filled-out version. I chose the numbers that my parents play in the Florida lotto each week. Don’t get me started on that…)
The payoffs are as follows:
(This doesn’t even account for the times where people win and are too drunk to remember to cash in their tickets.) Note that the Keno card authors were even nice enough to even give some of the relevant odds. Unfortunately, it’s not really enough to tell whether this is a bad bet. Given that it wouldn’t be profitable for the Keno people to offer a good bet, it’s fair to assume that none of the options are good bets, but how bad a bet is this? So of course I posed this question to Excel. First, the probabilities of matching numbers are the following: (click for full size):
(That one cell is in red because it’s off from the Keno card a tiny bit. Who do I have to call about this?) The cell in yellow is highlighted because choosing 2 numbers gives you the best odds of winning anything at all, but this doesn’t take into account differences in the amounts you can win. So, given these probabilities, what are the expected returns you get for each choice of numbers to pick?
Well, those expected returns are less than stellar. (What, you expected them to be positive?) Again, the least bad is to pick 5 numbers. This of course doesn’t account for any risk aversion or diminishing marginal utility of money. Maybe we can do better by adding on the bonus option?
Yeah, I didn’t think so. I’m actually surprised that the bonus option wasn’t significantly worse than the regular option (and can actually be a little better), since I think of it as insurance in blackjack, which is a notoriously bad bet.
So, my overall conclusions:
1. Keno is a terrible bet. The expected return on your investment hovers around -30%. Go play blackjack instead.
2. The odds on the pick 3 are off on the last digit compared to what is on the Keno card.
3. The pick 2 has the best odds of winning anything at all.
4. The best you can do in terms of return percentage is to pick 5 and do the bonus. But the return percentages aren’t that different regardless of what you do.
5. The pick 5, despite having the best expected return, has unusually unfavorable odds for winning anything at all.
6. Despite what I originally guessed, the pick 1 is a terrible choice.
7. I am a huge nerd.
8. See point 1.
If you want to play around with this analysis, you can find the file here. So why did I do all this? (besides the obvious fact that I get my jollies from playing with Excel) I think I’ve mentioned before that Nudge by Richard Thaler and Cass Sunstein is one of my favorite general interest economics books. One of the points that the authors make is that there are some markets (eg. extended warranties) where competition doesn’t lead to a favorable outcome for consumers. They argue that extended warranties are overpriced, but there is no incentive for the companies offering them to disclose this to potential customers. Furthermore, it is difficult (due to the fact that extended warranties are usually purchased at point of sale) for third parties to compete for this business. Theoretically, I should be able to profit by setting up a stand outside Best Buy and offering to educate paying customers on the unprofitability of extended warranties. I am now somehow picturing the therapy booth that Lucy sets up in Peanuts:
Hm. I do have a purple dress…but instead, I’ll just spout off here and ask you to donate or buy some stickers. 🙂 In fairness, the Keno bet is actually less bad than a traditional lottery, since in most state lotteries only about 50 percent of the money taken in is paid out to players. (Betcha you didn’t know that before!) This would imply that your expected return is in the neighborhood of -50% with the lotto rather than -30% with Keno.
Given these terrible numbers, why do people take these bets? Some economists would just say that these people are risk loving, or alternatively that they have increasing marginal utility of money. However, in order to be making a proper decision, people have to know what the expected return on their investment is. If they can’t tell you, then if they are making a good choice then it must be purely by accident. Furthermore, I highly doubt that most gamblers go through the exercise that I did above before making a bet. Other economists argue that lottery players are myopic, or short-sighted, and they act in risk-loving ways when it involves small-scale investment even though when posed with a collection of similar bets they would turn the option down. Coming back to the parents example, they are more than happy to spend a dollar a week on a “bad” bet (and in fact get super irritated if they forget to buy a ticket, since they are convinced* that their numbers will come up the one week they don’t play), and have done so for about 15 years now, but I doubt they would take me up on a lotto style bet where the buy in was $780…even though that is what they have spent in total playing the lotto. (I imagine that they must have won something over the years, but I don’t know how much.) Turns out a lot of people think the same way when it comes to gambling, especially since, at these stakes, the alternative is to buy one extra latte a month instead.
* My parents are very smart people, and I would probably play the lotto too, if for the anticipation of seeing whether I win if nothing else. Then again, I’m probably also too lazy to actually go and buy the tickets. Luckily, my parents said they would split it with me if they ever won.