Gambler's Gain #002: Gambling for attention
Welcome to issue #002 of Gambler’s Gain—a weekly newsletter that brings you a selection of news and math related to gambling.
Gambler’s Gain is put together by the team at BtrBet, who are also building tools you can use to bet better. And if it’s more your style, you can find us on Twitter at @btrbetapp.
News
Learning about climate change through gambling
Gambling gets people’s attention. We’ve written before about this being used to increase awareness of the Special Olympics World Games and the WNBA. Now, a new article based on two studies from 2018 and published in Nature Climate Change, has shown that when given the ability to bet on climate-related events in a climate prediction market, bettor’s concern regarding climate change, their support for remedial climate action, and knowledge about climate issues were all positively changed in statistically significant ways, regardless of whether or not the bettor was climate skeptic. Some of the prediction markets that participants bet on were questions like “Would the global warming index reach +1.14°C by October 1, 2018?” and “Will the word pair ‘Climate change’ appear on the front page of the Fox News website in the coming 3 days?”.
In the first of two conducted studies, participants who were identified as “climate believers” placed larger bets on average than “climate skeptics”. This makes sense to us gamblers because we know from experience and from the Kelly Criterion that the fastest way to grow your bankroll is to place a larger wager when you have an edge (i.e., when you are more confident in a favorable outcome).
Since we know that gambling gets people’s attention, and with the results of this study showing that bettor’s opinions on politically polarized topics such as climate change can be impacted by participating in a prediction market, it begs the question as to what other politically polarizing topics can use this teaching mechanism to shift people’s opinions. Take any political line of demarcation, such as NIMBY v YIMBY or “cancel federal student debt” v “don’t cancel federal student debt”, and there is an opportunity to make people rethink their views by participating in a prediction market. As a locally-elected legislator for a village of ~3500 people in Ohio, where most of the political division I see is around municipal spending, I’d like to see more public education on how government budgeting and the legislative process thereof work.
If podcasts are your thing, NPR’s The Indicator from Planet Money has an interview with one of the study’s authors that is worth a listen as they discuss how participating in prediction markets in this way can get people to “bet their beliefs” by demonstrating what they really believe about a topic.
Raising football players, not gamblers
The NCAA has updated its penalties for student-athletes who violate its gambling rules—it’s now a tiered system based on the amount of money wagered. Under the new guidelines, a student-athlete who bets or provides information to bettors about a game that they’re personally involved in will permanently lose their collegiate eligibility. If a student-athlete bets on their own sport, but on a game not involving their school, they can lose eligibility for up to half of one season. For any other gambling activity, the student-athlete will lose some portion of a season of eligibility based on the total amount wagered.
There has always been lots of money surrounding college sports, and now that student-athletes can financially benefit from their name, image, and likeness (NIL), these young adults have even more money surrounding them. Given that the NFL has now suspended 10 players in less than two years — most recently three Indianapolis Colts players and one Tennessee Titans player — for violating its gambling policies, it seems like the NCAA is doing the right thing to encourage student-athletes to behave ethically and conform to the expectations that will be had of them should they go pro. College is a time and place where impressionable young adults find themselves without the support network that they grew up having, and the NCAA is well-positioned to provide guardrails here to help support student-athletes. Remember, you should only ever bet what you are willing and able to lose, and for student-athletes, the cost of even a single bet can be huge if losing eligibility also means losing NIL deals.
Math
Last week we talked about the expected profit or loss from betting on rolling at least one six in N rolls of die using the true odds. We posed the question of if you were making this into a casino game, what odds would you offer?
Typically, a casino’s house odds are set so that they look simpler but give the casino a definite edge. If I were running this game as a casino, I would want to minimize the likelihood of having to pay out a win to a player. Since we know that rolling at least one six in four rolls of fair die is likely to happen more often than not, it would make sense to set the rules of the casino game to offer the player up to a maximum of three rolls. Because the chances of rolling at least one six increase as the number of die rolls increases, the lowest payout should occur on the maximum number of die rolls, and the largest payout should occur on the fewest number of rolls.
Giving even money odds (1:1) on rolling at least one six in three rolls implies that the player expects to break even on such bets. “Breaking even” means that the expected value of such bets is $0. However, we know from our calculation of the true odds that, in reality, they should expect to lose $0.16 for every $1 wagered on such an outcome. The house has a clear advantage using 1:1 odds for rolling a six in at least three rolls of a fair die.
If we offer 2:1 odds for rolling at least one six in two rolls, then the player wins $3 total (a $2 profit plus their original $1 wager) if they roll at least one six in two rolls of the die. The expected value of such odds, however, is (1/3)-(2/3) = (-2/3). This implies that under these house odds, players should expect to lose $0.67 for every dollar wagered. Because the expected loss is larger than the expected loss using the true odds (-$0.39 for every $1 wagered), the house has an advantage using these house odds of 2:1 instead of using the true odds of 25:11.
The true odds of rolling a six in one roll of a fair die are 5:1. As we saw last week, players should expect to lose $0.67 of every dollar wagered using these odds. We don’t need to modify them for our casino game since there is already a house advantage and since they are already simple odds made up of small integers.
Playing with house odds in mind
In general, if the house pays out at odds less than the true odds, the player has an advantage that they can exploit by playing the game many times and letting the law of large numbers work in their favor. To convert our true odds into simpler-looking house odds that preserve the house advantage, all that you have to do is reduce the fractional representation of the odds to a new fraction that is greater than the true odds but has a smaller denominator. Consider our house odds of 2:1 for rolling at least one six in two rolls of a fair die. The true odds of that outcome are 25:11. Since 1/2 > 11/25, our house advantage is retained. If we instead offered 3:1 odds, then we would lose our house advantage since 1/3 < 11/25, and players would have an advantage - not good for our casino.
Casinos do this kind of odds “simplification” for all of their games in order to preserve and enhance their edge. Consider the following table of true and typical house odds for American roulette from Edward Packel’s The Mathematics of Games and Gambling
| Type of bet | True odds | House odds |
|---|---|---|
| Color (red or black) | 20:18 | 1:1 |
| Parity (even or odd) | 20:18 | 1:1 |
| 18 #s (1–19 or 19–36) | 20:18 | 1:1 |
| 12 #s (columns or dozens) | 26:12 | 2:1 |
| 6 #s (any 2 rows) | 32:6 | 5:1 |
| 4 #s (any 4 number square) | 34:4 | 8:1 |
| 3 #s (any row) | 35:3 | 11:1 |
| 2 #s (adjacent) | 36:2 | 17:1 |
| Single #s | 37:1 | 35:1 |
For each of these bets, the house’s odds are greater than the true odds and thus, the house ensures that they have an edge on every wager placed by the player.
Using the house odds of American roulette, every wager has an expected value of -$0.0526, which means that bettors should expect to lose just over $0.05 for every dollar wagered. This is typically reported as a 5.26% house edge. You can check this by calculating the expected value of a bet on black: (18/38)x($1)+(20/38)(-$1) = -2/38x($1) or -$0.0526 or 5.26% Likewise, betting on a single number will yield $35 one time out of 38 spins and -$1 37 times out of 38 spins. This gives an expected value of (1/38)x($35) + (37/38)x($1) = -2/38x($1) or -$0.0526 or 5.26%.
If Dostoevsky’s The Gambler wasn’t enough to convince you of the futility of roulette, then let this math convince you that the smart bettor will avoid the roulette table. Unless, of course, you’re Claude Shannon (the father of the branch of mathematics called Information Theory) or Edward O. Thorp (applied probability pioneer), who built the world’s first wearable computer in the early 1960s to accurately predict the outcome of roulette to great financial success before such devices were made illegal in the 1980s. Fortune’s Formula by William Poundstone is a great read on Shannon and Thorp’s many exploits in putting brains and probability to work in their favor across gambling in casinos and in the stock market.
American roulette has both a green 0 and 00, while European roulette only has a single green 0. What does this change in true odds imply about the expected value of bets placed playing European roulette versus American roulette? Can you calculate the house edge for European roulette using the methodology outlined above?
Next week, we’ll talk about craps and how the “free odds” offered by many casinos allow players to considerably lessen the house’s edge.
Product updates
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