Tag Archives: Mathematics

Choice Architecture: Even in “Heads or Tails,” It Matters What’s Presented First

If you’re familiar with behavioural economics, then the results of this study will be right up your alley.

The researchers set out to determine whether there was a “first-toss Heads bias.” Meaning, when flipping a coin and the choices are presented “Heads or Tails,” there would be a bias towards people guessing “Heads” (because it was presented first). Through running their tests, they found something else that surprised them [Emphasis Added]:

Because of stable linguistic conventions, we expected Heads to be a more popular first toss than Tails regardless of superficial task particulars, which are transient and probably not even long retained. We were wrong: Those very particulars carried the day. Once the response format or verbal instructions put Tails before Heads, a first-toss Tails bias ensued.

Even in something as simple as flipping a coin, something where the script “Heads or Tails” is firmly engrained in our heads, researchers discovered that by simply switching the order of the choices, the frequency with which people chose one option or the other changed. That’s rather incredible and possibly has implications from policy to polling. However:

There is, of course, no reason to expect that, in normal binary choices, biases would be as large as those we found. In choosing whether to start a sequence of coin tosses with Heads or Tails, people ostensibly attach no importance to the choice and therefore supposedly do not monitor or control it. Since System 1 mental processes (that are intuitive and automatic) bring Heads to mind before Tails, and since there is no reason for System 2 processes (which are deliberative and thoughtful; see, e.g., Kahneman & Frederick, 2002) to interfere with whatever first comes to mind, many respondents start their mental sequence with Heads. However, in real-life questions people often have preferences, even strong ones, for one answer over another; the stronger the preference, the weaker the bias. A direct generalization from Miller and Krosnick (1998) suggests that in choices such as making a first-toss prediction, where there would seem to be no good intrinsic reason to guide the choice, order biases are likely to be more marked than in voting. At the magnitude of bias we found, marked indeed it was. Miller and Krosnick noted with respect to their much smaller bias that “the magnitude of name-order effects observed here suggests that they have probably done little to undermine the democratic process in contemporary America” (pp. 291–292). However, in some contexts, even small biases can sometimes matter, and in less important contexts, sheer bias magnitude may endow it with importance.

OK, so maybe these results don’t add too much to “government nudges,” but it can — at a minimum — give you a slight advantage (over the long haul) when deciding things by flipping coins with your friends. How?

Well, assuming that you are the one doing the flipping, you can say to your friend: “Tails or Heads?” (or “Heads or Tails?”) and then be sure to start the coin with the opposite side of what your friend said, facing up. A few years ago, Stanford math professor Persi Diaconis showed that the side facing up before being flipped is slightly more likely to be the side that lands facing up.

ResearchBlogging.orgBar-Hillel M, Peer E, & Acquisti A (2014). “Heads or tails?”–a reachability bias in binary choice. Journal of experimental psychology. Learning, memory, and cognition, 40 (6), 1656-63 PMID: 24773285

Wanna Make a Name for Yourself: Answer One of These Questions

In The Guardian today, there’s an article that lists “20 big questions in science.” If you want to be famous (at least in some circles), answer one of the questions. Of course, there are some ‘answers’ to the questions already. Or maybe it’d be more accurate to say that there are some hypotheses or that there is some ‘general knowledge’ in the domain of the question. However, there don’t seem to be any definitive answers, yet.

Here are the questions with a few thoughts after some of them:

1. What is the universe made of?

2. How did life begin?

3. Are we alone in the universe?

If pressed to give an answer on number three, I’d probably say something to the effect of: given how big the universe is, mathematically speaking, isn’t it more likely that there is other life out there somewhere than isn’t?

4. What makes us human?

5. What is consciousness?

On number five, I remember reading a very intriguing article in The Atlantic this past winter that explored the question: what does it mean to be conscious? It approached this question in the context of anesthesia. If this question interests you, this is one way to delve into the topic.

6. Why do we dream?

While there are many theories on why we dream, one of my favorite ways for interpreting dreams is through Jeremy Taylor’s method. This method also outside the context of dreaming.

7. Why is there stuff?

8. Are there other universes?

9. Where do we put all the carbon?

10. How do we get more energy from the sun?

Number ten, while also making you famous, would likely also make you extremely wealthy unless you went the route of Jonas Salk and polio.

11. What’s so weird about prime numbers?

12. How do we beat bacteria?

13. Can computers keep getting faster?

14. Will we ever cure cancer?

15. When can I have a robot butler?

16. What’s at the bottom of the ocean?

On number sixteen: when you realize that 95% of the ocean is unexplored, it sort of gets you curious about what might be down there. More than that, 99% of the Earth is water. There’s a lot we don’t know about the planet we inhabit.

17. What’s at the bottom of a black hole?

18. Can we live for ever?

19. How do we solve the population problem?

20. Is time travel possible?

On number twenty: if this turns out to be true, that would make for some interesting ethical and moral dilemmas.

Every Game Counts The Same: Does It Really?

In most sports, there is a “regular” season and a “post” season. That is, the teams play against it each other for a set number of games to jockey for position in the playoffs. As I write this, I’m thinking about in particular, as it is getting very near to the end of their season. As the season comes to a close, many teams are either jockeying for position in the playoffs or they are struggling to remain one of the teams that will get to play in the playoffs.

I was having a conversation with someone the other day about the relative importance of each game, ie. “every game counts.” Some people like to say that games at the end of the season “count more” than games at the beginning of the season. They’ll tell you quite a fancy story about how and why the games at the end mean more to a team than the games at the beginning of the season. And I want to believe them. I want to believe that there’s a formula that accounts for “time” in the relative importance of games. To my knowledge, there isn’t and a game won in the beginning of the season is equal to a game won at the end of the season.

Looking at it mathematically: there are 162 games in a season. So, every game is worth 1/162nd of a team’s record. If a team wins a game on May 6th, that game is worth 1/162nd of that team’s record. If a team loses on June 12th, that game is still worth 1/162nd of that team’s record. And if a team wins the last game of the season (!) that game is still worth 1/162nd of that team’s record.

I think where a lot of people get confused or misled when it comes to games at the end of the season meaning more is because of the cultural bias. It is often written of and spoke of that games at the end of the season mean more than games at the beginning of the season. As a result, people begin to believe this and say it themselves (creating a bit of an ). At the end of the day (literally), the last game of the season has the same weight on a team’s record as a game at the beginning of the season.

Note 1: this line of thinking doesn’t apply to those sports that use a more sophisticated way of measuring the success of their teams. For instance, some sports, like soccer, often use “goal differential” as a way of distinguishing the relative placement of their teams.

Note 2: for sports that have such relatively “short” seasons like the NFL, one could argue that a game later in the season is worth more because of the various tiebreakers that are used for Winning percentage, etc., but the sentiment of every game counting the same still holds.