prior probability

Review of Chapter 2 of “Thinking in Bets” by Annie Duke

Chapter 2 of Annie Duke’s beautiful book contains so many powerful ideas that I cannot do it justice in a short blog post, but I will nevertheless try. Among other things, this chapter explores the ways in which our beliefs can disrupt or mess with our ability to make sound decisions. To summarize, our beliefs, even our most cherished ones, are dangerous in two different ways. One is belief formation: we often acquire our beliefs without testing their accuracy or probing their truth value. Worse yet, the other potential danger is inertia; that is, once we acquire a given belief, it becomes next to impossible to revise or modify. Put another way, beliefs tend to become fossilized or frozen in our minds, and this mental inertia is dangerous because it leads to motivated reasoning, or in the eloquent words of Annie Duke (p. 56): “Instead of altering our beliefs to fit new information, we do the opposite, altering our interpretation of that information to fit our beliefs.”

To sum up, motivated reasoning can impede the search for truth and impair our ability to make sensible decisions, but does this mental mischief have a viable solution? Is it really possible to stop motivated reasoning in its tracks or at least ameliorate this temptation? Here is where “Thinking in Bets” (literally!) comes into play. Simply put, Annie Duke’s brilliant solution to the sin of motivated reasoning is to reframe our decisions in probabilistic terms, i.e. convert our decisions into bets. Why wagers? Because bets or wagers encourage us to engage in Bayesian reasoning, to question or reconsider the accuracy of our beliefs, to calibrate or adjust our beliefs based on the available evidence. Ok, but how does the act of placing a simple bet produce such a skeptical or questioning Bayesian attitude? By forcing one to have “skin in the game,” so to speak. After all, a bet costs money, and no one likes to lose money, or as Annie Duke puts it (p. 43), “when you are betting, you have to back up your belief by putting a price on it.” (Or as my colleague and friend Alex Tabarrok is so fond of saying, “a bet is a tax on bullshit.”)

Alas, motivated reasoning is not the only form of mental mischief that deforms our good judgments and destabilizes our ability to make sound decisions. In addition to the sin of motivated reasoning, we must also beware the temptation of self-serving bias, as we shall see in our next post.

Review of Chapter 1 of “Thinking in Bets” by Annie Duke

Chapter 1 of “Thinking in Bets” concludes with this pithy phrase: “… you’ve been betting all along.” These five words not only capture one of Annie Duke’s main ideas; they also sum up why probabilistic thinking is so essential to every aspect of our lives. Simply put, the world is rarely black and white. Why? Because of the problem of incomplete information, which introduces various levels of uncertainty into almost every decision we make. Unlike a game of complete information like chess, where all the pieces are in plain sight and where the rules are well-defined and apply equally to both sides, real life strategic situations (such as business competition, romance, or career choices) are more like the game of poker, a game of incomplete information. In the popular Texas hold’em version of poker, for example, each player is dealt two private cards, while the distribution of the five community cards (such as those pictured below) is entirely probabilistic. (If you have never played poker before, here is a fun video tutorial via pokernews.com.)

As an important aside, there is another big reason why real life is less like chess and more like poker, i.e. a game involving various levels of uncertainty. Whenever we break a rule or attempt to engage in cheating in real life, there is some positive probability that we won’t get caught, or if we do get caught, that we won’t get punished. Sometimes the rules are unclear, or even if they are clear, they might be applied unevenly or unfairly for reasons beyond our control. The point is that the law itself is often highly probabilistic, so a decision to cheat or break the rules is very much like placing a bet. Although the particulars of each of our bets will depend on various factors, such as our level of uncertainty and the stakes involved, the main point Annie Duke is making here is that life is more like poker than chess: whenever we make a decision (big or small), we are in reality placing a bet on the future. Stay tuned; we will proceed to Chapter 2 in our next post …

Below is a small subset of the best books, essays, and online courses that I have been reading or watching during this quarantine period:

1. “Thinking in Bets” by Annie Duke (Portfolio/Penguin, 2019), available here via Amazon. Suffice it to say, I have now added Annie Duke to my pantheon of intellectual heroes. She was not only a PhD student at Penn; she is also a world-class professional poker player as well, and her beautiful book “Thinking in Bets” is one of the best works I have read thus far about the virtues of probabilistic thinking. This tome is so good that I will be writing up and posting on this blog a chapter-by-chapter review in the days to come. (I discovered this book by accident last month while I was working on another project. Thanks Google!)

2. “The Economics of Maps” by Abhishek Nagaraj & Scott Stern (Journal of Economic Perspectives, Vol. 34 (2020), pp. 196-221), available for free here via the American Economics Association (AEA). How much does it cost to make a map? Who pays for these costs, and how are they recovered? Among other things, this paper identifies five economic and institutional factors shaping the data and design choices made by mapmakers. Shout out to that polymath of polymaths, Tyler Cowen, for bringing this beautiful paper to my attention many months ago.

3. “Math on trial: how numbers get used and abused in the courtroom” (pictured below) by Leila Schneps & Coralie Colmez (Basic Books, 2013), available here via Amazon. This beautiful book–which contains ten chapters, each one devoted to a leading criminal or civil case involving the use (or misuse) of probability theory, both by prosecutors and plaintiffs as well as by defense attorneys–is dedicated “to all those who have suffered miscarriages of justices, and to all victims of crimes whose perpetrators went unpunished, due to the misuse or misunderstanding of [probability theory] in the legal process.” In other words, judges must avoid false positives (punishing the innocent) and false negatives (allowing the guilty to go unpunished). Among those cases dissected in this tome are the infamous Dreyfus Affair (a tragic example of a false positive) and the Amanda Knox trial (a false negative). Shout out to Professor Karl Schmedders for bringing this excellent book to my attention.

In addition to these readings, I have also enrolled in Professor Charles Fried’s online course on “Contract Law: From Trust to Promise to Contract” (via edX) to compare notes, so to speak. Professor Fried, who teaches contracts at Harvard Law School and who I had the pleasure of meeting in March of 2011 at Suffolk Law School, is one of my academic role models and intellectual heroes.

In addition to the six philosophical “houses” pictured below, we recently posted this fun example (involving literary figures) of conflict-prone groups.

hat tip: @Philosophymeme0

As you may have heard by now, there are more cases of coronavirus in the USA than anywhere else in the world. But with apologies to the late Darrell Huff, most coronavirus tables and charts I have seen thus far tend to focus on the absolute or total number of infections or deaths per country or region, instead of adjusting these dreadful data for population, like the chart below does. Notice how, once we adjust for population size, the USA is not (yet) the most lethal location for this virus.

Finally, a coronavirus chart that adjusts the data for population!

John Conway (pictured below) is one of my mathematical heroes, along with Martin Gardner and John Nash. Among other things, Conway invented a simple zero-player game called “The Game of Life,” for which he will always be remembered by generations of geeks like me. Sadly, Conway died on April 11 from the coronavirus. Here is a beautiful tribute by my colleague and friend Steven Landsburg.

Updated 4/16: Hey, where is my mask? In the mail? If the Mayor of Los Angeles or the Governor of New York is going to make the wearing of masks compulsory (see here and here), then shouldn’t City Hall or Albany be able provide masks to anyone who needs one (like they do in Singapore)?