prior probability

Having now finished the first half of Cheryl Misak’s intellectual biography of Frank Ramsey (“A Sheer Excess of Powers”), I am especially struck by what Ramsey (pictured below) accomplished before reaching his 19th birthday. In particular, in the space of one month (January of 1922), Ramsey published a devastating critique of left-wing “social credit” proposals, akin to the “universal basic income” schemes of our times; he published a detailed review of John Maynard Keynes’s Treatise on Probability, a review that demolished Keynes’s approach to probability; and he not only translated Ludwig Wittgenstein’s now-famous Tractatus–which is considered to be one of the most important works of modern philosophy–his translation was approved by the demanding and ornery Wittgenstein himself!

Over a century ago, the legal giant Oliver Wendell Holmes invited us to look at the law through the lens of probability theory, or in Holmes’s own immortal words: “The prophecies of what the courts will do in fact, and nothing more pretentious, are what I mean by the law.” Ironically, few legal scholars have taken up Holmes’s intriguing invitation. During the last ten years (2011-2020), however, I authored the following papers in which I applied Bayesian probability to various aspects of law and culture:

1. A Bayesian Model of the Litigation Game (2011), in which I develop a Bayesian model of litigation.
2. Visualizing Probabilistic Proof (2013), in which I use Bayesian methods to solve the “blue bus problem” in evidence law.
3. Finding Santiago (2015), in which I focus on Hemingway’s hero in “The Old Man and the Sea” and explore the inner workings of the old man’s mind through a probabilistic or Bayesian lens.
4. Judge Hercules or Judge Bayes? (2016), in which I use Bayesian methods to solve Newcomb’s Problem.
5. Probabilistic Interpretation II: The Case of the Speluncean Explorers (2017), in which I examine Lon Fuller’s famous fictional case from a Bayesian perspective.
6. A Bayesian Analysis of the Hadley Rule (2019), one of the papers in this fine collection of essays, in which I examine the rule of Hadley v. Baxendale from a Bayesian perspective.
7. The Case for Bayesian Judges: Putting Posner and Vermeule into Practice (in press), in which I develop a simple Bayesian model of adjudication.

The formal paper pictured above consists of two succinct sentences and is the shortest-known paper published in a serious math journal; more details are available here. File under: proof by contradiction (hat tip: @pickover).

We presented the basic mechanics of Bayesian voting in one of our previous posts and showed how this simple and intuitive method of voting combines the best of both worlds: the ability of voters to express the intensity of their preferences along with the simplicity of one-man, one-vote. In this post, I want identify and discuss several additional virtues of Bayesian voting. Since Warren Smith has already compiled a comprehensive list of the advantages of Bayesian or “score” voting here, I will limit this particular blog post to the following three virtues:

1. The virtue of resistance to strategic voting. One of the most important lessons in the literature on voting methods is that all systems of voting, no matter how exotic or how complex, can be gamed or manipulated; the problem of strategic voting plagues all voting rules. Bayesian voting, by contrast, is not only simple and easy to use; as long as each voter is allocated only one ballot, it is also immune to most forms of strategic voting, or in the words of Warren Smith, “Your score for candidate C in no way affects the battle between A vs. B. Hence, you can give your honest opinion of C without fear of ‘wasting your vote’ or hurting A. You never have an incentive to betray your favorite candidate by giving a higher score to a candidate you like less.” (This is such an important point in favor of Bayesian voting that I will devote a future blog post to it.)

2. The virtue of familiarity. Furthermore, unlike quadratic voting and other exotic or complex forms of voting, such as Borda counts, Condorcet ranked pairs, instant runoffs, etc. (as an aside, for an excellent overview of different voting methods check out this helpful entry in the Stanford Encyclopedia of Philosophy), most people are already familiar with and have ample experience in Bayesian voting. Think of Yelp reviews for restaurants and TripAdvisor reviews for hotels, or Rotten Tomatoes reviews for movies or Amazon reviews for products and books (see example pictured below), just to name a few.

3. The virtues of flexibility and adaptability. Bayesian voting is so simple and easy to use that it can be used in a wide variety of settings–not just political elections but also jury trials (questions of fact) and appellate cases (questions of law). For the sake of brevity, I won’t describe these possible novel applications of Bayesian voting in this post, but for more information about these possibilities, check out my discussion of “Bayesian verdicts” [here] and my description of Bayesian judges [here].

4. The virtue of intellectual cross-fertilization. Last but not least, for me the chief virtue of Bayesian voting is that it makes explicit the intellectual link between degrees of belief and intensity of preferences. This point is such an important one for me that I will devote a separate blog post (my next one) to it …

I interrupt my series of blog posts on Bayesian voting to share this video clip with my loyal followers. It’s a video I shot on my phone of Tim Tebow’s first-ever Spring Training home run …

Now that we have described how quadratic voting works (see my blog post dated 23 Feb. 2020) and have presented several salient objections to this complicated method of voting (see my previous blog post), in this post I shall present a simpler alternative procedure of collective decision-making, one that I have described on many previous occasions (see here and here, for example, via The Journal of Brief Ideas). For the record, my alternative method of voting is inspired by the work of Dr. Warren Smith and goes by various names, including cardinal voting, range voting, score voting, and utilitarian voting, just to name a few. But in my case, because I wish to make an explicit connection between intensity of preferences and degrees of belief, I prefer to call this technique “Bayesian voting.” In brief, under this method of collective decision-making voters are allowed express the intensity of their preferences on every issue or candidate on the ballot. The winner is the candidate that receives the highest cumulative score.

That’s it! That’s all there is to it. There is no need to distribute a set number of “voice credits” to the voters beforehand or to any crunch any obscure mathematical equations beyond simple addition. To see for yourself how Bayesian voting might work in practice, in a real-world election, check out the sample Bayesian ballot pictured below and which I have borrowed from this helpful website:

Image credit: Ciaran Dougherty

Or, better yet, let’s return to the very same example that we already used in one of our previous posts to illustrate how quadratic voting works: the upcoming South Carolina presidential primary. Assuming once again (for the sake of simplicity) that only five candidates are competing in this early primary, a Bayesian ballot might thus look like this:

Under the traditional one-man, one-vote rule, each voter is allowed to choose only one candidate from this five-person slate of candidates. Under Bayesian voting, by contrast, each voter would be allowed to rank or score every candidate along some uniform scale, such as the familiar 0 to 10 scale. (I have used a five-point scale in my sample South Carolina ballot above, since many voters will already be familiar with the five-star rating system of yore used by Netflix to rank films.) As I mentioned above, the winner would be the candidate that receives the highest cumulative score. Bayesian voting thus combines the best of both worlds: it captures the simplicity of the “one-man, one-vote” rule, and it also allows voters to express the intensity of their beliefs and preferences. Furthermore, as Warren Smith and others have shown, in addition to its simplicity and intuitive appeal, Bayesian voting has many additional virtues. I will identify and describe these virtues in my next blog post.

I explained how quadratic voting works in my previous post, where I presented a simple quadratic voting procedure in which voters are allocated an equal number of “vote credits” before going to the polls. Yet, whenever we are evaluating a proposal for reform like quadratic voting, we must not only consider the benefits of the proposed reform; we must also weigh the costs of the reform as well. Does quadratic voting pass this cost-benefit test? With this test in mind, I will identify several potential problems with this method of voting.

1. The “Rube Goldberg” Objection. The most serious objection to quadratic voting is that it is too damn complicated to operationalize at an acceptable cost. Like a fabled Rube Goldberg Machine, Quadratic Voting introduces a non-trivial amount of complexity into the simple act of voting. For starters, how many “vote credits” should each voter receive? May voters accumulate unused vote credits for future elections? Will voters be allowed to transfer some or all of their vote credits to other voters? What would a “quadratic ballot” look like? What happens if a voter makes an arithmetical error when allocating his vote credits among his preferred candidates? And so on. For all its faults and imperfections, the “one-man, one-vote” rule is easy to understand and easy to operationalize, while even the simplest quadratic voting schemes, by contrast, resemble an elaborate and cumbersome Rube Goldberg machine, a device designed to perform a simple task in an indirect and overly-cumbersome way (like the elaborate “Self-Operating Napkin” contraption pictured below).

2. The “Tyranny of the Minority” Problem. Another potential problem with quadratic voting is that, despite its complexity, it is not immune to strategic voting. Specifically, a large-enough minority of intense zealots could end up overriding the will of the majority of voters by allocating their “vote credits” in a strategic manner. To see this, consider the South Carolina example from my previous blog post. To keep it simple, let’s assume that x number of Democrat voters go to the polls to choose their preferred candidate, where x = 10 voters, and let’s further assume that six of these voters prefer a moderate candidate like Amy Klobuchar, Pete Buttigieg, or Joe Biden, while the remaining four voters prefer a more radical candidate like Bernie Sanders or Elizabeth Warren. If the six moderate voters end up distributing their “vote credits” evenly among the three moderate candidates–while the four zealous voters agree to assign all of their “vote credits” to one of the radical candidates–, the radical candidate could end up winning the primary by a landslide(!), even though he or she is supported by only 40% of the voters! In fairness, this objection applies to other forms of voting as well, including the “one-man, one-vote” system, but at least with one-man, one-vote, the lack of majority support is transparent in the final vote count. Quadratic Voting, by contrast, could produce a false picture of reality, with the radical/minority candidate receiving the largest number of total votes. File under: WTF!

3. The Buterin Problem. In his excellent primer on Quadratic Voting, Vitalik Buterin identifies an important gap in all Quadratic Voting schemes: who decides what issues and which candidates get to go on the ballot in the first place? In fairness, this objection applies equally to all methods of voting: whoever has the power to set the agenda (or to decide which candidates may run for office) can strategically manipulate the agenda or ballot to achieve his preferred outcome. But this objection is especially salient in the context of quadratic voting. Why? Because even the simplest version of quadratic voting is far more complicated and elaborate that the “one-man, one-vote” rule (see Objection #1 above), so why should we adopt a more complex procedure voting if the complex method is just as amenable to manipulation as “one-man, one-vote” schemes?

Nevertheless, despite these objections, ideally we would still like to find a viable way of taking the intensity of voter preferences into account. To this end, I will present an alternative — and far more simpler — method of achieving this goal in my next post, a method I have christened “Bayesian voting.”