I mentioned in my previous post that Frank Ramsey was one of the founding fathers of the subjective interpretation of probability, and I also noted how Ramsey’s approach forever changed my view of the world. Before Ramsey (BR), I used to think that beliefs in propositions were binary or “all-or-nothing” entities–either a proposition was true or false. After Ramsey (AR), I now realize that beliefs can come in many shades of grey or degrees. Take one’s belief in God, for example. People often discuss or debate the question of God’s existence in binary terms: either “God exists” or “He does not exist,” right? But what if you are not 100% sure whether such a supreme being exists, or what if you find the arguments for or against the existence of an all-powerful deity equally compelling or persuasive? In a word, why can’t we assign a probability of 1/2 to this proposition? Indeed, why can’t we assign any other intermediate value between 0 and 1, with 0 representing a Richard Dawkins-like dogmatic atheism and 1 representing a Pope Francis-like faith in God?
On this subjective or “Ramsian” view of probability, one’s personal belief in the likelihood of God’s existence (or in any other proposition) is simply a numerical representation of one’s individual degree of confidence or “degree of belief” in God. Thus stated, one’s belief in God can take any value from 0 to 1, depending on how strong or weak one’s belief in God is (as the case may be). More importantly, even if two individuals have access to the same pieces of evidence or to the same arguments, they could still end up assigning different probabilities to the same proposition! After all, facts are often open to a wide variety of competing interpretations, or maybe they assign different weights to the evidence available to us, or maybe my belief in X is an essential part of my self-identity, etc., etc.
Furthermore, the possibility of degrees of belief–the idea that beliefs can be partial–not only sheds lights on our understanding of probability; this idea also illuminates the concept of “preferences.” Simply put, if beliefs and probabilities can come in degrees, so too can preferences. [As an historical aside, the idea of preference intensity was first introduced and formalized by two European economists, Ragnar Frisch and Vilfredo Pareto, in the mid-1920s (see, e.g., Farquhar & Keller (1989), p. 205), around the same time when Frank Ramsey was developing his model of subjective probability!] This insight, in turn, has important implications for politics and law–for any domain in which voting is used to engage in collective decision-making, whether it be a population of registered voters, a group of legislators, a panel of appellate judges, or even a 12-man jury. Voting in all of these various domains (ballot box, legislatures, and courts) are based on the general principle of “one-man, one-vote,” and these votes are binary in nature: either the liberal or the conservative candidate is elected; either the legislation is enacted or not; either the defendant is guilty or not guilty. But what if we replaced the traditional method of binary voting with a more nuanced method of “Bayesian voting”–i.e. what if we allowed voters, legislators, judges, and jurors to express their degrees of belief (or the intensity of their preferences) when they are casting their votes?
It turns out that the leading proponents of “Quadratic Voting” (QV) such as Glen Weyl, Vitalik Buterin, and many others have finally recognized this virtue, but alas their proposed QV methods have many deep flaws when compared to pure “Bayesian voting.” I will identify and discuss these flaws in my next few blog posts.
Reference: Peter H. Farquhar and L. Robin Keller, “Preference intensity measurement,” Annals of Operations Research, Vol. 19 (1989), pp. 205-217.