prior probability

As I mentioned at the end of my previous post, I gave up game theory for good when I rediscovered Adam Smith during the summer of 2020. As it happens, my Smithian “aha moment” was part of a larger trend in my scholarship, for I had already published a handful of legal history papers, The Pamphlet Wars: The Original Debate over Citizenship in the Insular Territories (1999), Deconstructing Darwin (2005), and Gödel’s Loophole (2014). But beginning in 2019 — the same year I wrote my last formal game theory paper — I returned to history yet again with my paper Domestic Constitutional Violence, which revisited two obscure Little Rock cases that unsuccessfully attempted to challenge the legality of President Eisenhower’s decision to send paratroopers to Arkansas to desegregate Central High School.

Then, in 2020, I published Guaranteed Minimum Income: Chronicle of a Political Death Foretold, where I retold the story of “The Family Assistance Act of 1970″, a precursor to contemporary calls for universal basic income or UBI. (Had this historic bill been enacted into law, it would have provided every poor family with children a guaranteed minimum income!) And in 2021, I wrote a paper originally titled the “The Leibniz Conspiracy” (published in 2022 as The Leibniz Conspiracy) about a little-known conspiracy theory championed by the mathematical logician Kurt Gödel.

In short, my turn to legal history may have been what primed my eventual rediscovery of Adam Smith in 2020. Stay tuned, for I will write about my newfound fascination with and scholarly interest in the life and ideas of the Scottish philosopher-economist in a future post …

As I was saying in my previous two posts, I began teaching Roman and Constitutional Law in 1998, but one thing that I failed to mention is that I soon became frustrated with traditional legal analysis. Why? Because most legal scholars, like most economists, are prone to the Nirvana fallacy (if you know, you know!), and worse yet, most legal scholarship is tedious, normative (i.e. ideological), and non-falsifiable.

I therefore turned to game theory, a branch of mathematics, and by 2008 I had published my first formal paper, A Game-Theoretic Analysis of the Impasse over Puerto Rico’s Status, in which I model the legal and political debate about Puerto Rico’s constitutional status as a “truel” or three-man showdown.

And then, over the next 12 years, I extended the methods of game theory to a wide variety of legal and political questions, including the Coase theorem, litigation strategy, and the strategic decision whether to evade or comply with the law. Though only half of my body of formal work ever got published (indicated by an asterisk below), I ended up writing up an average of one game-theory paper per year during this span of time, 2008 to 2019:

1. A game-theoretic analysis of public-private contracts in the water sector (2009). I presented this paper at the National University of Singapore in July of 2009.
2. *El caso de Puerto Rico: a game-theoretic analysis of the Puerto Rican status debate (2010). I presented this paper at a LatCrit conference at American University in October of 2010.
3. **Modelling the Coase Theorem (2012). This was my second peer-reviewed research article, which was published in Volume 5, Issue 2 of The European Journal of Legal Studies.
4. Evade or comply? (2013). This work in progress models the strategic decision whether to evade or comply with the law.
5. *The evolutionary path of the law (2014). Not really a game theory paper — it’s a review of a book about a theoretical biologist who made many contributions to game theory: Ullica Segerstråle’s beautiful biography of W. D. (Bill) Hamilton.
6. *Does the prisoner’s dilemma refute the Coase Theorem? (2014). This paper, co-authored with my friend and colleague Orlando Martinez, relaxes some assumptions about the prisoner’s dilemma in order to allow Coasian bargaining between the prisoners.
7. The poker-litigation game (2015). This paper presents a simple game-theoretic model of litigation.
8. Law is a battlefield: the Colonel Blotto litigation game (2016). This draft paper presents a more complex game-theoretic model of litigation.
9. Condorcet’s Paradox and Puerto Rico Status (2018). This draft paper models the Puerto Rico status debate as a voting paradox.
10. **So long suckers: bargaining and betrayal in Breaking Bad (2019). My last game theory paper presents a four-player bargaining game called “So long suckers”.

* = published paper; ** = refereed

Why did I decide for all practical purposes to abandon game theory after 2019? In two words: I rediscovered Adam Smith … (To be continued.)

Postscript: If you want to look “under the hood” and learn about the nuts and bolts of game theory, check out this online course on “Game Theory” led by Professor Ben Polak (Yale) or this online course on “Model Thinking” led by Scott Page (Michigan). Enjoy!

Nota bene: I meant to do this survey of my previous work during my sabbatical last fall. Better late than never!

I began teaching Roman and Constitutional Law in 1998, travelled every summer (the law school where I taught had a summer study abroad program in Toledo, Spain), and eventually wrote up a handful of papers, including Deconstructing Darwin (2005); Domestic Violence, Strategic Behavior, and Ideological Rent-Seeking (2006); and The Most Senile Justice? (2007). Alas, most of my work during this first phase of my scholarly life (1998 to 2007), including two of the three papers mentioned above, went unpublished. Although I was reading and writing every single day, I did not like editing, so I published very little work during this time. But after discovering Thomas Schelling (pictured below) and his 1960 book The Strategy of Conflict by chance (circa 2007), I started to learn the nuts and bolts of game theory, build simple game theory models, and write up my results instead of publishing traditional law review articles. I will survey my game theory years (2008 to 2019) in my next post.

I will return to Part 2 of David Hume’s essay “Of Miracles” and to Adam Smith’s treatment of taxes in Book 5, Chapter 2 of The Wealth of Nations next month. In the meantime, I want to survey the first 20 years or so of my scholarly life, something I have been meaning to do since my sabbatical last fall. In summary, after living in Paris in the summer of 1998, I began teaching Roman and Constitutional Law at the Pontifical Catholic University of Puerto Rico in the fall of that same year and published my first paper in La Revista de Derecho Puertorriqueño in the spring of 1999. Since then, I have visited almost 40 countries across five continents, published over 60 more scholarly papers, contributed five chapters to various books, and co-authored two college textbooks. In my next post, I will survey my first few published papers.

I have been meaning to reblog this post from my fellow francophile and friend Sheree, so here it is!

Happy Pi Day 3.14! I am happy to report that I was invited to attend a research colloquium on “Data Law & AI Ethics” at the University of Pennsylvania (Wharton School) this weekend, where I will be presenting my work-in-progress on “Self-Regulation, ‘Constitutional A.I.,’ and Gödel’s Loophole.” (FYI: A draft of my paper is available here. I will post an updated version after I digest the feedback I receive at this weekend’s colloquium.) Below the fold is the full colloquium agenda:

I will conclude my survey of Part 1 of David Hume’s famous essay “Of Miracles” by proposing a better approach to the problem of miracles: that of Frank Ramsey and Bruno de Finetti.

To the point, Ramsey and de Finetti, building on the ideas of Thomas Bayes and Richard Price, developed a subjective theory of probability (see here and here). Broadly speaking, subjective probability represents an individual’s personal assessment of, or “degree of belief” in, the likelihood of an event happening or the truth of a proposition or hypothesis, e.g. that a given miracle really happened. In other words, whenever we estimate the probability of some event or proposition, our estimate is almost always based on our personal perceptions and experience, i.e. on gut feelings instead of formal calculations. So why not extend the concept of subjective probability as well as Bayesian reasoning more generally to reports of miracles?

Before proceeding any further, two digressions are in order. One is the relationship between subjective probability and political philosophy, for what I find most compelling about the idea of subjective probability is that it is consistent with the classical liberal ideal in favor of natural liberty: people have different beliefs about the world, and those beliefs should be tolerated — nay, celebrated! — so long as no one is harmed. My other digression is that one does not have to express one’s subjective probability or degree of belief in a miracle using numerical values. (See here, for example. See also John Maynard Keynes, Treatise on probability, Macmillan (1921), pp. 20-22.) It is sufficient to use such qualitative formulations as “highly likely, almost certain, or virtually impossible” to describe the likelihood of some event or proposition, given a body of evidence. (See James Franklin, “The objective Bayesian conceptualisation of proof and reference class problems,” Sydney Law Review, Vol. 33 (2011), pp. 545-561, especially pp. 547-548.)

In any case, once we accept the subjective nature of probability, we can easily apply Bayesian reasoning to miracles as follows: given a report of miracle M, you should assign a subjective probability value to the likelihood, however remote, that M really took place, and then you should incrementally update your subjective prior up or down as new evidence about M becomes available. (Moreover, my Bayesian approach to miracles is especially apt given that Bayes’s famous theorem may have originated in response to Hume’s argument against miracles!) For a step-by-step explanation of Bayesian reasoning, see my 2013 paper “Visualizing probabilistic proof” or my 2011 paper “A Bayesian model of the litigation game“. For now, however, I will conclude with one final observation: Bayes > Hume.

Thus far, I have surveyed “the good” and “the bad” sides of David Hume’s famous argument against miracles (see here and here). That leaves “the ugly”: Hume’s circular definition of what a miracle is.

To the point, for Hume “[a] miracle is a violation of the laws of nature” (Hume, Of Miracles, para. 12; cf. Voltaire 1764/1901, p. 272). Alas, the problem with Hume’s definition is that it is circular! Why is this definition a circular one? Because unlike human laws, which are defied and disregarded all the time, “laws of nature” consist of patterns or regularities that can never be set aside or suspended. In other words, a violation of a law of nature has, by Hume’s own definition, a probability of zero!

To see why, ask yourself two deeper questions: (1) what is a “law of nature” (see here, for example), and (2) what does it mean to “violate” or disobey such a thing? Although Hume himself uses the term “laws of nature” in several different senses (see here), at a minimum a law of nature in the traditional Newtonian sense (see, e.g., “Newton’s Three Laws of Motion“) generally refers to some uniform or regular pattern of behavior. On this view of “natural law”, a miracle must consist of an unforeseen departure from or an unexpected interruption of such a pattern.

By way of illustration, consider Newton’s First Law: “Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.”(Translation: “Every object perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.”) In plain English, this law tells us that objects tend to “keep on doing what they’re doing” until they are acted upon by an outside force. The dirty clothes in your hamper, for example, will remain there until someone grabs them and throws them in the washing machine.

Maybe you agree with Hume’s circular definition of miracles. After all, what if your clothes did wash themselves? That would be a true miracle! But from a purely logical perspective, the problem with Hume’s definition is that he’s rigged the game ex ante. To see why, recall Hume’s two-part probability test for miracles: 1st, given some evidence that a miracle took place, Hume would have us assign two separate probability values to the evidence — the probability p₁ that the miracle really happened and the probability p₂ that the evidence is either mistaken or fraudulent or otherwise defective — and 2nd, Hume would have us compare p₁ and p₂: you should believe in the miracle only if p₁ > p₂.

The problem, however, is this: under Hume’s natural-law-violation definition of miracles, p₁ = 0, no amount of evidence of whatever kind would ever be sufficient to establish the occurrence of a miracle, a truly dogmatic and closed-minded position to take, especially when we have so many reports of miracles from so many different sources! Nevertheless, that said, it takes a theory to beat a theory, so in my next post, I will present an alternative method for evaluating incredible claims of miracles.

***

As I mentioned at the end of my previous post, David Hume’s argument against miracles has two big blind spots. One is the unknown probability problem; the other is the reference class problem. To appreciate the significance of these logical troubles, recall Hume’s two-part probability test for miracles: 1st, given some evidence that a miracle took place, Hume would have us assign two separate probability values to the evidence — the probability p₁ that the miracle really happened and the probability p₂ that the evidence is either mistaken or fraudulent or otherwise defective — and 2nd, Hume would have us compare p₁ and p₂: you should believe in the miracle only if p₁ > p₂.

But what if the probability values for p₁ and p₂ don’t add up to 1? By way of illustration, what if p₁ = 0.01 and p₂ = 0.1? (Or in plain English, what if you believe there is only a 1% chance that the miracle in question really took place, and at the same time, you also believe there is only a 10% chance the evidence of the miracle is reliable?) In this case, what do we do with the missing 89% of the probability space in this example, i.e. the “unknown probability”?! Alas, Hume does not say, and Hume’s silence, in a nutshell, is what I mean by the unknown probability problem.

The reference class problem, however, is an even bigger problem for Hume, for how do we figure out what numerical values to assign to p₁ and p₂ in the first place? Stated formally, in deciding whether a particular miracle X is real or not, what is the reference class of X? After all, X could be a member of a wide variety of classes or groups, and the probability of X will thus differ depending on how the class or group of X‘s is defined. (See, e.g., James Franklin (2011), The objective Bayesian conceptualisation of proof and reference class problems, Sydney Law Review, Vol. 33, pp. 545–561.)

Airplane crashes, for example, have been in the news lately, especially since the fatal midair collision at Reagan National Airport on January 30th of this year, but at the same time, we are also being told that aviation safety is at an all-time high. This disparity between the number of news reports and the true level of aviation safety provides a textbook example of the reference class problem, for to estimate the probability of an aviation disaster, one might use the frequency of crashes of all aircraft (civilian and military), the frequency of crashes of a particular model or type of aircraft (e.g. helicopters), or the frequency of airplane accidents in the last five, ten, or y years.

To recap, Hume’s argument against miracles is caught between the Scylla of the unknown probability problem — i.e. the possibility that p₁ and p₂ do not add up to 1 — and the Charybdis of the reference class problem — i.e. the difficulty of deciding what class or group of events to use when calculating p₁ and p₂. Can either of these Humean probability monsters be tamed? Stay tuned, for before I try to rescue Hume from this metaphorical Strait of Messina, I want to take the Scottish essayist to task for his circular definition of “miracles”.