prior probability

I will continue my survey of my previous scholarly work next week. In the meantime, I will be presenting a new work-in-progress on “Adam Smith and Geneva: Some Long-Lasting Encounters” — along with my friend, colleague, and co-author Alain Alcouffe — at the University of Salento this weekend, where the next meeting of the International Adam Smith Society (IASS) will take place. Here is the full conference program.

In addition to Thomas Bayes and Ronald Coase (cf. my previous two posts), another “beautiful mind” that has captured my imagination is the mathematical philosopher Kurt Gödel. (Shout out to my colleague and friend Orlando I. Martinez-Garcia, who pointed me in Gödel’s direction in the early 2000s.) My most cited paper is Gödel’s Loophole, which revisits the story of Gödel’s discovery of a deep logical contradiction in the U.S. Constitution. I wrote that landmark paper in 2012 and published it in 2014, and in the last few years, I have returned to the Austrian logician three more times:

1. The Leibniz Conspiracy (2022). This paper explores a little-known conspiracy theory that Kurt Gödel himself, the greatest logician since Aristotle, believed in!
2. Gödel’s Loophole: A Prequel (2024). This paper surveys three “self-coups” that occurred in Central Europe during the interwar period — Yugoslavia in 1929, Austria in 1933, and Romania in 1938 — that Gödel, who lived in Vienna during this time, may have been familiar with and that may have informed his studies of the U.S. Constitution in the 1940s.
3. Gödel’s Loophole 2.0 (forthcoming). This paper extends the general logic of “Gödel’s loophole” to an otherwise promising new method of AI safety called “Constitutional AI”.

One of my intellectual heroes is the English economist Ronald Coase. More specifically, a large chunk of my scholarly work can be traced back to two of his landmark papers: The Federal Communications Commission (1959) and The Problem of Social Cost (1960). I wrote my first “Coase paper” in the summer of 2004 — though I did not get around to publishing it until 2010 (see item #10 below) — and have since published nine more papers extending Coase’s ideas to many different domains:

1. Outer Space Auctions? (2023/2024). This paper presents a Coasian solution to the problem of space congestion and orbital debris in Low Earth Orbit.
2. Coase’s parable (2023). This paper explores the origins of Coase’s idea of reciprocal harms.
3. Coase and the Corleones (2022). This book chapter extends Coase’s idea of reciprocal harms in the context to the famous wedding scene in the original Godfather movie.
4. Of Coase and Copyrights: The Law and Economics of Literary Fan Art (2020).
5. Does the Prisoners Dilemma Refute the Coase Theorem? (with Orlando I. Martinez Garcia, 2014). This paper extends the Coase theorem to the Prisoner’s Dilemma.
6. Trolley Problems (2014). This paper presents a Coasian solution to the trolley problem.
7. Modelling the Coase Theorem (2012). This paper formalizes the logic of the Coase Theorem.
8. Clones and the Coase Theorem (2011). This paper extends Coase’s idea of reciprocal harms to the conflict in the original Blade Runner film.
9. Coase and the Constitution (2011). This paper proposes the creation of “federalism markets” in which governmental powers and functions would be allocated to Congress, the states, or even private firms through decentralized auction mechanisms and secondary markets.
10. Coase’s Paradigm (2010). I wrote this homage to Coase during the summer of 2004 but did not get around to publishing it until six years later!

My first foray into probability theory was my 2011 paper Chance and Litigation, which explores whether litigation outcomes are random. Although I had not yet rediscovered Bayes’ theorem when I began writing my first probability paper (circa 2008 or 2009), my interest in probability theory inevitably led me to the ideas of Thomas Bayes, and I soon began exploring the logic of inverse probability and extending Bayesian reasoning and subjective probability to several different areas of law, including litigation outcomes, proof problems, jury voting, and judge voting. Most (but not all) of my probability papers were published from 2011 to 2020:

1. **A Bayesian Model of the Litigation Game (2011). I wrote this paper in Amsterdam during the summer of 2011, and it is my first refereed paper.
2. *Visualizing Probabilistic Proof (2014). This paper, which I wrote at my in-laws’ house in Tarpon Springs during the summer of 2013, revisits the Blue Bus Case, a famous thought-experiment in law involving probabilistic proof, and presents simple Bayesian solutions to different versions of the blue bus problem.
3. Judge Hercules or Judge Bayes? (2015). This unpublished paper explores two possible connections between hard cases in law and Newcomb’s Paradox in philosophy. One is that Newcomb’s Problem is like a “hard case” in law — i.e. a choice problem with conflicting and equally logical solutions. The other is that the superior being in Newcomb’s Problem and the mythical Judge Hercules in Ronald Dworkin’s theory of law are the same person.
4. *Why Don’t Juries Try Range Voting? (2016). This paper builds on the idea of “subjective probability” to propose the use of “range voting”, which I later christened Bayesian voting, by juries. (Jurors would score the evidence presented by the parties at trial on a scale of one to ten or some other specified scale.)
5. **A Bayesian Analysis of the Hadley Rule (2018). This paper was published in a collection of essays on the law of contracts (see pp. 925-928). Although this is one of the shortest formal papers I have ever written (only nine paragraphs), it underwent ten revisions during the editing process.
6. *The Case for Bayesian Judges (2019) Like my 2016 jury paper (see item #4 above), this paper builds on the idea of “subjective probability” to propose a simple method of judicial voting.
7. Subjective Probability and Legal Proof (2020). This unpublished paper presents a Bayesian critique of Ron Allen and Mike Pardo’s “relative plausibility” theory of legal proof. (Among other things, their approach to proof can be “Dutch-booked”.)
8. Weyl Versus Ramsey: A Bayesian Voting Primer (2020). This unpublished paper compares and contrasts two methods of Bayesian voting or Bayesian preference-aggregation that allow voters not only to rank their preferences but also to express their degree: Quadratic Voting (QV) and Ramsian Voting (RV).

* = published paper; ** = refereed

The title of this beautiful ballad translates to “fireplace”; the TikTok remix is below the fold:

My last few posts have revisited some of my previous scholarly work, including my first few scholarly papers, my game theory years, and my turn to legal history. I will hit the following autobiographical topics in the days ahead:

1. Papers I have written inspired by Bayes, Coase, and Gödel
2. Pedagogical papers
3. Book reviews
4. Books and book chapters
5. My rediscovery of Adam Smith

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.