Beatrice Cherrier has asked me to put together a post on Martin Shubik’s informal tripartite classification of work in game theory:

- High church
- Low church
- Conversational

Shubik discussed this classification in two retrospective, reflective articles:

- M. Shubik, “What is an Application and When is Theory a Waste of Time?”
*Management Science*33 (1987): 1511–1522. - Martin Shubik, “Game Theory and Operations Research: Some Musings 50 Years Later,”
*Operations Research*50 (2002): 192–196.

Neither article is exactly what you would call philosophical. They belong to the ill-organized, often repetitious genre of commentary addressing 1) the epistemological status of formal modeling, and 2) the always-dicey relationship between theory development and practical application. Beatrice, I should note, is something of a connoisseuse of this important, under-respected literature, at least as it pertains to economics. Follow her on Twitter for occasional dispatches from its labyrinths.

Shubik’s commentary in these two pieces is meandering but candid, covering a range of issues crossing between mathematics, computation, management, and economics. He emphasizes the disconnect between formal inquiry and the unstructured problems of decision in management and everyday life. While Shubik is fairly critical of theoreticians’ ability to engage with real-world styles of decision-making (a *constant* if perhaps overworked complaint within the operations research community), he nevertheless makes it reasonably clear that the theoreticians and practical advisers in different fields all work on distinct kinds of problems, employing distinct styles of work, and abiding by distinct sets of social and intellectual norms. Each kind of work has fostered its own kinds of success, and the challenge has been to create environments—Shubik offers 1950s-era RAND as an example—in which different kinds of work can build productively on each other.

For Shubik, game theoreticians have operated according to different rules depending on the style of work they employed.

**High church game theory **develops a coherent, rigorous, and evocative vocabulary of strategic behavior. However, the theorists who work in it value it more because it explores a unique set of mathematical/logical structures than because it serves as a guide to human behavior or prescribes firm rules of action. John Nash’s foundational work on equilibrium in noncollaborative games certainly fits the bill. As Shubik (1987) puts it:

High church game theory is the domain of mathematics, axiom systems, formal systems and solution concepts. Much of it can verge on “art for art’s sake.” There are subindustries of mathematics such as the exhaustive exploration of all four-person games in characteristic function form; or quasi-philosophical occupations such as the search for a single perfect noncooperative equilibrium point as a normative solution to a game in strategic form.

Who are the sponsors, producers and consumers of this work? What are the motivations? What defines application? Probably the most honest answer is that to the “high church game theorist” the work is fun and the ability to have the time to do it is provided by academia and a few foundations.

Shubik is certainly aware of military patronage of game theory, but I think he is correct to characterize support for “high church” work as an indulgence, which military (and other) patrons extended to communities—not even necessarily to the same individuals—who also worked on things like weapons analysis. Patrons probably did not expect high church work to yield any sort of immediate practical result. As for weapons analysis,^{1} Shubik actually characterizes it as an example of “low church” game theory.

Shubik is less clear about the nature of **lo****w church game theory**; in his 2002 piece he identifies the “new industrial organization [theory]” [pdf] as an example. It seems safe to say that it is a style of game theory that makes extensive use of the formal apparatus that the theory provides, but without the exhaustive logical rigor of the work done in the high church. We might suppose the term “modeling,” though applicable to high-church and conversational game theory, most comfortably describes the work of the low church.

Unfortunately, Shubik does not tell us whether he would consider R. Duncan Luce and Howard Raiffa’s seminal book *Games and Decisions *(1957) to be low church. Although containing considerable formalism, I would tend to think the discussion is sufficiently divorced from analysis of the mathematical structure of game theory to designate it as low church.

Interestingly, Shubik does explicitly identify numerical game theory as falling under his definition of low church, which, he notes, “involves sponsorship to work on a specific application producing, if only for illustrative purposes, actual calculations and possibly a parametric sensitivity analysis” (Shubik 1987). This is the aspect of the low church that encompasses weapons analysis, but also any other aspect of game theory in which quantitative solutions to game theoretical problems are produced.

Incidentally, the small number of practical numerical applications of game theory has long dogged the theory’s reputation. Shubik (1987) generally concurs with the assessment: “I believe that these applications have been of some, but nevertheless relatively modest, worth, but nowhere near the applied value of linear programming.”^{2}

According to Shubik (1987) **conversational game theory** “consists of advice, suggestions and counsel as to how to think strategically.” Discarding most or all of the formal content of the theory, it retains core concepts—think “zero-sum game” and the ubiquitous Prisoner’s Dilemma—which are deployed in conceptual instruction to a wide variety of audiences, such as business students. Indeed, as Shubik (2002) explains, it is the conversational facet of game theory that “has made its way into the language of every consultant and has caught the imagination of the public.” As such, Shubik suggests that it is perhaps conversational game theory that has had the most value in practice.

Shubik (2002) does not explicitly identify Thomas Schelling as a conversational (or possibly low church) game theorist, but his comments on Schelling’s work illustrates the tension and complementarity of the high church with the lower two echelons:

When consulting at RAND, I read the notes of Tom Schelling, which led to his book on the strategy of conflict [1960]. I was deeply opposed to it at the time because it was (and still is) loaded with basic errors and a misunderstanding of elementary game theory. But what I failed to appreciate at that time was that it was the work of a social scientist willing to take the mindset of game theory seriously but not willing to accept the rules of the game as given. It is precisely the concern for context and the “games within the game,” where the fuzzy meld of strategic—as well as tactical—modeling took place in “the old operations research.”

^{3}Strategic analysis in application has no neat and tidy rules to turn over to the boys writing the algorithms to solve everything (Shubik 2002)

Although often pursued independently, according to Shubik (1987) the different facets of game theory do indeed reinforce each other, intellectually and institutionally:

Without high church game theory the concepts, illustrations and stories of conversational game theory would hardly exist and certainly would not have a coherent intellectual basis. Without conversational and high church game theory, sponsorship for low church game theory would hardly exist.

That’s probably true enough, but I would also point out that Shubik’s classification is hardly official or even very well known. While the sorts of distinctions and interrelations that Shubik draws out roughly describe how the theoretical community operates, there is still plenty of room for confusion and lack of consensus about the nature of any given theoretical achievement.

One of the things that struck me when John Nash recently died was how his decidedly “high church” achievements were routinely boiled down into a “conversational” essence. A number of obituaries and reflections cast the Nash equilibrium for the Prisoner’s Dilemma, mutual betrayal, as if it were a kind of behavioral law rather than one of the simplest possible illustrations of what a Nash equilibrium is. Conversational style may be unavoidable in the obituary genre, but, in this case, not only was the technical achievement of Nash not well communicated (which is forgivable), its purpose and method was not well communicated either.

At the same time, such confusion has often afflicted critics of game theory as well. The mathematical sophistication of high church (and, for that matter, low church) game theory has sometimes been regarded as offering specious authorization to theorists to offer rigid prescriptions for strategic action (as in nuclear weapons policy), as well as crucial rationalization that encourages “our society” to engage in a self-interested, transactional, calculating approach to social and political interaction (echoing concerns expressed by much earlier figure such as Max Weber).

Attending to distinctions such as the one Shubik describes helps us to articulate a more balanced and realistic understanding of theorists and their work.

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- Weapons analysis concerns comparing the efficacy of different weapons designs in anticipated combat scenarios by integrating the engineering specifications of the weapon (e.g., accuracy, rate of fire, etc.) with formal expressions describing the tactics for using the weapon and the vulnerabilities—and counter-tactics—of enemy targets to derive probabilities of successful outcomes. ↩
- Linear programming is a highly influential computational method that was developed concurrently with game theory beginning in the late 1940s, and employs structures that are mathematically related to game theory. ↩
- This is a reference to the distinction between an early manifestation of operations research that was largely geared toward the provision of advice to decision makers, and its later manifestation as a field primarily concerned with finding and exploring methods of mathematical optimization and efficient computation. ↩