Reinhard Selten and the making of modern game theory

Reinhard Selten, it is no exaggeration, is a founding father of two massive branches of modern economics: experiments and industrial organization. He passed away last week after a long and idiosyncratic life. Game theory as developed by the three co-Nobel laureates Selten, Nash, and Harsanyi is so embedded in economic reasoning today that, to a great extent, it has replaced price theory as the core organizing principle of our field. That this would happen was not always so clear, however.

Take a look at some canonical papers before 1980. Arrow’s Possibility Theorem simply assumed true preferences can be elicited; not until Gibbard and Satterthwaite do we answer the question of whether there is even a social choice rule that can elicit those preferences truthfully! Rothschild and Stiglitz’s celebrated 1976 essay on imperfect information in insurance markets defines equilibria in terms of a individual rationality, best responses in the Cournot sense, and free entry. How odd this seems today – surely the natural equilibrium in an insurance market depends on beliefs about the knowledge held by others, and beliefs about those beliefs! Analyses of bargaining before Rubinstein’s 1982 breakthrough nearly always rely on axioms of psychology rather than strategic reasoning. Discussions of predatory pricing until the 1970s, at the very earliest, relied on arguments that we now find unacceptably loose in their treatment of beliefs.

What happened? Why didn’t modern game-theoretic treatment of strategic situations – principally those involve more than one agent but less than an infinite number, although even situations of perfect competition now often are motivated game theoretically – arrive soon after the proofs of von Neumann, Morganstern, and Nash? Why wasn’t the Nash program, of finding justification in self-interested noncooperative reasoning for cooperative or axiom-driven behavior, immediately taken up? The problem was that the core concept of the Nash equilibrium simply permits too great a multiplicity of outcomes, some of which feel natural and others of which are less so. As such, a long search, driven essentially by a small community of mathematicians and economists, attempted to find the “right” refinements of Nash. And a small community it was: I recall Drew Fudenberg telling a story about a harrowing bus ride at an early game theory conference, where a fellow rider mentioned offhand that should they crash, the vast majority of game theorists in the world would be wiped out in one go!

Selten’s most renowned contribution came in the idea of perfection. The concept of subgame perfection was first proposed in a German-language journal in 1965 (making it one of the rare modern economic classics inaccessible to English speakers in the original, alongside Maurice Allais’ 1953 French-language paper in Econometrica which introduces the Allais paradox). Selten’s background up to 1965 is quite unusual. A young man during World War II, raised Protestant but with one Jewish parent, Selten fled Germany to work on farms, and only finished high school at 20 and college at 26. His two interests were mathematics, for which he worked on the then-unusual extensive form game for his doctoral degree, and experimentation, inspired by the small team of young professors at Frankfurt trying to pin down behavior in oligopoly through small lab studies.

In the 1965 paper, on demand inertia (paper is gated), Selten wrote a small game theoretic model to accompany the experiment, but realized there were many equilibria. The term “subgame perfect” was not introduced until 1974, also by Selten, but the idea itself is clear in the ’65 paper. He proposed that attention should focus on equilibria where, after every action, each player continues to act rationally from that point forward; that is, he proposed that in every “subgame”, or every game that could conceivably occur after some actions have been taken, equilibrium actions must remain an equilibrium. Consider predatory pricing: a firm considers lowering price below cost today to deter entry. It is a Nash equilibrium for entrants to believe the price would continue to stay low should they enter, and hence to not enter. But it is not subgame perfect: the entrant should reason that after entering, it is not worthwhile for the incumbent to continue to lose money once the entry has already occurred.

Complicated strings of deductions which rule out some actions based on faraway subgames can seem paradoxical, of course, and did even to Selten. In his famous Chain Store paradox, he considers a firm with stores in many locations choosing whether to price aggressively to deter entry, with one potential entrant in each town choosing one at a time whether to enter. Entrants prefer to enter if pricing is not aggressive, but prefer to remain out otherwise; incumbents prefer to price nonaggressively either if entry occurs or not. Reasoning backward, in the final town we have the simple one-shot predatory pricing case analyzed above, where we saw that entry is the only subgame perfect equilibria. Therefore, the entrant in the second-to-last town knows that the incumbent will not fight entry aggressively in the final town, hence there is no benefit to doing so in the second-to-last town, hence entry occurs again. Reasoning similarly, entry occurs everywhere. But if the incumbent could commit in advance to pricing aggressively in, say, the first 10 towns, it would deter entry in those towns and hence its profits would improve. Such commitment may not possible, but what if the incumbent’s reasoning ability is limited, and it doesn’t completely understand why aggressive pricing in early stages won’t deter the entrant in the 16th town? And what if entrants reason that the incumbent’s reasoning ability is not perfectly rational? Then aggressive pricing to deter entry can occur.

That behavior may not be perfectly rational but rather bounded had been an idea of Selten’s since he read Herbert Simon as a young professor, but in his Nobel Prize biography, he argues that progress on a suitable general theory of bounded rationality has been hard to come by. The closest Selten comes to formalizing the idea is in his paper on trembling hand perfection in 1974, inspired by conversations with John Harsanyi. The problem with subgame perfection had been noted: if an opponent takes an action off the equilibrium path, it is “irrational”, so why should rationality of the opponent be assumed in the subgame that follows? Harsanyi assumes that tiny mistakes can happen, putting even rational players into subgames. Taking the limit as mistakes become infinitesimally rare produces the idea of trembling-hand perfection. The idea of trembles implicitly introduces the idea that players have beliefs at various information sets about what has happened in the game. Kreps and Wilson’s sequential equilibrium recasts trembles as beliefs under uncertainty, and showed that a slight modification of the trembling hand leads to an easier decision-theoretic interpretation of trembles, an easier computation of equilibria, and an outcome that is nearly identical to Selten’s original idea. Sequential equilibria, of course, goes on to become to workhorse solution concept in dynamic economics, a concept which underscores essentially all of modern industrial organization.

That Harsanyi, inventor of the Bayesian game, is credited by Selten for inspiring the trembling hand paper is no surprise. The two had met at a conference in Jerusalem in the mid-1960s, and they’d worked together both on applied projects for the US military, and on pure theory research while Selten visiting Berkeley. A classic 1972 paper of theirs on Nash bargaining with incomplete information (article is gated) begins the field of cooperative games with incomplete information. And this was no minor field: Roger Myerson, in his paper introducing mechanism design under incomplete information – the famous Bayesian revelation principle paper – shows that there exists a unique Selten-Harsanyi bargaining solution under incomplete information which is incentive compatible.

Myerson’s example is amazing. Consider building a bridge which costs $100. Two people will use the bridge. One values the bridge at $90. The other values the bridge at $90 with probability .9, and $30 with probability p=.1, where that valuation is the private knowledge of the second person. Note that in either case, the bridge is worth building. But who should pay? If you propose a 50/50 split, the bridge will simply not be built 10% of the time. If you propose an 80/20 split, where even in their worst case situation each person gets a surplus value of ten dollars, the outcome is unfair to player one 90% of the time (where “unfair” will mean, violates certain principles of fairness that Nash, and later Selten and Harsanyi, set out axiomatically). What of the 53/47 split that gives each party, on average, the same split? Again, this is not “interim incentive compatible”, in that player two will refuse to pay in the case he is the type that values the bridge only at $30. Myerson shows mathematically that both players will agree once they know their private valuations to the following deal, and that the deal satisfies the Selten-Nash fairness axioms: when player 2 claims to value at $90, the payment split is 49.5/50.5 and the bridge is always built, but when player 2 claims to value at $30, the entire cost is paid by player 1 but the bridge is built with only probability .439. Under this split, there are correct incentives for player 2 to always reveal his true willingness to pay. The mechanism means that there is a 5.61 percent chance the bridge isn’t built, but the split of surplus from the bridge nonetheless does better than any other split which satisfies all of Harsanyi and Selten’s fairness axioms.

Selten’s later work is, it appears to me, more scattered. His attempt with Harsanyi to formalize “the” equilibrium refinement, in a 1988 book, was a valiant but in the end misguided attempt. His papers on theoretical biology, inspired by his interest in long walks among the wildflowers, are rather tangential to his economics. And what of his experimental work? To understand Selten’s thinking, read this fascinating dialogue with himself that Selten gave as a Schwartz Lecture at Northwestern MEDS. In this dialogue, he imagines a debate between a Bayesian economist, experimentalist, and an evolutionary biologist. The economist argues that “theory without theorems” is doomed to fail, that Bayesianism is normatively “correct”, and the Bayesian reasoning can easily be extended to include costs of reasoning or reasoning mistakes. The experimentalist argues that ad hoc assumptions are better than incorrect ones: just as human anatomy is complex and cannot be reduced to a few axioms, neither can social behavior. The biologist argues that learning a la Nelson and Winter is descriptively accurate as far as how humans behave, whereas high level reasoning is not. The “chairman”, perhaps representing Selten himself, sums up the argument as saying that experiments which simply contradict Bayesianism are a waste of time, but that human social behavior surely depends on bounded rationality and hence empirical work ought be devoted to constructing a foundation for such a theory (shall we call this the “Selten program”?). And yet, this essay was from 1990, and we seem no closer to having such a theory, nor does it seem to me that behavioral research has fundamentally contradicted most of our core empirical understanding derived from theories with pure rationality. Selten’s program, it seems, remains not only incomplete, but perhaps not even first order; the same cannot be said of his theoretical constructs, as without perfection a great part of modern economics simply could not exist.