Rationality, Economics and Games

1. The “Center for Rationality”

“Founded in 1991, the Hebrew University’s Center for the Study of Rationality  [at first it was simply called "Center for Rationality"] is a unique venture in which faculty, students, and guests join forces to explore the rational basis of decision-making. Coming from a broad sweep of departments — mathematics, economics, psychology, biology, education, computer science, philosophy, business, statistics, and law — its members apply game- theoretic tools to examine the processes by which individuals seeking the path of maximum benefit respond to real-world situations where individuals with different goals interact.” 

Game theory was always strong at the Hebrew University of Jerusalem, and a nice aspect of it is the combination of mathematics and debating. As an undergraduate I was quite interested in game theory along with combinatorics and convexity, and my first published paper was on game theory, with Michael Maschler and Guillermo Owen. Later I moved in other directions, but more recently, in part because of my membership in the Center for the last ten years and in part because of my collaboration with economists Ariel Rubinstein (who was my classmate in my undergraduate years) and Rani Spiegler, I am trying to do research and write papers in theoretical economics. Not having the basic instincts of an economist, and lacking some basic background, makes it especially difficult.

Let me also mention that there are very interesting connections between computer science and economics and a very large emerging research community.  

2. Many many controversies

Among the many issues discussed and debated in seminars at the Center (the regular ones are the “Game Theory Seminar” on Sundays and the “Rationality on Friday” seminars on… Fridays,)  roundtables, the annual retreat, Sunday’s sandwich gatherings, and ample debates over e-mail were:

The controversy over expected utility theory (we will come back to it below); (Little updates: May, 21)

The role of psychology in economics;

The relevance of “neuroeconomy”;

Economics and the law and, in particular, judicial activism;

Privacy and surveillance;

Labor unions in general and the university professors labor union in particular;

The controversies over governance of Israeli universities, differential salaries for professors, and higher tuition for students;

Various issues related to the Center itself, like the fierce struggle with the university administration to get more offices in the late 90s, and the role, advantages, and disadvantages of this and similar research centers in university life;

Issues regarding the Israeli-Arab conflict, and war and peace in general.

 

3. Expected utility and rationality 

Let’s first briefly talk about expected utility theory and one controversy arising from it, to earn the right to move later to a few anecdotes.

Expected utility is a beautiful mathematical theory dealing with choices under uncertainty. Suppose there are n alternatives and we want to understand the preference of a rational agent between lotteries involving the alternatives. A lottery is a scenario of the following form: you have with probability 1/3 alternative A and with probability 2/3 alternative B. The preference relations are required to be “rational,” or in other words, to form an order relation. Lotteries of lotteries are also considered. Under a few natural axioms regarding the decision-maker’s behavior you reach the conclusion that you can associate to every alternative X a utility which is a real number, such that the preference relation between lotteries is derived by the order relation between their expected utilities. 

The debate on expected utility theory is old, and there are well-known experiments showing that individual choices deviate systematically from the predictions based on the expected utility model. (There is also evidence that individual behavior deviates from “rationality,” which asserts that the preferences are transitive, and that preferences may depend on factors that are not represented at all in this model.) Even if you use the expected utility model, identifying an individual’s utility function is very difficult. On top of this, identifying the probabilities involved in cases of uncertainty is a major issue in and of itself.

Quite recently, Matthew Rabin pointed out that expected utility theory leads to very counterintuitive conclusions. “Suppose that from any initial wealth level an expected-utility maximizer turns down gambles where she loses $100 or gains $110, each with 50% probability,  then she will turn down 50-50 bets of losing $1000 or gaining any sum,” and suppose that from any lifetime wealth level of less than $350,000 an expected-utility maximizer turns down gambles where she loses $100 or gains $105, each with 50% probability, then from an initial wealth level of $340,000 she will turn down 50-50 bets of losing $4000 or gaining $635,670. Rabin’s surprising discovery (related also to a 1963 paper by Samuelson) drew a lot of attention, and Rabin’s own interpretation was that this is strong evidence against expected utility theory. 

Ignacios Palacios-Huerta and Roberto Serrano’s response is that people with large initial wealth level will accept the small gambles in Rabin’s example. Their response can be regarded as claiming that the correct interpretation of Rabin’s observation is that the expected utility theory is overly inclusive rather than incorrect. The absurd conclusion refers to irrelevant regions of the theory, they claim, and they support their claim with empirical data. The tension between a theory or a model being incorrect or too narrow and it being too wide occurs in many scientific controversies and these two possibilities are often alternative interpretations of the same piece of evidence.

Zvi Safra and Uzi Segal pointed out that Rabin’s difficulty arises also in more general notions of utility. (I regard Safra and Segal’s results as weakening the interpretation of Rabin’s result as an argument for rejecting expected utility theory). Ariel Rubinstein also refers to Rabin’s findings in the section “The Dilemma of Absurd Conclusions” in a (rather provocative) paper where he describes various dilemmas facing the economic theorist. This controversy had a role in triggering the recent papers of Aumman and Serrano  and of Foster and Hart on riskiness.

This is a very nice controversy to watch.   I am weakly leaning toward Serrano’s side of the debate (and in general to the more “classical” economics theory,) but I have to mention that I have been exposed more to this side. More precisely, I view Rabin’s observation as one describing an important small effect regarding deviation from expected utility theory on (one or a few) small bets. This small effect needs to be corrected (or avoided) when trying to practically apply expected utility theory (which, as I said, is extremely difficult anyway), but I see no reason to believe that it nullifies expected utility theory and its prominent role in economics.

“Small???” you may ask. Turning down a 50-50 bet of losing $4000 or gaining $635,670 is a small matter? No; but note that this is a mathematical conclusion that follows from one’s behavior toward a few small bets that deviates from expected utility theory.  Once you see the behavior of the decision-maker as expressed by a (reasonable) utility function plus a rather small error term regarding her behavior for  one or a few small bets, the absurd conclusion does not apply.

Here is an example of what I mean: suppose you claim that a function f of your dollar income is linear while in reality it is only linear in the value rounded up to the next dollar. If you try to compute f(2000) based on a linear function through f(23.75) and f(24.12) you will make a huge mistake and yet the linear approximation is good.

Update (May 21): Uzi Segal wrote me:  ”I’m not too sure about the ‘correct’ interpretation of our results. certainly it is true that whatever the problem is, it is not EU. but whether the conclusion is ‘there is no real problem’ or ‘the problem is with having a global preference relation’ — I don’t know. I’m afraid it is more likely the latter than the former.” Safra and Segal’s paper will be published in “Econometrica” and the final version can be found on the journal’s site. Piero La Mura wrote me that he “recently proposed a tractable generalization of expected utility (inspired by quantum information theory) which seems to avoid all the main anomalies, including Rabin’s.”

 

4. A few anecdotes

Game theory B.C. In the little workshop celebrating the Center’s inauguration Aumann gave a talk entitled “Game Theory in Jerusalem B.C.”  The title of the lecture was quite puzzling: “rationality in Jerusalem” already seemed a little contradictory, but “rationality in Jerusalem B.C.”? What could it mean? Aumann was interested in the early appearance of game theory in the Talmud, but we could not recall any examples from life in ancient Jerusalem, and Aumann’s only response to our questions about it  was: “Come to the lecture”.  Well, as it turned out in the lecture, by B.C. he meant ”Before Center,” so he actually talked about Game theory in Jerusalem up to that time.    

Maschler and the game theory exam. Game theory was the first course I took as a high-school-on-strike student in the 1970-1971 academic year. (School teachers were on strike that year for several months.) Michael Maschler was a very impressive teacher and the lecture hall was filled with students. He also always came to class in a jacket and tie which in Israel is quite unusual. The following year I was already a university student and I took ”Advanced Game Theory,” where Maschler suggested an open problem about certain dynamical processes leading to the bargaining set solution in cooperative games. (These dynamical processes were introduced by Stearns and Billera.) I thought I could solve one direction of one problem and I wrote Maschler about it and indeed my solution worked (but in the the other direction — not the one I thought) and Maschler and Owen proved the other direction and we wrote a paper together. In 1974 I had to finish my degree and I had no grade for “Game Theory” so I took the exam. To my surprise that year our theorem was part of the material and there was a question on it. Rather than answering the question I referred to my paper and, while in this mood, I continued to give sloppy answers to the other questions just to indicate that I knew the answers.  The TA and Maschler were not very impressed by my exam and the grade I got was 19/100. (Later, Maschler and Bezalel Peleg gave me a make-up exam and this time I answered the questions appropriately and got a good grade.) This was a good lesson.

A ride from Tel Aviv with Aumann. After getting my undergraduate degree I went to the army and part of my service was in the Tel Aviv area. From time to time I went to the Tel Aviv University game theory seminar that Aumann (who was a HU professor but also taught at TAU) ran. Among the regular participants were Abraham Neyman (Merale), who was known as a legendary mathematics problem-solver in my undergraduate years, Yair Tauman, Sergiu Hart, and Dubi Samet. One day Aumann gave me a lift from Tel Aviv to Jerusalem. He asked me to take out a piece of paper and to write down a matrix. It was a 2-by-2 payoff matrix of a non-zero-sum game, so each entry was a vector of payoffs for the two players. Then he asked me to do some calculations, made his point, suggested another matrix, asked for more calculations, made another point, and so on. Quite often he took his eyes off the road and pointed to my piece of paper and said something like: Here you wrote (2,3), but it should be (4,3), or here you made the worng calculation - the plus sign should be a minus. This divided attention between the road and my piece of paper was scary enough, but on top of this, Aumann used the following strategy for driving: any time he saw an opportunity to gain an advantage by changing lanes he did so, and these changes were quite frequent and often performed while he was looking at my piece of paper and pointing out some required modification of what was written there. While Auman was in control of both the driving and the mathematics, this was certainly one of the scariest car rides I ever took.

  

 

Maschler (1978), Peleg (1980), and Aumann (1977) (Oberwolfach pictures collection)

(May, 15: By mistake an unedited draft version was uploaded first. Fixed.) 

Tags: Controversies, Economics, Expected utility, Game theory, stories and anecdotes