Wednesday, December 28, 2005

On the first night of Chanukah, Yisrael Aumann delivered his Nobel Lecture in Hebrew University. He discussed, in popular terms, some consequences of the ideas for which he won the prize.

Imagine two players playing a game in which either can choose to cooperate or act selfishly. If both cooperate, they each get 3000. If both act selfishly, they each get 1000. If one cooperates and the other acts selfishly, the selfish one gets 4000 and the cooperative one gets 0. Note that for each player (A), regardless of what the other player (B) does, A makes 1000 more by acting selfishly than by cooperating. For that reason, the only "equilibrium point" of the game is the one in which both act selfishly. The result is that both end up with 1000 rather than 3000. Obviously, this is not optimal for either one. (Formally, we say that this result is not "Pareto optimal", i.e., there are solutions in which both players can do better.) If the players could negotiate an agreement and trust that such an agreement would be enforced (say, by a third party), then they obviously could do better.

Now let's imagine that there is no third party enforcer but rather the same players are doomed to play the same game over and over again. It turns out that in this scenario, cooperative behavior is self-reinforcing, since a player is prevented from acting selfishly by the threat that the other player will react to selfish behavior by acting selfishly himself in subsequent games. Thus, in repeated games, players will ultimately cooperate even without an enforcer.

There is, however, one limitation to this self-reinforcement. Players must value profits in the future almost as much as profits now. If they don't, the threat of future losses as punishment for short-term profits is an inadequate threat. Aumann repeated at least three times that the practical consequence of this is that those who place too high a premium on "peace now" will delay peace indefinitely, while those who establish a credible threat of retaliation have a chance to achieve peace sooner.

After the lecture, there was a reception for friends, family, colleagues and a variety of bessere mentschen. A classical cellist played while hors d'ouevres were served; bekitzur, the organizers were aiming for a somewhat hoity toity atmosphere. Except that Aumann insisted on first getting together a minyan for maariv, then lighting chanukah lecht and then singing Maoz Tzur -- all six stanzas until the bitter end. Gevaldig.

Saturday, December 17, 2005

A number of people have asked me to write something about Nobel laureate Yisrael Aumann's work on game theoretic ideas in the gemara. Here's a very brief synopsis of his most important work in that area. It concerns the Mishnah in Kesubos 93a.

The Mishnah considers a case in which a man has left behind three wives (or, more generally, creditors) to whom he owes 100, 200 and 300, respectively. The estate, however, is inadequate to cover all the debts, so that some mechanism is needed for distributing the limited funds equitably. The Mishnah offers three examples of what it regards as the proper distribution. If the estate consists of 100, then it is divided equally, that is, each woman receives 33 1/3. If it consists of 300, it is divided proportionally, that is, each woman receives exactly half of her claim. If it consists of 200, then the woman whose claim is 100 receives 50 and the other two women receive 75 each.

The gemara notes that the Mishnah reflects the view of Rabbi Nathan but that Rabbi Yehuda HaNasi disagrees. Some opinions in the gemara further argue that even Rabbi Nathan’s rules only apply in special cases. Nevertheless, we might reasonably ask if there is some discernible principle that underlies the three cases. On the face of it, either equal distribution (as in the case of an estate of 100) or proportional distribution (as in the case of an estate of 300) makes good sense, but it is not clear why the one method should be used in one case and the other method in the other. Moreover, the case of the estate of 200 follows neither method and appears altogether mysterious.

Aumann noted that game theory provides the appropriate tools for analyzing the three cases of the Mishnah in order to find some general principle that could be applied to any size estate and any number of claimants with any size claims. In his original paper, written with Michael Maschler and published in an economics journal, Aumann shows that the solution offered by the Mishnah uniquely satisfies a particular set of constraints and can best be described in terms of a game-theoretic concept known as the nucleolus. In a later paper, written for an audience knowledgeable in Talmud but not necessarily in mathematics, Aumann sidesteps the game-theoretic technicalities that motivated the earlier results and explains the same ideas in terms of well-known Talmudic concepts.

The central idea, as presented in the latter paper, can be easily grasped. There is a well-known principle for dividing a disputed sum between two claimants. According to this method, described at the very beginning of Bava Metzia, each claimant first gets that part conceded by the other claimant and then what remains is divided equally. Thus, in the famous case of one claiming ½ and the other claiming the whole, the second claimant receives the ½ conceded by the first and the remaining half is split. This leaves the first claimant with ¼ and the second with ¾, as indicated in the Mishnah there. This same principle appears in several other places in the gemara.

The question that arises is: can this principle be generalized to more than two claimants? Aumann notes a remarkable fact about the three cases in the Mishnah in Kesubos: In each case, if one isolates any two of the three women, the total that the Mishnah gives the two collectively is divided between them according to the two-claimant principle of Bava Metzia. Consider, for example, the difficult case of the estate of 200 and the women who claim 100 and 200, respectively. The Mishnah gives the two women 125 collectively (50 and 75, respectively). Of this 125, the first concedes 25 (she only claims 100) and the second concedes nothing (her claim is greater than the total), so the second gets the conceded 25 and the remaining 100 is divided equally. This leaves the first with 50 and the second with 75, exactly as stated in the Mishnah. The same holds for all other pairs of women in all the cases discussed in the Mishnah.

Aumann concludes that the Mishnah in Kesubos is a generalization of the two-claimant principle in Bava Metzia. Most remarkably, Aumann proves that this generalization is unique. No solution to the three-claimant problem of Kesubos, other than the one offered, would be consistent in this manner with the two-claimant solution offered in Bava Metzia. In fact, there is only one such solution – and a constructive method for computing it – for any size estate and any number of women with any size claims.

Aumann’s generalization of the Mishnah’s problem is by no means the only possible one, (see R. Brody, “A Talmudic Principle of Distribution" in Higayon 1 for a different generalization) but it certainly serves as an outstanding example of the use of modern mathematical tools in the service of Torah study.

Tuesday, December 13, 2005

So we're a religious family in a frontier town. My younger daughter's favorite bedtime reading is Little House on the Prairie and my kids don't watch TV but rather DVDs of old episodes of The Waltons.

I've got to run off now to do me chores.

Thursday, December 08, 2005

I hate to bicker about the pig-headedness of people who basically share my views but you write what you know.

Miki Eitan is one of the few really excellent parliamentarians in the Knesset. He is intelligent, fair-minded and knows the rules. He is also one of the few politicians who really appreciates the extent to which the Supreme Court has become a runaway train. Finally, he is the only politician who is on top of the whole constitution issue. In my opinion, it's important that he get a safe slot on the Likud list and I've promised to help him with that.

He promised me that he would make a big fuss about the unprecedented severity in the treatment of arrested anti-disengagement protesters by police and prosecutors. He has called a special meeting of the Knesset Law Committee, which he chairs, for the purpose of raking the attorney general and the Justice Minister over the coals. He asked me to help him make as strong a case as possible.

My partner set off to get as many of the files of the 700(!) cases that remain open in order to help out with this. The people who have the most information about these cases, and who do terrific work, happen to live in Hevron. You might imagine that they'd jump at this opportunity. It is truly dinei nefashos, since most of these cases involve young people whose potential careers could be seriously jeopardized by a conviction.

But, no. They decided that by helping out, they'd be doing Miki a favor and for this they must exact a price. There is an endless case involving the shuk hasitonai in Hevron in which the attorney general is screwing over the Jewish community. (The details are not important for now, but suffice it to say that the AG is literally asking the court to steal Jewish property and hand it over to Arabs.) So the geniuses with the files aren't handing them over until Miki launches a pre-emptive offensive against the AG for a brief he is allegedly going to present in court next week.

How does one begin to explain the insanity of all this? These 700 kids whom these people have promised to help are being held hostage to an unrelated case. Miki is trying to do the right thing, and is indeed one of the only people in a position to help, and is being treated like he's asking them for a handout. And where is the sense of proportion in trading in dinei nefashos for dinei mamanos?

The answer to this is, unfortunately, all too clear.

Tuesday, December 06, 2005

In doing politics one encounters two very different kinds of political animals. There are those who view politics as a form of business: the object is to get the best possible deal. And there are those who view politics as a form of therapy: the object is to draw attention to one's purity of ideology and commitment.

The business types sometimes lose sight of exactly what it is that they're trying to maximize in all the deal-making. When all else is forgotten, the default is to maximize one's own glory. And the ideologues sometimes forget quite how much damage they do to their own cause by not knowing when it's time to cut a deal.

Most politicians are actually at their best precisely when they act against type. For example, Michi Ratzon is a Likud MK of the business variety, who took a principled stand against Sharon and, for a fleeting moment, actually sounded like a statesman. Uzi Landau, another Likud MK but of the ideologue variety, was trailing badly in the race to head the party and decided to cut a deal with Bibi Netanyahu. A number of good friends of mine who were very active in his campaign were devastated. But the truth is that it was a brief transcendent moment for Landau in which he actually sounded like a good politician.

Here ends the symmetry. The people I hang with are almost uniformly ideologues. This is partly because of the places I hang and partly because most people who don't actually have to get anything done have the luxury of ideological purity. (Conversely, there are those types who choose politics as a career from a young age -- think Ehud Olmert or Ofer Penis-Putz -- who regard principles as a liability.) Again and again, I run into people whose sole political objective is to shrei gevald, regardless of the price. Examples abound. Today, I offer one topical example. More will follow.

The fence/wall that is going up mostly along the green line has reached my little town (MLT). The whole fence is one giant useless populist boondoggle that serves the single purpose of delineating a border in all the wrong places. This is not the place to explain everything that's wrong with it. The question is what should be done about it. Given the fact that about a million miles of fence have already been built and a trillion dollars have already been invested in it, you've got to figure that stopping it cold in its tracks is a long shot.

So Shaul Goldstein, the "mayor" of Gush Etzion, has invested a good deal of time lobbying the makhers to at least revise the route of the fence in those places where it is most damaging. For this he has been attacked by purists, who prefer: i. "issuing a clear statement against the very idea of a fence and winning public support" (this from a guy who brought down Bibi for being too soft, so that we got Barak instead; he remains convinced that the masses are behind him) and ii. "sabotaging the bulldozers and lying down in front of the tractors".

To be sure, there are circumstances in which I myself would lie down in front of the tractors. But the first line of defense has always got to be to try to cut the best deal possible because statements and sabotage don't ever bring results; they are purely therapeutic measures. Unfortunately, though, some ideologues need to first and foremost vilify anybody who actually tries to get something done. They view a deal as the least satisfying possible outcome since it offers neither the ecstasy of victory nor the kind of self-righteous soul-cleansing catharsis that only a truly unambiguous screwing over can elicit.

Tomorrow (bli neder) I'll discuss an astounding tale of misplaced priorities in Hevron.