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Did e almudi Know of e
By Roman A. Ohrenstein Ph.D

Last month, Hebrew Univeristy's Robert Auman won the Noble Prize for Economics, for his contribution to the modern Game Theory. This essay, part one of two, will explore how the Talmudic sages, 2000 years ago, utilized this theory in their Talmudic dialectic.

The purpose of this essay is to demonstrate that within the purview of the Talmudic dialectics, the game method was often utilized to solve intricate questions, theoretical as well as practical.  This is particularly evident in Talmudic discussions of transactional issues involving conflicting business interests, where complicated decisions have to be made under conditions of uncertainty.  In the course of those discussions, the scholars reason in categories, which are similar to some of those employed in modern game theory.

Games as such, whether of chance or of strategy (upon which game theory is based), have been practiced since time immemorial, including among the ancient Hebrews.1

The Talmudic scholars were familiar with their characteristics, and they also utilized games of strategy tactics both in their scholarly discourses and in their solving of transactional disputes.  The historical background, methodological process, and approach to the solution of conflict situations in the Talmud combined to promote employment of game method in the quest for the solution of certain economic problems.

 The Minimax Principle

Game theory is one of the youngest contributions to modern economic analysis.  It is a theory of conflict situations that likens economic behavior to games of strategy, such as poker, chess, and even war.  The modern origins of the theory may be traced to the second decade of the past century, but it was only clearly established as a scientific discipline through the publication, in 1944, of the Theory of Games and Economic Behaviour by John Von Neumann and Oskar Morgenstern.2

In essence, game theory is concerned with competitive economic behavior, and as in a real game, it is characterized by the presence of common factors, such as conflicting interests, incomplete information, and the interplay of rational decisions under conditions of uncertainty.  This, in turn, subjects the decision maker to certain risks.  Game theory may, therefore, help the risk-taker to select an optimum strategy.3

For our purpose, it will suffice to mention that the basic tool of game theory is a payoff matrix with alternative choices or strategies for parties involved in a conflict of interests, in which the actions of one party influences its rival.  For example, A and B, two opponents, are pursuing their self-interest.  Seen from A’s perspective, if he is to succeed, he must try to guess how his moves will be countered by his rival.  To put this colloquially, “what does he think I think he will do?”  A wrong guess might prove too costly.  How is the game resolved?  The simplest way is to list the worst possible result that an opponent could inflict, and find the strategies that realize the best outcome from the list.

Thus, in a contest between A and B, A will pursue a strategy of maximizing the minimum gain, (abb. Max-mim), while B will attempt to minimize his maximum loss (abbr. Min-max).  The optimum strategy (or mix of strategies) is then determined by the so-called minimax principle.  Initially, game theory was too complex for full-scale analysis of real-world cases.  But nowadays, it is commonly employed a.o. by various industries to solve intricate transactional problems.  In fact, game theory is now viewed as one of the most important developments in modern economics.

Background to the Talmudic Game Method

In order to prepare the ground for the discussion of the Talmudic game method, it is relevant to note something of both the historical background and the methodological approach, which characterizes the Talmudic debate.

Historically, games of chance were well known to the ancient world.  In fact, some Oriental gambling games go back as far as 2100 B.C.E.  The Lydians, Herodotus relates, were adept in the invention of games.  And gambling with dice was particularly fashionable at the Persian court during the reign of Cyrus the Young, at the end of the fifth century B>C>E>  As for the “early tribes of Germany,” Tacitus writes, they “make games of hazard a serious occupation even when sober….,” while in less sober moments, they even gambled themselves into slavery.4  The ancient Hebrews, too, were acquainted with gambling.  There is ample reference to guessing games in the Bible (Judg. 14:12ff: I Kings 10:1-3)  However, it was from the Mishnaic times onward that the rabbis described gambling as a form of robbery.  As for the characteristics of the Talmudic game method, three motifs distinguish the Talmudic debate: the transactional, the strategic and the playful.

The Transactional Motif

Historically, the transactional motif has its origin in the early annals of Talmudic literature.  Thus, in the Midrashic volume Sifrai (32:25), which dates back to ca. 200 C.E., the term Nosse V’noten (“give and take”) is used to designate scholarly controversies.  And in the Talmud, the two synonymous expressions Masso Umatan and Nosse V’noten are applied idiomatically to ordinary business dealings like “buying and selling.”  The Aramaic equivalent for “nosse V’noten” is Shakla V’taria, meaning “give and take.”5  These terms were probably derived from the barter practice prevalent in antiquity.

Characteristically, those transactional expressions were used interchangeably in Talmudic parlance, both for business dealings and scholarly discourse.  Nowadays, too, we speak of “selling an idea.”  And in the contemporary State of Israel, the term Masso U’matan is frequently employed to describe intricate diplomatic negotiations.

In view of the common historical origin and characteristics of those terms, as well as of their frequent usage among both the ancients and modern alike, it is evident that the transactional motif was an important feature of the Talmudic analytical thought process and, as we shall show, of practical application.

The Strategic Motif

 The strategic metaphor figures prominently in Talmudic parlance.  Military terms were frequently employed to describe the animated debates among the scholars.  The ingenious arguments and rebuttals, the sharp moves and countermoves, the paradigms, the questions and methods, were often portrayed in the Talmud as if they took place in a war zone on the battlefield.

Even before Amoraic period (ca. 200 CE), the rabbis referred to those debates in military terms.  Inasmuch as sharpness, lucidity, and precision were central in their discussions, the scholars used the military metaphors to highlight the virtues of the battles of the mind.  They referred to them as Mil-hamtah-shell-Torah, “the war for the sake of the Torah.”  It was a.o., a battle for clarity and understanding (Sanh. lllb).

Indeed, there are many references in the Talmud to this effect.  Suffice it to mention that the rabbis interpreted biblical verses that glorify military prowess as referring to the battles of the mind.  (Kidd 30b).  For example, the biblical verse in I Samuel 16:18

In which King David’s qualities are praised, is interpreted by R. Judah in the name of Rav (3rd cent. C.E.) as follows:…that is cunning in playing- “knowing the correct answers”; a man of war-knowing how to ‘give and take’ in the battles of the Torah”; prudent in matters-“knowing how to deduce one thing from another”; and comely person-“who demonstrates the proofs for his opinions”; and the Lord is with him-“the ruling is always in accordance with his views.” (Ber. 93b)6

As can be seen, the leitmotif of these interpretations is that warfare and debate have strategic similarities, and that in the war of ideas, “strategic depth” is the name of the game.

The Playful Motif

An interesting feature of this intellectual duel is that the discussions among the scholars were frequently gamelike.  The debaters, like seasoned players, were out looking and trying to one-up each other in a clash of wits to test the opponent’s acumen.  This gamelike method was purposely cultivated in order to elicit twinkles of insight and flashes of brilliance.  As the Talmud puts it: Lehaded bo et Ha-Talmidim-“to sharpen the intellect of the disciples” (Megillah 15b).

In fact, the Talmud relates on numerous occasions that the great masters purposely used faulty reasoning in order to test their disciple’s keenness of mind.  They wanted to know whether they would be sufficiently alert to spot mistakes.7  And since a brisk scholarly mind was called harif, “sharp,” they expressed their preference by quoting the following proverb:….”one grain of sharp (harif) pepper is worth more than a bucket full of pumpkins….” (Megillah 7a).  The harif was extolled because of his brilliance to resolve difficulties and his ability to solve problems.

In light of these motifs, it becomes apparent how the Talmudic sages came to extend this threefold method to all sorts of problem solving, especially when dealing with complicated questions of economic conflict.

Kubbiyah-A Zero- Sum Game

Games of strategy as well as of chance are frequently the object of discussion in the Talmud. Among the various games mentioned by the rabbis is a form of dice called kubbiyah.  Its name is derived from the Greek Kybeia, meaning cube.  It consists of small wooden, mostly painted, cubes and used in games of dice.  Inasmuch as in dice the outcome depends entirely on chance and cannot be affected by the cleverness of the players, it is deemed a game of pure chance.  The rabbis characterize kubbiyah as a gambling game where one man’s loss is another’s gain.  Today, such games are classified as a zero-sum game because the winnings are just offset by the losses, and like in roulette on an unbiased wheel, there is no “system” for playing dice.8

Although there is no system in such games, they are not completely devoid of some strategic elements.  For instance, amateurs who gamble for fun might resort to a randomized strategy.  Thus, in a coin-matching play, they can agree to flip the coin X number of times for a game sequence, with each choosing heads and tails half the time.  In this way, they can hope, according to the law of probability, to break even.  The same goes for the roulette wheel as well as for dice.  Still, here no one can individually affect the outcome of the game, unless the dice are loaded.

It is interesting to note that the Talmud )Sanh. 24b-25a) makes a distinction between the social status of the “professional gambler” and of the amateur who derives his livelihood from another profession.  Whereas the former is categorized as a “robber,” the latter is viewed by the sages more benignly.  The rationale for their opposition toward professional gamblers is explained in socioeconomic terms of Yishuv ha’Olam, i.e., as being detrimental to “social welfare.”  Thus, if we should apply the game theory criteria to the Talmudic distinction between different categories of gamblers, we may find another reason for the more lenient attitude toward amateur players than to professional gamblers.  The former who mostly play for fun may randomize the game to break even, while the latter are always out to “make a killing.”

However, games such as chess and poker, in which the players can utilize ingenuity to affect the outcome, are in the category of strategic games.  The essential difference between games of strategy and games of (pure) chance-according to McKinsey-“lies in the circumstances that intelligence and skill are useful in playing the former but not the latter.”9

Here, we shall concern ourselves with games of strategy.  We shall present two typical examples-the Asmakhta, and the highly sophisticated Talmudic minimax, called Pesharah.  They illustrate conflicting business situations in which strategic calculations are of central importance to the outcome of the business venture.

Asmakhta: A Positive-Sum Game

One way of understanding economic activity is to think of it as a game initially involving two single participants, like buyer and seller, producer and consumer, or lender and borrower.  Such exchanges are usually termed positive-sum games, because both parties expect positive gains from a given transaction.  The Talmudic Asmakhta meets those criteria.

What is an Asmakhta?  It is a legal concept with economic ramifications.  Generally, it pertains to a business deal between two parties, based on a promise but lacking in complete resolve or firm commitment.  For example, A pays a portion of his indebtedness to B, leaves the bill of debt with C as a security, and agrees to pay the full amount on the bill if at a stipulated time he should fail to pay the balance.  This transaction includes two obligations; one relating to the outstanding balance; the second, to the payment of the penalty (B.B. 168a).

Temporarily, we will ignore the legal question of whether or not such a promise constitutes a firm resolve on the part of A and under what circumstance an Asmakhta does or does not confer title,10 for our concern is mainly with the conflicting nature of the problem-its “strategic” significance and analytical relevance.  Accordingly, here is a two-party bargaining situation in which both participants expect a positive gain from this transaction.  The borrower expects to use the loan productively, make a profit, and repay the loan in due time.  Whereas the lender with the collateral on deposit, might calculate that in the event the borrower will not meet the deadline, he will forfeit the full amount of the bill in addition to the portion already paid.  Clearly, there are here both risks and opportunities for both of the parties involved.  But that is not all.  Suppose the borrower, for whatever reason, becomes destitute or ill.  This might well cause additional risks and complications (see Ned. 27b), and require additional precautionary measures as well as careful strategic planning.

In this instance we have an example of a conflict situation in which the outcome is controlled partly by one side and partly by the opposite side.  It is one of many Asmakhtot discussed in the Talmud that illustrates that even in such a simple “positive-sum” game, a proper strategy must be selected for the purpose of minimizing the losses and maximizing the gains.  What is the consensus of the sages concerning an Asmakhta?  The consensus is that an Asmakhta does not confer title.  The reasoning here is that when one of the parties binds himself to an exaggerated penalty, it indicates a lack of “perfect intention” on the part of that person.  As summarized by Marcus Jastrow,  “It gives the claimant no rights, because the law presumes that he who made such a promise could not have meant it seriously, but had in view only to give the transaction the character of good faith and solemnity.”11

The consensus of the sages, it must be emphasized, cannot be construed as providing an economic solution to what was a legally defective business deal in its foundations.  Further, in the above example, Asmakhta is merely illustrative of a positive-sum game, and does not enter into the technical elements of game theory.  It is notable however, because of the direction it indicates.  More directly relevant, in terms of game theory, is the rabbinic concern with Pesharah.

The Talmudic Minimax: Pesharah

In the course of daily events, people may encounter personal or business situations of a conflicting nature for which the law makes no provision.  Sometimes the situation is so entangled that even the law may find it difficult to disentangle.  When in such a predicament, it is prudent to resort to a compromise.

The rabbis examined such conflict situations analytically and suggested a rational strategy for their resolution by means of Pesharah,.  The term Pesharah, as used by the Tannaim, denotes compromise or conciliation as well as arbitration.  As an illustration, they used relevant models that show that the most prudent way to resolve such conflicts is through a strategic compromise.

Accordingly, the Mishnah in Ketubot X:6 discusses three typical conflict situations that necessitate a policy of strategic rationality.  For a pointed example, we shall select one of those models that illustrate its gamelike nature and solution.

 (Continued next week).



  1. Consult “Games,” Encyclopedia Judaica (Jerusalem: Keter, 1971), Vol. 7, pp. 303-304 and “Gambling” ibid., pp. 299-303.
  2. J. von Neumann and O. Morgenstern, Theory of Games and Economic Behaviour (Princeton, N.J.;Princeton University Press, 1944).
  3. A recent discussion is, Mohammed Dore et all, (eds.), John Von Neumann and Modern Economics (Oxford: Clarendon, 1989), pp. 151-162.  See in particular, Thinking Strategically, by Avinash Dixit and Barry Nalebuff (New York-London: W.W. Norton & Co., 1991).  It is an interesting popularly written book.  For an excellent review of this subject, see Andrew Schotter & Gerhard Schwodiauer, “Economics and Game Theory:  A Survey,” Journal of Economic Literature (June 1980, Vol. XVIII, No. 2), pp. 479-527.  See also, Maskin, Eric, Recent Developments in Game Theory (England: Edward Elgar Publ. Inc., 1999).
  4. Encyclopedia Britannica, Vol. 9, pp. 1119-1121 offers further detail.
  5. Yoma 86a, Shabbat 120a, Tossefta, Gittin V:1, Baba Kama 72a, Sotah 7b, and in many other places.
  6. See Louis Jacobs, The Talmudic Argument (Cambridge University Press, 1984), p.8-11.
  7. See Taan, 7a; Nazir 59b; Zev. 13a; Hulin 43b; and other places.
  8. See  Sanh. 24b, along with Rashi’s explanation, loc. Cit.  Note: in describing the various categories of games, we are concerned here only with their characteristics for purposes of illustration and as a point of departure for further analysis.
  9.  J.C.C. McKinsey, Introduction to the Theory of Games (New York: McGraw-Hill, 1952), pp. 1-5 ff.
  10. For a discussion of this subject, see B. Bathra 168a, B. Metz. 66a; Ned. 27b, along with the various commentaries and codes.
  11. Dictionary of Talmudic and Midrashic Literature, p. 94.
  12. In this quotation, the author has designated the money sums in terms of dollars for the sake of clearer explication of the dilemma.
Posted on November 24, 2005
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