aumann-lecture

Prize Lecture1, December 8, 2005by

RobertJ. Aumann

Center for the Study of Rationality, and Department of Mathematics, TheHebrew University, Jerusalem, Israel.

“Wars and other conflicts are among the main sources of human misery.”Thus begins the Advanced Informationannouncement of the 2005 Bank ofSweden Prize in Economic Sciences in Memory of Alfred Nobel, awarded forGame Theory Analysis of Conflict and Cooperation. So it is appropriate todevote this lecture to one of the most pressing and profound issues that con-front humanity: that of War and Peace.

I would like to suggest that we should perhaps change direction in our ef-forts to bring about world peace. Up to now all the effort has been put intoresolving specific conflicts: India–Pakistan, North–South Ireland, variousAfrican wars, Balkan wars, Russia–Chechnya, Israel–Arab, etc., etc. I’d like tosuggest that we should shift emphasis and study war in general.

Let me make a comparison. There are two approaches to cancer. One isclinical. You have, say, breast cancer. What should you do? Surgery?Radiation? Chemotherapy? Which chemotherapy? How much radiation? Doyou cut out the lymph nodes? The answers are based on clinical tests, simplyon what works best. You treat each case on its own, using your best informa-tion. And your aim is to cure the disease, or to ameliorate it, in the specificpatient before you.

And, there is another approach. You don’t do surgery, you don’t do radia-tion, you don’t do chemotherapy, you don’t look at statistics, you don’t lookat the patient at all. You just try to understand what happens in a cancerouscell. Does it have anything to do with the DNA? What happens? What is theprocess like? Don’ttry to cure it. Just try to understandit. You work with mice,not people. You try to make them sick, not cure them.

Louis Pasteur was a physician. It was important to him to treat people, tocure them. But Robert Koch was not a physician, he didn’t try to cure people.He just wanted to know how infectious disease works. And eventually, hiswork became tremendously important in treating and curing disease.

War has been with us ever since the dawn of civilization. Nothing has beenmore constant in history than war. It’s a phenomenon, it’s not a series of iso-

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A very lightly edited version of the 40-minute lecture actually delivered at the Royal SwedishAcademy of Sciences in Stockholm. We are grateful to Professor Nicolaus Tideman for pointingout an error in a previous version.

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lated events. The efforts to resolve specific conflicts are certainly laudable,and sometimes they really bear fruit. But there’s also another way of goingabout it – studying war as a general phenomenon, studying its general, defin-ing characteristics, what the common denominators are, what the differencesare. Historically, sociologically, psychologically, and – yes – rationally. Whydoes homo economicus– rational man – go to war?What do I mean by “rationality”? It is this: A person’s behavior is rationalif it is in hisbest interests, given hisinformation.With this definition, can war be rational? Unfortunately, the answer is yes;it can be. In one of the greatest speeches of all time – his second inaugural –Abraham Lincoln said: “Both parties deprecated war; but one would makewar rather than let the nation survive; and the other would accept war ratherthan let it perish. And the war came.”

It is a big mistake to say that war is irrational. We take all the ills of theworld – wars, strikes, racial discrimination – and dismiss them by calling themirrational. They are not necessarily irrational. Though it hurts, they may berational. If war is rational, once we understand that it is, we can at least some-how address the problem. If we simply dismiss it as irrational, we can’t ad-dress the problem.

Many years ago, I was present at a meeting of students at Yale University.Jim Tobin, who later was awarded the Prize in Economic Sciences in Memoryof Alfred Nobel, was also there. The discussion was freewheeling, and onequestion that came up was: Can one sum up economics in one word? Tobin’sanswer was “yes”; the word is incentives. Economics is all about incentives.So, what I’d like to do is an economic analysis of war. Now this does notmean what it sounds like. I’m not talking about how to finance a war, or howto rebuild after a war, or anything like that. I’m talking about the incentivesthat lead to war, and about building incentives that prevent war.

Let me give an example. Economics teaches us that things are not always asthey appear. For example, suppose you want to raise revenue from taxes. Todo that, obviously you should raise the tax rates, right? No, wrong. You mightwant to lowerthe tax rates. To give people an incentive to work, or to reduceavoidance and evasion of taxes, or to heat up the economy, or whatever.That’s just one example; there are thousands like it. An economy is a game:the incentives of the players interact in complex ways, and lead to surprising,often counter-intuitive results. But as it turns out, the economy really worksthat way.

So now, let’s get back to war, and how homo economicus– rational man – fitsinto the picture. An example, in the spirit of the previous item, is this. Youwant to prevent war. To do that, obviously you should disarm, lower the levelof armaments. Right? No, wrong. You might want to do the exact opposite. Inthe long years of the cold war between the US and the Soviet Union, whatprevented “hot” war was that bombers carrying nuclear weapons were in theair 24 hours a day, 365 days a year. Disarming would have led to war.

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The bottom line is – again – that we should start studying war, from all view-points, for its own sake. Try to understand what makes it happen. Pure, basicscience. Thatmay lead, eventually, to peace. The piecemeal, case-based ap-proach has not worked too well up to now.

Now I would like to get to some of my own basic contributions, some ofthose that were cited by the Prize Committee. Specifically, let’s discuss repeat-ed games, and how they relate to war, and to other conflicts, like strikes, andindeed to all interactive situations.

Repeated games model long-term interaction. The theory of repeatedgames is able to account for phenomena such as altruism, cooperation, trust,loyalty, revenge, threats (self-destructive or otherwise) – phenomena that mayat first seem irrational – in terms of the “selfish” utility-maximizing paradigmof game theory and neoclassical economics.

That it “accounts” for such phenomena does not mean that people deliber-ately choose to take revenge, or to act generously, out of consciously self-serv-ing, rational motives. Rather, over the millennia, people have evolved normsof behavior that are by and large successful, indeed optimal. Such evolutionmay actually be biological, genetic. Or, it may be “memetic”; this word derivesfrom the word “meme,” a term coined by the biologist Richard Dawkins toparallel the term “gene,” but to express social, rather than biological, hered-ity and evolution.

One of the great discoveries of game theory came in the early seventies,when the biologists John Maynard Smith and George Price realized thatstrategic equilibrium in games and population equilibrium in the livingworld are defined by the same equations. Evolution – be it genetic or mem-etic – leads to strategic equilibrium. So what we are saying is that in repeatedgames, strategic equilibrium expresses phenomena such as altruism, co-operation, trust, loyalty, revenge, threats, and so on. Let us see how that worksout.

What do I mean by “strategic equilibrium”? Very roughly, the players in agame are said to be in strategic equilibrium(or simply equilibrium) when theirplay is mutually optimal: when the actions and plans of each player are rationalin the given strategic environment – i.e., when each knows the actions andplans of the others.

For formulating and developing the concept of strategic equilibrium, JohnNash was awarded the 1994 Prize in Economics Sciences in Memory of AlfredNobel, on the fiftieth anniversary of the publication of John von Neumannand Oskar Morgenstern’s Theory of Games and Economic Behavior. Sharing thatPrize were John Harsanyi, for formulating and developing the concept ofBayesianequilibrium, i.e., strategic equilibrium in games of incomplete infor-mation; and Reinhard Selten, for formulating and developing the concept ofperfectequilibrium, a refinement of Nash’s concept, on which we will say morebelow. Along with the concepts of correlatedequilibrium (Aumann 1974,1987), and strongequilibrium (Aumann 1959), both of which were cited inthe 2005 Prize announcement, the above three fundamental concepts consti-tute the theoretical cornerstones of noncooperative game theory.352

Subsequent to the 1994 prize, two Prizes in Economic Sciences in Memoryof Alfred Nobel were awarded for applicationsof these fundamental concepts.The first was in 1996, when William Vickrey was awarded the Prize posthu-mously for his work on auctions. (Vickrey died between the time of the Prizeannouncement and that of the ceremony.) The design of auctions and of bid-ding strategies are among the prime practical applications of game theory; agood – though somewhat dated – survey is Wilson 1992.

The second came this year – 2005. Professor Schelling will, of course, speakand write for himself. As for your humble servant, he received the prize forapplying the fundamental equilibrium concepts mentioned above to repeatedgames. That is, suppose you are playing the same game G, with the same play-ers, year after year. One can look at this situation as a single big game – the so-called supergameof G, denoted G∞– whose rules are, “play Gevery year.” Theidea is to apply the above equilibrium concepts to the supergame G∞, ratherthan to the one-shot game G, and to see what one gets.

The theory of repeated games that emerges from this process is extremelyrich and deep (good – though somewhat dated – surveys are Sorin 1992,Zamir 1992, and Forges 1992). In the few minutes that are available to me, Ican barely scratch its surface. Let me nevertheless try. I will briefly discuss justone aspect: the cooperative. Very roughly, the conclusion is that

Repetition Enables Cooperation.

Let us flesh this out a little. We use the term cooperativeto describe any pos-sible outcome of a game, as long as no player can guaranteea better outcomefor himself. It is important to emphasize that in general, a cooperative out-come is notin equilibrium; it’s the result of an agreement. For example, inthe well-known “prisoner’s dilemma” game, the outcome in which neitherprisoner confesses is a cooperative outcome; it is in neither player’s best inter-ests, though it is better for both than the unique equilibrium.

An even simpler example is the following game H: There are two players,Rowena and Colin. Rowena must decide whether both she and Colin will re-ceive the same amount – namely 10 – or whether she will receive ten timesmore, and Colin will receive ten times less. Simultaneously, Colin must decidewhether or not to take a punitive action, which will harm both Rowena andhimself; if he does so, the division is cancelled, and instead, each player getsnothing. The game matrix is

Acquiesce

Divide Evenly

10

Divide Greedily

100

1

010

0

0Punish

0

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The outcome (E,A), yielding 10 to each player, is a cooperative outcome, asno player can guarantee more for himself; but like in the prisoner’s dilemma,it is not achievable in equilibrium.

Why are cooperative outcomes interesting, even though they are notachievable in equilibrium? The reason is that they are achievable by contract– by agreement – in those contexts in which contracts are enforceable. And thereare many such contexts; for example, a national context, with a court system.The Talmud (Avot 3, 2) says,

.ʥʲʬʡʭʩʩʧʥʤʲʸʺʠʹʩʠ ,ʤʠʸʥʮʠʬʮʬʠʹ ,ʺʥʫʬʮʬʹʤʮʥʬʹʡʬʬʴʺʮʩʥʤ

“Pray for the welfare of the government, for without its authority, man wouldswallow man alive.” If contracts are enforceable, Rowena and Colin canachieve the cooperative outcome (E,A) by agreement; if not, (E,A) is forpractical purposes unachievable.

The cooperative theory of games that has grown from these considerationspredates the work of Nash by about a decade (von Neumann andMorgenstern 1944). It is very rich and fruitful, and in my opinion, has yield-ed thecentral insights of game theory. However, we will not discuss these in-sights here; they are for another Prize in Economic Sciences in Memory ofAlfred Nobel, in the future.

What I do wish to discuss here is the relation of cooperative game theory torepeated games. The fundamental insight is that repetition is like an enforce-ment mechanism, which enables the emergence of cooperative outcomes inequilibrium– when everybody is acting in his own best interests.

Intuitively, this is well-known and understood. People are much more co-operative in a long-term relationship. They know that there is a tomorrow,that inappropriate behavior will be punished in the future. A businessmanwho cheats his customers may make a short-term profit, but he will not stay inbusiness long.

Let’s illustrate this with the game H. If the game is played just once, thenRowena is clearly better off by dividing Greedily, and Colin by Acquiescing.(Indeed, these strategies are dominant.) Colin will not like this very much – heis getting nothing – but there is not much that he can do about it.Technically, the onlyequilibrium is (G,A).

But in the supergame H∞, there issomething that Colin can do. He canthreatento Punish Rowena for ever afterwards if she ever divides Greedily. Soit will not be worthwhile for her to divide greedily. Indeed, in H∞this is actu-ally an equilibrium in the sense of Nash. Rowena’s strategy is “play Efor ev-er”; Colin’s strategy is “play Aas long as Rowena plays E; if she ever plays G,play Pfor ever afterwards.”

Let’s be quite clear about this. What is maintaining the equilibrium inthese games is the threat of punishment. If you like, call it “MAD” – mutually as-sured destruction, the motto of the cold war.

One caveat is necessary to make this work. The discount rate must not betoo high. Even if it is anything over 10% – if $1 in a year is worth less than 90354

cents today – then cooperation is impossible, because it’s still worthwhile forRowena to be greedy. The reason is that even if Colin punishes her – andhimself! – for ever afterwards, then when evaluated today, the entire eternalpunishment is worth less than $90, which is all that Rowena gains today by di-viding greedily rather than evenly.

I don’t mean just the monetary discount rate, what you get in the bank. Imean the personal, subjective discount rate. For repetition to engender co-operation, the players must not be too eager for immediate results. The pre-sent, the now, must not be too important. If you want peace now, you may wellnever get peace. But if you have time – if you can wait – that changes the wholepicture; thenyou may get peace now. t’s one of those paradoxical, upside-down insights of game theory, and indeed of much of science. Just aweek or two ago, I learned that global warming may cause a cooling of Europe,because it may cause a change in the direction of the Gulf Stream. Warmingmay bring about cooling. Wanting peace now may cause you never to get it –not now, and not in the future. But if you can wait, maybe you will get it now.The reason is as above: The strategies that achieve cooperation in an equi-librium of the supergame involve punishments in subsequent stages if co-operation is not forthcoming in the current stage. If the discount rates aretoo high, then the players are more interested in the present than in the future, and a one-time coup now may more than make up for losses in the sequel. This vitiates the threat to punish in future stages.

To summarize: In the supergame H∞ of the game H, the cooperative out-come (E,A) is achievable in equilibrium. This is a special case of a muchI

more general principle, known as the Folk Theorem, which says that anyco-operative outcome of anygame Gis achievable as a strategic equilibrium out-come of its supergame G∞– even if that outcome is not an equilibrium out-come of G. Conversely, every strategic equilibrium outcome of G∞is acooperative outcome of G. In brief, for any game G, we have

THEFOLKTHEOREM:The cooperative outcomes of Gcoincide with the equi-librium outcomes of its supergameG∞.

Differently put, repetition acts as an enforcement mechanism: It makes co-operation achievable when it is not achievable in the one-shot game. Ofcourse, the above caveat continues to apply: In order for this to work, the dis-count rates of all agents must be low; they must not be too interested in thepresent as compared with the future.

There is another point to be made, and it again relates back to the 1994Prize. John Nash got the Prize for his development of equilibrium. ReinhardSelten got the Prize for his development of perfectequilibrium. Perfect equi-librium means, roughly, that the threat of punishment is credible;that ifyouhave to go to a punishment, then after you punish, you are still in equilibrium– you do not have an incentive to deviate.

That certainly is notthe case for the equilibrium we have described in thesupergame H∞of the game H. If Rowena plays Gin spite of Colin’s threat,

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then it is notin Colin’s best interest to punish forever. That raises the ques-tion: In the repeated game, can (E,A) be maintained not only in strategicequilibrium, but also in perfectequilibrium?

The answer is yes. In 1976, Lloyd Shapley – whom I consider to be thegreatest game theorist of all time – and I proved what is known as the PerfectFolk Theorem; a similar result was established by Ariel Rubinstein, indepen-dently and simultaneously. Both results were published only much later(Aumann and Shapley 1994, Rubinstein 1994). The Perfect Folk Theoremsays that in the supergame G∞of any game G, any cooperative outcome of Gis achievable as a perfectequilibrium outcome of G∞– again, even if that out-come is not an equilibrium outcome of G. The converse of course also holds.In brief, for any game G, we have

THEPERFECTFOLKTHEOREM:The cooperative outcomes of Gcoincide withthe perfect equilibrium outcomes of its supergameG∞.

So again, repetition acts as an enforcement mechanism: It makes coopera-tion achievable when it is not achievable in the one-shot game, even whenone replaces strategic equilibrium as the criterion for achievability by themore stringent requirement of perfect equilibrium. Again, the caveat aboutdiscount rates applies: In order for this to work, the discount rates of allagents must be low; they must not be too interested in the present as com-pared with the future.

The proof of the Perfect Folk Theorem is quite interesting, and I will illus-trate it very sketchily in the game H, for the cooperative outcome (E,A). Inthe first instance, the equilibrium directs playing (E,A) all the time. IfRowena deviates by dividing Greedily, then Colin punishes her – plays P. Hedoes not, however, do this forever, but only until Rowena’s deviation becomesunprofitable. This in itself is still not enough, though; there must be some-thing that motivates Colin to carry out the punishment. And here comes thecentral idea of the proof: If Colin does not punish Rowena, then Rowenamust punish Colin – by playing G– for not punishing Rowena. Moreover, theprocess continues – any player who does not carry out a prescribed punish-ment is punished by the other player for not doing so.

Much of society is held together by this kind of reasoning. If you arestopped by a policeman for speeding, you do not offer him a bribe, becauseyou are afraid that he will turn you in for offering a bribe. But why should henot accept the bribe? Because he is afraid that you will turn him in for accept-ing it. But why would you turn him in? Because if you don’t, he might turnyou in for not turning him in. And so on.

This brings us to our last item. Cooperative game theory consists not onlyof delineating all the possible cooperative outcomes, but also of choosingamong them. There are various ways of doing this, but perhaps best known isthe notion of core, developed by Lloyd Shapley in the early fifties of the lastcentury. An outcome xof a game is said to be in its “core” if no set Sof play-ers can improve upon it – i.e., assure to each player in San outcome that is bet-356

ter for him than what he gets at x. Inter alia, the concept of core plays a cen-tral role in applications of game theory to economics; specifically, the coreoutcomes of an economy with many individually insignificant agents are thesame as the competitive (a.k.a. Walrasian) outcomes – those defined by a sys-tem of prices for which the supply of each good matches its demand (see,e.g., Debreu and Scarf 1963, Aumann 1964). Another prominent applicationof the core is to matchingmarkets (see, e.g., Gale and Shapley 1962, Roth andSotomayor 1990). The core also has many other applications (for surveys, seeAnderson 1992, Gabszewicz and Shitovitz 1992, Kannai 1992, Kurz 1994, andYoung 1994).

Here again, there is a strong connection with equilibrium in repeatedgames. When the players in a game are in (strategic) equilibrium, it is notworthwhile for any one of them to deviate to a different strategy. A strongequilibrium is defined similarly, except that there it is not worthwhile for anysetof players to deviate – at least one of the deviating players will not gainfrom the deviation. We then have the following

THEOREM(AUMANN1959):The core outcomes of Gcoincide with the strongequilibrium outcomes of its supergameG∞.

In his 1950 thesis, where he developed the notion of strategic equilibriumfor which he got the Prize in Economic Sciences in Memory of Alfred Nobelin 1994, John Nash also proposed what has come to be called the NashProgram– expressing the notions of cooperative game theory in terms ofsome appropriately defined noncooperative game; building a bridge betweencooperative and noncooperative game theory. The three theorems presentedabove show that repetition constitutes precisely such a bridge – it is a realiza-tion of the Nash Program.

We end with a passage from the prophet Isaiah (2, 2–4):

.םיוגהלכוילאורהנו ,תועבגמאשינו ,םירההשארבייתיברההיהיןוכנ ,םימיהתירחאבהיהויכ ;ויתוחרואבהכלנו ,ויכרדמונרויו ,בקעייהלאתיבלא ,יירהלאהלענווכל ,ורמאוםיברםימעוכלהו ,םיתיאלםתוברחותתיכו ;םיברםימעלחיכוהו ,םיוגהןיבטפשו .םלשורימיירבדו ,הרותאצתןויצמ

.המחלמדועודמליאלו ,ברחיוגלאיוגאשיאל ;תורמזמלםהיתותינחו

“And it shall come to pass ... that ... many people shall go and say, ... let us goup to the mountain of the Lord, ... and He will teach us of His ways, and wewill walk in His paths. … And He shall judge among the nations, and shall re-buke many people; and they shall beat their swords into ploughshares, andtheir spears into pruning hooks; nation shall not lift up sword against nation,neither shall they learn war any more.”

Isaiah is saying that the nations can beat their swords into ploughshareswhen there is a central government – a Lord, recognized by all. In the ab-sence of that, one canperhaps have peace – no nation lifting up its swordagainst another. But the swords must continue to be there – they cannot bebeaten into ploughshares – and the nations must continue to learnwar, in or-der notto fight!

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REFERENCES

Anderson, R. M., 1992, “The Core in Perfectly Competitive Economies,” in Aumann andHart 1992, 413–457.

Aumann, R. J., 1959, “Acceptable Points in General Cooperative n-Person Games,” inContributions to the Theory of Games IV, Annals of Mathematics Study 40, edited by A. W.Tucker and R. D. Luce, Princeton: at the University Press, 287–324.

Aumann, R. J., 1964, “Markets with a Continuum of Traders,” Econometrica32, 39–50.

Aumann, R. J., 1974, “Subjectivity and Correlation in Randomized Strategies,” Journal ofMathematical Economics1, 67–96.

Aumann, R. J., 1987, “Correlated Equilibrium as an Expression of Bayesian Rationality,”Econometrica55,1–18.

Aumann, R. J. and Hart, S. (eds.), 1992, 1994, 2002, Handbook of Game Theory, with economicapplications, Vols. 1, 2, 3, Elsevier, Amsterdam.

Aumann, R. J. and Shapley, L. S., 1994, “Long-Term Competition: A Game-TheoreticAnalysis,” in Essays in Game Theory in Honor of Michael Maschler, edited by N. Megiddo,Springer, New York, 1–15.

Debreu, G. and Scarf, H., 1963, “A Limit Theorem on the Core of an Economy,”International Economic Review 4,235–246.

Forges, F., 1992, “Repeated Games of ncomplete nformation: Non-Zero-Sum,” inAumann and Hart 1992, 155–177.

Gabszewicz, J. J. and Shitovitz, B., 1992, “The Core in Imperfectly Competitive Economies,”in Aumann and Hart 1992, 459–483.

Gale, D. and Shapley, L. S., 1962, “College Admissions and the Stability of Marriage,”

American Mathematical Monthly 69,9–15.Kannai, Y., 1992, “The Core and Balancedness,” in Aumann and Hart 1992, 355–395.

Kurz, M., 1994, “Game Theory and Public Economics,” in Aumann and Hart 1994,1153–1192.

Peleg, B., 1992, “Axiomatizations of the Core,” in Aumann and Hart 1992, 397–412.

Roth, A. and Sotomayor, M., 1990, Two-Sided Matching: A Study in Game-Theoretic Modelingand Analysis, Econometric Society Monograph Series, Cambridge: at the University Press.Rubinstein, A., 1994, “Equilibrium in Supergames,” in Essays in Game Theory in Honor ofMichael Maschler, edited by N. Megiddo, Springer, New York, 17–28.

Sorin, S., 1992, “Repeated Games with Complete Information,” in Aumann and Hart 1992,71–107.

von Neumann, J., and Morgenstern, O., 1944, Theory of Games and Economic Behavior,Princeton: at the University Press.

Wilson, R., 1992, “Strategic Analysis of Auctions,” in Aumann and Hart 1992, 227–279.Young, H. P., 1994, “Cost Allocation,” in Aumann and Hart 1994, 1193–1236.

Zamir, S., 1992, “Repeated Games of Incomplete Information: Zero-Sum,” in Aumann andHart 1992, 109–154.Portrait photo of Robert J. Aumann by photographer D. Porges.

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