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The goal of this paper is to analyze the relationship between the two notion of rationality devised by John von Neumann in his classic Theory of Games and Economic Behavior, namely, the minimax and the expected utility rule. It is argued that, when

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DO PRUDENT AGENTS PLAY LOTTERIES?VON NEUMANN’S CONTRIBUTION TO THETHEORY OF RATIONAL BEHAVIOR
BY
NICOLA GIOCOLI
I. INTRODUCTION
The year 2003 marked the 100th anniversary of the birth of John von Neumann (1903-1957), one of greatest geniuses of the last century. Beyond contributing to ﬁelds asdiverse as set theory, quantum mechanics, atomic energy, and automatic computing,von Neumann has also had a decisive inﬂuence upon modern economics. From theinvention of game theory to the axiomatization of expected utility, from the introduc-tion of convex analysis and ﬁxed-point techniques to the development of the balancedgrowth model, the von Neumann heritage can be clearly traced in several areas of ourdiscipline. The aim of this paper is to clarify the relationship between the two conceptsof rationality he devised in his classic 1944 book
Theory of Games and Economic Behavior
, written with the collaboration of the Austrian economist Oskar Morgenstern(von Neumann and Morgenstern 1953).The
Theory of Games and Economic Behavior
contains both a notion of rationalbehavior under conditions of strategic—or non-parametric—uncertainty, the
minimax criterion
, and a notion of rational behavior under parametric uncertainty,the
expected utility rule
. According to the former, whenever a player is involved ina particular kind of strategic situation, called two-person zero-sum game, it is rationalto choose the strategy that gives her the maximum payoff among the worst possibleoutcomes that may arise due to the other player’s choice, i.e., that minimizes themaximum loss that she may suffer as an effect of the other player’s behavior. Accord-ing to the latter, whenever an agent is asked to select an action among a set of alterna-tives whose eventual outcomes are uncertain, it is rational to choose the action thatmaximizes a particular function called the expected utility function, that is, the sum
Department of Economics, University of Pisa, via Curtatone e Montanara 15, 56126, Pisa, Italy. E-mail: gioco-li@mail.jus.unipi.it. I thank Marco Dardi, Robert Leonard, Phil Mirowski, Aldo Montesano, Ivan Moscati, SalimRashid, Roy Weintraub, the two referees, and all the participants to the “John von Neumann centenary session” atthe 2003 meeting of the
History of Economics Society
. The ﬁnancial support of MIUR PRIN 2002 “Mathematicsin the History of Economics” is gratefully acknowledged. I bear full responsibility of any remaining error.ISSN 1042-7716 print; ISSN 1469-9656 online/06/010095-15
#
2006 The History of Economics SocietyDOI: 10.1080/10427710500509714
Journal of the History of Economic Thought,Volume 28, Number 1, March 2006
of the utilities of the outcomes associated with a certain action, each weighted with therespective probability.All this sounds standard and safe, so much so that, given the overwhelming roleplayed by von Neumann in writing the analytical parts of the
Theory of Games and Economic Behavior
, it seems straightforward to view him as the inventor of the tworules that feature in, respectively, section 17 (the minimax) and section 3 (expectedutility theory) of the 1944 book.
1
Yet, while there can be no doubt that the maingoal of von Neumann and Morgenstern was to provide a new characterization of rational behavior valid for strategic situations,
2
so that the minimax truly representsthe logical pillar of their whole theory, it is by no means certain that they alsoaimed at devising a rule of rational choice under parametric uncertainty. On the onehand, in the ﬁrst chapter of the book they explicitly downplayed the theoretical andempirical relevance of choice situations involving just one decision-maker. On theother hand, it turns out that the role of expected utility theory in the
Theory of Games and Economic Behavior
was simply to provide a pragmatic solution to a neces-sity that at least the economic side of the co-authorship felt as extremely urgent,namely, that of building a bridge between the traditional characterization of rationalityand the brand new game-theoretic approach, in order to better capture the attention of orthodox neoclassical economists and persuade them of the superiority of the latter.Hence, no real decision theory is developed in the
Theory of Games and Economic Behavior
and it would be wrong to ascribe to von Neumann
two
characterizationsof rationality. Indeed, the expression “rational behavior” has one and only onemeaning throughout the
Theory of Games and Economic Behavior
, that is, to abideby the minimax criterion.
3
As I will detail in the ﬁnal section, this conclusion has important consequences for acorrect reconstruction of the history of post-World War II neoclassical economics andfor the proper placement of John von Neumann in such a history. As far as the charac-terization of rationality is concerned, von Neumann-the-mathematician should in factbe considered as closer in spirit to those economists who defended a view of the econ-omic agent as a real human being, endowed with her motives and passions and capableof justifying her choice criteria, than to those who have reduced the individual to apurely formal entity, whose pattern of behavior is identiﬁed by a set of axiomaticproperties.The structure of the paper is as follows. In the next section I employ a particularclassiﬁcation of rationality notions to clarify the difference between the minimaxand the expected utility rule. In section 3, I tackle the issue of the kind of rationality
1
The minimax rule had been ﬁrst proposed by von Neumann in his pioneering 1928 paper on game theory: seevon Neumann (1928). The axiomatic derivation of the expected utility rule was added as an Appendix only in thesecond (1947) edition of the
Theory of Games and Economic Behavior
.
2
See, for example, von Neumann and Morgenstern (1953, pp. 1, 9, 31, 46).
3
Abstraction being made of course of the very special case of the decision problem tackled by an isolated Robin-son Crusoe; see below, section 3. Note that even the analysis of games with more than two players and
/
or withoutthe zero-sum constraint, as well as the whole theory of the stable set, which cover the last 400 pages of the
Theoryof Games and Economic Behavior
, are based upon the minimax: not only is every player’s choice to join a par-ticular coalition driven by minimax rationality, but also every coalition is modeled by von Neumann and Mor-genstern as being engaged in a two-person zero-sum game against all the other players, so that it has to stick to the minimax rule.
96 JOURNAL OF THE HISTORY OF ECONOMIC THOUGHT
suggestedinthe
TheoryofGamesand EconomicBehavior
.The role ofexpected utilitytheory in von Neumann’s analysis is explained in section 4. In the following section Ishow why the modern Bayesian approach togame and decision theory fails to meet therequirements of rationality of
Theory of Games and Economic Behavior
. Finally, I willargue in favor of the historical relevance of a proper assessment of von Neumann’sapproach to rationality.
II. A CLASSIFICATION OF RATIONALITY NOTIONS
In a 1998 paper, the Italian economist Aldo Montesanohas proposeda classiﬁcation of the notions of rationality that are commonly employed in modern economics. Inparticular, he distinguishes between the “rationality of actions” and the “rationalityof the preference system.”The expression “
rationality of actions
” (also called “rationality of agents”) refersto the circumstance that agents choose according to a criterion; it is the existence of such a criterion that makes an agent’s action rational (Montesano 1998, pp. 290–91).In principle there can be many such criteria, though the one that is universally adoptedin economics is the maximization rule, i.e., “choose the action that gives you themaximum satisfaction,” or, equivalently, “choose according to a complete and tran-sitive preference system.” Yet, Montesano observes that neither the agent’s prefer-ences nor their properties are really necessary for the deﬁnition of the “rationalityof actions.”
4
To declare that rational agents are those who choose according to acriterion is in fact equivalent to saying that they are those who behave
consistently
.Thus, a formal condition of consistency—better known under the name of weak axiom of revealed preference—is all we need to develop all the implications of rational economic behavior, including general equilibrium theory (Montesano1998, p. 291).There is however another notion of rationality which, though unnecessary for theresults of general equilibrium analysis, is highly convenient from the viewpoint of many applications of modern economics. It is a notion descending from the attributionof particular properties to the agent’s preferences, so that it is properly called byMontesano “
rationality of the preference system
”: “The preference system onactions is said to be rational if it is rationally based on the consequences of actions”(Montesano 1998, p. 292). The truly interesting case is, of course, the general onewhere actions and consequences do
not
coincide, so that the preferences over the con-sequences do not simply overlap those over the actions. In such a case the “rationalityof the preference system” requires the validity of three different rationality features:the rationality of the preference system over the outcomes, the rationality of the expec-tations over the outcomes and the rationality of the rule that employs the previous twofeatures to determine the preference system over the actions (Montesano 1998,p. 292). Leaving the reader to refer to Montesano’s paper for the details, what
4
One might say that the only “preference” the decision-maker is required to hold is that for the choice criterionitself, i.e., either for the maximization rule or, in the case of von Neumann’s games (see below), for prudence. Yetthis “preference” for a speciﬁc criterion is not a preference in the formal sense of modern decision theory.
DO PRUDENT AGENTS PLAY LOTTERIES? 97
really matters for us here is to note that the latter rule forms the subject matter of stan-dard decision theory, where the main problem is precisely that of determining a ruleallowing the agent to make her choices, that is, to pass from the preference and expec-tations over the outcome of her actions to the preference over the actions themselves.As I will explain below, such a rule is said tobe rational if it satisﬁes a pre-assigned listof axioms.Let us now apply Montesano’s classiﬁcation to the notions of rational behaviordeveloped by John von Neumann. Being nothing but a criterion of choice to befollowed in particular strategic situations called two-person zero-sum games, theminimax rule is clearly a case of “rationality of actions.”
5
It is the rule that answersthe main analytical question of the
Theory of Games and Economic Behavior
,namely, that of providing the content of an imaginary “handbook for the goodplayer” capable of instructing the agent on how to behave rationally when affectedby strategic uncertainty.The idea itself of characterizing rational behavior in terms of a “good” or “best”way of playing
6
shows that von Neumann’s 1944 goal was clearly
prescriptive
.This in turn explains why he aimed at an “
objective
” characterization of strategicrationality, that is, one that depended exclusively on the pre-deﬁned structure of thegame payoffs and that were autonomous of the players’ mental processes and of their opinion on the other players’ mental processes. A true “handbook” notion of rationality requires in fact that no reference be made to the agents’ expectations,beliefs, etc., lest its general prescriptive validity be undermined. The minimax satisﬁesthe “objectivity” requirement as it allows the player to warrant herself (at least) acertain payoff—called security value—regardless of the rival’s behavior.To show that the minimax is indeed a case of “rationality of actions” we just have toobserve that the notion of “choosing according to a criterion” entails that the propertyof rationality can be attributed to a certain behavior only if the agent can provide a justiﬁcation for her choice. The minimax allows
two
such justiﬁcations. First, aplayer may explain her minimax behavior in terms of the legitimate motive of
prudence
, that is, of the desire to avoid incurring in large losses; alternatively, shemay refer to her willingness to regain the
full control
over the outcome of heractions even in a game setup, that is, to sterilize the impact of strategic uncertainty.
7
Thus, the minimax rule is a very particular case of “rationality of actions,” where thecriterion to be followed is not “choose so that to maximize your satisfaction,” butrather “choose so that to obtain at least a certain outcome” or “choose so that toachieve the full control over the outcome of your choice.” Finally, much like the stan-dard notion of “rationality of actions,” the minimax can also easily meet the require-ment of consistency. Indeed, one could conceive of a suitably modiﬁed version of the
5
Indeed, it is not difﬁcult to also characterize the minimax as a case of “rationality of the preference system”: justtake a player to be so risk adverse that she subjectively assigns probability one to her rival’s choice of the actionthat would cause her the maximum loss. However, as I show below (section 5), such a characterization wouldclearly clash with von Neumann’s refusal to employ subjective probabilities to model strategic uncertainty.
6
See, for example, von Neumann and Morgenstern (1953, p. 100).
7
More on this below. Note that both the theme of achieving an “objective outcome” of the game and that of theplayers’ “control” over their payoff were already the central ones in von Neumann’s 1928 paper.
98 JOURNAL OF THE HISTORY OF ECONOMIC THOUGHT
weak axiom of revealed preference capable of capturing the idea that a rational playeris consistent in her use of the minimax rule.
8
What about the expected utility rule? Again, there is little doubt that the criterionembodies a theory of rationality and, in particular, a theory of the rationality of the rulethat, starting from the agent’s expectations and preferences over the outcomes, deter-mines her preferences over the actions. Indeed, in modern decision theory the latterpreferences are said to be rational if and only if they can be represented with theexpected utility rule, that is, if and only if they satisfy the axioms introduced byvon Neumann in section 3 and the Appendix of the
Theory of Games and Economic Behavior
(basically, the continuity and independence axioms). Hence, the axiomsof expected utility theory are necessary and sufﬁcient for the rationality of theagent’s preferences over the actions, that is, for rational decision theory.Of course, alternativesets of axioms—andthus alternativenotions of“rationality of preferences”—can be suggested in order to account for, say, the non-availability of objective probability distributions, as in Leonard J. Savage’s subjective expectedutility theory (Savage 1954), or the non-desirability of some of the strongest propertiesof these distributions, like in recent approaches based upon non-additive probabilities(Gilboa 1987, Schmeidler 1989). As remarked by Montesano (1998, p. 294), this mul-tiplicity simply reﬂects the circumstance that when we move from the “rationality of actions” to the “rationality of preferences” the role of the model-maker is enhanced, asit is up to her to select the list of axioms that the modeled preferences need to satisfy tobe called “rational.” Obviously, the criterion underlying the “rationality of actions” isalso imposed by the model-maker, but while such a criterion has to be immediatelyand directly justiﬁable in terms of the agent’s motives and desires (for example, sat-isfaction, prudence, etc.), the axioms supporting the “rationality of preferences” neednot be so, as they can be merely technical properties required just for analyticalconvenience.
9
Once more, it seems safe to conclude that in 1944 von Neumann bestowed on econ-omics two novel notions of rationality: one was a “rationality of actions” notion, theminimax rule, the other was a “rationality of preferences” notion, the expected utilityrule. The point is that, as I argue in the next sections and differing from the way it ispresented in modern textbooks, von Neumann and Morgenstern’s own expected utilitytheory was
not
a theory of the “rationality of preferences.”
8
Moreover, as most other solution concepts, the minimax also warrants the existence of an
interpersonal
kind of consistency: if bothplayers followthe minimax rule, theirstrategiesare mutually consistentandno player hasanyreason to change her behavior, so that the game is actually “solved.” Yet, von Neumann was not so much inter-ested in solving games, but rather in prescribing strategic rationality. This is demonstrated by the self-imposedrequirement that the criterion of rational play be robust to the possibility that the rival behaved
irrationally
(see von Neumann and Morgenstern 1953, p. 32). Hence the later defense of the minimax in terms of itsbeing the best strategy against the rival’s own minimax choice (see, for example, Luce and Raiffa 1957, pp.62–63), while remarkable under both an analytical and a historical viewpoint (especially because of its represent-ing the game-theoretic counterpart of the ﬁxed-point technique which was to become so crucial in postwar econ-omics: see Giocoli 2003b) and while being somehow anticipated in the
Theory of Games and Economic Behavior
itself (cf. pp. 147–48), does not capture the main motive behind von Neumann’s endorsement of that criterion.
9
See,e.g.,the far-from-convincingdefenseoftheirownaxiomsoffered byvonNeumannandMorgenstern(1953,pp. 28–29, 630–32) who, as I argued elsewhere, were hardly in favor of a purely formal approach to the axio-matic method (Giocoli 2003b, p. 31).
DO PRUDENT AGENTS PLAY LOTTERIES? 99

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