Maurice Allais: a review of his major works, a memoriam, 1911-2010.

Ramrattan, Lall ; Szenberg, Michael

Introduction

Maurice Felix Charles Allais was a French economist, engineer, historian, and physicist. According to Paul Samuelson, "he was a fountain of original and independent discoveries" and a "part of a Paris renaissance in economic theory. Had Allais' earliest writings been in English, a generation of economic theory would have taken a different course" (Samuelson, V. 5, 1986, 83-85). Indeed, a large portion of his work is still not available in English. Nevertheless, Allais' translated literature is more than enough to paint a portrait of his genius.

In this memoriam, we focus on the work for which he won the Nobel Memorial Prize in economics in 1988, namely, "for his pioneering contributions to the theory of markets and efficient utilization of resources." Allais interprets this contribution in a way that is synonymous with the popular definition of the economic problem: "I should like to interpret this motivation in its broadest sense, that is to say, as relating to all those conditions which may ensure that the economy satisfies with maximum efficiency the needs of men given the limited resources they have at their disposal" (Allais, AER, 1997, 3). He identified his fundamental contribution to economics in five areas: "the theory of economic evolution and general equilibrium, of maximum efficiency, and of the foundations of 'economic calculus'; the theory of intertemporal processes and maximum capitalistic efficiency; the theory of choices under uncertainty and the criteria to be considered for rational economic decisions; the theory of money, credit, and monetary dynamics; and probability theory, as well as the analysis of time series and their exogenous components" (Ibid., 1997, 4).

Allais came to economics with a strong background in physics and history. For experiments in physics he received the Galabert Prize from the French Astronautical Society, and a laureate prize from the Gravity Research Foundation of the U. S., both in 1959. In the area of history, he authored a book entitled Rise and Fall of Civilizations-Economic Factors. From 1933 to 1987, he received fourteen scientific prizes, the most distinguished being the Gold Medal from the National Center for Scientific Research in 1978 (Allais, 1992, 20-21).

Methodology

Allais had a broad methodological approach to economics. "Analysis of societies obviously requires a synthesis of all the social sciences: political economics, law, sociology, history, geography, and political science," he wrote (Allais, 1992, 31). In his economic writings, he took inspiration from the philosophy of Alexis de Tocqueville, Leon Walras, Irving Fisher, Vilfredo Pareto, and John Maynard Keynes. He pursued theory with facts, maintaining that his goals were "first, to constantly found [theories] on in-depth theoretical investigations; second, to always provide accompanying quantitative estimates" (Ibid., 29).

One manifestation of Allais's methodological approach can be illustrated by his well-known paradox. Peter Fishburn (1991, 28) classified its place in decision theory as "experiments that refute expected utility's ability to describe actual behavior" and as "non-linear alternatives to expected utility." Allais (1990, 8) claimed that it was not a mere counter-example, but that it was based on a general theory of random choice. Generally, examples show either that something makes sense or that something does not make sense (Gelbaum and Olmsted, 1964, V). Allais believed that a fundamental theory about the psychology of risk is missing in the expected utility hypothesis. It was a reaction also to an axiomatic theory--something that is accepted without proof and allows us to make logical (deductive) conclusions. Allais stated that "Mathematics is merely a tool for transforming statements. Real importance attaches only to

the discussion of the premises adopted and the result obtained" (Allais, 1979, 37).

In his counter-example, Allais did not treat the axiomatic method as a dead science (Samuelson, 1972, V. 3, 316). He organized his early works into five axioms--probability, ordered field of choice, absolute preference, composition, and an index of psychological values (Ibid., 457, 460). After revisiting his early studies, he added two more axioms--homogeneity and invariance of the index of psychological value, and cardinal isovariations (Ibid., 1979, 480-481). He also incorporated an axiom of Ole Hagen to the effect that "a constant increase in the utility of every outcome increases the utility of the entire prospect by the same amount" (Fishburn, 1987, 835).

In terms of mathematical tools, Allais indicated a partiality for the calculus over set theoretic approaches in economics. In developing his market economy concept, he wrote, "from an economic point of view, reasoning based on marginal equivalences and surpluses is very fruitful; it provides a better understanding of the underlying nature of economic phenomena than the demonstration, under very restrictive conditions, of the existence of a price vector sufficient for the equilibrium of a market economy" [Italics original] (Allais, 1978, 150). Overall, his view of mathematics in economics is that: "Formal rigor is of little value if it is accompanied by a serious distortion of the true nature of reality, and it is better to have an approximative theory that corresponds to actual reality than a formally rigorous theory that can only be built by seriously distorting the facts" (Ibid., 1978, 149). Furthermore, "Every theory is necessarily approximative, and the approximative nature of a theory is not a defect in itself. The only imperative that can justifiably be demanded of a theory is that it should not distort reality sufficiently to modify its nature" (Ibid., 1978, 148).

In his discussions on market economy, Allais changed the DNA, so to speak, of general equilibrium. Just as scientists found that arsenic can replace phosphorus at the core of DNA structure, Allais found that the pressure of free competition on human beings can replace price taking. Only in a stable state are input prices uniquely defined, because efficiency results from the pressure that free competition puts on human beings rather than on the price system (Munier, 1995, 20-21).

Major Works

Allais' first major book, In Quest of an Economic Discipline, 1943 enunciated an equivalence theorem about any state of equilibrium and any state of maximum efficiency. He defined four new concepts relating to "the surface of maximum possibilities in the hyperspace of preference indices of the consumption units; the concept of distributable surplus corresponding to a feasible modification of the economy from a given situation; the concept of loss, defined as the maximum distributable surplus for all feasible modifications of the economy which leave the preference indices unchanged; and the related concept of surfaces of equal loss in the hyperspace of preference indices" (Allais, 1997, 4).

In his second major book, Economy and Interest, 1947, we find precursors to many well-known models in modern economics. His intergenerational model was an extension of his notion of maximum efficiency. He touched on the theory of productivity of capital, which foreshadowed the modern contribution of the golden rule of optimal growth theory. We also find discussions on the transaction demand for money, behavioral economics, and his famous Allais Paradox. What follows is a sample of some of his major theories.

The Allais Paradox

This paradox relates to utility, preference, and probability. Allais was reacting to the axiomatic expected utility hypothesis that started with the Swiss mathematician and physicist Daniel Bernoulli (1700-1782). If one is offered an equal [50:50] chance of getting 0 or 20,000 ducats, the mathematical expectation theory yields a value of: 0.5(0) + 0.5(20,000) = 10,000 ducats. In general, if [x.sub.i] is the amount, and [p.sub.i] is the probability, the expected value is the average outcome, E([x.sub.i]) = [bar.x], and so, E([x.sub.i] - [bar.x]) = 0 is considered a fair price for the lottery. Let P be the person's expectation of profits. We can then compare P with [bar.x]. If E([x.sub.i] - P) > 0, the lottery will be favorable, and if E([x.sub.i] - P) < 0, then the lottery will be unfavorable (Jensen, 1967, 164).

Daniel Bernoulli (1954, 24) pointed out that "all men cannot use the same rule to evaluate the gamble ... the determination of the value of an item must not be based on its price, but rather on the utility it yields. The price of the item is dependent only on the thing itself and is equal for everyone; the utility, however, is dependent on the particular circumstances of the person making the estimate. Thus there is no doubt that a gain of one thousand ducats is more significant to a pauper than to a rich man though both gain the same amount."

Essential to the Bernoulli doctrine is the distinction between a physical fortune, x, and a moral fortune or utility, y. A change in utility dy is not only proportional to changes in physical fortune, dx, but inversely proportional to x as well. We can therefore write dy = k(dx/x), where k is the proportional constant. Integrating that equation yields: y = k log(x) + C. The constant of integration depends on a person's initial wealth defined as = [x.sub.0], which is estimated as C = -k log([alpha]) by setting y = 0. According to A. Marshall (1982, 693), Bernoulli thought of [alpha] as the "income which affords the barest necessaries of life." Utility is therefore represented by a log arithmetic function: y = k log(x) - k log([alpha])=k log(x/x) (Keynes, 1973, 350). This equation describes a curve that shows diminishing marginal utility of money and that people do not have linear utility (Samuelson, V. 5, 1986, 135). As the marginal utility of money declines, "it follows that the mathematical expectation of utility (rather than of money) in the game was finite, so that the individual would be willing to pay only a finite stake" (Arrow, 1984, V. 3, 23).

Daniel Bernoulli put his expected utility hypothesis to the task of explaining the St. Petersburg paradox, proposed by his cousin, Nicholas Bernoulli. The paradox is that "Peter tosses a coin and continues to do so until it should land 'heads' when it comes to the ground. He agrees to give Paul one ducat if he gets 'heads' on the very first throw, two ducats if he gets it on the second, four if on the third, eight if on the fourth, and so on, so that with each additional throw the number of ducats he must pay is doubled" (Bernoulli, 1954, 31). The payoff for the nth toss is [2.sup.n]. The mathematical expectation is [[SIGMA].sup.[infinity].sub.n=1] ([2.sup.n])(1/[2.sup.n]), which represents an infinite amount of money. To help illuminate this solution, we use the equivalent relation proposed by Allais, namely, cardinal utility function, V ~ a + [log C/log 2], where V is the psychological monetary value of the prospect, a is a proportional constant, and C is the player's capital (Allais, 1990, 3). We recall that 1/2 and 2/4 etc. belong to the same equivalent class of rational numbers. We test their equivalence by setting them equal and cross multiplication. Similarly, we can have u, v, w, etc. as equivalent classes in U. An equivalent, say v in U, is defined by a set of events. The events [x.sub.1], [x.sub.2] belong to an equivalent class if the agent is indifferent between them, i.e., [x.sub.1]/[x.sub.2] (Malinvaud, 1952, 679). With the Allais' formulation, V ~ S18 when a = 0.942, and C = $100,000, indicating that a player is not willing to pay much for the coin toss prospect (Allais, 1990, 3).

The Bernoulli model is concerned with risks, using objective probability such as the tossing of a coin to value an outcome. His model can be looked at as an objective expected hypothesis (OEH) that preserves the mathematical expectation by introducing a non-linear utility function which was extended by von Neumann and Morgenstern (1944). Thomas B ayes (1763) seems to have reversed the process, defining probability in terms of mathematical expectation, which is described as the inverse probability problem (Keynes, 1973, 192; 413-414). It is similar to Ramsey's development of the subjective expected utility (SEU) model, which was later expanded by Savage (Neumann, 1990, 191). Ramsey started with measuring a person's degree of belief by proposing "a bet, and [seeing what were] the lowest odds which he [would] accept" (Ramsey, 1960, 172). The difference is that the OEH was probability based and therefore espoused risk analysis, whereas the SEU model was concerned with uncertainty as well as risk (Machina, 2005, 2-3; Neumann, 1990, 190).

The axiomatic literature on utility theory is vast. We need to specify some objectives to navigate our way through it. First, we need to show that the logarithmic function is not necessarily bounded as Bernoulli thought. Second, we need to develop the theory to a level that is clearly axiomatic. Third, we need to distinguish between a risk measure of a random toss of a fair coin, and uncertainty such as the concern with the state of nature where the coin is bent for instance. Fourth, we need to show the axiomatic formulation in a form that is illustrative of the Allais paradox. Then some implication and discussion of Allais's positions would be necessary. The rest of this section discusses those objectives in turn.

Objective 1: Bounded vs. Unbounded Utility Function

Karl Menger argued that the addition to wealth formula, dy, described above "implies that the subjective expectation in the Petersburg Game is finite, but by changing the game slightly, one can stipulate a similar game for which not only the mathematical but also the subject expectation based on the logarithmic value function is infinite and yet no one in his right mind would risk a substantial amount" (Menger, 1967, 217). For example, Jensen 1967, 168) has shown that if the reward to the coin toss was [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] instead of just [2.sup.n] then the payoff would be Ln2(1 + 1 + ...), an infinite sum. Menger's position is stated in a general theorem:

Karl Menger (Unbounded Utility Theorem): "For any evaluation of additions to a fortune by an unbounded function, there exists a game related to the Petersburg Game in which subjective expectation of the risk-taker on the basis of that value function is infinity" [Italics original] (Menger, 1967, 218).

Following Karl Menger's insight, one hope is to obtain a bound for the utility function. Intuitively, Menger followed the suggestion that money wealth, W, can be thought of as bounded if the value of it does not increase beyond a certain amount. For instance, "The subjective value f(W) of an amount of money W is equal to W, if W is smaller than $10 million, and is equal to $10 million whenever W exceeds that amount" (Menger, 1967, 219). The aim here is to obtain a bounded function f(W) < M < [infinity], such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], otherwise, we will open a Mengerian Super Petersburg Paradox (Samuelson, V. 5, 1986, 135, 141). If the utility function is not bounded, then problems arise with the axioms such as transitivity and continuity, which are necessary for the axiomatic formulation of the utility function. For instance, two prospects might both have E[f(w)] = [infinity] while being non-indifferent, undermining the transitive argument (Ibid., V. 5, 145).

Blackwell and Girshick (1954) have extended the NM utility function to consider conditions that bound the utility function from above and below. Consider two lotteries, [p.sub.1] ..., [p.sub.u], ...; [q.sub.1] ..., [q.sub.u], ... with probabilities [[alpha].sub.1] ..., [[alpha].sub.u], ... If we let I. [[alpha].sub.1][q.sub.1] + [[alpha].sub.2][q.sub.2] + ..., and [[alpha].sub.1][p.sub.1] + [[alpha].sub.2][.sub.2] + ..., then for all "ues" [q.sub.u] [greater than or equal to] [p.sub.u] implies II [greater than or equal to] I. Also if for some "u", [q.sub.u] > [p.sub.u] and [[alpha].sub.u] > 0 then II > I. Blackwell and Girschik (1954, Ch. 4) build a utility function in this way that is order-preserving, linear in the probabilities, and is bounded (Ibid., 109-110).

Objectives 2 & 3: Axiomatic Works on the Bernoullian Utility Model in terms of Risk and Uncertainty

The works of Ramsey (1960), von Neumann and Morgenstern (1944) (NM), and Savage (1954) continue the axiomatization of the Bernoulli expected utility hypothesis, while alternating between their consideration of risk and uncertainty. Frank Ramsey, whose model includes uncertainty, added to our degree of belief, proper guide to conduct, and the finite number of alternatives to random gains (Ramsey, 1960, 183). He expounded the first axiomatic bases of the expected utility hypothesis based on moral propositions where full belief is represented by a probability of 1, the opposite probability being 0, and equal belief in the two is represented by the probability of .5 (Ibid., 175). Ramsey's model, as summarized by Fishburn (1989, 388), considers events, E, denoted by A and B; outcomes of events denoted by x, y, z, w; utility denoted by u; and probabilities denoted by [pi]. The Ramsey axioms model is as follows:

RM1 : {x if A, y if not A} is preferred to {z if B, w if not B} implies and is implied by

RM2: [pi](A)u(x) + [1 - [pi](A)]u(y) > [pi](B)u(z) + [1 - [pi](B)]u(w).

The interpretation of his model has four steps. "First identify an 'ethically neutral' event E with [pi](E) = 1/2. Second, use E to assess u on outcomes, largely by indifference comparisons between acts of the form {x if E, y if not E}. Third, use u to measure [pi](A), the person's degree of belief that A obtains, as follows: if x is preferred to y, and y to z, and if y is indifferent to {x if A, z if not A}, then [pi](a) = [u(y) - u(z)]/[u(x) - u(z)]. The final step extends the third to assess conditional probabilities" (Fishburn, 1989, 388).

Von Neumann and Morgenstem [NM] were concerned with the measurability of utility only in the second edition of their book in 1947. They approached economics through the lenses of rational behavior, particularly from the belief that a numerical approach to utility will supersede the ordinal approach to utility. They "proved that the Bernoulli principle can be derived as a theorem from a few simple assumptions" (Botch, 1967, 197). They characterized their contribution this way: "We have assumed only one thing ... that imagined events can be combined with probabilities" (NM, 1953, 20). They call the event imagined because they locate it in the future, mainly because they did not want to complicate their analysis in dealing with the past, present and future. The probability number that combines the events is a real number between 0 and 1. Events aka entities, objects or abstract utilities, when combined with probabilities are also events entities, objects or abstract utilities.

By the measurability of utility, von Neumann and Morgenstern wanted "a correspondence between utilities and numbers" (Ibid., 24). In other words, the goal is that a preference relation between two events, and a probability operation on two events should correspond to a number. To achieve that goal, some properties of the relationship and operation must be postulated. These postulates are of three types--complete ordering, ordering and combining, and the algebra of combining, which is a purely mathematical task (Ibid., 26). The hypothesis of completeness is necessary because if "the preferences of the individual are not all comparable, the indifference curves do not exist" (Ibid., 19-20). An individual should be able to rank his preference in a trichotomous way using the signs <, >, and =, and also be consistent in his ranking by following the transitive rule. Such ordering allows combination of the form that if an event is preferable to another, then even a chance of the event is preferred to the other. Combination of this sort implies that the indifference curve would be linear and parallel (MasColell, 1995, 178). The algebra of combining appeals to continuity: "However desirable [an event] may be in itself, one can make its influence as weak as desired by giving it a sufficiently small chance. This is a plausible 'continuity' assumption" (NM, 1953, 27). The algebra does not require an order in which events are combined, and allowed for combinations to occur in steps as well. The three types of postulate allowed NM to form the following two major axioms:

NM1: x [??] [left and right arrow] u(x) [greater than or equal to] u(y).

NM2: u[(1 - [pi])x + [pi]y] = (1 - [pi])u(x) + [pi]u(y).

The convention followed in the interpretation of these axioms is that the right-hand side of the equations corresponds to utilities, and the left-hand side to numbers, because "utilities are numerically measurable quantities" (Ibid., 16). NM1 says that if event x is preferred to event y then the utility which is a number for x is at least greater than the utility for y. NM2 says that one can distribute the utility over a mixture of the events.

How do the NM axioms work? NM1 and NM2 determine the utility of x and y. To find the utility of another event, z, we will consider u(1 - [pi])u(x) + [pi]u(y) as a standard lottery. We would then have to determine the probability, either through an interview or behavioral observations, which would make us indifferent between the standard lottery and the new event, z, namely: u(z)= (1 - [pi])u(x) + [pi]u(y)(Baumol, 1965, 518; Dixon, 1980, 207-212). In other words, given our preference for a glass of tea to a cup of coffee, if we introduce a third object, such as a glass of milk, a person must now decide whether "he prefers a cup of coffee to a glass the content of which will be determined by a 50%-50% chance device as tea or milk" (NM, 1953, 18).

The two NM axioms are equivalent to a complete and transitive preference relationship that satisfies the Archimedean and the independence axioms (Kani and Schmeidler, 1991, 1770). Samuelson was the first to show that the NM axioms satisfy the independence axiom. The independence axiom holds that "Whether heads or tails come up, the A lottery ticket is better than the B lottery ticket; hence, it is reasonable to say that the compound (A) ticket is definitely better than the compound (B) ... This is simply a version of what Dr. Savage calls the 'sure-thing principle'" (Samuelson, V. 1, 1966, 139). Leonard Savage uses this principle to establish probabilities and utility functions (Kani and Schmeidler, 1991, 1767). The Archimedean property is a standard mathematical concept which states that if x is preferred to y, then a multiple of x is preferred to y, namely nx > y. We demonstrate how the Archimedean and independence axioms strengthened the axiomatic method of expected utility following Herstein and Milnor work (1953).

Because of the lengthiness of their presentation, practitioners choose simpler systems such as the Herstein-Milnor axioms discussed below to demonstrate NM axioms. I.N. Herstein and John Milnor (HM) axioms are necessary for the existence for the John yon Neumann and Oskar Morgenstern (1944) utility on a mixture space S. Mixture refers to probability weighting, order means preference, and mixture space is a set of prospects. As an example, they gave us a, b [member of] S; [lambda], [mu] [member of] [0, 1]. We can mix a, b to get [mu]a + (1 - u)b [member of] S. This operation is possible because of three mixture axioms (Herstein and Milnor, 1953, 265):

I. 1a + (1 - 1)b = a, [left and right arrow] (1 - 0)a + 0b = a,

II. [mu]a + (1 - [mu])b = (1 - [mu])b + [mu]a, and

III. [lambda][[mu]a + (1 - [mu])b] + (1 - [lambda])b = ([lambda]u)a + (1 - [lambda]u)b.

With these mixtures, we can show that [lambda]a + (1 - [lambda])a = a. This is proven by putting a = b; [mu] = 0 in III to get: [lambda][0a + (1 - 0)a] + (1 - [lambda])a. By I, the square bracket items = a, and by II we can switch terms around, yielding: [lambda]a + (1 - [lambda])a = a. More complicated examples can be done, but it is more interesting to point out that the HM axioms are "at least necessary conditions" for the existence of an NM utility (Ibid., 266). Their assumptions are featured as follows:

HM1: (Completeness). The space of lotteries, S, is completely ordered by the preference relation [??]. For lotteries a, b, complete ordering means that 1. Either a [??] b, or b [??] a, 2. The reflective property is a [??] a and 3. For lotteries a, b, c, the transitive property: a [??] b, and b [??] c, implies a [??] c.

HM2: (Continuity). For some elements in S, a, b, c [member of] S, there exists a probability [member of] [0.1], such that a mixture of a, b will be preferred to c and vice versa. This is expressed as (A). {[alpha]|[alpha]a - (1 - [alpha])b [??] c} and (B).{[alpha]|c [??] [alpha]a + (1 - [alpha])b}.

Using NM1 and NM2 we can apply the utility concept to get (A') x {[alpha]|[alpha]u(a)+(1 - [alpha])u(b) [greater than or equal to] u(c)} and (B') x {[alpha]|u(c) [greater than or equal to] [alpha]u(a) + (1 - [alpha])u(b)}, respectively. These are closed sets as the probability lies in the [0,1] closed interval. From (A'), we can find [alpha] [greater than or equal to] [u(c) - u(b)]/[u(a) - u(b)] for u(a) > u(b), and [alpha] [less than or equal to] [u(c) - u(b)]/[u(a) - u(b)] for u(a) < u(b). If we set u(a) = u(b) we get zero or the whole interval.

The idea of continuity implies that we can perturb the probability [alpha] without changing the ranking of the lotteries. Herstein and Milnor (267) preserved continuity by the limiting concept: lim [[alpha].sub.i[right arrow][infinity] = [alpha]. If we are given two sequences of points such as [p.sub.n], [q.sub.n], then for all n we can state that [p.sub.n] [??] [q.sub.n] [left and right arrow] lim [p.sub.n] [??] lim[q.sub.n] (MasColell et al., 1995, 46). Continuity helps us avoid such phenomena as infinitely favorable or unfavorable outcomes of a lottery. We want to purge such outcomes because they would create a lexicographical ordering which would make the indifference curve non-existent (Ibid., 171).

HM3: Given a, a' [member of] S, a ~ a', then for every b [member of] S, 1/2 a + 1/2 b ~ 1/2 a' + 1/2b. If one is indifferent between a and a', then one is indifferent between a 50:50 chance of getting a or b, and a 50:50 chance of getting a' or b. This is the Herstein-Milnor way of simplifying the independence axiom (Fishburn, 1983, 303).

Originally, the NM axioms did not explicitly show the independent axiom. They used abstract operations or an abstract utility concept (Karni and Schmeidler, 1770). HM generalized the outcome to a mixture set using a weaker independence axiom (HM3) and a stronger mixed continuity axiom than what traditionally goes by the name Archimedean axiom. It is a traditional student exercise to show that the HM axioms are necessary for the NM utility. HM1 follows from the well-ordering of the real line, HM2 follows from NM1 and NM2. HM3 follows from a ~ a' [left and right arrow] u(a) = u(a'). This is a mathematical venture, but the steps in the march toward a NM utility function is worth noting. The completeness axiom gives the best and worst outcomes. The continuous axiom allows us to get an indifference curve. It tells us that there exists a probability [alpha] [member of] [0.1] such that for the lottery c, we can write u(c) = [alpha], which is a construction of

the utility function (Jehle, 1991, 198; Laffront, 1989, 11). We will describe this function to a greater extent later, but for now it is worth noting also that the utility function is better described as a "kind of function, with certain specific mathematical property," rather than a function that represents preference in the ordinal senses. It is a mapping from the gamble to the real line possessing the expected utility property (Jehle, 1991, 197).

Objective 4: Allais' Reaction to the Axiomatic Model

Using the standard NM example, u(z) = (1 - [pi]) u(x)+[pi]u(y), we can show indifference with an example such as u($40) = 1/2u($100) - l/2u(0), indicating "indifference between a $40 gain with certainty and an even-chance gamble between a gain of $100 and no gain. The same algebraic expression, rewritten as u($40)- u($0)= u($100) u($40), has the Bernoullian interpretation that, apart from any consideration of chance, the individual's degrees of preference for $40 over $0 and for $100 over $40 are equal" (Fishburn, 1989, 390).

Table 1 below illustrates the Allais Paradox. Since U(A) = U(100), U(B) = .89U(100) +. 1U (500) + .01U(0), U(C) = .89U(0) +. 11U(100), and U(D) = .9U(0) + 1U(500), U(A) - U(B) = U(C) - U(D) by simple arithmetic.

If one prefers A to B, then U(A) > U(B). Doing the arithmetic, we get: U(100) > .89U(100) + .1U(500) + .01U(0) or U(100) - .89U(100) > .1U(500) + .01U(0)or .llU(100) > .1U(500) + .01U(0).

Similarly if one prefers D to C as Allais found, then U(D) > U(C). Doing the math yields: .9U(0) +. 1U(500) > .89U(0) +. 11U(100) or .9U(0) - .89U(0) + .lU(500) > .l 1U(100) or .11U (100) < .01U(0) + .1U(500), where the less than sign contradicts the above greater than sign (Machina, 2003, 24; Munier 1995a, 192; Resnik, 1987, 104). Allais (1990, 5), therefore, found through experiment that the preference A to B is matched with the preference D to C which contradicts the NM axioms.

An attempt to cope with the Allais paradox was provided by Savage who participated in Allais' experiment and agreed first with his conclusion. But after further reflection, particularly on the logic of the Independence Axiom or the sure-thing principle, he saw a flaw in his original decision and opted to make a correction to his original choice. Such a reflection is obtained from an experiment that asks us to draw tickets labeled 1 to 100 at random. This is given as the heading in Table 2 below.

The first column of Table 2 corresponds with the four situations given by Allais in Table 1. Situation A in the first row in Table 2 indicated a guaranteed $100M payoff irrespective of any drawing of a ticket. Situations B, C, and D are gambles. To interpret Situation B in Table 2, we note that the probability of S0 is given as .01 in Table 1 for Situation B, which means 1 ticket in 100, therefore, we place the $0 under Ticket 1. Similarly we place $500 under Tickets 2-11 for Situation B, because Table 1 shows its probability is .1, which is 10 of 100. The same logic makes us place $100 for Situation B under Ticket 12-100 for its probability in Table 1 is .89 or 89 of 100. The rest of Table 2 is filled in the same manner.

Savage reflected that payoffs of tickets 12-100 would not have any influence in choosing A over B or C over D and can therefore be omitted in the decision making. This observation is referred to as the "common effect." Now, the matrix of payoff in section A is the same as the matrix of payoff in section B for tickets 1-11. Upon reflection, Savage now is willing to state that the original choice he made, which was inconsistent with the NM axioms, was an error. The preference of 3 to 4 can be reversed only if one makes an error in choice. Such errors are normative and can be corrected should it be pointed out to the person.

Allais rejoined the debate by pointing out that Savage's experiment had destroyed the certainty part of the experiment. We get the inputs for Row A in Table 2 by breaking up a certainty of winning $100M with three probabilities: .01+.1+.89 = 1 (Sugden, 2004, 696). This procedure had eliminated the "complementarity effect operating in the neighborhood of certainty" (Allais, 1979, 535). In other words, the certainty of $100M cannot be so factored. This is the general argument Allais made against the independence axiom, where a third prospect is put in complementary relations with two others. When two events are mixed under the independence axiom, they are considered as being mutually exclusive, that is, complementarity is not allowed. Another way of looking at it is to observe that one can have only one of the two events, one with a probability of [alpha], and the other with a probability of (1 - [alpha]), but the two events do not occur together (Ibid., 141). Students who learn choices of bundles in consumer theory can appreciate that two commodities in a bundle can be jointly consumed, which contrasts with two outcomes in choice under risk where the outcomes are mutually exclusive.

Allais claims that the independence axiom will fall apart if we can "find case in which the complementarity relations ... may change the order of preference" [Italics original] (Allais, 1979, 90). Fishburn (1988, 85-86) feels that it is in the nature of empirical analysis that such findings may occur. "The empirical fact is that the nature of r and the size of [lambda] can make a difference in the preference between [lambda]p + (1 - [lambda])r and [lambda]q + (1 - [lambda])r, and it is hard to ignore this in assessing the normative adequacy of independence ... The point is that there are certain patterns of preferences, held by reasonable people for good reasons, that simply do not agree with the axioms of expected utility theory."

To clinch the complementarity argument, we adapt Fishburn's tabular illustration (1979, 248-249). If one chooses the first row, a chance device will determine that his payoff will be [lambda]p(x) or (1 - [lambda])r(x). Traditionally, we can argue that if the payoffs of the first row dominate the payoffs of the second row, p will be chosen over q, and r will be chosen over s. But "Allais' criticism lies in the assertion that ... an individual's preference judgment ... is properly based on a comparison of these two gambles in their full perspectives and not on a comparison of separate parts such as p versus q and r versus s" (Please see Table 3).

The techniques we use to reason out our choices "involve a combination of the three basic techniques, namely, rule-based decision, probabilistic inference, and analogies" (Gilboa and Schmeidler, 2001, 2) John Conlisk (1989) was concerned with three tests that lay bare the independence axiom. MacCrimmon and Larson (1979, 349-351) listed 23 rules. Rule 6 for instance states that when one alternative is certain, select it even if you are giving up a chance of winning a bigger amount with a lower probability. Rule 6 can be seen as a composite of two other rules--Rule 10, which takes the prospect with the higher probability when two prospects have payoffs that are desirable, and Rule 2, which takes the prospect with the larger payoff when their probabilities are similar.

In responding to the Allais paradox, Oskar Morgenstern (1979, 178) pointed out that the domain of axioms should be restricted, meaning that the probabilities used should not "go to 0.01 or even less than 0.001 ... a normal individual would have some intuition of what 50:50 or 25:75 means." On the theoretical side, "if our preferences are only partially ordered--which means, grossly speaking, that they are in considerable disarray--then there is no presently known guiding principle for optimal allocation" (Ibid., 1979, 182).

Experiments revealed other causes of violation of the expected utility hypothesis beside the common effect cited above. According to Michael Weber (1978, 100) even when one disregards the effect of the last column in Table 2, one is likely to experience more negativity in choosing B over A and lose, than in choosing D over C and lose. In Table 2, F1 < F2 < 0, analogous to - 12 < -5 < 0, indicates such negativity. The reason for the terrible feeling of F1 is that A is certain, and to lose something one has for certain would create more negativity than losing something one is not sure about. As explained by Kahneman and Tversky, "people underweigh outcomes that are merely probable in comparison with outcomes that are obtained with certainty. This tendency, called the certainty effect, contributes to risk aversion in choices involving sure gains and to risk seeking in choices involving sure losses. In addition, people generally discard components that are shared by all prospects under consideration. This tendency, called the isolation effect, leads to inconsistent preferences when the same choice is presented in different forms" (Kahneman and Tversky, 1979, 263). Only when such feelings are taken into account would the paradox be in line with the prediction of the expected utility theory. The problem becomes more complicated when Disappointment and Regrets are considered, making the Allais paradox resilient (Weber, 1998).

Analysts frequently use the simplex method to demonstrate violations of the expected utility hypothesis. Here we illustrate the common effects and the fanning-out process. It involves the construction of two types of indifference curves, one for the expected utility hypothesis and one for the mathematical expectation hypothesis, and to show that the latter is steeper than the former. This concept is best explained by a rectangular isosceles triangle first presented by Jacob Marschak (1950) and popularized by Mark Machina (1990).

Figure 1 is a simplex representing the Allais payoffs and probabilities on the three axes. The rectangular box at the origin shows the coordinate of points such as ([p.sub.1], [p.sub.2], [p.sub.3]). Points such as A and B can be similarly coordinated by other boxes. The prizes are [x.sub.3] = $500M representing the best outcome, [x.sub.2] = $100M the second best outcome, and [x.sub.1] = $0M $0 the worst. These outcomes form a set W = [x.sub.3] > [x.sub.2] > [x.sub.1]. The probability set is D = [0, M], which is represented in this case as a set with three probability elements, D = {[p.sub.1], [p.sub.2], [p.sub.3]} that sums to 1. We write A as preferred to or indifferent to B as A [greater than or equal to] B. By the NM axioms, the ordering yields a utility function A [greater than or equal to] B [left and right arrow] U, U(A) [greater than or equal to] U(B). Although we show the payoff amount on the axes, it is common practice to scale the axes to unity for analysis (MasCollel et al., 1995, 169).

The next step is to obtain indifference curves by cutting the simplex with a hyperplane, H(p) (Conlon, 1995, 637). These cuts yield triangle figures such as I(p), which are the indifference curves for this simplex configuration. The indifference curve will be increasing upwards as the best outcome is measured upwards, which is analogous to the three dimensional representation of consumer choice over bundles of commodities in standard microeconomics. Abusing the geometry of simplex somewhat, we can think to the linear indifference curves in a two dimensional surface by collapsing the intermediate payoff between the best and worse payoff. Linear indifference curves and the Iso-value curves, respectively, can be drawn for a constant level of satisfaction, as in K and L: [p.sub.1]U([x.sub.1]) + [p.sub.2]U([x.sub.2]) + [p.sub.3]U([x.sub.3]) = K; [p.sub.1]([x.sub.1]) + [p.sub.2]([x.sub.2]) + [p.sub.3]([x.sub.3]) = L. Taking the derivative of the indifference and Iso-value curves allows us to find their slopes. Whenever the slope of the Iso-value curves exceeds the slope of the indifferent curve, fanning-out is present.

Fanning-out occurs as the probability in the D-distribution of probabilities changes. "Intuitively, if the distribution ... involves very high outcomes, I may prefer not to bear further risk in the unlucky event that I don't receive it ... But if (the distribution) ... involves very low outcomes, I may be more willing to bear risk in the event that I don't receive it" (Machina, 1987, 129-130). Fanning-out is more pronounced at the outer edges of the collapsed triangle, and can exhibit linear as well as nonlinear utility curves. The curvature is captured by the Arrow-Pratt ratio that shows changes in the slope of the curve measured by the ratio of their second partial to the first partial derivative. The "nonlinearity in a preference functional is to specify how the derivative (i.e. the local utility function) of the functional varies as we move about the domain D[0, M]. Our formal hypothesis ... as we move from one probability distribution in D[0, M] to another ... [is that] the local utility function becomes more concave at each point x ... in terms of the Arrow-Pratt ratio" (Machina, 1983, 282).

[FIGURE 1 OMITTED]

As mentioned above, the two dimensional representation on the unit triangle, the side of [p.sub.2], will collapse to the origin. Following Starmer (2000, 340), two parallel lines that would represent Allais' A and B and C and D prospects can be indicated by arrows in the best vs. worst plane as shown in Figure 1. With [p.sub.2] now collapsed to the origin, the origin will represent situation A, and the situation B is given by the probability coordinates (.01, .1). In a similar way the second arrow in the parallelogram would represent situations C and D. The common consequence criterion requires that the slopes of the two arrows be the same.

The arrow labeled prob. = 1 is a multiple of the arrow labeled prob. < 1, and suggest what is known as the common ratio effect. As the former arrow decreases, we will move towards the right on lower indifference curves. This ratio will also show inconsistent choices.

Such linear and parallel lines would reflect the predictions of the independence axiom (Sugden, 2004, 695). But we will find that "the individual is most sensitive to changes in the probability of [x.sub.1] relative to changes in the probabilities of [x.sub.2] and [x.sub.3] (i.e. MRS ([x.sub.2] [right arrow] [x.sub.3], [x.sub.2] [right arrow] [x.sub.1]; F) is the highest near the left edge of the triangle, or in other words precisely when [x.sub.1] is a low probability event (i.e. [p.sub.1] is low)" (Machina, 1983, 285).

In 1988, Allais presented an expansion of his model, which accentuated the difference as well as illustrated how to reconcile his position with the expected utility model. He introduced a special probability distorting function [theta](x), and a utility of a sure monetary payoff function, u(x) (Munier, 1995, 38; Stigum 2003, 464). The former function measures attitude towards risks. The latter function can have any shape such as convex or concave. Both functions can be continuous and strictly increasing. The distorted function representing the utility of a prospect can be written as:

Z(P) = [u.sub.1] +[theta](1 - [p.sub.1])[u([x.sub.2]) - u([x.sub.1])] + [theta](1 - [p.sub.1] - [p.sub.2])[u([x.sub.2]) - u([x.sub.1])] + ... + [theta]([p.sub.n])[u([x.sub.n] - u([x.sub.n-1])

When there is no probability distortion represented by [theta], then the model reverts back to the expected utility hypothesis. This extended formulation allows a variety of utility models (Sigum, 2003, op. cit.,). Gathered "under the heading of the "anticipated utility" hypothesis ... This could be the ultimate result of Allais' contribution to decision under risk" (Munier, 1995, 39).

Besides Allais' generalized model, Machina (1995b) has attempted another generalization that has sparked some controversy. His publication of two errors in Allais' impossibility theory starts the problem. In his 1982 and 1983 articles, Machina discussed the impossibility of local and generalized utility functions. The controversy is about defining a Bernoullian index that simultaneously satisfies three conditions. As Allais puts it: "This Impossibility Theorem shows the impossibility of simultaneously meeting the three following conditions: definition of the local neo-Bernoullian index in the discrete case ... its validity over the whole interval (0, M) ... and its definition up to within a linear transformation" [Italics original] (Allais, 1995, 264). Disagreement centers on the appropriate definition of a local discrete utility function.

By way of summary, the Allais paradox picked up on the marginal valuation of income/wealth in the original Bernoulli specification, and added a person's attitude toward risk to it. While attitudes toward risk are built into the curvature of a person's utility function in the NM axiomatic model, it appears only as different between one person and another. In the Allais' model, however, attitude toward risk is generalized to account for changes in the same person. This change is said to be systematic, and not due to mere randomness or illusion. The Allais paradox holds, therefore, that "attitudes towards risk change not only from an individual to another, but also for a given individual between different patterns of risk" [Italics original] (Munier, 1995, 36).

Allais (1990, 8) demonstrated that his paradox has some novelty, which he posits to "basic psychological realities" that would not identify monetary with psychological values, and the distribution of risks in valuing cardinal utilities. Risks show up in the standard VM lottery described above, where expected utility hypothesis will yield the same values for many combinations of two prospects.

Overlapping Generation Model (OLG)

In his Economy and Interest (1947), Allais presented a model of consumption for individuals in time periods that overlap for successive generations. This model is said to precede Paul Samuelson's (1958) popularization of the subject by 11 years (Malinvaud, 1995, 111). Some differences need to be pointed out between the two models. Allais studied interaction between the production and consumption sectors, while Samuelson studied trade between different generations. While the data that Allais used for production and preferences were not sufficient to determine the rate of interest and allocation of resources, Samuelson developed a demographic theory of the interest rate equal to the rate of increase of the population. Yet another difference is that Allais used two time periods, and Samuelson used three. We have discussed Samuelson's contribution in this area elsewhere (Szenberg et al., 2006). In Allais, however, government intervention leads to different interest rates (Malinvaud, 1994, 126-127).

Allais' framework, consumers provide a fixed quantity of labor in the first period, and do not work in the second period. Consumers are of the same type and they consume in both periods. One can write the production functions using Q for consumption goods, K for production goods, [L.sub.1] for employment in the production goods sector, [L.sub.2] for employment in the consumption goods sector, U for land, [alpha] is a constant, and all values are equal to or greater than zero. The two production functions and their employment restrictions are:

Q = [square root of [L.sub.2]][square root of (K + U)] (1)

K = [alpha] [L.sub.1] (2)

L = [L.sub.1] + [L.sub.2] (3)

Considerable degrees of freedom are allowed in the model. Malinvaud (1995, 116) distinguishes three typical cases where the young consumers' wealth is either their labor income, or the national income, or the sum of rent and labor income (Malinvaud, 1987, 104-105). A golden rule condition requires a maximum output of consumer goods at a zero interest rate. A stationary equilibrium condition would require specification of consumer choices to work with the production plans in the two period setting.

The predictions of Allais' OLG model are not unique because of the numerous degrees of freedom required for a stationary equilibrium. Some important variables to be specified include distribution rights, technical feasibility, psychological preferences, and consumption plans. Consumption plans have resource restriction discounted at the youthful stage. Distribution rights have to be specified intergenerationally. Malinvaud shows different predictions for resources as an exogenous datum, as work only revenues, as aggregate income, as rents distributed to the young, and as rents distributed to the old (Malinvaud, 1995, 121-125).

Allias' OLG is an alternative to traditional general equilibrium models. Practitioners have tried to reconcile differences between Allais and Samuelson versions, on the one hand, and the Arrow-Debreu general equilibrium model on the other. The works of Allais and Samuelson "would have complemented each other, because they brought to light different effects of the overlapping generation's structure" (Malinvaud, 1987, 105).

Attempts to reconcile OLG with other Walrasian-type general equilibrium models are still being studied (Genakoplos, 1987; Geanakoplos and Polemarchakis, 1991). Extensions of the Allais-Samuelson model "permit generations to live longer, and even be immortal, include many commodities in each period and introduce uncertainty" (Geanakoplos, 1991). We find that "Walras law need not hold for economies of overlapping generations ... and ... the model of overlapping generations has been interpreted as "lack of market clearing at infinity" (Ibid., 1901). In general, to get market clearing for OLG models, we may require that consumption bundles exceed initial endowment, that prices do not signal aggregate scarcity, and that competitive allocations are not Pareto Optimal (Ibid., 1902).

On the empirical side, the OLG model is "a workhorse of macroeconomics, monetary theory, and public finance" (MasCollel, 1995, 769). For instance, Kotlikoff's work has given rise to a new term in the expansion and articulation of the OLG model particularly in generational accounting. Both Kotlikoff and Diamond take up current and future concerns of the Social Security problem, a good indication of the relevance of the model for the 21st Century (see Szenberg et al., 2006).

Monetary Theory

The quantity theory of money has a long history. We find Keynes turning it into a demand for money function, based on transaction, speculation, and precautionary motive (Keynes, 1936, Ch. 15). We can then write the demand for money function in an operational way, representing a liquidity preference function that varies with wealth, income, and expected returns, and the expected return from the broad spectrum of assets that can be held as wealth. In the post WWII period, accelerating prices accounted solely as the determinant of inflation. As one researcher in the monetarists' school puts it, "The astronomical increases in prices and money dwarf the changes in real income and other real factors" (Cagan, 1956, 25).

In the early 1950s, Cagan and Allais were simultaneously evolving demand for cash balance models that would make the quantity theory falsifiable in situations of rapidly increasing prices. As Allais explained, "Cagan's research was brought to my attention by Friedman in a discussion we had in July 1954 when I described to him the interesting results I had reached ... in my research on the theory of the business cycle" (Allais, 1966, 1123).

The predictions of Allais' and Cagan's models are essentially the same, namely that the demand for cash balances depends on the rate of change in prices. Allais made a correspondence of the variables, showing that the different choices lead to the same goal (Allais, 1966). Allais explained his unique approach this way: "My theory of monetary dynamics is based on the introduction of new concepts which have no equivalent in the earlier literature; the concepts of the psychological rate of interest, the rate of forgetfulness, and the reaction time, whose values vary according to the economic situation; the concept of the coefficient of psychological expansion which represents the average appraisal of the economic situation by all economic agents; the concept of psychological time, the referential of psychological time being such that the laws of monetary dynamics remain invariant therein" (Allais, 1997, 6). In this formulation, heredity makes the present depend on the past and relativity makes the dependent relation unchanging or invariant when we use the psychological time in place of physical time.

Model of a Market Economy

Basically, Allais forged a general equilibrium model that depends on the efficient use of surplus in the economy. In this model, economic agents make transactions that generate surplus and distribute them in the economy to reflect optimality and stability. In outlining his contributions, Allais stated, "My work on economic evolution and general equilibrium, maximum efficiency, and the foundations of economic calculus has developed in two successive phases, from 1941 to 1966, and from 1967 to the present day" (Allais, AER, 1997, 4). Allais thought that through the effective distribution of surplus, the economy would tend toward a state of maximum efficiency.

Allais' concept of surplus dominates the role of prices in traditional general equilibrium.

"A surplus can be realized when the marginal equivalences of consumption and production units differs" (Allais, 1977, 122). "The maximum distributable surplus of a given good is the largest quantity of that good that can be made available by a better organization of the economy which leaves all preference indexes unchanged" (Ibid., 133).

Allais' concern for the market economy has created two research programs in literature. The programs are in the direction of probing Pareto optimal condition, and stability of equilibrium. Pareto optimality means "a situation where any one preference index is maximal for given values of the other preference indexes" (Ibid., 134). Allais asserts a concept of general equilibrium where "there is no potential surplus for any good" (Ibid., 134). Allais's concept of equilibrium differs from the Walrasian concept where the overall demand is equal to the overall supply, and from Edgeworth's concept where one preference index is maximal for given values of the other preference indexes. His definition, however, still depends on multiple, convergent, and stable equilibrium.

In developing the Pareto and stability conditions for the economy, Allais shows a partiality for the calculus-based approach and eschews concerns with topology and convex sets. Some extensions of his model in modern literature, however, use both tools. We discuss the two aspects of his model further as follows.

Pareto Optimal Conditions

In his article "Economic Surplus and the Equimarginal Principle" in the The New Palgrave Dictionary of Economics, Second Edition, 2008, Allais gave a utility frontier illustration of the Pareto Optimal conditions. "A situation of maximum efficiency can be defined as a situation in which it is impossible to improve the situation of some people without undermining that of others." Allais' illustration corresponds to points on the utility frontier, which demarcates points above that are impossible, and points below that are possible. Allais emphasized that this definition of maximum efficiency is made independent of the assumptions of continuity, differentiability, or convexity, except only for a common (nummaire) good. Following Allais' 2008 presentation, this can be fleshed out using a function [f.sub.i]([U.sub.i], [V.sub.i], ... [W.sub.i]) for consumers, and [f.sub.i]([U.sub.j], [V.sub.j], ... [W.sub.j) for producers. The goods U vary continuously, and it enters all the production and consumption functions. The utility frontier is defined to represent a state where the producer index is equal to zero, i.e., [f.sub.j]([U.sub.j], [V.sub.j], ... [W.sub.j]) = 0.

Modern researchers have been able to estimate this optimal condition with the use of a benefit function. Following David Luenberger, a benefit function that measures changes from the utility function in a reference bundle, g, can be made. The benefit function has three elements, b(g; x, u), which "measure the amount that an individual is willing to trade, in terms of a specific reference commodity bundle g, for the opportunity to move from utility level u to a consumption bundle x" (Luenberger, 1992a, 461) "The concept of a benefit equilibrium is a natural modification of that of a competitive equilibrium. Utility is just replaced by individual benefit" (Luenberger, 1992, 234).

In Allais' model, "a necessary and sufficient condition for X* to be Pareto efficient is that the distributable surplus be negative or zero for all feasible X (i.e., X* is zero maximal), and his statement is correct, in general, except for edge pathologies" (Ibid.,232). One can picture values of X* as points on the Allais utility frontier, and all feasible @@points, X, as points below the frontier. Like the production possibility curve one observes in elementary economics, the challenge is to find correspondence between the points of X and points of X*. We have adapted the standard consumer maximization and Edgeorth Box following Munier (1995, 21-22) and others in Figures 2a and 2b below to illustrate the Allais equilibrium conditions.

[FIGURE 2 OMITTED]

Figure 2a shows paths from the feasible point X approaching the Allais utility frontier, which is non-convex. One can imagine a series of allocations starting from X and converging to different utility points such as [X.sub.1] and [X.sub.2]. The allocation vector is thought of as a series of points {x}n = {[x.sub.1], [x.sub.2], ... [x.sub.n]}, and on the utility frontier are a series of utility functions, [U.sub.i](X), which is also a series [{U}.sub.n] = {[u.sub.1], [u.sub.2], ... [u.sub.n]}. Allocations are consumption bundles of commodities. The allocation set is usually assumed to be convex, closed, and bounded from below (Luenberger, 1995, 161). A set of feasible allocations is defined as the situation where the sum of the allocation is equal to the sum of the traders' endowments (Courtault and Tallon, 2000, 478).

For equilibrium, we want to know if the utility sequence will converge to a maximum as each allocations sequence converges. The optimal point X* is such an equilibrium point in standard analysis where the price line is tangent to a convex utility curve. Extending the argument to a general benefit function would make the oval shaped benefit region tangent to X* as demonstrated by Briec and Garderes (2004, 106).

Figure 2b shows the feasible set of allocation, IR, as the hatched lense-shaped area in the Edgeworth Box. The core is the segment AB of the Pareto Optimal (PO) curve. Both IR and PO are determined by the endowment, e, of the traders. Allais' equilibrium is indicated by the curved arrow on the core. Walrasian equilibrium is attained where the straight arrow intersects the core. While both equilibrium conditions are in the core, Allias' equilibrium arises from many paths leaving an initial state, while only one equilibrium arises in the Walrasian model.

Allais' Stable States

The initial state of the economy, [E.sub.1], is characterized by the consumer and producer functions. A finite change in [E.sub.1], designated by [delta][E.sub.1] comes about by finite changes in the variables of the functions. A new state [E.sub.2] = [E.sub.1] + [delta][E.sub.1] will emerge from such changes. A third state [E.sub.3] that is made "isohedonous" with the state [E.sub.1] accounts for changes that return the preference indices to their initial values.

Commenting on Allais' model, Munier asserted that the set of stable states of the economy include the set of Walrasian equilibria of that economy. This condition is likened to the core concept of the Edgeworth Box with two traders. Traditional research that emphasizes a given price system establishes a unique Walrasian state in the core. For Allais, however, the stable state in the core need not be unique, and when more than two traders are involved, it may take on a larger core. Only in the stable states are input prices uniquely defined, because efficiency results from the pressure that free competition puts on human beings rather than on the price system (Munier, 1995, 20-21).

Conclusions

We found major precursors to modern theory in the works of Maurice Allais. We have touched on his paradox, overlapping generation model, and the market economy in this review. His paradox has steered research into a new direction in the economic literature embracing psychological experiments of a highly scientific nature. It was a springboard for leading research to diverge from the traditional expected utility model toward a more psychological paradigm.

Allais' market model has turned the core elements of the Walrasian or Edgeworthian general equilibrium research program from price towards the effect of competition on humans. As we showed in Figures 2a and 2b, more general equilibrium results are included, and uniqueness based on convex analysis or convergence of the core is not the main object of general equilibrium analysis.

This journal editor's personal recollection of gratitude is in place. Maurice Allais was a member of the committee which conferred upon him the Irving Fisher Award for the Economics of the Israeli Diamond Industry (Szenberg, 1973). The other members included Kenneth Boulding, Milton Friedman, Egon Neuberger, and Paul Samuelson.

Editing the volume on the Eminent Economists, Their Life Philosophies, which included the opening essay, "The Passion for Research," by Maurice Allais, provides lofty lessons in scholarship. Getting the final version of Allais' essay "took about eight submitted drafts, fifty letters and cables, and numerous [overseas] telephone calls." As is mentioned in the volume, "this is meticulousness of the highest order on part of the contributor." Students are made to understand that "inspiration in the words of Tchaikovsky, 'is a guest that does not visit lazy people.'"

How appropriate and timely to conclude this Memoriam with Allais' quote (1999, 74): "... without any exaggeration, the current mechanism of money creation through credit is certainly the 'cancer' that's irretrievably eroding market economies of private property."

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TABLE 1.
The Allais Paradox (in Millions of Dollars)

         First Pairs of Offers

 Situation A        Situation B

  Win     Prob.     Win     Prob.

 $100       1      $100     0.89
                   $500      0.1
                  Nothing   0.01

        Second Pairs of Offers

  Situation C        Situation D

  Win     Prob.     Win     Prob.

Nothing   0.89    Nothing    0.9
 $100     0.11     $500      0.1

Source: Allais, 1990, 5

TABLE 2.
The Allais Data in Savage Format (in Millions of Dollars)

                 Ticket: 1       Tickets: 2-11    Tickets: 12-100
              Allais (p =.01)   Allais (p =.10)   Allais (p=.89)

Situation A         $100             $100              $100
Situation B        $O+F1             $500              $100
Situation C         $100             $100                $0
Situation D        $O+F2             $500                $0

Adapted from Savage, 1972, 103

TABLE 3.
Combination of Gambles via Chance Device

                      Gambles                  Chance Device

                                          [lambda]   1 - [lambda]
You Chose   [lambda]p +  (1 - [lambda]r      p            r
            [lambda]q +  (1 - [lambda]s      q            s

Source: Adapted from Fishburn 1979, 248