Arrow's Result Research Articles

Product Innovation Incentives: Monopoly vs. Competition

In contrast to Arrow's result for process innovations, we show that the gain from a product innovation can be larger to a secure monopolist than to a rivalrous firm that would face competition from independent sellers of the old product. A monopolist incurs profit diversion from its old good but may gain more than a rivalrous firm on the new good by coordinating the prices. In a Hotelling framework, we find simple conditions for the monopolist's gain to be larger. We also explain why the ranking of innovation incentives differs under vertical product differentiation.

An Extension of Arrow's Result on Optimal Reinsurance Contract

AbstractWe consider the problem of finding reinsurance policies that maximize the expected utility, the stability and the survival probability of the cedent for a fixed reinsurance premium calculated according to the maximal possible claims principle. We show that the limited stop loss and the truncated stop loss are the optimal contracts.

From Arrow's Theorem to Incentives and Price Dynamics

Abstract Arrow's impossibility theorem has served as a dominant result within the social sciences for the last half-century. But, does it mean what we thought it did? Does it really require a dictator, or are there alternative, more benign interpretations? By developing an alternative interpretation, we discover that Arrow's result has interesting implications in a variety of topics ranging from the free rider problem of economics, incentives, and even an explanation for the chaotic behavior of price dynamics.

The Insurance of Lower Probability Events

INTRODUCTION When an individual faces a single source of risk, the optimal insurance contract contains full insurance above a deductible when the premium only depends upon the actuarial value of the contract. This result was established not only under expected utility (Arrow 1965; Raviv 1968) but also for any non-expected utility framework that satisfies the second order stochastic dominance (SSD) property (see Machina 1995). Besides Gollier and Schlesinger (1996) have shown that the superiority of the deductible policy results from its better ability to reduce the variability of final wealth for a given cost of insurance. More specifically they prove that any insurance contract of equal cost will be dominated, in the second order sense, by one with a deductible policy. Although this result turned out to be very important in many respects, its practical relevance is limited by the assumption that there is only one source of uncertainty. In practice, people and firms face many simultaneous sources of risk and they have to decide both on the amount of their total insurance budget and on its allocation among the different sources of risk. Recently, Gollier and Schlesinger (1995) showed that people would like to cover multiple risks by purchasing an contract covering all risks at the same time, with full insurance above a deductible on the aggregate loss. Unfortunately, such an umbrella contract is rarely offered in reality. As a consequence, decision-makers have to consider the optimal level of coverage for each risk separately. In this context, Gollier and Schlesinger prove that it is optimal to purchase contracts with full insurance above a deductible for each peril. In this paper, we extend all these results by considering rules that should be followed in choosing the deductible level for each risk in terms of its characteristics. To be more specific, we consider an individual who faces two pure risks, [Mathematical Expression Omitted] and [Mathematical Expression Omitted] which differ both in terms of the of the loss occurrence ([p.sub.1] and [p.sub.2]) and in terms of the level of potential loss ([L.sub.1] and [L.sub.2], respectively).(1) For instance, [Mathematical Expression Omitted] might be a low frequency risk ([p.sub.1] low, e.g. and earthquake risk or the risk of cancer in the year to come for a young healthy individual) with a high potential loss ([L.sub.1]). On the contrary, [Mathematical Expression Omitted] may stand for a risk likely to materialize ([p.sub.2] is high) but with moderate or mild severity ([L.sub.2] is not too large). Given that insurance is offered separately for each risk, how should the optimal deductibles ([[D.sub.1].sup.*] and [[D.sub.2].sup.*]) compare to each other? Intuition suggests that one should cut down the large losses and select deductible levels so that the net loss (the loss minus the indemnity) should be equal for each risk. After all, the Arrow's deductible result suggests that it is optimal not to indemnify any loss below a limit that is the same for any source of risk affecting the wealth of the insured person. If this extension of the Arrow's result were correct, one would select equal levels of deductible for each risk. In this paper we show that such a policy is not, in general, optimal despite it's a priori intuitive appeal. Reducing the deductible has two contradictory effects on expected utility. First, it increases the premium. Second, it increases the indemnity in case of loss. It is a priori not clear how a change in the of loss affects the optimal deductible which is a best compromise between these two adverse effects. In fact, in most cases, a strong probability effect will be at work inducing the individual to choose low deductibles (hence, high coverage) for those risks that are very unlikely. Hence, our paper tends to support the sometimes held view that insurance is the most appropriate risk management tool for low frequency risks. …

Insurance Demand without the Expected-Utility Paradigm

This article focuses on two cornerstone results in insurance economics: Mossin's Theorem on the optimality of full versus partial coverage, and Arrow's Theorem on the optimality of straight deductible policies. Both of these results are examined in a model assuming only risk aversion, and not necessarily expected-utility maximization. The results also are examined with the inclusion of a noninsurable background risk. Arrow's result is robust enough to hold in all of these situations. Mossin's result is shown to hold with a slight weakening, to account for possible nonsmoothness of preferences in non-expected-utility models.

A NOTE ON THE EXISTENCE OF AN ISLAMIC SOCIAL WELFARE FUNCTION

This paper aims at examining the possibility of constructing an Arrow‐social welfare function (SWF) in an Islamic community, that is abide by the Islamic tradition (i.e. the Shari'ah), where the domain of the individuals is defined in the context of Islamic framework. This domain is defined by the Shari'ah, where part of that domain is constrained, such that those actions that lie in the wajib (i.e. obligatory) or haram (i.e. prohibited), where the individual has no choice, must be thoroughly explored with respect to such a welfare function to consistency with the Shari'ah. Individual choice functions only within that segment of the domain that Islamic law leaves to the individual; the individual makes choices concerning actions in which they will not lose any praise or reward if they do or do not act one way or another. My concern in this regard will be the Mubah (i.e. permissible) range of the individual domain. In the process of examining the possibility of constructing an Arrow form (SWF) in an Islamic community, I have added a new Axiom to those suggested by Arrow (1951, pp.22–31). Kenneth J. Arrow, in his book Social Choice and Individuals Values (1951), has argued that five requirements of ‘fairness’ must characterize an acceptable social welfare function. He finds that these are inconsistent with each other. In fact, no welfare function exists that will satisfy all of those conditions simultaneously. As laid out by Arrow, these five conditions are, respectively: (1) universal domain; (2) positive association of individuals' values; (3) independence of irrelevant alternatives; (4) citizens' sovereignty; (5) nondictatorship. These conditions, characterized by Arrow as “seemingly innocuous” together rule out the possibility of deriving a complete and consistent (SWF) (Luce and Raiffa, 1957, p.328). The new axiom which I will refer to as Islamically imposed axiom such that, in the process of formulating an Arrow form social welfare function, it is important to be consistent with the Shari'ah. The Islamically imposed condition will thus guarantee the consistency with the Shari'ah in the process of formulating a social welfare function. This axiom implies that to maintain consistency the following must hold: When ti, (which refers to actions that individuals must do) is an element of the choice set (xi, ti), ti will always be chosen over xi (which refers to the feasible set of alternatives that individuals can choose from), and when tio is an element of the choice set (which refers to actions that individuals must not do), it will never be chosen. I came to the conclusion that when the choice set contains alternatives that the individual must do (wajib) or must not do (haram), the former will be chosen and the latter will never be chosen. Thus, for the general case, Arrow's negative results follow. If these elements are not in the choice set (i.e. all elements in the choice set belongs to Mubah activities), then Arrow's results follow in a straightforward fashion.

Impossibility and Infinity

Kenneth Arrow's General Possibility Theorem virtually created the academic field of specialization known as social choice theory and can reasonably be claimed as one of the handful of basic new ideas produced in the last thirty years, along with generative grammar, the quark theory, the structure of DNA, and not too many more. But, although there is no difficulty in seeing that the theorem is an extraordinary intellectual tour de force, it took a long time before it was accepted that it could not be evaded by a slight reformulation of the axioms, and even now it is hard to keep clear in one's mind just what it establishes and why it works. The two articles comprising this symposium mark, in my view, a major step forward in making the significance of Arrow's results plainer and throwing light on his proof. The article by MacKay shows how a crucial part of the proof of Arrow's theorem can be cast in a way that links it to the more familiar and perspicuous notion of infinite regress. Hardin's companion piece sets MacKay's article in a broader context and gives a slow-motion demonstration of the key move in Arrow's own version of the proof. It is my hope that even readers who have not previously acquainted themselves with this kind of work will find these articles both accessible and rewarding.

The Concept of `Choice' and Arrow's Theorem

Kenneth J. Arrow has argued that any procedure for summing individual choices into social policy either must violate basic and noncontroversial criteria of rationality or else must be dictatorial or imposed. A procedure is dictatorial if it always allows the choices of one individual to become social policy regardless of the choices of other individuals. A method is imposed if it allows social policies to be formed without regard to the choices of the individuals in the system. Arrow's result, the General Possibility Theorem, has been widely understood to demonstrate that no constitution, political process, or social welfare (or preference) function can satisfy the minimal requirements of political legitimacy implicit in any democratic political philosophy.' There have been many attempts to show that one or more of Arrow's premises (axioms and conditions) are too strong or unwarranted. None of these has been generally successful in removing the doubts or dispelling the worries which were occasioned by Arrow's initial discussion. Little attention has been focused on the transitivity axiom. There seems to be a very broad consensus that Arrow is correct in claiming it to be a criterion of rational choice. This axiom states that, for all alternative courses of action or items of choice, x, y,

An Axiomatic Theory of Tournament Aggregation

An axiomatic theory for aggregation of individual preferences is developed. Many authors, since K. J. Arrow, have studied the case where every individual preference is an ordering. We study here the case where every individual preference is a tournament (for instance, in “paired comparisons”). The original results obtained can be compared to those of the classic theory. For example, we prove, in this context, a generalisation of Arrow's theorem and we emphasize duality between Arrow's results and Black-Inada-Sen's results (technically by means of a Galois connection between two lattices). We used social functions defined by means of “families of majorities” (simple games),

The existence of group preference functions

Although Kenneth Arrow's impossibility theorem in social choice theory ([1]) is well-known and famous, it seems to be less common knowledge that this result does not hold for the case of an infinite number of voters. As far as I know, the first time this fact has been mentioned in print is in Peter Fishburn's [3]. There he gave a proof of the existence of what Arrow called a "social welfare function", using a special kind of probability measure. In a letter Peter Fishburn has informed me that Julian Blau knew about the infinite voter result in 1960 already. He has, however, not published anything on it. In this paper I will prove that not only do such functions exist, but they exist in great numbers. In fact, the cardinal number of functions satisfying the conditions posed by Arrow equals the total number of functions from the same domain. Arrow's theorem will follow as a corollary when the main theorem is combined with a simple mathematical fact. The roles played by the different conditions are spelt out more clearly than usual when Arrow's result is proved in this way. The technique of the proof is also applicable to the case of so-called "quasi-transitive" social preference. Especially Amartya Sen [8] and Frederic Schick [7] have argued that a possible solution to the problem of social choice would be to relax the condition that social preference be transitive and require only

Some Negative Results on the Existence of Comparative Statics Results in Portfolio Theory

Consider an investor who has a certain amount of wealth to invest in a riskless security and several risky securities. The investor's optimal portfolio will depend on his attitudes towards risk, his wealth and the probability distribution of the security returns. An interesting question to ask is how the investor's optimal portfolio is affected by changes in his wealth, given that all other things remain constant. For example, does the total amount invested in risky securities increase as wealth increases? Does the proportion of wealth invested in risky securities decrease as wealth increases? Questions such as these have been investigated by Arrow [1, Chapter 3] in the case of one riskless security and one risky security. Arrow showed that if the investor's von Neumann-Morgenstern utility function exhibits decreasing absolute risk aversion and increasing relative risk aversion, the amount invested in the risky security is an increasing function of wealth and the proportion of wealth invested in the risky security is a decreasing function of wealth. More recently, Cass and Stiglitz [3] have shown that Arrow's results do not generalize to the case of many risky securities. They give an example where an investor who can purchase one riskless security and two risky securities invests a greater proportion of his wealth in the two risky securities when his wealth increases, even though his utility function exhibits increasing relative risk aversion. Cass and Stiglitz note, however, that Arrow's results do generalize for an important, if highly restrictive, class of utility functions-those for which the mix of risky securities in the investor's optimal portfolio is independent of the investor's wealth for all probability distributions of security returns. Such utility functions are said to possess the separation property. The purpose of this paper is to prove that the separation property is a necessary condition as well as a sufficient condition for the generalization of Arrow's results to the case of many risky securities. We will show that given more than one risky security and a utility function which does not possess the separation property, it is always possible to pick probability distributions for the returns of the risky securities so that the directions of change which Arrow established for the single risky security case are reversed; that is, for some probability distributions of security returns, the total amount invested in risky securities decreases as wealth increases, and for other probability distributions of security returns, the proportion of wealth invested in risky securities increases as wealth increases. In fact, we will show that there always exist probability distributions of security returns such that the amount (proportion of wealth) invested in every risky security decreases (increases) as wealth increases. It should be emphasized, moreover, that this is the case

Income redistribution and social choice: A pragmatic approach

The conclusions of modern formal welfare analysis can be summarized in two statements. First, although there exists an infinite number of Pareto optimum positions which exhaust all possibilities of improving the welfare of one individual without diminishing the welfare of others, there is no objective criterion for selecting one of these optima as the most preferred or best position.1 Second, if individuals' preferences are to count and other apparently reasonable conditions are to be met, no voting system will be able to achieve a consistent ordering of these op t imum positions, i.e., to pick the most preferred position under all circumstances. This is Arrow's famous Impossibility Theorem.2 These conclusions seem to rule out the possibility of defining a unique welfare maximum and designing public policies to achieve it. Movements along the utility possibility frontier result in some individuals gaining while others lose thus altering the distribution of income and welfare. Ranking alternative positions on the frontier implies comparisions of individuals' utilities or welfare ccontributions which economists deny the capacity to make on behalf of society. Furthermore Arrow's results suggest the individuals would not be able to express their views on these comparisions accurately and consistently through voting.