Over the past few years, we have explored a conceptually deep, simple, change of perspective that leads to a novel approach to economics. Much of current economic theory is based on early work in probability theory, performed specifically between the 1650s and the 1730s. This foundational work predates the development of the notion of ergodicity, and it assumes that expectation values reflect what happens over time. This is not the case for stochastic growth processes, but such processes constitute the essential models of economics. As a consequence, nowadays expectation values are often used to evaluate situations where time averages would be appropriate instead, and the result is a “paradox,” “puzzle,” or “anomaly.” This class of problems, including the St. Petersburg paradox and the equity-premium puzzle, can be resolved by ensuring the following: the stochastic growth process involved in the problem needs to be made explicit; the process needs to be transformed to find an appropriate ergodic observable. The expectation value of the new observable will then indeed reflect long-time behavior, and the puzzling essence of the problem will go away. Here we spell out the general recipe, which we phrase as the solution to the general gamble problem that stood at the beginning of the debate in the 17th century. We hope that this recipe will resolve puzzles in many different areas.

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For clarity of exposition, we limit ourselves to the context of an individual evaluating gambles in situations where any attendant circumstances other than wealth, x, expressed in money can be disregarded. Currently, the dominant formalism for treating this problem is utility theory. Utility theory was born out of the failure of the following behavioral null model: individuals were assumed to optimize changes in the expectation values of their wealths. We argue that this null model is a priori a bad starting point because the expectation value of wealth does not generally reflect what happens over time. We propose a different null model of human behavior that eliminates, in many cases, the need for utility theory: an individual optimizes what happens to his wealth as time passes.

1 Boltzmann and statistical mechanics ,” in Boltzmann's Legacy 150 Years After His Birth, Atti dei Convegni Lincei ( Accademia Nazionale dei Lincei , Rome , 1997), Vol. 131, pp. 9– 23 ; available at 1. E. G. D. Cohen, “,” in Boltzmann's Legacy 150 Years After His Birth, Atti dei Convegni Lincei (, 1997), Vol. 131, pp. 9–; available at http://arXiv.org/abs/cond-mat/9608054v2 stochastic processes, stationary independent increments. Being stationary and independent, these observables have many ergodic properties, of which the following specific property is relevant here. The question whether the time average of an observable is well represented by an appropriate expectation value dates back to the 19th-century development of statistical mechanicsand is the origin of the field called “ergodic theory.” In the following, we will identify, for differentstationary independent increments. Being stationary and independent, these observables have many ergodicof which the following specificis relevant here.

Ergodic property (equality of averages)

The expectation value of the observable is a constant (independent of time), and the finite-time average of the observable converges to this constant with probability one as the averaging time tends to infinity.

Whether an observable possesses this property is crucial when assessing the significance of its expectation value. We will refer to observables with this property as “ergodic observables.”

decision theory. Decision theory studies mathematical models of situations that create an internal conflict and necessitate a decision. For instance, we may wish to model the situation of being offered a lottery ticket. The conflict is between the unpleasant certainty that we have to pay for the ticket, and the pleasant possibility that we may win the jackpot. It necessitates the decision whether to buy a ticket or not. Although economics deals with many types of decisions, not all of which are monetary, the quantitative treatment of the gamble problem is central to many branches of economics including utility theory, decision theory, game theory, and asset pricing theory which in turn informs macroeconomics, as has been argued convincingly. 2 2. J. H. Cochrane, Asset Pricing ( Princeton University Press , 2001). Gambles are the formal basis ofstudies mathematicalof situations that create an internal conflict and necessitate a decision. For instance, we may wish tothe situation of being offered a lottery ticket. The conflict is between the unpleasant certainty that we have to pay for the ticket, and the pleasant possibility that we may win the jackpot. It necessitates the decision whether to buy a ticket or not. Although economics deals with many types of decisions, not all of which are monetary, the quantitative treatment of the gamble problem is central to many branches of economics including utilityand asset pricingwhich in turn informs macroeconomics, as has been argued convincingly.

We will be dealing with mathematical models but use a common suggestive nomenclature. In this section, we write in small capitals those terms of everyday language that in the following will refer to mathematical objects and operations.

A gamble is a set of possible net payouts D ( n ) with associated probabilities p ( n ) , where n is an integer designating an outcome . For convenience, we order outcomes such that D ( n + 1 ) > D ( n ) . Different gambles are compared, the decision being which to subject one's wealth to, and more generally to what extent.