Abstract
This book is about “uncertainty” in decision science and economic science and its application to everyday problems we face.
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Notes
- 1.
In mathematics, a proposition is a statement that is either always true or always false. For example, the statement that a Martian lives on Mars is a proposition, and the statement that living on Mars is not comfortable is not.
- 2.
When we take “time” into consideration explicitly, we use S to denote a state space for one period and then use its “self-product” to denote the state space over the entire time span; that is, the “true” state space according to the context of the main text. Another common symbol for the state space is \(\Omega \).
- 3.
Mathematically, a single state, often denoted by s, is an element of the state space S. With the symbols, we write \(s \in S\).
- 4.
A family of subsets satisfying these conditions is named the algebra or the \(\sigma \) -algebra. See Chap. 2.
- 5.
If we state it with more mathematical precision, it is given by \(\{ \phi , \{``\text {sunny''}\}, \{``\text {rainy''}, ``\text {snowy''} \}, S \}\) because the empty set and the whole state space are always included by any information structure . This is because some whether must take place tomorrow, leading to our always knowing that S has occurred and that \(\phi \) has never occurred. See the next paragraph for discussion of “occurred”.
- 6.
The mathematical requirements for a subset of S to be qualified as an event have natural interpretations. They are important not only mathematically, but also in terms of economics. See Chap. 2 for the precise statement of these requirements and see Sect. 3.12 for further discussions on this “eventness”.
- 7.
By “learning,” we may become more informed about the true state of the world. As such, the information structure may change over time. As a result of some “learning,” say, by a new possibility of observing the stock price of a firm that produces snow tires, the information structure may become “finer” and it may become represented by singleton events: \(\{``\text {sunny''} \}\) , \(\{``\text {rainy''} \}\), and \(\{``\text {snowy''} \}\). We consider this sort of “learning” process in Chaps. 13 and 15.
- 8.
For the reason why we say “basically,” see the above footnote and Sect. 3.12.
- 9.
Note that the event given by the question is defined well enough in light of the preceding two subsections.
- 10.
With the full rigor of mathematics, the limit might not exist. To avoid such an issue, we may replace “\(\lim \)” with, say, “\(\limsup \)”.
- 11.
Some attempts have been made toward incorporating the frequentist approach into a framework of decision-making. For example, see Gilboa and Schmeidler (2001) . Their theory is known as case-based decision theory. It develops a concept of the similarity function that gauges the frequency of an events.
- 12.
Note that 0 is special because it is the unique number satisfying \(0 + 0 = 0\). The “additivity” we mention in the next subsection forces the probability of an event that never happens to be 0.
- 13.
An extremely famous example of an axiom is Euclid’s fifth axiom in geometry: “Two parallel lines never intersect.” While there is no argument about the axiom’s legitimacy and infallibility, we may deny this axiom to use non-Euclidean geometry, e.g., Riemannian geometry.
- 14.
The precise definitions of these mathematical terms are given in Chap. 2.
- 15.
This suggests that there might exist a probability that is not additive, which turns out to be true later.
- 16.
This fact follows from the additivity axiom because the probabilities of H and T need to add up to unity.
- 17.
The additivity of the probability defined as the “limit” of ratios needs to be proven. This could be a big issue although it has never been discussed to our knowledge. We do not address this issue in this book.
- 18.
The existence of randomizing devices is essential in some areas of economics. For example, in defining the so-called “mixed strategy” in game theory, a randomizing device plays a central role.
- 19.
Knight (1921).
- 20.
Savage (1954).
- 21.
Savage proved much more than this. According to Savage’s theorem, if her behavioral pattern complies with the axioms, she behaves as if she calculates the mathematical expectation of her prospects of random utility by using the subjective probability. If she chooses a specific action, she should maximize her expected utility thus calculated. In the main text, we focus on the existence of the single probability that is included among the claims of his theorem. Furthermore, if we only care about the existence of the single additive subjective probability, Savage’s axioms can be weakened. See Machina and Schmeidler (1992) and Sect. 3.6 of this book.
- 22.
“Knightian uncertainty” means the situation where a probability does not yet exist; “ambiguity” means that the decision-maker is endowed with a set of probabilities; “maxmin” behavior is a decision-maker’s specific attitude toward ambiguity.
- 23.
- 24.
- 25.
The distinction between a charge and a measure is truly mathematical. The details are shown in Chap. 2.
- 26.
- 27.
Roughly speaking, a capacity is convex if the capacity of the union of the mutually disjoint events is larger than or equal to the sum of the capacities of each single event. For the precise definition, see Chap. 2.
- 28.
Mathematical expression is helpful when we analyze dynamic economic models with Knightian uncertainty. See, for example, Sect. 2.5.
- 29.
We can consider a “hybrid” of MEU theory and CEU theory. We may assume that the agent is endowed with a set of probability capacities that are not necessarily additive. Although interesting, we do not pursue this line of research in this book.
- 30.
The unique equilibrium price in a dynamic economic model is quite often “well-behaved” in the sense that it is “stationary” and “Markovian.” For example, it depends on the weather on that day rather than on the date or the weather of the past week. Hence, the booms and crashes of the equilibrium prices are difficult to explain by such a model. See Epstein and Wang (1995) in this regard.
- 31.
- 32.
In the definition of the mean-preserving spread , the meaning of being “more disperse” is defined precisely via mathematics. Importantly, this sense of dispersion does not coincide with the mathematical “variance” in many cases.
- 33.
- 34.
- 35.
Among others, Klibanoff et al. (2005) is noteworthy. However, they assume that uncertainty can be parameterized and that the distribution of the parameter is known, rather than unknown. In this book, uncertainty is so deep that even the family of distributions is unknown.
- 36.
- 37.
- 38.
- 39.
- 40.
It should be noted that a constant is the only “measurable” function when no information is available, since a constant can be evaluated correctly without knowing any information about whichever event has occurred actually.
- 41.
The conditional mathematical expectation of a random variable is the “best” approximation of it by another random variable, i.e., by a “measurable” function, which is “measurable” with respect to some partial information that is coercer than the information with respect to which the original random variable is measurable. These simple mathematical facts are often overlooked. This may be because both the definition and the existence-proof of the conditional mathematical expectation are formally conducted through the Radon–Nikodym derivative in standard textbooks of mathematics.
- 42.
If we could find a function that gauges an error when approximating one random variable by another, which we call error function, and if the “best” approximate of the former random variable by a constant via this error function were the given certainty equivalent, we might use this error function to define a conditional certainty equivalent given some coercer information. If this conditional certainty equivalent, which itself is a function like a conditional mathematical expectation, were uniquely defined for any coercer information, we might define it as the conditional certainty equivalent.
Ozaki (2009) proves that the conditioning scheme proposed in the preceding paragraph actually works “nicely” for some class of certainty equivalents called the implicit means, leading to an invention of the conditional implicit mean.
- 43.
Recall that the normal distribution is completely characterized by its mean and variance.
- 44.
See, for example, DeGroot (1970). The conjugate family of the normal distribution with an unknown mean and an unknown variance turns out to be the bivariate distribution. The conditional distribution of the mean given a variance is a normal distribution, and the marginal distribution of the reciprocal of the variance is a Gamma distribution.
- 45.
For these conditions, see DeGroot (1970).
- 46.
Basic ideas of this chapter were first developed in Nishimura and Ozaki (2002).
- 47.
In other words, it is a “nonlinear” extension of a transition probability, or Markov chain with a finite state space.
- 48.
See Sect. 1.2.1 for the term “behavioral” axiom.
- 49.
To see an example of the “if-and-only-if”-type statement, consider ten objects of choice among which the agent must choose the best alternative. Two possible “behavioral” axioms may look like (1) she can name the best object, the second-best object and the worst object given any three objects in this set of ten objects, and (2) once she names one object better than another, then she always names so in any situation of choice. An “if-and-only-if”-type statement should be as follows: she satisfies these two axioms if and only if she makes a choice about these ten objects as if she puts a mutually distinct number to each of the ten objects and she always chooses the object that is labeled by a higher number in any situation of choice.
- 50.
The precise meaning of the terms of “robust” and “shake” are given in Chap. 6.
- 51.
An important exception is Rinaldi (2009).
- 52.
The issue of dynamic consistency arises even in the two-period model. It is of extreme importance when we consider an infinite horizon because such consistent structure of a preference is fully exploited when we analyze the models by some mathematical method named dynamic programming .
- 53.
We use the terms of “dynamical consistency” and “time consistency” interchangeably.
- 54.
Chapter 10 considers a continuous-time model and the claim that the contents of Chap. 7 concerning dynamic programming are used in that chapter stands only heuristically. Furthermore, searching a firm mathematical foundation for dynamic programming techniques for continuous-time optimization problems is still ongoing. The results currently available in the literature for continuous-time models are basically derived by the “variational method.” The authors believe that the assumption necessary for solving the problems can be significantly weakened if we invoke dynamic programming techniques by assuming a dynamically consistent preference like the one in Chap. 10.
- 55.
See, for example, the theory and its applications of quasi-hyperbolic discounting, which leads to dynamically inconsistent preferences (Laibson 1997).
- 56.
See Epstein and Le Breton (1993).
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Nishimura, K.G., Ozaki, H. (2017). Overall Introduction. In: Economics of Pessimism and Optimism. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55903-0_1
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