bounded rationality

Examples of bounded rationality in the following topics:

• Problems with the Rational Decision-Making Model

  • Critics of rational choice theory—or the rational model of decision-making—claim that this model makes unrealistic and over-simplified assumptions.
  • Their objections to the rational model include:
  • The more complex a decision, the greater the limits are to making completely rational choices.
  • The theory of bounded rationality holds that an individual's rationality is limited by the information they have, the cognitive limitations of their minds, and the finite amount of time they have to make a decision.
  • Bounded rationality shares the view that decision-making is a fully rational process; however, it adds the condition that people act on the basis of limited information.

• Behavioral Economics: Irrational Actions

  • Behavioral economics focuses on the bounds of rationality of economic agents.
  • Market inefficiencies: include the study non-rational decision making and incorrect pricing.
  • This graph shows the three stages of rational decision making that was devised by Herbert Simon, a notable economist and scientist.

• Polynomial and Rational Functions as Models

  • Polynomials and rational functions are both relatively accurate and relatively easy to use.
  • For lots of datasets, their are no asymptotes and data is more or less bounded.
  • To deal with the asymptotic problems of polynomials, we also use rational functions:
  • A rational function is the ratio of two polynomial functions and has the following form:
  • Discuss the advantages and disadvantages of using polynomial and rational functions as models

• The Intermediate Value Theorem

  • For each value between the bounds of a continuous function, there is at least one point where the function maps to that value.
  • It is false for the rational numbers Q.
  • However there is no rational number x such that f(x) = 0, because √2 is irrational.
  • A graph of a rational function, .

• Asymptotes

  • A rational function can have at most one horizontal or oblique asymptote, and many possible vertical asymptotes; these can be calculated.
  • Vertical asymptotes are vertical lines near which the function grows without bound.
  • A rational function has at most one horizontal or oblique asymptote, and possibly many vertical asymptotes.
  • In other words, vertical asymptotes occur at singularities, or points at which the rational function is not defined.
  • Explain when the asymptote of a rational function will be horizontal, oblique, or vertical

• Urban Planning in the Greek High Classical Period

  • Hippodamus of Miletus is considered the "father" of rational city planning, and the city of Priene is a prime example of his grid planned cities.
  • Before rational city planning, cities grew organically and often radiated out from a central point, such as the Acropolis and Agora at the center of Athens.
  • He is seen as the originator of the idea that a town plan might formally embody and clarify a rational social order.
  • Its colonnaded stoa bounded the public space to the north.
  • Instead, the rational plan of Priene allowed for access to multiple sites of the city and easy navigation through the city.

• Intermediate Value Theorem

  • The intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value.
  • It is false for the rational numbers $\mathbb{Q}$.
  • However there is no rational number $x$ such that $f(x) =0$, because $\sqrt 2$ is irrational.

• Zeroes of Polynomial Functions With Rational Coefficients

  • A real number that is not rational is called irrational.
  • The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers.
  • In mathematics, the adjective rational often means that the underlying field considered is the field Q of rational numbers.
  • Rational polynomial usually, and most correctly, means a polynomial with rational coefficients, also called a "polynomial over the rationals".
  • However, rational function does not mean the underlying field is the rational numbers, and a rational algebraic curve is not an algebraic curve with rational coefficients.

• Series and Sigma Notation

  • The numbers to be summed (called addends, or sometimes summands) may be integers, rational numbers, real numbers, or complex numbers.
  • Where i represents the index of summation; $x_i$ is an indexed variable representing each successive term in the series; $m$ is the lower bound of summation, and $n$ is the upper bound of summation.
  • Informal writing sometimes omits the definition of the index and bounds of summation when these are clear from context, as in:

• Introduction to Rational Functions

  • A rational function is one such that $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$; the domain of a rational function can be calculated.
  • Note that every polynomial function is a rational function with $Q(x) = 1$.
  • For a simple example, consider the rational function $y = 1/x$.
  • Factorizing the numerator and denominator of rational function helps to identify singularities of algebraic rational function.
  • Graph of a rational function with equation $\frac{(x^2 - 3x -2)}{(x^2 - 4)}$.