P5 Maths Heuristics
Gabriel Litke
A heuristic (/hjʊˈrɪstɪk/; from Ancient Greek εὑρίσκω (heurskō) 'method of discovery' [1]) or heuristic technique (problem solving, mental shortcut, rule of thumb)[2][3][4][5] is any approach to problem solving that employs a pragmatic method that is not fully optimized, perfected, or rationalized, but is nevertheless "good enough" as an approximation or attribute substitution.[6][7] Where finding an optimal solution is impossible or impractical, heuristic methods can be used to speed up the process of finding a satisfactory solution.[8][9] Heuristics can be mental shortcuts that ease the cognitive load of making a decision.[10][11][12]
Heuristic reasoning is often based on induction, or on analogy[.] [...] Induction is the process of discovering general laws [...] Induction tries to find regularity and coherence [...] Its most conspicuous instruments are generalization, specialization, analogy. [...] Heuristic discusses human behavior in the face of problems [...that have been] preserved in the wisdom of proverbs.[13]
A heuristic is a strategy that ignores part of the information, with the goal of making decisions more quickly, frugally, and/or accurately than more complex methods (Gigerenzer and Gaissmaier [2011], p. 454; see also Todd et al. [2012], p. 7).[15]
Heuristics are strategies based on rules to generate optimal decisions, like the anchoring effect and utility maximization problem.[16] These strategies depend on using readily accessible, though loosely applicable, information to control problem solving in human beings, machines and abstract issues.[17][18] When an individual applies a heuristic in practice, it generally performs as expected. However it can alternatively create systematic errors.[19]
The most fundamental heuristic is trial and error, which can be used in everything from matching nuts and bolts to finding the values of variables in algebra problems. In mathematics, some common heuristics involve the use of visual representations, additional assumptions, forward/backward reasoning and simplification.
In psychology, heuristics are simple, efficient rules, either learned or inculcated by evolutionary processes. These psychological heuristics have been proposed to explain how people make decisions, come to judgements, and solve problems. These rules typically come into play when people face complex problems or incomplete information. Researchers employ various methods to test whether people use these rules. The rules have been shown to work well under most circumstances, but in certain cases can lead to systematic errors or cognitive biases.[21]
George Polya studied and published on heuristics in 1945.[67] Polya (1945) cites Pappus of Alexandria as having written a text that Polya dubs Heuristic.[68] Pappus' heuristic problem-solving methods consist of analysis and synthesis.[69]
The study of heuristics in human decision-making was developed in the 1970s and the 1980s, by the psychologists Amos Tversky and Daniel Kahneman,[81] although the concept had been originally introduced by the Nobel laureate Herbert A. Simon. Simon's original primary object of research was problem solving that showed that we operate within what he calls bounded rationality. He coined the term satisficing, which denotes a situation in which people seek solutions, or accept choices or judgements, that are "good enough" for their purposes although they could be optimised.[82]
Rudolf Groner analysed the history of heuristics from its roots in ancient Greece up to contemporary work in cognitive psychology and artificial intelligence,[83] proposing a cognitive style "heuristic versus algorithmic thinking", which can be assessed by means of a validated questionnaire.[84]
Heuristics, through greater refinement and research, have begun to be applied to other theories, or be explained by them. For example, the cognitive-experiential self-theory (CEST) is also an adaptive view of heuristic processing. CEST breaks down two systems that process information. At some times, roughly speaking, individuals consider issues rationally, systematically, logically, deliberately, effortfully, and verbally. On other occasions, individuals consider issues intuitively, effortlessly, globally, and emotionally.[91] From this perspective, heuristics are part of a larger experiential processing system that is often adaptive, but vulnerable to error in situations that require logical analysis.[92]
In 2002, Daniel Kahneman and Shane Frederick proposed that cognitive heuristics work by a process called attribute substitution, which happens without conscious awareness.[93] According to this theory, when somebody makes a judgement (of a "target attribute") that is computationally complex, a more easily calculated "heuristic attribute" is substituted. In effect, a cognitively difficult problem is dealt with by answering a rather simpler problem, without being aware of this happening.[93] This theory explains cases where judgements fail to show regression toward the mean.[94] Heuristics can be considered to reduce the complexity of clinical judgments in health care.[95]
A good example is a model that, as it is never identical with what it models, is a heuristic device to enable understanding of what it models. Stories, metaphors, etc., can also be termed heuristic in this sense. A classic example is the notion of utopia as described in Plato's best-known work, The Republic. This means that the "ideal city" as depicted in The Republic is not given as something to be pursued, or to present an orientation-point for development. Rather, it shows how things would have to be connected, and how one thing would lead to another (often with highly problematic results), if one opted for certain principles and carried them through rigorously.
Heuristic is also often used as a noun to describe a rule of thumb, procedure, or method.[98] Philosophers of science have emphasised the importance of heuristics in creative thought and the construction of scientific theories.[99] Seminal works include Karl Popper's The Logic of Scientific Discovery and others by Imre Lakatos,[100] Lindley Darden, and William C. Wimsatt.
In legal theory, especially in the theory of law and economics, heuristics are used in the law when case-by-case analysis would be impractical, insofar as "practicality" is defined by the interests of a governing body.[101]
The present securities regulation regime largely assumes that all investors act as perfectly rational persons. In truth, actual investors face cognitive limitations from biases, heuristics, and framing effects. For instance, in all states in the United States the legal drinking age for unsupervised persons is 21 years, because it is argued that people need to be mature enough to make decisions involving the risks of alcohol consumption. However, assuming people mature at different rates, the specific age of 21 would be too late for some and too early for others. In this case, the somewhat arbitrary delineation is used because it is impossible or impractical to tell whether an individual is sufficiently mature for society to trust them with that kind of responsibility. Some proposed changes, however, have included the completion of an alcohol education course rather than the attainment of 21 years of age as the criterion for legal alcohol possession. This would put youth alcohol policy more on a case-by-case basis and less on a heuristic one, since the completion of such a course would presumably be voluntary and not uniform across the population.
Heuristics refers to the cognitive shortcuts that individuals use to simplify decision-making processes in economic situations. Behavioral economics is a field that integrates insights from psychology and economics to better understand how people make decisions.
Anchoring and adjustment is one of the most extensively researched heuristics in behavioural economics. Anchoring is the tendency of people to make future judgements or conclusions based too heavily on the original information supplied to them. This initial knowledge functions as an anchor, and it can influence future judgements even if the anchor is entirely unrelated to the decisions at hand. Adjustment, on the other hand, is the process through which individuals make gradual changes to their initial judgements or conclusions.
Anchoring and adjustment has been observed in a wide range of decision-making contexts, including financial decision-making, consumer behavior, and negotiation. Researchers have identified a number of strategies that can be used to mitigate the effects of anchoring and adjustment, including providing multiple anchors, encouraging individuals to generate alternative anchors, and providing cognitive prompts to encourage more deliberative decision-making.
Other heuristics studied in behavioral economics include the representativeness heuristic, which refers to the tendency of individuals to categorize objects or events based on how similar they are to typical examples,[104] and the availability heuristic, which refers to the tendency of individuals to judge the likelihood of an event based on how easily it comes to mind.[105]
Stereotyping is a type of heuristic that people use to form opinions or make judgements about things they have never seen or experienced.[106] They work as a mental shortcut to assess everything from the social status of a person (based on their actions),[12] to classifying a plant as a tree based on it being tall, having a trunk, and that it has leaves (even though the person making the evaluation might never have seen that particular type of tree before).
Stereotypes, as first described by journalist Walter Lippmann in his book Public Opinion (1922), are the pictures we have in our heads that are built around experiences as well as what we are told about the world.[107][108]
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