An Axiomatic Approach to the Law of Small Numbers

Jawwad Noor; F. Payró

An Axiomatic Approach to the Law of Small Numbers

Jawwad Noor, F. Payró

Boston University (BU)

Producció científica: Document de treball

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Resum

Amb creences sobre els resultats de les tirades de moneda com a primitiu, formalitzem la Llei dels Petits Nombres (Tversky i Kahneman (1974)) amb un axioma que expressa una creença que la mitjana mostral de qualsevol seqüència tendirà cap al biaix percebut de la moneda al llarg del camí. L'agent es representa per una creença que el biaix de la moneda depèn del camí i s'autocorregeix. El model és consistent amb l'evidència usada per a donar suport a la Llei dels Petits Nombres, com ara la Fal·làcia del Jugador. En un entorn d'inferència bayesiana, mostrem com l'aprenentatge es veu afectat per la interacció entre dues forces potencialment oposades: una creença en absència de ratxes i una creença que la mitjana mostral tendirà cap al biaix vertader. Mostrem que, a diferència dels altres resultats d'aprenentatge de la literatura (Rabin (2002), Epstein, Noor, and Sandroni (2010)), l'última força assegura que l'agent almenys admet el paràmetre vertader com a possible en el límit, si no aprèn amb certesa que és vertader. En un entorn evolutiu, mostrem que els agents que creuen en la Llei dels Petits Nombres mai són expulsats de la cursa evolutiva per agents "estàndards" que entenen correctament l'aleatorietat.

Idioma original	Anglès
Lloc de publicació	Barcelona
Editor	BSE Working Papers
Nombre de pàgines	26
Estat de la publicació	Publicada - 30 de set. 2022

Sèrie de publicacions

Nom	BSE Working Paper Series
Núm.	1364

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https://bse.eu/research/working-papers/axiomatic-approach-law-small-numbersLlicència: C In Copyright

Análisis teórico y empírico del comportamiento de individuos y empresas con especial énfasis en innovación, salud y el mercado digital
Perez Castrillo, J. D. (Investigador/a principal), Macho Stadler, I. (Investigador/a), Martínez Giralt, X. (Investigador/a), Olivella, P. (Investigador/a), Piolatto, A. (Investigador/a) & Stockler Barbosa, G. G. (Col.laborador/a)
Ministerio de Ciencia e Innovación (MICINN), Fons Europeu de Desenvolupament Regional (FEDER)
1/01/19 → 30/09/22
Projecte: Projectes i Ajuts a la Recerca

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title = "An Axiomatic Approach to the Law of Small Numbers",

abstract = "With beliefs over the outcomes of coin-tosses as our primitive, we formalize the Law of Small Numbers (Tversky and Kahneman (1974)) by an axiom that expresses a belief that the sample mean of any sequence will tend towards the coin{\textquoteright}s perceived bias along the entire path. The agent is represented by a belief that the bias of the coin is path-dependent and self-correcting. The model is consistent with the evidence used to support the Law of Small Numbers, such as the Gambler{\textquoteright}s Fallacy. In the setting of Bayesian inference, we show how learning is affected by the interplay between two potentially opposing forces: a belief in the absence of streaks and a belief that the sample mean will tend to the true bias. We show that, unlike other learning results in the literature (Rabin (2002), Epstein, Noor, and Sandroni (2010)), the latter force ensures that the agent at least admits the true parameter as possible in the limit, if not learn with certainty that it is true. In an evolutionary setting, we show that agents who believe in the Law of Small Numbers are never pushed out of the evolutionary race by “standard” agents who correctly understand randomness.",

keywords = "Law of Small Numbers, Belief Biases, Heuristics, Gambler{\textquoteright}s Fallacy, Learning, Misspecified beliefs, Evolution",

author = "Jawwad Noor and F. Payr{\'o}",

note = " The authors would like to thank seminar audiences at Boston University, UPenn, U of Edinburgh, Manchester and Durham, for helpful feedback.",

year = "2022",

month = sep,

day = "30",

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T1 - An Axiomatic Approach to the Law of Small Numbers

AU - Noor, Jawwad

AU - Payró, F.

N1 - The authors would like to thank seminar audiences at Boston University, UPenn, U of Edinburgh, Manchester and Durham, for helpful feedback.

PY - 2022/9/30

Y1 - 2022/9/30

N2 - With beliefs over the outcomes of coin-tosses as our primitive, we formalize the Law of Small Numbers (Tversky and Kahneman (1974)) by an axiom that expresses a belief that the sample mean of any sequence will tend towards the coin’s perceived bias along the entire path. The agent is represented by a belief that the bias of the coin is path-dependent and self-correcting. The model is consistent with the evidence used to support the Law of Small Numbers, such as the Gambler’s Fallacy. In the setting of Bayesian inference, we show how learning is affected by the interplay between two potentially opposing forces: a belief in the absence of streaks and a belief that the sample mean will tend to the true bias. We show that, unlike other learning results in the literature (Rabin (2002), Epstein, Noor, and Sandroni (2010)), the latter force ensures that the agent at least admits the true parameter as possible in the limit, if not learn with certainty that it is true. In an evolutionary setting, we show that agents who believe in the Law of Small Numbers are never pushed out of the evolutionary race by “standard” agents who correctly understand randomness.

AB - With beliefs over the outcomes of coin-tosses as our primitive, we formalize the Law of Small Numbers (Tversky and Kahneman (1974)) by an axiom that expresses a belief that the sample mean of any sequence will tend towards the coin’s perceived bias along the entire path. The agent is represented by a belief that the bias of the coin is path-dependent and self-correcting. The model is consistent with the evidence used to support the Law of Small Numbers, such as the Gambler’s Fallacy. In the setting of Bayesian inference, we show how learning is affected by the interplay between two potentially opposing forces: a belief in the absence of streaks and a belief that the sample mean will tend to the true bias. We show that, unlike other learning results in the literature (Rabin (2002), Epstein, Noor, and Sandroni (2010)), the latter force ensures that the agent at least admits the true parameter as possible in the limit, if not learn with certainty that it is true. In an evolutionary setting, we show that agents who believe in the Law of Small Numbers are never pushed out of the evolutionary race by “standard” agents who correctly understand randomness.

KW - Law of Small Numbers

KW - Belief Biases

KW - Heuristics

KW - Gambler’s Fallacy

KW - Learning

KW - Misspecified beliefs

KW - Evolution

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BT - An Axiomatic Approach to the Law of Small Numbers

PB - BSE Working Papers

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ER -

An Axiomatic Approach to the Law of Small Numbers

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Análisis teórico y empírico del comportamiento de individuos y empresas con especial énfasis en innovación, salud y el mercado digital

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