The term Stochastic dominance is used in decision theory to refer to situations where one lottery (a probability distribution over outcomes) can be ranked as superior to another, with only limited knowledge of preferences. Decision theory is an interdisciplinary area of study, related to and of interest to practitioners in mathematics, statistics, economics, philosophy, management and psychology. ...
The simplest case, statewise dominance arises when one prospect gives a better outcome than another in every possible state of nature (more precisely, at least as good an outcome in every state, with strict inequality in at least one state). For example, if a dollar is added to every prize in a lottery, the new lottery statewise dominates the old one. Anyone who prefers more to less (in the standard terminology, anyone who has monotonic preferences) will always prefer a statewise dominant lottery. In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ...
The canonical case of stochastic dominance, referred to as first-order stochastic dominance arises when for any outcome x, the dominant distribution gives a higher probability of receiving an outcome equal to or better than x than the dominated distribution.
The other commonly used case of stochastic dominance is second-order stochastic dominance. All risk-averse expected-utility maximizers prefer a second-order stochastically dominant lottery to a dominated lottery. The same is true for non-expected utility maximizers with concave local utility functions.
Higher orders of stochastic dominance have also been analyzed, as have generalizations of the dual relationship between stochastic dominance orderings and classes of preference functions.
PS: On the violation of stochasticdominance cited by Kahneman and Tversky (1979), if this is the example you intended: that violation required two choices and a theory of joint receipt to show that it is a violation.
In that case, 58% of a sample chose the dominated gamble, but 58% was not significantly different from 50% by the standard two-tailed test of significance at the.05 level.
In that case, the dominated gamble was supposedly made to seem better because for each named "event" (color of marble drawn from an urn), the dominated gamble gave at least the same or a higher consequence.
Investigations of decision-making concerning stochasticdominance and violations of stochasticdominance are conducted through experiments presented in a questionnaire form, containing choices, such as gambles similar to the choice above with gambles E and F. These experiments in the past have presented gambles in text format, such as the gambles below.
In the pie chart format, violations of stochasticdominance were expected to be less than ½ because visual aids (the pie charts) were anticipated to assist in the mental process of deduction and comparison of percentages and values of the gambles, therefore influencing decision-making and reducing violations of stochasticdominance.
These models are able to predict violations of stochasticdominance and furthermore, they are able to correlate coalescing as a factor that increases probabilities of violations of stochasticdominance and event-splitting as a factor that eliminates violations of stochasticdominance.
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