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Random utility model
In economics, a random utility model (RUM), also called stochastic utility model, is a mathematical description of the preferences of a person, whose choices are not deterministic, but depend on a random state variable.
A basic assumption in classic economics is that the choices of a rational person choices are guided by a preference relation, which can usually be described by a utility function. When faced with several alternatives, the rational person will choose the alternative with the highest utility. The utility function is not visible; however, by observing the choices made by the person, we can "reverse-engineer" his utility function. This is the goal of revealed preference theory.[citation needed]
In practice, however, people are not rational. Ample empirical evidence shows that, when faced with the same set of alternatives, people may make different choices. To an outside observer, their choices may appear random.
One way to model this behavior is called stochastic rationality. It is assumed that each agent has an unobserved state, which can be considered a random variable. Given that state, the agent behaves rationally. In other words: each agent has, not a single preference-relation, but a distribution over preference-relations (or utility functions).[citation needed]
Block and Marschak presented the following problem. Suppose we are given as input, a set of choice probabilities Pa,B, describing the probability that an agent chooses alternative a from the set B. We want to rationalize the agent's behavior by a probability distribution over preference relations. That is: we want to find a distribution such that, for all pairs a,B given in the input, Pa,B = Prob[a is weakly preferred to all alternatives in B]. What conditions on the set of probabilities Pa,B guarantee the existence of such a distribution?[citation needed]
Falmagne solved this problem for the case in which the set of alternatives is finite: he proved that a probability distribution exists iff a set of polynomials derived from the choice-probabilities, denoted Block-Marschak polynomials, are nonnegative. His solution is constructive, and provides an algorithm for computing the distribution.
Barbera and Pattanaik extend this result to settings in which the agent may choose sets of alternatives, rather than just singletons.
Block and Marschak proved that, when there are at most 3 alternatives, the random utility model is unique ("identified"); however, when there are 4 or more alternatives, the model may be non-unique. For example, we can compute the probability that the agent prefers w to x (w>x), and the probability that y>z, but may not be able to know the probability that both w>x and y>z. There are even distributions with disjoint supports, which induce the same set of choice probabilities.
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Random utility model AI simulator
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Random utility model
In economics, a random utility model (RUM), also called stochastic utility model, is a mathematical description of the preferences of a person, whose choices are not deterministic, but depend on a random state variable.
A basic assumption in classic economics is that the choices of a rational person choices are guided by a preference relation, which can usually be described by a utility function. When faced with several alternatives, the rational person will choose the alternative with the highest utility. The utility function is not visible; however, by observing the choices made by the person, we can "reverse-engineer" his utility function. This is the goal of revealed preference theory.[citation needed]
In practice, however, people are not rational. Ample empirical evidence shows that, when faced with the same set of alternatives, people may make different choices. To an outside observer, their choices may appear random.
One way to model this behavior is called stochastic rationality. It is assumed that each agent has an unobserved state, which can be considered a random variable. Given that state, the agent behaves rationally. In other words: each agent has, not a single preference-relation, but a distribution over preference-relations (or utility functions).[citation needed]
Block and Marschak presented the following problem. Suppose we are given as input, a set of choice probabilities Pa,B, describing the probability that an agent chooses alternative a from the set B. We want to rationalize the agent's behavior by a probability distribution over preference relations. That is: we want to find a distribution such that, for all pairs a,B given in the input, Pa,B = Prob[a is weakly preferred to all alternatives in B]. What conditions on the set of probabilities Pa,B guarantee the existence of such a distribution?[citation needed]
Falmagne solved this problem for the case in which the set of alternatives is finite: he proved that a probability distribution exists iff a set of polynomials derived from the choice-probabilities, denoted Block-Marschak polynomials, are nonnegative. His solution is constructive, and provides an algorithm for computing the distribution.
Barbera and Pattanaik extend this result to settings in which the agent may choose sets of alternatives, rather than just singletons.
Block and Marschak proved that, when there are at most 3 alternatives, the random utility model is unique ("identified"); however, when there are 4 or more alternatives, the model may be non-unique. For example, we can compute the probability that the agent prefers w to x (w>x), and the probability that y>z, but may not be able to know the probability that both w>x and y>z. There are even distributions with disjoint supports, which induce the same set of choice probabilities.