The classic decision-theory problem of evaluating a gamble is treated from a modern perspective using dynamics . Linear and logarithmic utility functions appear not as expressions for the value of money but as mappings that result in ergodic observables for purely additive and purely multiplicative dynamics, the most natural stochastic processes to model wealth. This perspective is at odds with the boundedness requirement for utility functions in the dominant formalism of decision theory. We highlight conceptual and mathematical inconsistencies throughout the development of decision theory, whose correction clarifies that the modern perspective is legitimate and that boundedness of utility functionsis not required .
Gambles are random variables that model possible changes in monetary wealth. Classic decision theory transforms money into utility through a utility function and defines the value of a gamble as the expectation value of utility changes. Utility functions aim to capture individual psychological characteristics, but their generality limits predictive power. Expectation value maximizers are defined as rational in economics, but expectation values are only meaningful in the presence of ensembles or in systems with ergodic properties, whereas decision-makers have no access to ensembles and the variables representing wealth in the usual growth models do not have the relevant ergodic properties. Simultaneously addressing the shortcomings of utility and those of expectations, we propose to evaluate gambles by averaging wealth growth over time. No utility function is needed, but a dynamic must be specified to compute time averages . Linear and logarithmic "utility functions" appear as transformations that generate ergodic observables for purely additive and purely multiplicative dynamics, respectively. We highlight inconsistencies throughout the development of decision theory, whose correction clarifies that our perspective is legitimate . These invalidate a commonly cited argument for bounded utility functions .
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The classic decision-theory problem of evaluating a gamble is treated from a modern perspective using dynamics
Gambles are random variables that model possible changes in monetary wealth. Classic decision theory transforms money into utility through a utility function and defines the value of a gamble as the expectation value of utility changes. Utility functions aim to capture individual psychological characteristics, but their generality limits predictive power. Expectation value maximizers are defined as rational in economics, but expectation values are only meaningful in the presence of ensembles or in systems with ergodic properties, whereas decision-makers have no access to ensembles and the variables representing wealth in the usual growth models do not have the relevant ergodic properties. Simultaneously addressing the shortcomings of utility and those of expectations, we propose to evaluate gambles by averaging wealth growth over time. No utility function is needed, but a dynamic must be specified to compute time averages
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utility functions appear not as expressions for the value of money but as mappings that result in
"utility functions" appear as transformations that generate
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the most natural stochastic processes to model wealth. This perspective is at odds with the boundedness requirement for utility functions in the dominant formalism of decision theory. We highlight conceptual and mathematical
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the modern
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and that boundedness of utility functionsis not required
. These invalidate a commonly cited argument for bounded utility functions