Home > Uncategorized > Solving the fundamental problem of decision theory (wonkish)

Solving the fundamental problem of decision theory (wonkish)

from Lars Syll

Currently the dominant formalism for treating the [general gamble] problem is utility theory. Utility theory was born out of the failure of the following behavioral null model: individuals were assumed to optimize changes in the expectation values of their wealth. We argue that this null model is a priori a bad starting point because the expectation value of wealth does not generally reflect what happens over time. We propose a different null model of human behavior that eliminates, in many cases, the need for utility theory: an individual optimizes what happens to his wealth as time passes …

Our method starts by recognizing the inevitable non- ergodicity of stochastic growth processes, e.g. noisy multiplicative growth. The specific stochastic process implies a set of meaningful observables with ergodic properties, e.g. the exponential growth rate. These observables make use of a mapping that in the tradition of economics is viewed as a psychological utility function, e.g. the logarithm …

The dynamic approach to the gamble problem makes sense of risk aversion as optimal behavior for a given dynamic and level of wealth, implying a different concept of rationality. Maximizing expectation values of observables that do not have the ergodic property … cannot be considered rational for an individual. Instead, it is more useful to consider rational the optimization of time-average performance, or of expectation values of appropriate ergodic observables. We note that where optimization is used in science, the deep insight is finding the right object to optimize … The same is true in the present case  — deep insight is gained by finding the right object to optimize — we suggest time-average growth.

Ole Peters & Murray Gell-Mann

Although the expected utility theory is obviously both theoretically and descriptively inadequate, colleagues in economics, game theory and decision theory, gladly continue to use it, as though its deficiencies were unknown or unheard of.  

In the neoclassical theory of expected utility, one thinks in terms of ‘parallel universe’ and asks what is the expected return of an investment, calculated as an average over the ‘parallel universe’? It is as if one supposes that various ‘I’ are tossing a coin and that the loss of many of them will be offset by the huge profits one of these ‘I’ does. But this ensemble-average does not work for an individual, for whom a time-average better reflects the experience made in the ‘non-parallel universe’ we live in.

Since we cannot go back in time — entropy and the ‘arrow of time’ make this impossible — and the bankruptcy option is always at hand (extreme events and ‘black swans’ are always possible) we have nothing to gain from thinking in terms of ensembles and ‘parallel universe.’

Actual events follow a fixed pattern of time, where events are — as underlined by Peters and Gell-Mann — often linked in multiplicative processes (as e. g. investment returns with ‘compound interest’) which are basically non-ergodic.

Instead of arbitrarily assuming that people have a certain type of utility function – as in the neoclassical theory – we can obtain a less arbitrary and more accurate picture of real people’s decisions and actions by basically assuming that time is irreversible. When the bankroll is gone, it’s gone. The fact that in a parallel universe it could conceivably have been refilled, are of little comfort to those who live in the one and only possible world that we call the real world.

The works of people like Peters and Gell-Mann show that much of the high esteem that economists today have  of expected utility theory, game theory and decision theory, is questionable.

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