A deterministic event tree approach to uncertainty, randomness and probability in individual chance processes

Abstract

Both Popper and Good have noted that a deterministic microscopic physical approach to probability requires subjective assumptions about the statistical distribution of initial conditions. However, they did not use such a fact for defining an a priori probability, but rather recurred to the standard observation of repetitive events. This observational probability may be hard to assess for real-life decision problems under uncertainty that very often are - strictly speaking - non-repetitive, one-time events. This may be a reason for the popularity of subjective probability in decision models. Unfortunately, such subjective probabilities often merely reflect attitudes towards risk, and not the underlying physical processes.

In order to get ‘as objective as possible’ a definition of probability for one-time events, this paper identifies the origin of randomness in individual chance processes. By focusing on the dynamics of the process, rather than on the (static) device, it is found that any process contains two components: observer-independent (= ‘objective’) and observer-dependent (= ‘subjective’). Randomness, if present, arises from the subjective definition of the ‘rules of the game’, and is not - as in Popper's propensity - a physical property of the chance device. In this way, the classical definition of probability is no longer a primitive notion based upon equally possible cases, but is derived from the underlying microscopic processes, plus a subjective, clearly identified, estimate of the branching ratios in an event tree. That is, equipossibility is not an intrinsic property of the system object/subject but is forced upon the system via the rules of the game/measurement.

Also, the typically undefined concept of symmetry in games of chance is broken down into objective and subjective components. It is found that macroscopic symmetry may hold under microscopic asymmetry. A similar analysis of urn drawings shows no conceptual difference with other games of chance (contrary to Allais' opinion). Finally, the randomness in Lande's knife problem is not due to ‘objective fortuity’ (as in Popper's view) but to the rules of the game (the theoretical difficulties arise from intermingling microscopic trajectories and macroscopic events).