Simple Scalability Research Articles

Decision making under uncertainty: a comparison of simple scalability, fixed-sample, and sequential-sampling models.

The purpose of this article is to investigate the learning and memory processes involved in decision making under uncertainty. In two different experiments, subjects were given a choice between a certain alternative that produced a single known payoff and an uncertain alternative that produced a normal distribution of payoffs. Initially this distribution was unknown, and in the first experiment it was learned through feedback from past decisions, whereas in the second experiment it was learned by observing sample outcomes. In the first experiment, a response deadline was used to limit the amount of time available for making a decision. In the second experiment, an observation cost was used to limit the number of samples that could be purchased. The mean and variance of the uncertain alternative and the value of the certain alternative were factorially manipulated to study their joint effects on choice probability, choice response time (Experiment 1), and number of observations purchased (Experiment 2). Algebraic-deterministic theories developed for decision making with simple gambles fail to explain the present results. Two new models are developed and tested--fixed- and sequential-sampling models--that attempt to describe the learning and memory processes involved in decision making under uncertainty.

Stochastic transitivity and the unidimensional control of choice

A concurrent-chains paradigm was used to test three models of preference for schedules of reinforcement. Each model assumes choice to be a monotonic function of values on a single dimension (“simple scalability”). For choices involving variable-interval and fixed-interval schedules, this dimension is assumed to be an harmonic average of the interreinforcement intervals; for choices involving fixedinterval and fixed-ratio schedules, the dimension is the time to reinforcement irrespective of response rate; for choices involving different combinations of rate and duration of reinforcement, the dimension is the rate of “reinforcement-time”, or the product of the two parameters. Each model implies a strong form of transitivity (“functional equivalence”), which held for half the subjects (that half exclusively sensitive to the reinforcement variable specified by the model). An alternative unidimensional model which implies a weaker form of transitivity (“weak binary utility model”) was consistent with the preferences of all subjects. This model states that a single factor controls the direction of preferences but not necessarily the exact choice probability, as simple scalability assumes.

Difference measurement and simple scalability with restricted solvability

Let A, B be two sets, with B ⊂ A × A, and ≤ a binary relation on B. The problem analyzed here is that of the existence of a mapping u: A → R, satisfying: (a,b) ℓ ( a ́ , b ́ ) iff ∨∧ μ(b) − μ(a) ⩽ μ( b ́ ) − μ( a ́ ) whenever ( a, b), ( a′, b′) ∈ B. In earlier discussions of this problem, it is usually assumed that B is connected on A. Here, we only assume that B satisfies a certain convexity property. The resulting system provides an appropriate axiomatization of Fechner's scaling procedures. The independence of axioms is discussed. A more general representation is also analyzed: (a,b) ℓ ( a ́ , b ́ ) iff ∨∧ F[μ(b), μ(a)] ⩽ F[μ b ́ ] , where F is strictly increasing in the first argument, and strictly decreasing in the second. Sufficient conditions are presented, and a proof of the representation theorem is given.