Eighty subjects were recruited for the experiment, which was conducted in the EXEC Lab at the University of York. Subject ages ranged from 18 to 44 years. Seventy-eight subjects were students and two reported themselves as staff members at the University of York. There were 54 females and 26 males.
The experiment was conducted within subjects in two parts. Part I of the experiment tests the validity of Axioms 3, 4, and 5 from Section 2 in a perceptual task involving boxes containing red and blue dots. In Part II of the experiment, participants made choices between lotteries with different possible prizes and probabilities of winning money. The experiment permits a direct test for a link between properties of perception and behavior toward risk.
4.1 Part I of Experiment: Test of Axioms of Salience Perception
The axioms are designed to say when a pair of stimuli (x,y) is ‘more salient’ than another pair (x^',y'). To test these axioms as properties of perception, independent of any classical domain of choice behavior, we designed stimuli of pairs of boxes containing red and blue dots and incentivized participants to choose the box that is predicted by the axioms. If participants choose the box consistent with the axioms we cannot infer that the axioms are necessarily satisfied. But if participants select the box that is not predicted by one of the axioms (and for which they receive no monetary reward), one can infer that the difference was not salient and the axiom is violated.
Subjects were presented with 120 pairwise choice problems, each problem testing one of the three axioms (monotonicity in intervals, in ratios, or in differences). In each problem, subjects were presented with two boxes (Left and Right). Each box contained red and blue circles. The numbers of red and blue circles represent a pair of stimuli. Example screenshots from Part I and Part II of the experiment are given in Figure 1.
Subjects were incentivized by earning £10 if they chose correctly according to a question posed to them, and paying them nothing otherwise. Their total payment for Part I was the average payoff from all 120 problems.
For testing Axiom 3 (Monotonicity in Intervals) the question was:
Which box has the greatest difference between the number of blue balls and the number of red balls?
For testing Axiom 4 (Monotonicity in Ratios) the question was:
2. Which has the greatest ratio of blue balls to red balls?
To make this question a proper test of the axiom, both boxes had the same absolute difference.
For testing Axiom 5 (Monotonicity in Differences) the question was:
3. Which box has the greatest difference between the number of blue balls and the number of red
Figure 1: Screenshots from Parts I and Part II of experiment. The figure displays screens from the perceptual task (top panel) and the decision task (bottom panel) as they were presented to experimental participants.
To make the question a proper test of the axiom, both boxes had the same ratio. Although subjects were facing similar questions for Axiom 3 and Axiom 5, the underlying parameters (the number of blue balls and red balls) were designed differently. For Axiom 3, in one box ‒ the more salient box ‒ the number of blue balls and red balls is a super set of the other box - the less salient box. This suggests that, in the more salient box, there are more blue balls as well as less red balls compared to the less salient box. Hence, the subject’s task is to identify which box has more blue balls and fewer red balls.
For Axiom 5, we kept the ratio of the blue balls and red balls constant and varied the difference. Therefore, the subject’s task was to choose the box with the greatest difference, given a fixed ratio. The questions that asked ‘Which box has the greatest difference between blue balls and red balls?’ is sufficient for the conditions for both Axiom 3 and Axiom 5.
There were 16, 22, and 22 problems corresponding to Axioms 3, 4, and 5 respectively. These form 60 baseline problems; each of these was repeated twice, resulting in a total of 120 problems. Subjects were given written instructions which were read to them before starting the experiment. After the end of Part I of the experiment, subjects were given written instructions for Part II; these were also shown on their screens. Subjects had to wait for at least five minutes before they could start Part II.
4.2 Part II of Experiment: Choices under Risk
Part II of the experiment involved a set of pairwise lottery choice problems intended to investigate whether there is a relation between the salience axioms and behavior under risk. The lotteries in the problems were designed so that we could test the implications of the axioms. As the benchmark salience model assumes Axioms 3, 4, and 5 (through salience functions that satisfy ordering, DAS, and IPS), we tested the implications of the benchmark model in the lottery choice task.
There were 36 problems in Part II. The first 26 of these varied the ratios or differences of lottery payoffs to test implications of the axioms. In particular, the first 26 problems were designed to detect a preference for skewness. In these problems, DAS predicts risk-averse choices for moderate and high probability rewards while IPS predicts risk-seeking choices for low probability rewards. Five problems tested Axioms 3 and 5 jointly, and seven problems tested Axiom 5 independently. In these problems (problems 1-12), if Axioms 3, 4, and 5 represent the only source of deviations from maximizing a lottery’s expected value, a prototypical salience model predicts that subjects would choose lottery A. Eleven problems tested Axioms 3 and 4, and three problems tested Axiom 4 independently. If Axioms 3, 4, and 5 represent the only source of deviation from expected value maximization in these problems (problems 13-26), the salience model predicts that subjects would choose lottery B. The last 10 problems were designed to detect the common ratio effect.
Problems 1 through 12 (IPS for payoffs): The characteristics of IPS determined the design of problems 1 to 12. The benchmark salience model predicts a salience-based thinker would prefer the risky lottery over the certain payoff in these problems. For example, Problem 1 is shown in the top panel of Figure 2.
Figure 2. Sample Choices from Lottery Task
(i) Problem 1 testing IPS for Payoffs
A1, B1 PA1, PB1 A2, B2 PA2,PB2
A 100 0.01 0.01 0.99
B 1 0.01 1 0.99
(ii) Problem 26 testing DAS for Payoffs
A1, B1 PA1, PB1 A2, B2 PA2, PB2
A 100 0.99 0.01 0.01
B 99 0.99 99 0.01
(iii) Problem 27 testing IPS for Probabilities
A1, B1 PA1, PB1 A2, B2 PA2,PB2
A 9 0.90 0 0.10
B 18 0.45 0 0.55
Under salience theory, A is preferred to B if and only if σ(100,1)[v(100)-v(1)](0.01)>σ(0.01,1)[v(1)-v(0.01)](0.99). Assuming risk-neutrality, A is preferred to B if and only if 0.99σ(100,1)>(0.9801)σ(0.01,1). According to IPS and symmetry σ(100,1)>σ(0.01,1), and thus A is chosen over B.
Problems 13 through 26 (DAS for payoffs): The DAS and Ordering properties determined the design of problems 13 to 26. Salience theory predicts, assuming risk-neutrality, that a salience-based thinker would appear risk-averse in these problems. Problem 26 is shown in the middle panel of Figure 2. Under the benchmark salience model, DAS and ordering imply B is chosen over A.
Problems 27 through 36 (IPS for probabilities): The IPS property applied to probabilities determined the design of Problems 27-36. Each lottery has one non-zero payoff in which the amount of the payoffs was scaled according to the common ratio of their relative probabilities. Problem 27 is shown in the bottom panel of Figure 2.
There are two sets of common ratio effect problems. Problems 27-32 have a common ratio of 2 while 33-36 have a common ratio of 3. Subjects who conform to expected utility should not change their choice within a set. However, if probability perceptions satisfy IPS, then the salience models in Prelec and Loewenstein (1991) and Leland and Schneider (2018) predict a shift in choice behaviour as the probabilities are scaled down.
For Part I of the experiment, subjects were incentivized by earning £10 if they chose correctly according to a question posed to them, and earning nothing otherwise. Their total payment for Part I was their average payoff from all 120 problems.
We used the random lottery incentive mechanism for Part II. At the end of the experiment, each subject randomly selected a problem for their payment by drawing a disk from a bag containing 36 disks, numbered from 1 to 36. Their lottery choice in that problem was then played out for real by drawing from another bag containing 100 disks, numbered from 1 to 100. The total payment for the experiment is the sum of the payments from two parts of the experiment plus a £2.50 show-up fee. The average total payment to subjects was £12.85. The perceptual and decision problems for Parts I and II of the experiment are provided in the Appendix.