Summary

Imagine that you’ve invested some money into a stock market over the course of two years. In the first year, the market increases by 20%. In the second year, the market decreases by 20%. How well did your stock do overall?

The problem

Let \(I\) be the intial investment, \(p_{a}\) be the percentage change in year a and \(p_{b}\) be the percentage change in year b. The final investment value \(J\) is defined as:

\[J = I \times (1 + p_{a}) \times (1 + p_{b})\]

Rearranging terms, we find that the final value J is greater than the initial investment value I if and only if

\[p_{a} > \frac{-p_{b}}{1+p_{b}}\]

Because multiplication is commutative, we can swap the \(p_{a}\) and \(p_{b}\) terms:

\[p_{b} > \frac{-p_{a}}{1+p_{a}}\]

For the rest of this analyses, we will assume that one of the yearly changes is negative and one is positive. We’ll call the loss percengate \(p_{l}\) and the gain percentage \(p_{g}\)

We can graph this relationship as follows. Stimuli in the red region result in cumulative losses (J < I), while those in the blue region result in gains (J > I). Points on the black line indicate no change (J = I). Additionally, we’ve added points for the three stimuli in study 1

Study 1

Stimuli

All participants started with a $100 investment in each scenario. We originally had 4 stimuli in the experiment (see following table). However, we decided to exclude stimuli 4 from our analyses. The reason for this is because stimulus 4 was designed to have a true change of 0% - however, because the intended value of 33.33% was rounded to 33%, the true change was actually a very small loss of -0.25%. Because this very small change could lead to confusion, we ignored this stimulus.

Stimuli Percentage Up Percentage Down True Change
1 20% 10% +8%
2 50% 40% -10%
3 30% 30% -9%
4 (deleted) 33% 25% -0.25% (~0%)

8 Conditions

We had 2 between-participants independent variables resuling in 4 experimental conditions: 2 (Change order: Up-Down, Down-Up) x 2 (Numerican estimate: Yes, No).

Aggregate results from Study 1 are presented in Table XX

Response proportions from study 1
start_amount numEstCond change.cond s1.down s1.same s1.up s2.down s2.same s2.up s3.down s3.same s3.up
1 100 No DU 0.00 0.03 0.97 0.71 0.00 0.29 0.68 0.29 0.03
2 100 No UD 0.06 0.03 0.91 0.65 0.03 0.32 0.65 0.32 0.03
3 100 Yes DU 0.03 0.00 0.97 0.77 0.00 0.23 0.61 0.32 0.06
4 100 Yes UD 0.00 0.06 0.94 0.43 0.00 0.57 0.46 0.34 0.20

Percentages of correct choices are presented in Figure XX:

Correct Response proportions from study 1
numEstCond change.cond s1.cor s2.cor s3.cor
1 No DU 0.97 [0.86, 0.99] 0.71 [0.52, 0.85] 0.68 [0.49, 0.83]
2 No UD 0.91 [0.76, 0.97] 0.65 [0.44, 0.81] 0.65 [0.44, 0.81]
3 Yes DU 0.97 [0.83, 0.99] 0.77 [0.57, 0.9] 0.61 [0.39, 0.8]
4 Yes UD 0.94 [0.81, 0.98] 0.43 [0.22, 0.67] 0.46 [0.24, 0.69]

To see which variables affected choice quality, we conducted a binary logistic regression analysis on each stimulus

Correct Response proportions from study 1
stimulus order.hdi num.est.hdi
1 1.00 -0.45 [-0.57, -0.26] 0.5 [0.25, 0.63]
2 2.00 -97.7 [-195.76, -10.36] -38.01 [-124.07, 36.9]
3 3.00 -52.41 [-149.96, 32.59] -67.84 [-159.42, 31.78]

Study 2

Stimuli

Study 2 was identical to study 1 with one added condition. In addition to the $100 starting investment, we included a $137 starting investment

8 Conditions

We had 3 between-participants independent variables resuling in 8 experimental conditions: 2 (Starting Investment: $100 vs. $137) 2 (Change order: Up-Down, Down-Up) x 2 (Numerical estimate: Yes, No).

Response proportions from study 2
start_amount numEstCond changeorder_cond s1.down s1.same s1.up s2.down s2.same s2.up s3.down s3.same s3.up
1 100 No DU 0.04 0.17 0.78 0.59 0.04 0.37 0.59 0.41 0.00
2 100 No UD 0.00 0.00 1.00 0.57 0.04 0.39 0.61 0.36 0.04
3 100 Yes DU 0.11 0.04 0.82 0.68 0.04 0.25 0.61 0.36 0.00
4 100 Yes UD 0.02 0.02 0.95 0.72 0.00 0.28 0.67 0.33 0.00
5 137 No DU 0.05 0.14 0.78 0.57 0.08 0.32 0.57 0.38 0.03
6 137 No UD 0.03 0.03 0.94 0.34 0.06 0.60 0.46 0.43 0.11
7 137 Yes DU 0.03 0.08 0.89 0.65 0.05 0.30 0.57 0.43 0.00
8 137 Yes UD 0.03 0.06 0.91 0.40 0.03 0.57 0.43 0.54 0.03
Correct Response proportions from study 2
start_amount numEstCond changeorder_cond s1.cor s2.cor s3.cor
1 100 No DU 0.78 [0.62, 0.89] 0.59 [0.4, 0.75] 0.59 [0.4, 0.75]
2 100 No UD 1 [0.88, 1] 0.57 [0.34, 0.78] 0.61 [0.37, 0.8]
3 100 Yes DU 0.82 [0.62, 0.93] 0.68 [0.45, 0.84] 0.61 [0.37, 0.8]
4 100 Yes UD 0.95 [0.84, 0.99] 0.72 [0.54, 0.85] 0.67 [0.49, 0.82]
5 137 No DU 0.78 [0.6, 0.89] 0.57 [0.36, 0.75] 0.57 [0.36, 0.75]
6 137 No UD 0.94 [0.81, 0.98] 0.34 [0.14, 0.62] 0.46 [0.24, 0.69]
7 137 Yes DU 0.89 [0.74, 0.96] 0.65 [0.45, 0.81] 0.57 [0.36, 0.75]
8 137 Yes UD 0.91 [0.77, 0.97] 0.4 [0.19, 0.65] 0.43 [0.22, 0.67]

To see which variables affected choice quality, we conducted a binary logistic regression analysis on each stimulus

Correct Response proportions from study 1
stimulus startamount.hdi order.hdi num.est.hdi
1 1.00 -0.11 [-1.64, 1.32] 96.47 [39.1, 162.05] 13.16 [-51.37, 65.23]
2 2.00 -2.11 [-4.17, -0.54] -57.2 [-126.29, -0.86] 58.67 [0.48, 123.24]
3 3.00 -1.19 [-3.64, 0.12] -12.61 [-75.56, 36.27] 6.36 [-59.28, 65.96]