An experiment on case-based decision making

For a formal relation between expected utility theory and CBDT see Matsui (2000).

For a version of case-based decision theory that allows the agent to use such information, see Gilboa and Schmeidler (1997).

CBDT usually does not make any distinction between an action that resulted in zero utility and one that simply was not chosen, since zero utility is typically taken as the default aspiration level.

In such a scenario a case-based DM is assumed to randomly choose an action from the set of available actions that have not yet been chosen.

See Rubinstein (1988) and Sarin and Vahid (2004) for previous applications in economics.

In particular, several of the properties of the geometric models are consistently violated by experimental subjects. First, it has been shown that the identity property does not hold, i.e. subjects do not always perceive an object as identical to itself (see Podgorny and Garner 1979). Second, actual similarity evaluations are not always symmetric (see Holyoak and Gordon 1983; and Ortony et al. 1985). For instance, a subject reporting that domestic cats are very similar to tigers does not necessarily indicate that the same subject will report that tigers are very similar to domestic cats. Lastly, the triangle inequality often does not hold, nor does transitivity (Tversky and Gati 1982). Finding objects \(A\) and \(B\) very similar and objects \(B\) and \(C\) very similar does not necessarily indicate that the subject will find objects \(A\) and \(C\) very similar.

If one only knows if attributes are equal or not, one can represent it as the “city block” distance in a model where the new attribute \(a_{ij}\) is an indicator of the old attribute \(b_{i}\) and takes the value \(v_{i}\). This is standard encoding of qualitative variables as indicator ones in econometrics.

Screenshots of the other three treatments are given in Appendix 3 (Figs. 4, 5, 6).

We chose to work in a framework where a DM never encounters the same problem twice. CBDT can, however, be modified to study choices in repeated decision problems (see, e.g., Gilboa and Schmeidler 2001).

See Appendix 1 for a full set of the instructions.

We used a square, a triangle and a circle. To distinguish shapes easier, we colored them. Squares were always green, triangles blue and circles orange. Since the experimental interface included geometric forms and colors, we asked participants after the experiment how they interpreted those. In particular, participants were asked: “If it were the case that the three symbols you saw during the experiment [symbols again given here] stood for high, medium and low, which symbol would you think stood for which level?” We found no correlation between any symbol and level.

The market conditions displayed in the hypothetical scenarios are the same as those given in the marketing report so that similarity comparisons could be made between them.

To ensure comparability across treatments, the subjects faced the same payoffs function in the same order.

An example of a payoff function is \(\pi (.)=50q-0.009(3C_{1}C_{2}-C_{3}C_{4})q^{2}-1150\ln (q-48)\), where \(q\) is the quantity chosen and \(C_{i}\) indicates market condition \(i\).

Sessions varied in size from 4 to 18 participants.

The relatively coarse and simplified version of the feature based similarity function that we employ makes the same predictions as a more generalized similarity function used by Gilboa et al. (2006) in all but one market.

Note that subjects did not even know the range of payoffs. Any judgement regarding the satisfaction through the evaluation of obtained payoffs is highly subjective.

For the non-parametric tests used in this paper see Siegel and Castellan (1998).

For a first analysis we assume that the aspiration level of a case-based decision maker is zero. This aspiration level can be easily adapted as we discuss in section 6.2.

Such a calculation would lead to a MSD of 0.75. In general, the calculation of MSDs favors probabilistic models over point predictions (see Selten 1998 for an axiomatization of quadratic scoring rules).

Choices predicted by CBDT are different from those predicted by MAX in all but seven markets. Note that if only the rank of profit were available instead of the exact profit, then the results should not change according to the MAX heuristic. Since the focus of this paper is not the MAX heuristic, we do not vary the design to investigate its robustness.

As suggested by Karl Schlag, an alternative rule of thumb could be to prioritize similarity, i.e., choose the production value whose scenario has the most features in common with the current report (as long as its profit is positive). If there are more than one of those production values, choose the one with the highest profit from this set. If the production value whose scenario has the highest number of features in common with the current report has a negative profit associated with it, choose the one that has the next highest number of features in common. Again, if there are more of those, choose the one that has the highest profit associated with it. In our experimental setup predicted choices of such a heuristic coincide with predicted choices of CBDT in all but three markets. We therefore do not separately analyze its predictive power.

The specific test statistic is \(z=(p_{1}-p_{2})/S_{p_{c}}\), where \(p_{i}\) is the proportion in subsample \(i\), and \(S_{p_{c}}=\sqrt{p_{c}(1-p_{c})(\frac{1 }{N_{1}}+\frac{1}{N_{2}})} \) is an estimate of the standard error of the difference in proportions, \(p_{1}-p_{2}\). \(p_{c}\) is an estimate of the population proportion under the null hypothesis of equal proportions, \( p_{c}=(p_{1}N_{1}+p_{2}N_{2})/(N_{1}+N_{2})\), where \(N_{i}\) is the total number of subjects in subsample \(i\) (see Glasnapp and Poggio 1985).

Each subject is characterized by one number that corresponds to how many of her observed choices coincide with theoretically predicted choices.

These percentages are 100 % minus the percentage that is found at the intersection of the solid line with the vertical line at 1/4.

Given that our paper aims at explaining individual behavior, we do not simulate a probabilistic reinforcement learning model (e.g., Roth and Erev 1995) which would either lead to comparing choices with a probability vector or analyzing population means.

Given that there is not much variance in individual MSDs, we determine overall performance by calculating the average MSD for each treatment. Interestingly, when no current information is given, the model that fits “best” gives a weight of 0.1 to the obtained payoff independent of whether immediate feedback is given or not. Behavior seems quite “backward” looking. When current information is given, 0.2 returns the lowest average MSD when immediate feedback is given. However, when feedback is delayed the lowest average MSD is obtained with a weight of 0.9.