1 Introduction
Consider the following problem: You are asked to draw a red marble from either of two urns. Urn A contains 10 marbles, 1 of which is red. Urn B contains 100 marbles, 7 of which are red. Which urn do you choose? A rational actor maximizing the probability of choosing a red marble will choose Urn A. Psychologist Seymour Epstein and colleagues (1992, 1994, 1999) have documented that many individuals choose Urn B when presented with this choice or similar choices. Epstein named this the Ratio-Bias phenomenon, as it appears that individuals are biased toward choices with large numbers of potential successes, rather than large probabilities of potential successes.Footnote 1
The present investigation explores the Ratio-Bias phenomenon along two dimensions. First, we test for errors within this framework in a symmetric fashion. Referring to the example above, we not only present participants with decisions like that one, but also with decisions in which the urn with the larger number of marbles has the greater probability of success and is therefore the optimal choice. If we observe similar frequencies of errors in these two circumstances, then we conclude that there is no real Ratio-Bias phenomenon, but rather observation of random error in both directions. If the frequency of irrational decisions differs across these treatments, however, then we conclude that the Ratio-Bias phenomenon exists.
The second dimension of interest is that of incentives. The participant in a Ratio-Bias experiment confronts a decision that is well suited to the introduction of a small monetary incentive and the testing of its marginal effect. We implement a combination within- and between-subjects design to test the effect of monetary incentives on decision making, as well as varying the size of the incentive to test for the effect of incentive magnitude on decision optimality.
2 Previous research
Much extant research on the Ratio-Bias phenomenon has presented participants with decisions where the urn containing more potential successes (in absolute numbers) has a smaller probability of success. Denes-Raj and Epstein (1994) presented participants with a choice between a 1-in-10 chance of success and a 9-in-100 chance of success, and also a choice between 1-in-10 and 7-in-100. They found that many individuals (61% and 40%, respectively) preferred the latter choice (the large urn), while the former (the small urn) offered a greater probability of success. Kirkpatrick and Epstein (1992) presented participants with a choice between 1-in-10 and 10-in-100, and asked whether the participant would be willing to pay 10 cents for the privilege of choosing the larger urn (and winning $8 on a successful draw). By charging money to choose the large urn, Kirkpatrick and Epstein reduced the payoff to choosing the large urn and thus made the small urn the optimal choice. They found that a significant fraction of individuals chose the large urn. These experiments, while certainly interesting and suggestive, did not also test the symmetric decision in which the larger urn has a greater probability of success (or the larger reward). The current experiment tests (for instance) not only 1-in-10 against 7-in-100, but also 1-in-10 against 13-in-100. If we observe similar rates of errors in the two symmetric situations, then we conclude that the Ratio-Bias phenomenon does not really exist at all, and the previous results are merely a manifestation of the stylized fact that individuals sometimes choose suboptimally when making decisions with little or no incentive to choose optimally. If, however, we observe a significantly higher error rate for the 1-in-10 against 7-in-100 choice than for the 1-in-10 against 13-in-100 choice, then we conclude that the Ratio-Bias phenomenon exists.
The second treatment dimension for this experiment is the introduction of a small monetary incentive for success in some sessions and in some rounds. As mentioned above, Kirkpatrick and Epstein (Reference Kirkpatrick and Epstein1992) presented the participant with the choice of a 1-in-10 or 10-in-100 chance of winning $8, but charged the participant 10 cents for the privilege of choosing the large urn. By doing so, they offered a choice between an expected 80 cents for the small urn and an expected 70 cents for the large urn. They found that a substantial fraction of individuals choose the large urn in this situation, though the small urn is the optimal choice. However, they did not directly compare these results with an identical, but unincentivized control group, and were thus unable to test directly for the impact of the incentive.
The experimental economic literature on the impact of the existence and magnitude of monetary incentives is rich. See Camerer and Hogarth (Reference Camerer and Hogarth1999) for a general review. More specifically for our current interest, Blumenschein, et al. (Reference Blumenschein, Johanneson, Blomquist, Liljas and O'Conor1997) investigated hypothetical and real incentives in Vickrey auctions, and found a significant difference in behavior when real money is on the table. More recently, Holt and Laury (Reference Holt and Laury2002, Reference Holt and Laury2005) introduced real incentives to the seminal Kahneman and Tversky (Reference Kahneman and Tversky1979) framework, which relied entirely on hypothetical incentives, and obtained results significantly different from the earlier Kahneman-Tversky results. The Blumenschein et al. and Holt-Laury results suggest that the existence of a monetary incentive causes a substantial change in participant behavior. Other work has investigated varying the size of the monetary incentive. Thaler (Reference Thaler1986) argues that once a monetary incentive is introduced, raising the stakes will not necessarily induce more rational decisions. Attempting to test this, Roth et al. (Reference Roth, Prasnikar, Okuno-Fujiwara and Zamir1991) compared the results of several experiments across four countries with different standards of living and therefore different real values of the payoffs. They found significant differences in behavior, but this design leaves open the question of how much of the difference is attributable to the size of the incentive, and how much is due to varying cultural factors and norms across countries. Others have found similar results with similar methodologies and similar caveats — see Cameron (1990) or Slonim and Roth (Reference Slonim and Roth1998). The present experiment provides a convenient framework in which to test the differential impact of both the existence of a real incentive and its magnitude. We employ both within- and between-subjects treatments on participants drawn from the same population.
3 Experiment
In order to test for the existence of the Ratio-Bias phenomenon and the effect of incentives, we conducted an experimental investigation over the course of two semesters. Participants were recruited from the general population of students at a northeastern U.S. liberal arts college via word of mouth and announcement in psychology and economics classes. In sessions in which money was earned, this was not announced prior to the students' volunteering and appearing for the experiment.
3.1 Software
The authors created a custom software application to facilitate data collection.Footnote 2 The design of the experiment and the decision with which it presents the participant draws heavily upon the previous work by Epstein and colleagues, in order to facilitate comparison of results. The application presented the participant with 300 rounds of a binary decision. The participant's task in each round was to select the virtual urn from which a virtual marble will be randomly drawn. Urn A always contained ten virtual marbles, while Urn B always contained one hundred virtual marbles. Each urn was visually represented on the computer monitor, with the left-right location of the two urns randomly determined (each urn was equally likely to be on the left or right). Below each urn was a count of the total number of marbles and the number of red marbles. The upper left corner showed a count of the number of the round and a running total of the number of points the participant had earned throughout the session. The participant chose an urn by clicking the "Choose" button beneath it with the mouse.
The participant earned one point if the drawn marble was red. The number of red marbles in each urn was determined as follows: First, the number of red marbles in Urn A was randomly determined to be 1, 2, 3, or 4, with equal probabilities of each outcome. (The rest of the marbles were white). Then the number of red marbles in Urn B was determined to be ten times the number in Urn A, plus a deviation which we call epsilon. epsilon was a randomly generated parameter, with equal probabilities that it equaled -3, -2, -1, 1, 2, and 3. Figure 1a shows a screen capture of actual game play. In this case the base fraction of red marbles is 30% and epsilon = -2, and therefore the fraction of red marbles in the big urn is 28% (30% plus epsilon). In this case, Urn A is displayed on the left; as stated above, this is randomly determined.
Figure 1: Screen capture of game choice (a) and feedback (b).
After the participant chose the urn from which to draw a virtual marble, the marble was drawn and the result (red or white) displayed. At this point, the participant clicked a “Continue” button to proceed to the next round, as shown in Figure 1b. Note that the “Continue” button is located between the two “Choose” buttons on the screen; this forces the participant to return the mouse to the center of the screen, between the two urns, each round. Figure 1b shows a successful round, in which the participant has earned a point by drawing a red marble.
Participants were allowed to proceed through 300 rounds at their own pace, but were not dismissed (and paid, if applicable) until all participants had completed all rounds.
3.2 Sessions and treatments
Treatments varied across sessions, with all participants in a given session getting the same treatment. Each session consisted of 300 rounds for each participant. In Session 1, no monetary compensation was given. In Session 2, each successful draw earned the participant 5 cents for all rounds. In Session 3, each successful draw earned the participant 10 cents for all rounds. In Session 4, the participant earned nothing for points earned during the first 150 rounds, then earned 10 cents for each point earned from Round 151 through Round 300. Session 5 was the converse of Session 4: each participant earned 10 cents for each point earned during the first 150 rounds, then earned nothing for points earned from Round 151 through Round 300. In Session 6, participants were given $4 regardless of the number of points earned. For the purposes of our analysis, we treat session 6 as a control treatment with no incentive, since payment was not a function of performance. The payment structure of the particular treatment was announced at the beginning of each session. Participation also partially fulfilled course requirements in psychology for some participants. A session typically took about 45 minutes to complete. See Table 1 for a summary of the incentives offered by session.
Table 1: Incentives by session
3.3 Participants
Rather than arbitrarily limit the number of participants in any session, we allowed all volunteers who appeared for a particular session to participate (though participants were not allowed to participate in more than one session). As a result, the number of participants varied across sessions. See Table 2. (One participant was omitted for being an outlier, responding consistently incorrectly. Results are substantively the same with this subject included.)
Table 2: Participants by session
In a few cases, computer failures resulted in the loss of a few rounds of data. Specifically, in Session 3, three computer crashes resulted in the loss of the last 7, 9, and 6 rounds of data, and in Session 5, one computer crash resulted in the loss of the last 9 rounds of data.
3.4 On the magnitude of the incentive
In several treatments, participants were paid according to the number of red marbles drawn. The largest such incentive was 10 cents per point earned (red marble drawn). With the parameters used in the experiment, a participant choosing randomly between the two urns would have an average probability of drawing a red marble of 25%. A participant choosing optimally would increase this by an average of 1% to an expected 26% probability of success. Thus, the return to thought created by the 10 cents incentive is, on average, one-tenth of a cent per round, or a total of 30 cents over the entire 300 round experiment (if all rounds are incentivized). The nickel incentive is, obviously, half this amount.
4 Results and data analysis
In studies like this one, heterogeneity across subjects is a serious concern when estimating relationships in the data. In order to make our hypothesis testing as clear and as conservative as possible. we aggregated the fraction of optimal choices for each subject across each of the treatment conditions under discussion; we therefore have 162 observations for each treatment condition. All of the tests below compare within-subject differences in behavior across the treatment conditions.
4.1 Existence of ratio bias phenomenon
Our first salient result is confirmation of the existence of the ratio bias phenomenon. When ∊ is negative, a participant wishing to maximize the probability of drawing a red marble should choose the small urn; although the number of red marbles in the large urn is greater, the total number of marbles in the large urn is such that the probability of drawing a red marble from the large urn is smaller than the probability of drawing a red marble from the small urn. Table 3 shows decisions by subjects by the sign of ∊, aggregating across all rounds and all sessions. Each subject's average is treated as a data point here; the standard deviation of the subjects' optimal decision rates is given in parentheses.
Table 3: Optimal decisions by ∊ (across-subject standard deviations in parentheses)
When ∊ is negative, the small urn is the optimal choice; this is the treatment that has been tested by previous research. The large urn is the optimal choice when ∊ is positive; this is the heretofore untested treatment. Table 3 clearly shows that participants incorrectly choose the large urn when ∊ is negative far more frequently than they incorrectly choose the small urn when ∊ is positive. Indeed, participants correctly chose the small urn only 59.0% of the time when ∊ was negative; this is remarkably low, since as this is a binary decision, randomly choosing an urn would lead to a 50% rate of optimal decisions.\ Interestingly, participants chose optimally only 75.5% of the time when the large urn was optimal. This error rate indicates that previous studies lacking this control condition may have overstated the magnitude of the ratio-bias phenomenon if they attributed all errors to the bias.
Although the effect of increasing ∊ is significant for both negative and positive values of ∊ (respectively, t160=2.28, p=0.0238, and t160=2.49, p=0.0140), these effects were very small compared to the effect of whether ∊ was positive or negative, and they are ignored henceforth.
4.2 Practice effects
We can ask whether performance improved from the first half (rounds 1-150) to the second half (rounds 151-300). Ignoring sessions 4 and 5 (where incentive changed between the two halves), overall proportion of optimal choices did not change (means of .67 and .66 for the two halves, respectively, t104=-.37).
We can also ask whether the ratio bias effect changed, where bias is defined as the difference between optimal choices with positive and negative ∊. Here, the change between the two halves was significant. The mean bias was .12 for the first half and .08 for the second half (t104=2.68, p=0.0086). Thus, although overall accuracy did not increase, the tendency to choose the urn with more marbles did decrease over rounds.
4.3 No effect of base odds
One might also be interested in whether the manipulation of base odds affected decision optimality rates. In our data, this was not the case. Table 4 shows optimal decision rates by each value of base odds.
Table 4: Optimal Decisions by base odds (across-subject standard deviations in parentheses)
Simple t tests do not show a significant difference in optimal decision rates across these four treatment conditions.
4.4 Incentives affect accuracy
Our second salient result is that the presence of an incentive affects behavior. To test within-subject differences here, we confined our attention to sessions 4 and 5. Table 4 shows optimal decision rates by the presence of a monetary incentive.
For this test, we have only 56 data points. A t test of the interaction between half (first half vs\. second half) and session (session 4 with incentive in the second half, session 5 with incentive in the first half) was significant (t54=2.90, p=.0054, two tailed), indicating better performance with incentive.
Figure 2 and Figures 3 show the means by half (first 150 vs. second 150 rounds) for both optimal choice and bias, respectively. As is consistent with the large standard errors, no between-subject test of the incentive effect was significant. Moreover, comparison of sessions 4 and 5 on the bias measure shows that incentive, if anything, increased the bias, although the interaction between half (1st vs. 2nd) and session (4 vs. 5) was not significant (t54=1.62).
Figure 2: Proportion of optimal choices as a function of session and half: black bars represent no incentive (or fixed payment), white bars represent 10 cents, and gray bars, 5 cents. Error bars are standard errors of the means displayed.
Figure 3: Bias toward the higher number as a function of session and half: black bars represent no incentive (or fixed payment), white bars represent 10 cents, and gray bars, 5 cents. Error bars are standard errors of the means displayed.
5 Discussion and Conclusions
5.1 Confirmation of existence of Ratio Bias phenomenon
The first salient fact arising from this experiment and analysis is confirmation of the ratio bias phenomenon. In our data, suboptimal choices are far more frequent when the small urn is the optimal choice than when the big urn is optimal. This finding is robust to all of the other treatment conditions in the experiment - whether the participant is incentivized, the size of the incentive, the base odds of a successful draw, and the magnitude of the difference between the odds of a successful draw between the large and small urns.
5.2 Effect of incentive
Our second important result concerns the impact of incentives on these decisions. In our data, the introduction of a small incentive significantly reduces the frequency of suboptimal choices, although only in the (more sensitive) within-subject analysis. Interestingly, the incentive seems to matter despite its astoundingly small magnitude. In this experiment with our maximum incentive (10 cents per successful draw), an individual choosing urns completely randomly would expect to earn a total of 30 cents less than an individual who chooses optimally in every round.
5.3 Other findings
We found no effect of different base odds on the patterns of participant decisions. Both of our major findings (the existence of the ratio bias effect and the effect of the incentive) manifest themselves similarly when the base odds of drawing a red marble are 10%, 20%, 30%, and 40%. Moreover, changing the magnitude of the difference between the probability of success in the two urns had little impact on participant decisions in our data. Within the range we considered (-3% through +3%), observed behavior was roughly the same across all negative values of epsilon and across all positive values of epsilon.
The ratio bias itself, the tendency to choose on the basis of numerosity, thus leading to more optimal choices when these corresponded to the larger urn (and hence a difference between positive and negative epsilon), declined over rounds, suggesting that participants achieve some insight on their own. This bias was not reduced by incentive. Possibly the incentive leads people to fall back on a favored strategy.
The fact that incentives affected optimal choices but not the ratio bias suggests another possible locus of the incentive effect. It is interesting that only one subject chose the optimal response on every round. The failure of most subjects to optimize may be analogous to the (poorly named) research on “probability matching,” which finds a general failure to optimize in repeated plays (e.g., West & Stanovich, Reference West and Stanovich2003.) If this is true, then our results are somewhat consistent with those of Shanks et al. (2002), who found that performance improves substantially with incentive and practice combined. Incentive seems to help people learn to choose the option that is most likely to win.
5.4 Further Research
Several interesting questions remain, primarily related toward extending this methodology along two dimensions. First, how does the ratio bias phenomenon behave for larger values of ∊? There is certainly a point where the absolute number of red marbles in the big urn is smaller than that in the small urn, and our strong prior expectation would be 100% optimal choice at this level of ∊ (everyone would choose the small urn). But how does the ratio bias effect manifest itself for values of ∊ between this point and the values we examined here? Does the bias remain constant until abruptly ending at that point or some other point, or is there a more gradual reduction in the error rate?
Perhaps an even more interesting research problem is to learn more about the potential response to different size incentives within this framework. Along this dimension, there are interesting research questions in both directions. How small can the incentive get before individuals stop responding to it altogether? Moreover, what is the relationship between the size of the incentive and behavior for larger incentives than those examined here? Does increasing the incentive, and thus increasing the expected cost of an error, reduce the error rate? This research problem is of particular interest to experimental psychologists and behavioral economists seeking experimental methodology and results that generalize to situations where individuals are highly incentivized.
Appendix A: Experimental Protocol
Presented here are the oral instructions read to participants at the beginning of each session.
When I instruct you to begin, the computer monitor in front of you will display two groups containing different mixtures of white and red marbles. Your assignment is to accumulate the most red marbles possible. To do this, you will choose from which group of marbles you think you have the best chance of successfully getting a red one. After you make your choice, the computer will select a marble from the group you have chosen, just as if you had reached your hand into a bag containing all the marbles in your group and picked one at random.
[By session:]
1. (no instructions)
2. You will be compensated ten cents for every time the computer chooses a red marble from the group you select.
3. You will be compensated ten cents for every time the computer chooses a red marble from the group you select during Rounds 151 through 300.
4. You will be compensated ten cents for every time the computer chooses a red marble from the group you select during Rounds 151 through 300.
5. You will be compensated ten cents for every time the computer chooses a red marble from the group you select during Rounds 1 through 150.
6. You will be compensated $4 for your participation.
To make your selection click on the “choose” button directly under the group you want the computer to pick from. After you make a selection, the computer will tell you if it picked a red or white marble from the group you chose. Then, hit continue to choose from the next set of groups. You will be given 300 sets of groups to choose from. After the 300th choice you make, the computer will signal to you that your session is over. You have as much time as you need to complete this task. When everyone is done you will be debriefed and then dismissed. When I give the word hit “G” on your keyboard to begin. Are there any questions? Hit G and begin.
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