# Measuring and Disentangling Ambiguity and Confidence in the Lab

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Overestimation

#### 1.2. Overplacement

#### 1.3. Overprecision

## 2. Results

#### 2.1. Descriptive Statistics

#### 2.1.1. Attitude towards External Ambiguity

#### 2.1.2. Over/Under-Estimation and Perceived Competence

#### 2.1.3. Over/Under-Placement and Perceived Competence

#### 2.1.4. Over/Under-Precision and Perceived Competence When the Source of Ambiguity is “Internal”

#### 2.1.5. Over/Under-Precision and Perceived Competence when the Source of Ambiguity is “External”

#### 2.1.6. Investment Framing

#### 2.2. No Self Selection

#### 2.3. Principal Component Analysis

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Self Selection Treatment

_{1}if a yellow ball is drawn from the Bingo Blower or 0 if a pink or blue ball is drawn, and risky lottery Y, where they win the same amount, W

_{1,}with a (known) probability of ½ and 0 with probability ½. The Bingo Blower is located in the room where the experiment takes place and contains yellow, pink and blue balls. This choice is meant to measure subjects’ preference for BB-ambiguity (“BB” refers the Bingo Blower since ambiguity is measured here by means of the Bingo Blower7) vs. risk.

_{2}and W

_{3}tokens respectively if they are correct or if they over/underestimate the correct number by 1 unit.

_{4}and the others get zero—or another lottery (Beta) where they win either W

_{4}or zero with probability 1/n, where n is the number of participants in the session. This choice identifies subjects’ preference for internal ambiguity (based on relative precision, i.e., a measure of overprecision) in the high-competence task with respect to risk.

_{5}or get zero. This choice identifies subjects’ preference for internal ambiguity (based on placement, i.e., a measure of overplacement) in the low-competence task with respect to risk.

_{6}tokens if they are correct. This guess identifies subjects’ ex-post placement in the low-competence task.

_{7}or get zero. This choice captures subjects’ preference for external ambiguity (based on an exogenous source, i.e., a measure of overestimation) with respect to risk. Furthermore, subjects have to guess whether their estimation of NYC temperature is more, equally or less correct than the average temperature estimated in the session. This guess identifies subjects’ ex-ante estimation in case of external ambiguity.

_{7}or get zero. This choice captures another form of subjects’ preference for external ambiguity based on an exogenous source, the Bingo Blower, with respect to risk. This time, however, subjects could feel to have some kind of “control” and/or direct experience with the source of external ambiguity since the Bingo Blower was located in the room where the experimental sessions took place and subjects could observe it as long as they wanted and get as close to it as they liked in order to increase their perceived accuracy in estimating the number of yellow balls. This guess identifies subjects’ ex-ante estimation in case of external ambiguity. Furthermore, subjects have to guess whether their estimation of the number of yellow balls is more, equally or less correct than the average number estimated in the session.

_{i}, different for each subject i—they want to allocate between two pairs of lotteries (“Pair 1” and “Pair 2” respectively): they will decide the allocation of the tokens they earned for both pairs, but only one pair will be selected at random and actually played. The structure of this “investing gamble” is grounded on Gneezy and Potters (1997) [59].

- Investment Choice 1a: subjects have to decide how many tokens (G
_{i}) out of their endowment D_{i}that they want to allocate to a lottery where the probability of winning 2.5 G_{i}(instead of getting zero) corresponds to the percentile corresponding to their performance in task B. - Investment Choice 1b: subjects have to decide how many tokens (H
_{i}) out of their endowment D_{i}that they want to allocate to a lottery where they win 2.5 H_{i}(instead of getting zero) if a yellow ball is drawn from the Bingo Blower.

_{i}+ H

_{i}≤ D

_{i}must hold.

_{i}/H

_{i}in Pair 1 captures subjects’ preference for investing in internal ambiguity (based on placement) in the low-competence task instead of investing in external BB-ambiguity.

- Investment Choice 2a: subjects have to decide how many tokens (G
_{i}) out of their endowment D_{i}that they want to allocate to a lottery where the probability of winning 2.5 G_{i}is ½ and the probability of getting 0 is ½. - Investment Choice 2b: subjects have to decide how many tokens (H
_{i}) out of their endowment D_{i}that they want to allocate to a lottery where they win 2.5 H_{i}(instead of getting zero) if a yellow ball is drawn from the Bingo Blower.

_{i}+ H

_{i}≤ D

_{i}must hold.

_{i}/H

_{i}in Pair 2 captures subjects’ preference for investing in external BB-ambiguity vs. risk.

_{8}if they over/underestimate by one ball at the maximum. This is to capture how correct subjects are when evaluating external BB-ambiguity (BB-calibration) and serves as additional information to disentangle between internal and external BB-ambiguity.

#### 4.2. No Self Selection Treatment

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

**Instructions**

- In Stage 1 you will be asked to make a simple choice between two lotteries.
- In Stage 2 you will answer questions presented in two different Questionnaires with 20 questions each, and you will be asked to make an evaluative assessment.
- In Stage 3 you will be asked to make some forecasts and on this basis to choose between 4 related pairs of lotteries that will be presented in sequence to you.

- In Stage 4 you will have the possibility to allocate the tokens you accumulated in the previous Stages between two different investment opportunities.

**Stage 1**

**Stage 2**

**Stage 3**

**Round 1**

**Round 2**

**Round 3**

**Round 4**

**Stage 4**

**Warning: In each of these you should decide how to allocate your entire endowment. Only one of these will actually be carried out, which will be determined randomly on the computer.**

**Pair 1**

**Investment Opportunity 1a**—you invest G tokens from your endowment D, where G is the number of tokens of your choice: this will give you the chance to win 2.5* G with a probability corresponding to your ranking in Questionnaire B. For example, if your ranking in Questionnaire B implies that you did better than 80% of participants in your session, your probability of winning will be 80%, if you did better than 20% of participants, so that 80% of the participants did better than you, your probability of winning will be 20%.**Investment Opportunity 1b**—you invest H tokens from your endowment D, where H is the number of tokens of your choice: this will give you the possibility of winning 2.5* H if a yellow ball is drawn and nothing if it is a blue or red ball.- Importantly, also notice that you can decide not to invest any of your tokens in any of the two investment opportunities or you can decide to invest all of them in one or the other or you can decide to invest some tokens in one opportunity and some in the other (and perhaps also keep some without investing them). Thus, any division of the tokens between opportunity 1a, opportunity 1b, and not investing is allowed. The only rule that you have to respect is that the sum of tokens invested overall (G + H) does not exceed your endowment (D): G + H ≤ D.

**Pair 2**

**Investment Opportunity 2a**—you invest G tokens from your endowment D: this will give you the possibility of winning 2.5* G with a probability of 50% and nothing otherwise;**Investment Opportunity 2b**—you invest H tokens from your endowment D: this will give you the possibility of winning 2.5* H if a yellow ball will be drawn out and nothing if it is a blue or red ball.- As for Pair 1, you can choose any division of your tokens that you like between Investment Opportunity 2a, Investment Opportunity 2b, and keeping them, i.e., not investing. The only rule that you have to respect is that the sum of tokens invested overall (G + H) does not exceed your endowment (D): G + H ≤ D.

**FINAL QUESTIONNAIRE**

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1 | In Ellsberg’s (1961) [10] classical two-color paradox, the source of uncertainty can be represented by the color of a ball drawn randomly from an urn containing 50 black and 50 red balls (the known urn), or by the color of a ball drawn randomly from an urn with 100 black and red balls in unknown proportion (the unknown urn). Alternatively, other sources of uncertainty could be Stock indexes like the Dow Jones index or the Nikkei index (with the foreign index implying a higher level of uncertainty for a US resident due to the “home bias”). |

2 | Although Kwan et al. (2004) emphasized the need of distinguishing between the two manifestations [23]. |

3 | There are few, if any, documented reversals of overprecison, whereas there are many documented reversals of the other two varieties of overconfidence (overestimation and overplacement). |

4 | There is evidence (Speirs-Bridge et al., 2010 [37]) that people appear less confident when you give them an interval and ask them to estimate how likely it is that the correct answer is inside it, than if you specify a specific probability of being right and ask them for a confidence interval around it. In other words, people have higher confidence in probability estimates than confidence intervals. |

5 | At the end of Stage 3 each subject faces a screenshot resuming all the outcomes of choices made in Stages 1 to 3 separately. |

6 | Although the experiment seems at first sight quite cumbersome, this result confirmed that participants were able to understand it; this is also indirectly confirmed by the fact that all sessions lasted one hour and half without anybody delating the others. |

7 | The Bingo Blower is a rectangular-shaped, glass-sided: inside the glass walls are a set of pink, yellow, and blue balls in continuous motion being moved about by a jet of wind from a fan in the base (see the picture in Appendix A). In addition, during the sessions’ images of the Bingo Blower in action are projected via videocamera onto two big screens in the laboratory. Subjects are free at any stage to get closer to the Blower to examine it as much as they want. All the balls inside the Blower can at all times be seen by people outside, but, unless the number of balls in the Blower is low, the number of balls of differing colors cannot be counted because they are continually moving around: while objective probabilities do exist, subjects cannot know them. In this way, as noted by Hey and Pace (2014) [58], there is a “situation of genuine ambiguity which eliminates the problem of suspicion; the problem of using directly a second-order probability distribution; and the problem of using real events, therefore keeping the problem more similar to the original Ellsberg one”. |

Quest. A (High Competence) | Quest. B (Low Competence) | p-Value | Power | |
---|---|---|---|---|

Ex-ante estimation of performance | 14.07 out of 20 | 11.01 out of 20 | p < 0.000 | 0.999 |

Actual score | 10.14 out of 20 | 11.93 out of 20 | p < 0.000 | 0.819 |

Ex-ante overestimation of performance | 3.93 | −0.92 | p < 0.000 | |

% ex-ante overestimating subjects | 88% | 39% | p < 0.000 | |

Ex-post estimation of performance | 10.22 out of 20 | 11.13 out of 20 | p = 0.032 | 0.619 |

Ex-post overestimation of performance | 0.78 | -0.79 | p = 0.032 | |

% ex-post overestimating subjects | 47% | 30% | p = 0.013 |

Quest. A (High Competence) | Quest. B (Low Competence) | p-Value | |
---|---|---|---|

Overestimation of the degree of competition | 2.01 | n.a. | |

% subjects who declare to be “above the average” (ex-post) | 48% | 51% | p = 0.733 |

% ex-post overplaced subjects | 10% | 16% | p = 0.253 |

Quest. A (High Competence) | Quest. B (Low Competence) | ||
---|---|---|---|

# of subjects who bet on their precision | 58 out of 89 | # of subjects who bet on their precision | 49 out of 89 |

% subjects who bet on their precision | 65% | % subjects who bet on their precision | 55% |

# of subjects who wrongly bet on their precision (overprecise) | 50 out of 89 | # of subjects who wrongly bet on their precision (overprecise) | 23 out of 89 |

% of subjects who wrongly bet on their precision (overprecise) | 56% | % of subjects who wrongly bet on their precision (overprecise) | 26% |

Temperature of NY, 21 September (12 a.m.) | Number of Yellow Balls in the Bingo Blower | ||
---|---|---|---|

# of subjects who bet on their precision | 33 out of 89 | # of subjects who bet on their precision | 33 out of 89 |

% subjects who bet on their precision | 37% | % subjects who bet on their precision | 37% |

# of subjects who wrongly bet on their precision (overprecise) | 25 out of 89 | # of subjects who wrongly bet on their precision (overprecise) | 23 out of 89 |

% of subjects who wrongly bet on their precision (overprecise) | 28% | % of subjects who wrongly bet on their precision (overprecise) | 26% |

Quest. A (High Competence) | Quest. B (Low Competence) | p-Value | Power | |
---|---|---|---|---|

Ex-ante estimation of performance | 12.43 out of 20 | 9.97 out of 20 | p = 0.014 | 0.859 |

Actual score | 8.90 out of 20 | 10.93 out of 20 | p < 0.000 | 0.820 |

Ex-ante overestimation of performance | 3.52 | −0.95 | p < 0.000 | |

% ex-ante overestimating subjects | 64% | 43% | p < 0.000 | |

Ex-post estimation of performance | 9.84 out of 20 | 10.61 out of 20 | p = 0.074 | 0.673 |

Ex-post overestimation of performance | 0.93 | −0.31 | p = 0.032 | |

% ex-post overestimating subjects | 48% | 41% | p = 0.017 |

Quest. A (High Competence) | Quest. B (Low Competence) | p-Value | Power | |
---|---|---|---|---|

Overestimation of the degree of competition (average) | 2.20 | n.a. | ||

% subjects who declare to be “above the average” (ex-post) | 50% | 50% | p = 1.000 | 0.080 |

% ex-post overplaced subjects | 14% | 7% | p = 0.322 | 0.089 |

Variable Description | Mean | Std. Deviation | Factor Score |
---|---|---|---|

Internal Ambiguity | |||

Preference for high-competence-based ambiguity (vs. risk) | 0.684 | 0.466 | 0.684 |

Preference for low-competence-based ambiguity (vs. risk) | 0.526 | 0.501 | 0.274 |

Preference for no-competence-based ambiguity (BB # yellow balls) (vs. risk) | 0.360 | 0.402 | 0.672 |

Investment in competence-based-ambiguity (vs. risk) | 0.541 | 0.500 | 0.067 |

External Ambiguity | |||

Preference for BB-based ambiguity (vs. risk) | 0.511 | 0.501 | 0.703 |

Preference for no-competence-based ambiguity (temperature) (vs. risk) | 0.375 | 0.486 | 0.044 |

Investment in BB-ambiguity (vs. risk) | 0.684 | 0.466 | 0.709 |

SUR Model | |||
---|---|---|---|

Internal Ambiguity Score | External Ambiguity Score | ||

Treatment | 0.094 | Treatment | −0.356 |

(0.226) | (0.217) | ||

Score high-competence task | 0.031 ** | Score high-competence task | 0.044 |

(0.031) | (0.029) | ||

Score low-competence task | 0.025 | Score low-competence task | 0.014 |

(0.037) | (0.036) | ||

Earnings | 0.000 | Earnings | −0.001 * |

(0.001) | (0.001) | ||

Gender | 0.013 | Gender | −0.071 |

(0.184) | (0.191) | ||

Age | −0.005 | Age | 0.012 |

(0.044) | (0.046) | ||

Easy | 0.268 | Easy | 0.013 * |

(0.190) | (0.183) | ||

Constant | −0.979 | Constant | 0.235 |

(1.262) | (1.216) |

^{2}= 0.269, External Ambiguity Score: R

^{2}= 0.274. The dependent variables range from 0 to 1. Controls: geographical origin, past involvement in previous experiments, the main motivation driving the subjects’ choices during the experiment. ** significant at 5%; * significant at 10%.

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Di Cagno, D.; Grieco, D. Measuring and Disentangling Ambiguity and Confidence in the Lab. *Games* **2019**, *10*, 9.
https://doi.org/10.3390/g10010009

**AMA Style**

Di Cagno D, Grieco D. Measuring and Disentangling Ambiguity and Confidence in the Lab. *Games*. 2019; 10(1):9.
https://doi.org/10.3390/g10010009

**Chicago/Turabian Style**

Di Cagno, Daniela, and Daniela Grieco. 2019. "Measuring and Disentangling Ambiguity and Confidence in the Lab" *Games* 10, no. 1: 9.
https://doi.org/10.3390/g10010009