# Correcting for Random Budgets in Revealed Preference Experiments

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Measures of Choice Consistency

**Definition**

**1**

**.**A bundle ${x}^{t}$ is directly revealed preferred to bundle x, written ${x}^{t}{\u2ab0}^{R}x$ when ${p}^{t}\xb7{x}^{t}\ge {p}^{t}\xb7x$. A bundle ${x}^{t}$ is strictly directly revealed preferred to bundle x, written ${x}^{t}{\succ}^{R}x$, when ${p}^{t}\xb7{x}^{t}>{p}^{t}\xb7x$.

**Definition**

**2**

**.**A bundle ${x}^{t}$ is revealed preferred to x when there is a sequence of observations ${\left\{{x}^{{t}_{m}}\right\}}_{m=1}^{M}$ such that ${x}^{t}{\u2ab0}^{R}{x}^{{t}_{1}},\phantom{\rule{4pt}{0ex}}{x}^{{t}_{1}}{\u2ab0}^{R}{x}^{{t}_{2}},\dots ,\phantom{\rule{4pt}{0ex}}{x}^{{t}_{M}}{\u2ab0}^{R}x$.

**Definition**

**3**

**.**Dataset $\mathcal{D}$ satisfies GARP if for every pair of observations $({x}^{t},\phantom{\rule{4pt}{0ex}}{x}^{\tilde{t}})$, if ${x}^{t}$ is revealed preferred to ${x}^{\tilde{t}}$, then it is not the case that ${x}^{\tilde{t}}{\succ}^{R}{x}^{t}$.

**Proposition**

**1**

**.**Dataset $\mathcal{D}$ can be rationalized by a locally non-satiated utility function if and only if $\mathcal{D}$ satisfies GARP.

#### 2.1. Measures of Choice Consistency

#### 2.1.1. CCEI

**Definition**

**4**

**.**A bundle ${x}^{t}$ is relaxed directly revealed preferred to x, written ${x}^{t}{\u2ab0}^{R\left(e\right)}x$, when $e{p}^{t}\xb7{x}^{t}\ge {p}^{t}\xb7x$. A bundle ${x}^{t}$ is relaxed strictly directly revealed preferred to x, written ${x}^{t}{\succ}^{R\left(e\right)}x$, when $e{p}^{t}\xb7{x}^{t}>{p}^{t}\xb7x$. The relaxed revealed preference relation is the transitive closure of ${\u2ab0}^{R\left(e\right)}$.

#### 2.1.2. HMI

#### 2.1.3. Power of Revealed Preference Tests

## 3. Random Budget Method and Obstacles

- 1.
- Generate a set of random budgets using the same procedure as the experiment;
- 2.
- Simulate uniform choices over each budget;
- 3.
- Compute and record the measure of choice consistency;
- 4.
- Repeat steps 1–3 to create a distribution with S simulated individuals
- 5.
- Compare each individual’s measure of choice consistency to the simulated distribution from 4.

#### Correcting for Random Budgets

- 1.
- Generate budgets used by the ith individual;
- 2.
- Simulate uniform choices over each budget;
- 3.
- Compute and record the measure of choice consistency;
- 4.
- Repeat steps 2–3 to create a distribution with S simulated individuals
- 5.
- Compare the ith individual’s measure of choice consistency to the simulated distribution from 4.
- 6.
- Repeat steps 1–5 for each individual

## 4. Applications

- 1.
- The budget line crosses one axis at or above 50 tokens and;
- 2.
- The budget line crosses both axes at or below 100 tokens.

#### 4.1. Results for CFGK

#### 4.2. Results for CKMS

#### Correlation with Observable Characteristics

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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(1) | (2) | (3) | |
---|---|---|---|

CCEI | CCEI | CCEI | |

Sim CCEI Avg | 0.38 | 0.39 | |

(0.15) | (0.15) | ||

Female | −0.024 | −0.024 | |

(0.0089) | (0.0089) | ||

Age 35–49 | −0.016 | −0.017 | |

(0.011) | (0.011) | ||

Age 50–64 | −0.052 | −0.053 | |

(0.011) | (0.011) | ||

Age 65+ | −0.052 | −0.054 | |

(0.020) | (0.020) | ||

Medium education | 0.0090 | 0.0086 | |

(0.011) | (0.011) | ||

High education | 0.026 | 0.026 | |

(0.011) | (0.011) | ||

Income 2500–3499 | 0.026 | 0.025 | |

(0.012) | (0.012) | ||

Income 3500–4999 | 0.020 | 0.020 | |

(0.013) | (0.013) | ||

Income 5000+ | 0.033 | 0.033 | |

(0.014) | (0.014) | ||

Paid work | 0.028 | 0.027 | |

(0.018) | (0.018) | ||

House work | 0.046 | 0.045 | |

(0.021) | (0.021) | ||

Other | 0.037 | 0.036 | |

(0.019) | (0.019) | ||

Partner | −0.026 | −0.026 | |

(0.011) | (0.011) | ||

# Children | 0.00069 | 0.00039 | |

(0.0042) | (0.0042) | ||

Constant | 0.61 | 0.89 | 0.61 |

(0.11) | (0.022) | (0.11) | |

Observations | 1182 | 1182 | 1182 |

Adjusted ${R}^{2}$ | 0.00 | 0.06 | 0.06 |

(1) | (2) | (3) | |
---|---|---|---|

HMI | HMI | HMI | |

Sim HMI Avg | 0.50 | 0.47 | |

(0.17) | (0.16) | ||

Female | −0.39 | −0.37 | |

(0.14) | (0.14) | ||

Age 35–49 | −0.28 | −0.27 | |

(0.20) | (0.20) | ||

Age 50–64 | −0.93 | −0.91 | |

(0.19) | (0.19) | ||

Age 65+ | −0.75 | −0.71 | |

(0.31) | (0.31) | ||

Medium education | 0.31 | 0.33 | |

(0.17) | (0.17) | ||

High education | 0.69 | 0.72 | |

(0.18) | (0.18) | ||

Income 2500–3499 | 0.19 | 0.20 | |

(0.19) | (0.19) | ||

Income 3500–4999 | 0.11 | 0.097 | |

(0.20) | (0.20) | ||

Income 5000+ | 0.28 | 0.25 | |

(0.22) | (0.21) | ||

Paid work | 0.62 | 0.62 | |

(0.27) | (0.27) | ||

House work | 0.98 | 0.96 | |

(0.31) | (0.31) | ||

Other | 0.86 | 0.90 | |

(0.29) | (0.29) | ||

Partner | −0.39 | −0.36 | |

(0.18) | (0.18) | ||

# Children | −0.023 | −0.024 | |

(0.070) | (0.070) | ||

Constant | 12.6 | 22.2 | 13.1 |

(3.20) | (0.36) | (3.18) | |

Observations | 1182 | 1182 | 1182 |

Adjusted ${R}^{2}$ | 0.01 | 0.07 | 0.07 |

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Mahmood, M.A.; Rehbeck, J.
Correcting for Random Budgets in Revealed Preference Experiments. *Games* **2022**, *13*, 30.
https://doi.org/10.3390/g13020030

**AMA Style**

Mahmood MA, Rehbeck J.
Correcting for Random Budgets in Revealed Preference Experiments. *Games*. 2022; 13(2):30.
https://doi.org/10.3390/g13020030

**Chicago/Turabian Style**

Mahmood, Mir Adnan, and John Rehbeck.
2022. "Correcting for Random Budgets in Revealed Preference Experiments" *Games* 13, no. 2: 30.
https://doi.org/10.3390/g13020030