# Does Imperfect Data Privacy Stop People from Collecting Personal Data?

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. The Model

_{i}denotes the strategy of player i. All decisions are binary.

**Proposition**

**1.**

- (a)
- For $p=1$ and $M\le bI$$${s}_{1}=T\phantom{\rule{0ex}{0ex}}{s}_{2}\left(G\right)=X,\text{}{s}_{2}\left(B\right)=\overline{X}\text{}\mathrm{and}\text{}{s}_{2}\left(U\right)=\overline{X}$$
- (b)
- For $p=1$ and $M\ge bI$$${s}_{1}=\overline{T}\phantom{\rule{0ex}{0ex}}{s}_{2}\left(G\right)=X,\text{}{s}_{2}\left(B\right)=\overline{X}\text{}\mathrm{and}\text{}{s}_{2}\left(U\right)=X$$

**Proposition**

**2.**

- (a)
- For $p=0$$${s}_{1}=T,\text{}{s}_{1}\left(G|T\right)=D,\text{}{s}_{1}\left(B|T\right)={d}_{B}\text{}\mathrm{with}\text{}0\le {d}_{B}\le 1,\text{}\phantom{\rule{0ex}{0ex}}{s}_{2}\left(G\right)=X,{s}_{2}\left(B\right)=\overline{X}\text{}\mathrm{and}\text{}{s}_{2}\left(U\right)=\overline{X}$$
- (b)
- For $p=0$ and $M\ge bI$$${s}_{1}=\overline{T},\text{}{s}_{1}\left(G|T\right)={d}_{G\text{}}\text{}\mathrm{with}\text{}0\le {d}_{G}\le 1,\text{}{s}_{1}\left(B|T\right)=\overline{D},\text{}\phantom{\rule{0ex}{0ex}}{s}_{2}\left(G\right)=X,\text{}{s}_{2}\left(B\right)=\text{}\overline{X}\text{}\mathrm{and}\text{}{s}_{2}\left(U\right)=X$$

**Proposition**

**3.**

- (a)
- For $0<p<1$$${s}_{1}=T,\text{}{s}_{1}\left(G|T\right)=D,\text{}{s}_{1}\left(B|T\right)={d}_{B}\text{}\mathrm{with}\text{}0\le {d}_{B}\le 1,\text{}\phantom{\rule{0ex}{0ex}}{s}_{2}\left(G\right)=X,\text{}{s}_{2}\left(B\right)=\overline{X}\text{}\mathrm{and}\text{}{s}_{2}\left(U\right)=\overline{X\text{}}$$
- (b)
- For $0<p<1$ and $M\ge bI$$${s}_{1}=\overline{T},\text{}{s}_{1}\left(G|T\right)={d}_{G\text{}}\mathrm{with}\text{}0\le {d}_{G}\le 1,\text{}{s}_{1}\left(B|T\right)=\overline{D},\text{}\phantom{\rule{0ex}{0ex}}{s}_{2}\left(G\right)=X,\text{}{s}_{2}\left(B\right)=\overline{X\text{}}\text{}\mathrm{and}\text{}{s}_{2}\left(U\right)=X$$

#### 2.2. Robustness: Risk Aversion, Costs and Benefits from Testing and Equilibrium Refinements

#### 2.2.1. Risk Aversion

#### 2.2.2. Costs and Benefits from Testing

#### 2.2.3. Equilibrium Refinements

## 3. Experiment

#### 3.1. Experimental Design (Parameters and Treatments)

#### 3.2. Part 1 (One-Shot)

#### 3.3. Experimental Design for Part 2 (Repeated)

#### 3.4. Experimental Procedures

_{Perfect Privacy}= 76, N

_{Imperfect Privacy}= 102, N

_{Disclosure Duty}= 80). We computerized the experiment using z-Tree [30]. We recruited participants from the local subject pool using the online recruiting tool ORSEE [31]. In all treatments participants decided in one role, either as player 1 and or player 2 and kept that role for the whole experiment. To avoid testing out of general curiosity, players were informed ex ante that they will learn their type ex post (after decisions by both players were made). Further, participants were informed about all decisions and payoffs after every round. Procedures and parameters were common knowledge. Our experiment lasted 1 h. 10 points were equivalent to 6 euros. Participants received a 4 Euro show-up fee and earned 13.60 euros on average ($15.23 at that point in time). Participants also answered a short post-experimental questionnaire on their socio-economic background and their risk attitudes.

#### 3.5. Experimental Results

#### 3.5.1. Testing, Disclosing and Matching (Part 1: One-Shot)

^{2}-test, p-value = 0.001). Also, testing in Imperfect Privacy is significantly more likely than in Disclosure Duty (χ

^{2}-test, p-value = 0.001). Testing frequencies in Perfect Privacy and Imperfect Privacy do not significantly differ (χ

^{2}-test, p-value = 0.631). Hence, only if data loss is certain, a significant share of players stops collecting information. We summarize this finding in Result 1.

**Result**

**1**

**Result**

**2**

**Result**

**3**

^{2}-test, p-value = 0.267). Disclosure Duty yields the highest level of efficiency, since it results in the most matches but also fails to differ significantly from the other two institutions (Disclosure Duty vs. Imperfect Privacy, χ

^{2}-test, p-value = 0.124 and Disclosure Duty vs. Perfect Privacy, χ

^{2}-test, p-value = 0.693).

**Result**

**4**

#### 3.5.2. Testing, Disclosing and Matching (Part 2: Repeated)

^{2}= 18.25, p < 0.001). Further, we find if at all a weak positive time trend in testing frequencies (see also Figure 3). Model (1) further shows that participants who are generally more willing to take risks and older participants tend to test less frequently for their type.

^{2}= 20.61, p < 0.001). However, in part 2, matching with unknown types is also significantly more likely in Imperfect Privacy as compared to Perfect Privacy. Besides, there is a weak negative time trend in matching with unknown types Efficiency, measured in terms of unconditional matching in Model (4), does not significantly differ between Perfect Privacy and Imperfect Privacy and between Perfect Privacy and Disclosure Duty but is slightly less frequent in Disclosure Duty (0.59) compared to Imperfect Privacy (0.65, Wald-Test comparing coefficients of Imperfect Privacy and Disclosure Duty in model (4), χ

^{2}= 2.25, p < 0.097).

## 4. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Proofs of Propositions 1 to 3

**Proof of Proposition**

**1**

- (a)
- Assume player 2 will not match with unknown types ${s}_{2}\left(G\right)=X,\text{}{s}_{2}\left(B\right)=\overline{X},\text{}$ and ${s}_{2}\left(U\right)=\overline{X}$. If $p=1$, player 1 will test, i.e., ${s}_{1}=T$, because ${\pi}_{1}\left(T\right)=\left(1-b\right)M>{\pi}_{1}\left(\overline{T}\right)=0$. Player 2’s best response is ${s}_{2}\left(G\right)=X,\text{}{s}_{2}\left(B\right)=\overline{X},$ and ${s}_{2}\left(U\right)=\overline{X}$ if $M\le bI$, because ${\pi}_{2}\left(X|G\right)=M>0,\text{}{\pi}_{2}\left(X|B\right)=M-I0,$ and ${\pi}_{2}\left(X|U\right)=M-bI\le 0\iff \text{}M\le bI$.
- (b)
- Assume player 2 will match with unknown types ${s}_{2}\left(G\right)=X,\text{}{s}_{2}\left(B\right)=\overline{X},$ and ${s}_{2}\left(U\right)=X$. If $p=1,$ player 1 will not have herself tested, i.e., ${s}_{1}=\overline{T},\text{}\mathrm{sin}\mathrm{ce}\text{}\mathrm{a}\text{}\mathrm{tested}$ player 1 will automatically be disclosed and in case of a bad test result, player 1 would not receive a match. Player 2’s best response is ${s}_{2}\left(G\right)=X,\text{}{s}_{2}\left(B\right)=\overline{X},$ and ${s}_{2}\left(U\right)=X$ if $M\ge bI$ because ${\pi}_{2}\left(X|G\right)=M>0,\text{}{\pi}_{2}\left(X|B\right)=M-I0,$ and ${\pi}_{2}\left(X|U\right)=M-bI\ge 0\iff M\ge bI$.

**Proof of Proposition**

**2**

- (a)
- Assume player 2 will not match with unknown types ${s}_{2}\left(G\right)=X,\text{}{s}_{2}\left(B\right)=\overline{X},$ and $\text{}{s}_{2}\left(U\right)=\overline{X}$. If $p<1$, player 1 will disclose her type after a good test result $\text{}{s}_{1}\left(G|T\right)=D$ because ${\pi}_{1}^{G}\left(\overline{D}|T\right)=0<{\pi}_{1}^{G}\left(D|T\right)=M$. After a bad test result player 1 is indifferent whether to disclose her type $\text{}{s}_{1}\left(B|T\right)=D$ with $0\le {d}_{B\text{}}\le 1$ because ${\pi}_{1}^{B}\left(\overline{D}|T\right)=0={\pi}_{1}^{B}\left(D|T\right).\text{}$Player 2’s best response is $\text{}{s}_{2}\left(G\right)=X,\text{}{s}_{2}\left(B\right)=\overline{X},\text{}{s}_{2}\left(U\right)=\overline{X}\text{}$ because ${\pi}_{2}\left(X|G\right)=M>0,\text{}{\pi}_{2}\left(X|B\right)=M-I0,\text{}$and ${\pi}_{2}\left(X|U\right)=M-I\le 0$ for all M.
- (b)
- Assume $p$ = 0. Assume further that player 1 will never test and player 2 will match with unknown types. Clearly, player 1 cannot gain from testing if player 2 matches with unknown types. The same holds for player 2’s matching strategy ${s}_{2}\left(G\right)=X,{s}_{2}\left(B\right)=\overline{X},$ and $\text{}{s}_{2}\left(U\right)=X$ because ${\pi}_{2}\left(X|G\right)=M>0,\text{}{\pi}_{2}\left(X|B\right)=M-I0,$ and ${\pi}_{2}\left(X|U\right)=M-bI\ge 0\iff M\ge bI$.

**Proof of Proposition**

**3**

- (a)
- Analogous to proposition 2a.
- (b)
- Assume player 2 will match with unknown type, i.e., ${s}_{2}\left(G\right)=X,\text{}{s}_{2}\left(B\right)=\overline{X,}$ and ${s}_{2}\left(U\right)=X$. If $0<p<1,$ a tested player 1’s best response will be ${s}_{1}\left(G\right)={d}_{G\text{}}with\text{}0\le {d}_{G\text{}}\le 1\text{}$because ${\pi}_{1}^{G}\left(\overline{D}|T\right)=M={\pi}_{1}^{G}\left(D|T\right)$ and $\text{}{s}_{1}\left(B|T\right)=\overline{D}\text{}$ because ${\pi}_{1}^{B}\left(\overline{D}|T\right)=\left(1-p\right)M>{\pi}_{1}^{B}\left(D|T\right)=0$. It follows that$\text{}{s}_{1}=\overline{T}$. Player 2’s best response is $\text{}{s}_{2}\left(G\right)=X,\text{}{s}_{2}\left(B\right)=\overline{X,}\text{}$ and $\text{}{s}_{2}\left(U\right)=X$ if $M\ge bI$ because ${\pi}_{2}\left(X|G\right)=M>0,\text{}{\pi}_{2}\left(X|B\right)=M-I0,\text{}$and ${\pi}_{2}\left(X|U\right)=M-bI\ge 0\text{}\iff \text{}M\ge bI$.

## Appendix B. Incomplete Information Acquisition (Mixed Strategy Equilibria)

**Incomplete information acquisition under Disclosure Duty**

**Proof**

**Incomplete information acquisition under Perfect Privacy**

**Proof**

**Incomplete information acquisition under Imperfect Privacy**

**Proof**

## Appendix C. Instructions

**One-shot experiment (translated from German)**

**10 points = 6 Euros**. On the following pages, we will explain the procedures of Part 1. All participants received the same instructions. You will receive the instructions for Part 2 shortly after Part 1 has ended. Before the experiment starts, we will summarize the procedures verbally. After Part 2 we kindly ask you to answer a short questionnaire.

**The Experiment**

**Summary**

- An interaction yields additional 10 points for participant 1.
- How an interaction affects participant 2 depends on participant 1’s type. If participant 1 is a type A, participant 2 receives additional 10 points. If participant 1 is a type B, participant 2’s points are reduced by 5 points.
- If there is no interaction, points do not change.

**Procedure in Detail**

- One participant 1 and one participant 2 will be randomly assigned to each other. Participant 1 as well as participant 2 receive 10 points. Participant 1 does not know whether he is of type A or of type B. Participant 2 also does not know participant 1’ type.
- Participant 1 decides whether she wants to learn her type.
- [This bullet point was only included in Perfect Privacy]
- If participant 1 has decided to learn her type, she decides whether to inform participant 2.
**Please note**: If participant 1 knows her type and decided to inform participant 2, participant 2 will learn participant 1’s true type. If participant 1 knows her type but did not inform participant 2, participant 2 will not learn participant 1’s type. If participant 1 does not know her type, participant 2 will also not learn participant 1’s type. If participant 2 does not learn participant 1’s type, she will also not learn whether participant 1 herself knows her type. If participant 2 learns participant 1’s type, he also knows that participant 1 knows her type. - [This bullet point was only included in Imperfect Privacy]
- If participant 1 decided to learn his type, she decides whether to inform participant 2 about her type. If participant 1 decided to learn her type, but does not inform participant 2, a random mechanism determines whether player 2 learns player 1’s type nevertheless. In this case player 2 learns player 1’s type with a probability of 50%.
**Please note**: If participant 1 knows her type and decided to inform participant 2, participant 2 will learn participant 1’s true type. If participant 1 knows her type but did not inform participant 2, participant 2 will learn participant 1’s type with a probability of 50%. In both cases participant 2 does not know whether he was informed about the type randomly or directly by participant 1. In all other cases, participant 2 does not receive any information about participant 1’s type, i.e., if participant 1 does not know her type, participant 2 will also not learn participant 1’s type. If participant 2 does not learn participant 1’s type, he will also not learn whether participant 1 knows her type. If participant 2 learns participant 1’s type, he also knows that participant 1 knows her type. - [This bullet point was only included in Disclosure Duty]If participant 1 decides to learn her type, participant 2 will learn participant 1’s type too.
**Please note**: If participant 1 knows that she is type B, participant 2 will also learn that participant 1’s type is B. If participant 1 knows that she is type A, participant 2 will also learn that participant 1’s type is A. If participant 1 does not know her type, participant 2 will also not learn participant 1’s type. But participant 2 knows that participant 1 is of Type A with probability 2/3 (66%) and of Type B with probability 1/3 (33%) - Participant 2 decides whether he wants to interact with participant 1.
- If participant 2 decides to interact, participant 1 receives an extra 10 points. Participant 2’s points depend on participant 1’ type. If participant 1 is of type A, participant 2 receives an extra 10 points. If participant 1 is of type B, participant 2’s points are reduced by 5 points.
- If participant 2 does decides NOT to interact, both participants receive no extra points, so each of the participants has the 10 points received at the beginning.
- After all participants have made their decision you will receive information about your earnings. At the same time the type of participant 1 and whether an interaction took place will be shown to participants 1 and 2.

**Procedure on-Screen**

**Comprehension Questions:**(correct answers in parentheses, DD = Disclosure Duty, PP = Perfect Privacy, IP = Imperfect Privacy)

**True or False?**

**Further Questions:**

**Please fill in the Blanks:**

- If participant 2 decided to interact and participant 1 is of type A, participant 2 receives___ (10) points.
- If participant 2 decided to interact and participant 1 is of type B, participant 2 loses ___ (5) points.
- If participant 2 decided to interact, participant 1 receives an extra ___ (10) points.
- If participant 2 refused to interact, participant 1 receives an extra ___ (0) points and participant 2 an extra ___ (0) points.

**Repeated Experiment (Translated from German)**

**The Experiment—Summary**

- Experiment 2 consists of 10 periods.
- Every period has the same procedure and rules as Experiment 1.
- You have the same role (participant 1 or participant 2) as in Experiment 1 in all 10 periods.
- Participant 1 decides whether she wants to learn her type (A or B).
- [This bullet point was only included in Perfect Privacy and Imperfect Privacy]. If participant 1 decided to learn her type, she decides whether to inform participant 2 about her type.
- Participant 2 decides whether to interact with participant 1.

**Important:**

- In every period, you will be matched with another participant, i.e., with a participant you have not been matched with before (neither in Part 1 or Part 2).
- In every period for every participant 1 it will be randomly determined whether she is type A or B. The probability to be type A or B are the same as in Part 1.
- ⚪
- The probability of being a type A is 2/3 (or 66.66%).
- ⚪
- The probability of being a type B is 1/3 (or 33.33%).

- At the end of Part 2, i.e., after period 10, one period will be randomly determined to be payoff relevant for Part 2. For this, the participant with seat number 12 will roll a ten-sided die.
- Afterwards, the participant with seat number 12 will roll a six-sided die to determine whether participants will receive their earnings from part 1 or Part 2.

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1 | Apart from test based, type specific premiums, insurances nowadays also offer rebates, bonuses (or penalties) based on personal health data collected through health trackers or third parties (see e.g., [9]). |

2 | For a similar argument see also Hirshleifer [10]. |

3 | Consent Law describes the situation in which consumers “are not required to divulge genetic tests results. But, if they do, insurers may use this information” [12]. |

4 | In addition to Consent Law and Disclosure Duty, several other approaches have been discussed in the context of genetic testing. Barigozzi and Henriet [11] consider further the “Laissez-Faire approach”, under which insurers can access test results and require additional tests and “Strict Prohibition” of the use of test results. |

5 | |

6 | |

7 | Matthews and Postlewaite [25] focus on sellers’ testing behavior in the context of product quality when disclosure of test results is mandatory or voluntary and test results may be beneficial to consumers. For Perfect Privacy and Disclosure Duty our model mirrors the logic of their analysis and can be understood as a simplified version of their framework. However, our analysis differs in terms of who acquires information, what quality types are available and includes the additional environment of Imperfect Privacy. |

8 | Recently in a different setting, Bardey, De Donder and Mantilla [15] complement their theoretical analysis on different regulatory institutions for genetic testing with an experiment. However, their experimental design focuses on the joint decision of choosing a privacy institution and testing for one’s type using a series of individual lottery choice tasks. |

9 | In Section 2.2.2 we provide a robustness analysis on how psychological costs and prevention benefits affect the existence of equilibria derived in the simple model. |

10 | Our framework may also be interpreted as a situation in which the insurer offers two tariffs, one for good and one for bad health types. |

11 | While these results are derived by modelling patients and insurers as risk neutral and abstaining from modelling direct costs or benefits from testing, we discuss below whether these Proper Equilibria are robustness to common assumptions concerning risk aversion of patients and risk neutrality of insurers. Further, we discuss also the robustness of the different equilibria concerning costs and benefits from testing. |

12 | We refrain from modeling a partial internalization of the loss of utility (from a match with a bad type) of player 2 by player 1. Nevertheless modelling this internalization as a loss of I’ for player 1 does not change the model’s predictions as long as for player 1 I’ < M. We thank an anonymous referee for highlighting this aspect. |

13 | We discuss the robustness of our results with respect to risk aversion in Section 2.2.1. |

14 | We discuss the impact of explicit testing costs in Section 2.2.2. |

15 | We relegate formal proofs of all propositions as well as the derivation of mixed strategy equilibria to the appendix. |

16 | Assuming a risk neutral insurer but a risk averse consumer (as [13], equilibria with complete information or no information still exist for all institutions. Equilibria with incomplete information acquisition exist only for Disclosure Duty and Imperfect Privacy. |

17 | By doing so we implicitly deal with benefits from knowing to be the good type (which are in our model mathematically equivalent to costs from knowing to be the bad type) and costs from not knowing to be the good type (which are mathematically equivalent to benefits from knowing to be the bad type). |

18 | One randomly selected participant rolled a six-sided die to determine whether a test result was involuntarily displayed (depending on whether the number was odd or even). The participant was monitored and announced the number publicly. |

19 | |

20 | A copy of translated instructions can be found in Appendix C. |

21 | We cannot reject the hypothesis that disclosure behavior of tested good types is identical in Privacy and Imperfect Privacy (Fisher’s exact test, p-value = 0.926). |

22 | We cannot reject the hypothesis that disclosure behavior of tested bad types is identical in Perfect Privacy and Imperfect Privacy (Fisher’s exact test, p-value = 0.740). |

23 | More testing eventually reduces the number of mismatches. Engelhardt, et al. [32] for instance argue that on internet platforms for semi-anonymous encounters, provision of information about the own HIV status might result in a directed search and reduce the transmission rate by separating the uninfected and infected, e.g., through the use of condoms. |

24 | We carefully note that in the context of HIV testing, social preferences may matter strongly and many people may test and report their result, irrespective of the institutional setup. |

25 | Inequality aversion might also be the reason why some players 1 disclose their bad type. By this means they prevent player 2 from matching which would lead to an unequal allocation. |

**Figure 2.**Testing, disclosure, matching and frequency of matches in Part 1. (

**A**) Test frequencies across treatments in Part 1; (

**B**) Disclosure frequencies when tested; (

**C**) Matching with unknown type (# participants matching with an unknown type/# participants facing an unknown type); (

**D**) Total frequencies of matches.

**Table 1.**Random effects probit regressions (marginal effects), robust standard errors in parentheses (clustered on sessions), *** p < 0.01, ** p < 0.05, * p < 0.1.

Decision to… | ||||
---|---|---|---|---|

(1) | (2) | (3) | (4) | |

…Test | …Disclose | …Match with Unknown Type | …Match (Unconditionally) | |

Perfect Privacy | baseline | baseline | baseline | baseline |

Imperfect Privacy | −0.056 | −0.035 | 0.180 ** | 0.027 |

(0.040) | (0.029) | (0.076) | (0.032) | |

Disclosure Duty | −0.284 *** | n.a. | 0.426 *** | −0.019 |

(0.058) | (0.035) | (0.016) | ||

Period | 0.008 ** | 0.002 | −0.011 * | −0.000 |

(0.004) | (0.003) | (0.006) | (0.002) | |

Bad Type | −0.319 *** | |||

(0.040) | ||||

Imperfect Privacy x | 0.016 | |||

Bad Type | (0.030) | |||

Willingness to take | −0.011 * | 0.008 | 0.020 | 0.001 |

risks in general | (0.007) | (0.006) | (0.017) | (0.014) |

Male | −0.013 | −0.044 ** | 0.017 | 0.038 * |

(0.032) | (0.018) | (0.057) | (0.021) | |

Age | −0.004 ** | 0.001 | −0.025 | −0.024 *** |

(0.002) | (0.001) | (0.016) | (0.008) | |

Observations | 1290 | 849 | 373 | 1290 |

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**MDPI and ACS Style**

Schudy, S.; Utikal, V. Does Imperfect Data Privacy Stop People from Collecting Personal Data? *Games* **2018**, *9*, 14.
https://doi.org/10.3390/g9010014

**AMA Style**

Schudy S, Utikal V. Does Imperfect Data Privacy Stop People from Collecting Personal Data? *Games*. 2018; 9(1):14.
https://doi.org/10.3390/g9010014

**Chicago/Turabian Style**

Schudy, Simeon, and Verena Utikal. 2018. "Does Imperfect Data Privacy Stop People from Collecting Personal Data?" *Games* 9, no. 1: 14.
https://doi.org/10.3390/g9010014