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Several liability with sequential care: an experiment

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Abstract

By a laboratory experiment, we investigate the incentives of potential tortfeasors to make investments in order to reduce the probability of a given harm occurring. In our experiment, paired players make their decisions sequentially, i.e., a la Stackelberg and in case of a harm, they share liability, either symmetrically or asymmetrically. We vary their level of endowments so that one can become insolvent in case of harm, and show that subjects do not behave as predicted, due to considerations such as inequity aversion or reciprocity. We find that the first-acting tortfeasor does not exploit their advantage fully and invests much more than predicted. Our results also show that the second-acting tortfeasor acts reciprocally with the other, rather than rationally opting for their best response. Finally, we show that asymmetric liability apportionment may restore the incentives to invest for the first tortfeasor, and that insolvency of the second one pushes the first to overinvest in order to avoid harm.

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Notes

  1. Pittsburgh Reduction Co. v. Horton, 87 Ark. 576, 113 S.W. 647 (1908) described in (Grady, 1990).

  2. EWCA Civ 669, 9 June 2011.

  3. Examples of the State being involved arise in many contexts, one of which is environmental and technological risks. For instance, regarding technological accidents arising in French cities, the mayor, as an agent of the State, must act preventively by drawing up a technological risk prevention plan and in this respect must ensure that polluting companies, such as AZF in Toulouse or more recently Lubrizol in Rouen, have put in place effective protections to limit and prevent the toxic effects of the operation of the plant. In the AZF case, the State was found guilty for its negligence by an administrative court.

  4. Note that other rules exist, especially a joint and several liability rule which implies that in case of insolvency of one tortfeasor, the solvent one has to compensate for their share of the damage, plus the share that the other one could not bear. For an experimental comparison of joint and several liability with several (only) liability, see notably (Garcia et al., forthcoming).

  5. This can be referred to as the concept of dilution of liability, see notably (Dillbary, 2013; Dillbary et al., 2021).

  6. Note here that we chose to compare several apportionment rules for different levels of wealth, but we do not compare strict liability with negligence for instance. For papers with such a comparison but in context with solely one tortfeasor, see notably Deffains et al. (2019) and Angelova et al. (2014).

  7. Note here that in all treatments, we implement a fixed sharing rule (50–50 or 75–25), so that one’s investment only allows to reduce the probability of damage, but not one’s share of the damage which remains exogenous. This is a limitation of the paper as in reality, liability may depend upon relative investments. See notably Jacob et al. (2021) for an experimental comparison of per capita and proportional rules.

  8. Experimental instructions are available in Appendix A.

  9. Note that apart from this term of harm, the experiment is decontextualized in order to stay as general as possible.

  10. Tables 1 and 2 in the instructions, available in the appendix, respectively display a player’s benefit depending on her own decision and the probability of harm depending on both players’ decisions.

  11. Note that the cost here is either borne by the injurers if they are liable and solvable, or sunk and in this case not borne by any victim otherwise. Note that in reality, the victim may be a natural person as well as a legal entity. In the case of environmental disasters, the environment and other parties would be the victims. In order to remain as general as possible, we did not include a victim in the experiment. For papers including a victim, see e.g. Angelova et al. (2014) and Dillbary et al. (2021).

  12. Note here that this 75–25 apportionment liability is presented in this way to the subjects, but it is only theoretical: in fact, X really bears 75%, but Y does not bear 25% since her endowment is not sufficient. However, in case of a damage, Y has to give their entire endowment to repair the harm.

  13. It follows that: (i) increasing one investment (with the other one remaining unchanged) reduces the probability of causing harm, but (ii) also reduces the marginal efficiency of the other investment.

  14. We suppose that causality is proven: in case of harm, there is no doubt as to the fact that the agents’ activities caused the harm.

  15. For the USA, see the Comprehensive Environmental Response, Compensation, and Liability Act. In Europe, Directive 2004/35/CE on liability for harm to the environment enforces strict liability for the most dangerous activities (which are listed in Annex III). In England, Shavell (2018) p. 7 observes the enforcement of strict liability for “high risk” activities.

  16. An alternative rule we do not consider here would be joint and several liability: in case of insolvency of one tortfeasor, the remaining damages fall to the solvent tortfeasor (if any). So, the solvent agents can pay a part of damages that were initially to be paid by the insolvent ones.

  17. In the USA, for instance, a decision of Pennsylvania Supreme Court of \(19^{th}\) February 2020 states an equal sharing of liability between several tortfeasors in a case of asbestos disease, because of the impossibility of determining the contribution of each tortfeasor in the toxic tort suffered by the plaintiff. Many environmental harms are also indivisible, as documented by Ackerman (1973), Tietenberg (1989) or Daughety and Reinganum (2014).

  18. Details of all calculations are available in Appendix B.

  19. Laboratoire d’Économie Expérimentale de Strasbourg.

  20. ORSEE is a web-based Online Recruitment System for Economic Experiments developed by Greiner (2015). This experiment was programmed using z-tree (Fischbacher, 2007).

  21. One may argue that there is a dependence between observations at the session level, as the game is repeated with stranger matching. The low number of sessions per treatment does not allow us to perform statistical test on averages grouped by session. However, the reader can refer to Table 10 in Appendix C, which reports average participant decisions by type and by matching group.

  22. Such an end-of-game effect could have suggested that investment from participant X was mainly led by long-term strategic concerns. This was in any case unlikely to happen in our stranger-matching design, where X-Y pairs were reformed randomly at the beginning of each period.

  23. The full expression of the best response function can be found in appendix B—Determination of private equilibria.

  24. We include in those models a random term at the individual level and a random term at the session level. We do so to control for potential dependence between participants’ decisions within a session, as a consequence of our stranger-matching protocol. We also estimated standard random-effect models with only individual effects which lead to similar results and the same conclusions.

  25. In treatment T4, Y participants were investing less than their best response in 26.7% of the interactions. They invested more than their best response in 63.3% of the interactions. Finally they played their best response in 10% of the interactions.

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Funding was provided by CPER Ariane.

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Appendix

Appendix

1.1 Appendix A: Instructions of the experiment

Note to the reviewer: here are displayed the general instructions and instructions of part one: these are the same for all treatments. We will then display below the instructions for the second part (main game) of the experiment, which differ from one treatment to another. Note also that this version of the instructions has been translated from French to English.

1.1.1 General instructions—all treatments

General instructions

Thank you for participating in this decision-making experience. In this experiment, your monetary gains will depend on your decisions and random events. It is important that you read these instructions carefully. The experience consists of three independent phases. At the end of the experiment, you will receive the sum of your winnings from Phase 1 and Phase 2. In addition, you will receive 3 Euros for participating in the experiment. You will be paid in cash at the end of the session on an individual and confidential basis.

All your decisions are anonymous. You will never have to enter your name into the computer.

During the experiment, you are not allowed to communicate. If you have any questions, please raise your hand. One of us will come to answer your questions.

You have received the instructions for the first part of the experiment. You will receive the instructions for the second part once you have completed the first part of the experiment.

1.1.2 Part 1 instructions—all treatments

Part 1

You have only one decision to make in this first part. You must choose one lottery from six available lotteries. Your earnings will depend on the outcome of the lottery you choose. For each lottery, there are two available outcome: outcome A or outcome B.

Following your choice, the computer will run a coin to determine your earnings. Thus, both outcome A and outcome B have a 50% chance of being realized.

Below are presented the different lotteries available.

Lottey

Outcome A (50%)

Outcome B (50%)

1

2.80€

2.80€

2

3.60€

2.40€

3

4.40€

2.00€

4

5.20€

1.60€

5

6.00€

1.20€

6

7.00€

0.20€

For example, if you choose lottery 3, you will have as many chances to get 4.40€ as you have to get 2€.

You will not be informed of the outcome of this game until the end of the experiment.

Once all participants have chosen a lottery, we will distribute the instructions for part 2 of the experiment.

1.1.3 Part 2 instructions—treatments 1 and 2

Note to the reviewer: here are displayed the instructions for Treatments 1 and 2. We indicate in blue and bold letters the parts of instructions that are specific to Treatment 1 and in red and italic letters those that are specific to Treatment 2.

Second part of the experiment

During this second game, your payoffs are counted in ECUs. Your cash earnings for this game will be calculated using the following conversion rate: 60 ECUS \(=\) 1 €.

This game has 20 independent periods. You will have to make a decision in each period. At the end of the 20 periods, a randomly drawn participant will be designated to draw the three winning periods and will read them aloud to the other participants. The earnings for all participants in this second game will be calculated by adding these three periods together.

1.2 Description of the game

You are randomly assigned as Participant X or Participant Y at the beginning of this phase and you remain in this role for all 20 periods. You are a total of 20 participants, separated into 2 groups of 10 participants; these 2 groups will never interact with each other. Within each group, there are 5 X participants and 5 Y participants. At the beginning of each of the 20 periods, each participant X is matched with a participant Y. You will not know the identity of your pair. In addition, the pair you meet in each period is randomly determined before each new period.

1.3 Course of each period

Participants begin each period with an endowment of 150 ECUS. During each period, each participant will make a decision that will affect both his or her final gain and potentially, the gain of the other. Participant X is the first to make a decision: he must choose a number between 0 and 39. Once this decision is made, it is recorded and revealed to participant Y. Participant Y then makes a decision in turn by choosing a number between 0 and 39. This decision is revealed to participant X and the probability of occurrence of the damage corresponding to the decisions made by the two players is displayed on the computer screen.

Each participant’s gain is made up of three components: your initial endowment A determined at the outset, your benefit B entirely determined by your own decision, and the damage C, the probability of which is determined by both your choice and that of the other participant. More specifically:

  • A is your initial endowment of 150.

  • Your benefit B is in addition to your initial endowment and depends solely on the number you alone have chosen. The values of B based on the number chosen are shown in Table 8. B ranges from 60 to 0, depending on the number you choose. The higher the chosen number, the lower B is; the lower the chosen number, the higher B is. For example, if you choose the number 10, your benefit B equals 53.5; if you choose the number 30, your benefit B equals 25.2.

  • Depending on the decisions of the two participants, a damage C of 200 can occur with a given probability.

    If this damage occurs, both participants must assume half of this damage (i.e. 100 each); the part assumed by each is taken from their initial endowment A.

    If this damage occurs, the participants must reimburse this damage, but in different proportions: participant X must assume 150 while participant Y must assume 50. The portion assumed by each is taken from their initial endowment A.

    The numbers previously chosen by the two participants determine the probability of occurrence of this damage, which varies from \(0\%\) to \(100\%\) depending on the numbers chosen. The values of the probability of occurrence of the damage according to the numbers chosen by the two participants are presented in Table 9. For example, if participant X chooses the number 3 and participant Y chooses the number 7, then the harm occurs with a probability of \(58.4\%\). If participant X chooses the number 25 and participant Y chooses the number 10, then the harm occurs with a probability of \(12.2\%\). Note that if each participant chooses the number 39, the probability is \(0\%\); if each participant chooses the number 0, the probability is \(100\%\).

Table 8 Value of benefit depending on the number’s choice
Table 9 Probability of damage depending on both players' investments

The earnings of each participant are therefore as follows, depending on whether the damage occurs or not:

  • In case of occurrence of the damage:

    earnings of a player \(=\) initial endowment A (i.e. 150) \(+\) benefit Bassumed part of the damage C (i.e. 100)

    earnings of a player \(=\) initial endowment A (i.e. 150) \(+\) benefit Bassumed part of the damage C (i.e. 150 for X, 50 for Y)

  • In case of no damage:

    earnings of a player \(=\) initial endowment A (i.e. 150) \(+\) benefit B

Let’s take two examples at random. Please note that these two examples are only illustrations and are not intended to guide you in your decisions; for example, they are not intended to reflect the best possible situation for either or both of them.

Example 1: X chooses 35 then Y chooses 8: the damage occurs with a probability of \(7.6\%\) (Table 9)

Earnings of X:

  • If the damage occurs, which happens with a probability of \(7.6\%\):

    • X’s endowment is \(A=150\).

    • Component B of X is 12.5 (Table 8).

    • Damage C is equal to 200. X assumes half of C, i.e. 100.

    • In total, X gets \(150+12.5-100=62.5\).

    • Damage C is equal to 200. X assumes 150.

    • In total, X gets \(150+12.5-150=12.5\).

  • If no damage occurs, which happens with a probability of \(92.4\%\):

    • X’s endowment is \(A=150\).

    • Component B of X is 12.5 (Table 8).

    • There is no damage, so there is nothing to subtract from previous earnings.

    • In total, X gets \(150+12.5=162.5\).

Earnings of Y:

  • If the damage occurs, which happens with a probability of \(7.6\%\):

    • Y’s endowment is \(A=150\).

    • Component B of Y is 55.1 (Table 8).

    • Damage C is equal to 200. Y assumes half of C, i.e. 100.

    • In total, Y gets \(150+55.1-100=105.1\).

    • Damage C is equal to 200. Y assumes 50.

    • In total, Y gets \(150+55.1-50=155.1\)

  • If no damage occurs, which happens with a probability of \(92.4\%\):

    • Y’s endowment is \(A=150\).

    • Component B of Y is 55.1 (Table 8).

    • There is no damage, so there is nothing to subtract from previous earnings.

    • In total, Y gets \(150+55.1=205.1\).

Example 2: X chooses 2 then Y chooses 15: the damage occurs with a probability of \(36.1\%\) (Table 9)

Earnings of X:

  • If the damage occurs, which happens with a probability of \(36.1\%\):

    • X’s endowment is \(A=150\).

    • Component B of X is 58.9 (Table 8).

    • Damage C is equal to 200. X assumes half of C, i.e. 100.

    • In total, X gets \(150+58.9-100=108.9\).

    • Damage C is equal to 200. X assumes 150.

    • In total, X gets \(150+58.9-150=58.9\).

  • If no damage occurs, which happens with a probability of \(63.9\%\):

    • X’s endowment is \(A=150\).

    • Component B of X is 58.9 (Table 8).

    • There is no damage, so there is nothing to subtract from previous earnings.

    • In total, X gets \(150+58.9=208.9\).

Earnings of Y:

  • If the damage occurs, which happens with a probability of \(36.1\%\):

    • Y’s endowment is \(A=150\).

    • Component B of Y is 48.8 (Table 8).

    • Damage C is equal to 200. Y assumes half of C, i.e. 100.

    • In total, Y gets \(150+48.8-100=98.8\).

    • Damage C is equal to 200. Y assumes 50.

    • In total, Y gets \(150+48.8-50=148.8\).

  • If no damage occurs, which happens with a probability of \(63.9\%\):

    • Y’s endowment is \(A=150\)

    • Component B of Y is 48.8 (Table 8).

    • There is no damage, so there is nothing to subtract from previous earnings.

    • In total, Y gets \(150+48.8=198.8\).

Note: The value of B presented in Table 8, as well as the value of the probability of occurrence of the damage presented in Table 9are rounded values. It is therefore possible that your actual payoff may differ from your calculations by up to one unit.

Note that at the end of the period, you will not know whether or not a damage has occurred, the computer will only tell you the number chosen by your partner, as well as the probability of damage occurring as a result of your choice and that of the other participant.

Once the 1st period is over, you are paired with another randomly selected player and again you have to choose a number between 0 and 39. Earnings are calculated in the same way in each period.

At the end of this step, the computer will proceed for each of the 20 periods the draw to determine whether the damage actually occurred or not, according to the probability of occurrence of the damage obtained in each period. 3 periods to be paid among the 20 will then be drawn randomly by a participant. Whether there was damage or not, the method of calculating the earnings is always the same, since it is the sum of the earnings obtained during these 3 periods.

Before starting the experiment, we ask you to answer a few questions to test your understanding of the instructions. These questions will appear on your computer screen in a few minutes.

1.3.1 Part 2 instructions—treatments 3 and 4

Note to the reviewer: here are displayed the instructions for Treatments 3 and 4. We indicate in blue and bold letters the parts of instructions that are specific to Treatment 3 and in red and italic letters those that are specific to Treatment 4.

Second part of the experiment

During this second game, your payoffs are counted in ECUs. Your cash earnings for this game will be calculated using the following conversion rate: 60 ECUS \(=\) 1 €.

This game has 20 independent periods. You will have to make a decision in each period. At the end of the 20 periods, a randomly drawn participant will be designated to draw the three winning periods and will read them aloud to the other participants. The earnings for all participants in this second game will be calculated by adding these three periods together.

1.4 Description of the game

You are randomly assigned as Participant X or Participant Y at the beginning of this phase and you remain in this role for all 20 periods. You are a total of 20 participants, separated into 2 groups of 10 participants; these 2 groups will never interact with each other. Within each group, there are 5 X participants and 5 Y participants. At the beginning of each of the 20 periods, each participant X is matched with a participant Y. You will not know the identity of your pair. In addition, the pair you meet in each period is randomly determined before each new period.

1.5 Course of each period

Participant X begins each period with an endowment of 150 ECUS and participant Y begins each period with an endowment of 30 ECUS. During each period, each participant will make a decision that will affect both his or her final gain and potentially, the gain of the other. Participant X is the first to make a decision: he must choose a number between 0 and 39. Once this decision is made, it is recorded and revealed to participant Y. Participant Y then makes a decision in turn by choosing a number between 0 and 39. This decision is revealed to participant X and the probability of occurrence of the damage corresponding to the decisions made by the two players is displayed on the computer screen.

Each participant’s gain is made up of three components: your initial endowment A determined at the outset, your benefit B entirely determined by your own decision, and the damage C, the probability of which is determined by both your choice and that of the other participant. More specifically:

  • A is your initial endowment: it is equal to 150 if you are X and 30 if you are Y.

  • Your benefit B is in addition to your initial endowment and depends solely on the number you alone have chosen. The values of B based on the number chosen are shown in Table 8. B ranges from 60 to 0, depending on the number you choose. The higher the chosen number, the lower B is; the lower the chosen number, the higher B is. For example, if you choose the number 10, your benefit B equals 53.5; if you choose the number 30, your benefit B equals 25.2.

  • Depending on the decisions of the two participants, a damage C of 200 can occur with a given probability.

    If this damage occurs, both participants must, in principle, bear half of the damage (i.e. 100 each). However, since the part assumed by each participant is taken from their initial allocation A, player X can indeed assume his share of 100 but player Y cannot and therefore assumes only up to the amount of his allocation, i.e. 30. Note that there is a share of damage (\(200 - 100 - 30 =\)70) that is not assumed by anyone.

    If this damage occurs, the participants must reimburse this damage, but in different proportions: participant X must assume \(75\%\) of the damage, i.e. 150, while participant Y must in principle assume \(25\%\) of the damage, i.e. 50. However, the part assumed by each participant is taken from their initial allocation A: thus, participant X’s allocation of 150 does indeed allow him to assume his share of the damage (150), but participant Y’s allocation (30) does not allow him to assume all of his share (50) of the damage: the latter therefore assumes 30. Note that there is still a share of the damage \((200 - 150 - 30 = 20)\) that is not assumed by anyone.

    The numbers previously chosen by the two participants determine the probability of occurrence of this damage, which varies from \(0\%\) to \(100\%\) depending on the numbers chosen. The values of the probability of occurrence of the damage according to the numbers chosen by the two participants are presented in Table 9. For example, if participant X chooses the number 3 and participant Y chooses the number 7, then the harm occurs with a probability of \(58.4\%\). If participant X chooses the number 25 and participant Y chooses the number 10, then the harm occurs with a probability of \(12.2\%\). Note that if each participant chooses the number 39, the probability is \(0\%\); if each participant chooses the number 0, the probability is \(100\%\).

The earnings of each participant are therefore as follows, depending on whether the damage occurs or not:

  • In case of occurrence of the damage:

    earnings of a player \(=\) initial endowment A (i.e. 150 for X, 30 for Y) \(+\) benefit Bpart assumed of damage C (i.e. 100 for X, 30 for Y)

    earnings of a player \(=\) initial endowment A (i.e. 150 for X, 30 for Y) \(+\) benefit Bpart assumed of damage C (i.e. 150 for X, 30 for Y)

  • In case of no damage:

    earnings of a player \(=\) initial endowment A (i.e. 150 for X, 30 for Y) \(+\) benefit B

Let’s take two examples at random. Please note that these two examples are only illustrations and are not intended to guide you in your decisions; for example, they are not intended to reflect the best possible situation for either or both of them.

Example 1

X chooses 35 then Y chooses 8: the damage occurs with a probability of \(7.6\%\) (Table 9)

Earnings of X:

  • If the damage occurs, which happens with a probability of \(7.6\%\):

    • X’s endowment is \(A=150\).

    • Component B of X is 12.5 (Table 8).

    • Damage C is equal to 200. X assumes half of C, i.e. 100.

    • In total, X gets \(150+12.5-100=62.5\).

    • Damage C is equal to 200. X assumes 150.

    • In total, X gets \(150+12.5-150=12.5\).

  • If no damage occurs, which happens with a probability of \(92.4\%\):

    • X’s endowment is \(A=150\).

    • Component B of X is 12.5 (Table 8).

    • There is no damage, so there is nothing to subtract from previous earnings.

    • In total, X gets \(150+12.5=162.5\).

Earnings of Y:

  • If the damage occurs, which happens with a probability of \(7.6\%\):

    • Y’s endowment is \(A=30\).

    • Component B of Y is 55.1 (Table 8).

    • Damage C is equal to 200. Y cannot assume half of C, i.e. 100, he therefore assumes up to the amount of his endowment, i.e. 30.

    • In total, Y gets \(30+55.1-30=55.1\).

    • Damage C is equal to 200. Y cannot assume half of C, i.e. 50, he therefore assumes up to the amount of his endowment, i.e. 30.

    • In total, Y gets \(30+55.1-30=55.1\).

  • If no damage occurs, which happens with a probability of \(92.4\%\):

    • Y’s endowment is \(A=30\).

    • Component B of Y is 55.1 (Table 8).

    • There is no damage, so there is nothing to subtract from previous earnings.

    • In total, Y gets \(30+55.1=85.1\).

Example 2

X chooses 2 then Y chooses 15: the damage occurs with a probability of \(36.1\%\) (Table 9)

Earnings of X:

  • If the damage occurs, which happens with a probability of \(36.1\%\):

    • X’s endowment is \(A=150\).

    • Component B of X is 58.9 (Table 8).

    • Damage C is equal to 200. X assumes half of C, i.e. 100.

    • In total, X gets \(150+58.9-100=108.9\).

    • Damage C is equal to 200. X assumes 150.

    • In total, X gets \(150+58.9-150=58.9\).

  • If no damage occurs, which happens with a probability of \(63.9\%\):

    • X’s endowment is \(A=150\).

    • Component B of X is 58.9 (Table 8).

    • There is no damage, so there is nothing to subtract from previous earnings.

    • In total, X gets \(150+58.9=208.9\).

Earnings of Y:

  • If the damage occurs, which happens with a probability of \(36.1\%\):

    • Y’s endowment is \(A=30\).

    • Component B of Y is 48.8 (Table 8).

    • Damage C is equal to 200. Y cannot assume half of C, i.e. 100, he therefore assumes up to the amount of his endowment, i.e. 30.

    • Damage C is equal to 200. Y cannot assume half of C, i.e. 50, he therefore assumes up to the amount of his endowment, i.e. 30. In total, Y gets \(30+48.8-30=48.8\).

  • If no damage occurs, which happens with a probability of \(92.4\%\):

    • Y’s endowment is \(A=30\).

    • Component B of Y is 48.8 (Table 8).

    • There is no damage, so there is nothing to subtract from previous earnings.

    • In total, Y gets \(30+48.8=78.8\).

Note: The value of B presented in Table 8, as well as the value of the probability of occurrence of the damage presented in Table 9are rounded values. It is therefore possible that your actual payoff may differ from your calculations by up to one unit.

Note that at the end of the period, you will not know whether or not a damage has occurred, the computer will only tell you the number chosen by your partner, as well as the probability of damage occurring as a result of your choice and that of the other participant.

Once the 1st period is over, you are paired with another randomly selected player and again you have to choose a number between 0 and 39. Earnings are calculated in the same way in each period.

At the end of this step, the computer will proceed for each of the 20 periods the draw to determine whether the damage actually occurred or not, according to the probability of occurrence of the damage obtained in each period. 3 periods to be paid among the 20 will then be drawn randomly by a participant. Whether there was damage or not, the method of calculating the earnings is always the same, since it is the sum of the earnings obtained during these 3 periods.

Before starting the experiment, we ask you to answer a few questions to test your understanding of the instructions. These questions will appear on your computer screen in a few minutes.

1.5.1 Part 3 instructions—all treatments

We now ask you to answer a few questions about yourself, it will only take a few minutes. All your answers are anonymous and will remain confidential.

  1. 1.

    You are: a man—a woman (circle the correct answer).

  2. 2.

    In life, do you feel that you are a risk-taker or a cautious person? Rate on a scale of 1 to 10 where you think you stand, with 1 representing someone who is extremely cautious and 10 representing someone who loves to take risks.

    1        2        3        4        5        6        7        8        9        10

  3. 3.

    In life, would you say that most of the time you try to help others or are mainly concerned with your own interests? Rate on a scale of 1 to 10 where you think you are, with 1 representing someone who loves to help others and 10 representing someone who acts solely in their own best interest.

    1        2        3        4        5        6        7        8        9        10

  4. 4.

    During the main game, were you guided through each step solely by your win or also by your partner’s win?

    - only your earnings

    - your earnings and that of your successive partners

  5. 5.

    Generally speaking, do you feel that most people can be trusted or that you should be very careful with others?

    - Most people can be trusted

    - we have to be very careful

  6. 6.

    What criteria guided your decisions during the experiment?

  7. 7.

    What do you think was the objective of the experiment? What do you think we wanted to test?

1.6 Appendix B: theoretical developments

1.6.1 Details on the determination of private equilibria

The model is solved backwards. First, agent Y determines her optimal level of investment \(I_{Y}\), given the level of investment \(\bar{I_{X}}\) made by the agent X. As a consequence, Y has to determine the level of \(I_{Y}\) which is best-response to \(\bar{I_{X}}\). She has to solve:

$$\begin{aligned} \max _{I_{Y}} U_{Y}= W_{Y} + B_{Y}(I_{Y}) - p(\bar{I_{X}}, I_{Y}) D_{Y} \end{aligned}$$

with \(D_{Y} = \min \left\{ W_{Y},\lambda H\right\}\).

The best-response to \(\bar{I_{X}}\), \(I_{Y}^{*}(\bar{I_{X}})\), satisfies:

$$\begin{aligned} \frac{\partial U_{Y}}{\partial I_{Y}}=0 \Rightarrow B_{Y}'(I_{Y}) - p'_{I_{Y}}(\bar{I_{X}}, I_{Y}) D_{Y} = 0 \end{aligned}$$
(6)

Given the specifications introduced above, we find:

$$\begin{aligned} I_{Y}^{*}(\bar{I_{X}})=-\frac{\gamma }{\beta _{Y}+\gamma }\bar{I_{X}}+\left[ \frac{1}{\beta _{Y}+\gamma }\right] ln\left[ \frac{\gamma D_{Y}}{\delta _{Y}\beta _{Y}}\right] \end{aligned}$$
(7)

Then, knowing the best-response of agent Y, agent X determines her optimal level of \(I_{X}\), say \(I_{X}^{*}\). She thus maximizes the following program:

$$\begin{aligned} \max _{I_{X}} U_{X}= W_{X} + B_{X}(I_{X}) - p(I_{X},I_{Y}^{*}(I_{X})) D_{X} \end{aligned}$$

with \(D_{X} = \min \left\{ W_{X},(1-\lambda ) H\right\}\), and \(I_{Y}^{*}(I_{X})\) the best-response of agent Y for a given level of \(I_{X}\) (according to the function (7)).

\(I_{X}^{*}\) satisfies:

$$\begin{aligned}&\frac{\partial U_{X}}{\partial I_{X}}=0 \Rightarrow B_{X}'(I_{X}) - p'_{I_{X}}(I_{X},I_{Y}^{*}(I_{X})) D_{X} - \frac{\partial I_{Y}^{*}(I_{X})}{\partial I_{X}}p'_{I_{Y}}(I_{X},I_{Y}^{*}(I_{X}))D_{X} \nonumber \\&\quad \Rightarrow B_{X}'(I_{X}) + D_{X} \left[ - p'_{I_{X}}(I_{X},I_{Y}^{*}(I_{X})) - \frac{\partial I_{Y}^{*}(I_{X})}{\partial I_{X}}p'_{I_{Y}}(I_{X},I_{Y}^{*}(I_{X}))\right] = 0 \end{aligned}$$
(8)

Remind that \(B_{X}'(I_{X})\) is the marginal cost of investment of agent X, and \(- p'_{I_{X}}(I_{X},I_{Y}^{*}(I_{X}))D_{X}\) is her marginal benefit (in reducing the probability of causing harm and having to pay \(D_{X}\)). We can see that the expression: \(- \frac{\partial I_{Y}^{*}(I_{X})}{\partial I_{X}}p'_{I_{Y}}(I_{X},I_{Y}^{*}(I_{X}))\), reduces the marginal benefit from investing \(I_{X}\) (since \(\frac{\partial I_{Y}^{*}(I_{X})}{\partial I_{X}}<0\)). This is the Stackelberg effect: because of sequentiality and substitutability in investments, the agent X knows that if she reduces her level of investment, then the agent Y will have incentives to increase her own level of investment \(I_{Y}\), thus reducing the level of risk for both of them.

Given the specifications introduced above, we find:

$$\begin{aligned} \frac{\partial U_{X}}{\partial I_{X}}=0 \Rightarrow I_{X}^{*}=I_{X}(I_{Y}^{*}(I_{X}))=-\frac{\gamma }{\beta _{X}+\gamma } I_{Y}^{*}(I_{X}) + \frac{1}{\beta _{X}+\gamma } ln\left[ \frac{\gamma (1-\frac{\gamma }{\beta _{Y}+\gamma }) D_{X}}{\delta _{X}\beta _{X}}\right] \end{aligned}$$

with \(I_{Y}^{*}(I_{X})\) as given by (7), for a given level of \(I_{X}\). After replacing \(I_{Y}^{*}(I_{X})\) by (7) and rearranging terms we obtain:

$$\begin{aligned} I_{X}^{*}= & {} \left[ \frac{1}{1-\frac{\gamma ^{2}}{(\beta _{X}+\gamma )(\beta _{Y}+\gamma )}}\right] . \nonumber \\&\left[ \frac{1}{\beta _{X}+\gamma } ln\left[ \frac{\gamma (1-\frac{\gamma }{\beta _{Y}+\gamma })D_{X}}{\delta _{X}\beta _{X}}\right] - \frac{\gamma }{(\beta _{X}+\gamma )(\beta _{Y}+\gamma )} ln\left[ \frac{\gamma D_{Y}}{\delta _{Y}\beta _{Y}}\right] \right] \end{aligned}$$
(9)

and finally, after introducing the value of \(I_{X}^{*}\) [given by the expression (9)] in \(I_{Y}^{*}(I_{X})\) [given by (7)], we find the equilibrium level of \(I_{Y}\), \(I_{Y}^{*}\), such that:

$$\begin{aligned} I_{Y}^{*}= & {} \left[ \frac{1}{1-\frac{\gamma ^{2}}{(\beta _{X}+\gamma )(\beta _{Y}+\gamma )}}\right] . \nonumber \\&\left[ \frac{1}{\beta _{Y}+\gamma } ln\left[ \frac{\gamma D_{Y}}{\delta _{Y}\beta _{Y}}\right] - \frac{\gamma }{(\beta _{X}+\gamma )(\beta _{Y}+\gamma )} ln\left[ \frac{\gamma (1-\frac{\gamma }{\beta _{Y}+\gamma }) D_{X}}{\delta _{X}\beta _{X}}\right] \right] \end{aligned}$$
(10)

1.6.2 Comparison of private equilibria with social optima, and comparative statics on private equilibria

First, we provide a comparison of each private value with its corresponding social one.

By comparing (5) with (3), two remarks can be made:

  • \(\frac{1}{\beta _{Y}+\gamma } ln\left[ \frac{\gamma D_{Y}}{\delta _{Y}\beta _{Y}}\right] < \frac{1}{\beta _{Y}+\gamma } ln \left( \frac{\gamma H}{\delta _{Y}\beta _{Y}}\right)\) because \(D_{Y} < H\). Say this is effect 1.

  • \(- \frac{\gamma }{(\beta _{X}+\gamma )(\beta _{Y}+\gamma )} ln\left[ \frac{\gamma (1-\frac{\gamma }{\beta _{Y}+\gamma }) D_{X}}{\delta _{X}\beta _{X}}\right] > - \frac{\gamma }{(\beta _{X}+\gamma )(\beta _{Y}+\gamma )} ln\left( \frac{\gamma H}{\delta _{X}\beta _{X}}\right)\) because \(D_{X} < H\) and \((1-\frac{\gamma }{\beta _{Y}+\gamma }) < 1\). Say this is effect 2.

Effect 1 is the effect from insolvency (of agent Y): insolvency implies partial risk internalization, which leads to less incentives to invest in risk prevention measures. This tends to have: \(I_{Y}^{*}<I_{Y}^{**}\).

Effect 2 is the combination of two effects, which work together in the same way. \(D_{X} < H\) is the effect from agent X’s insolvency, and \((1-\frac{\gamma }{\beta _{Y}+\gamma }) < 1\) is the effect which comes from the fact that agent Y is a follower in a Stackelberg setting. More precisely, this latter effect comes from the fact that agent X takes into account agent Y’s best response to agent X’s decision, and this has then a consequence on agent Y’s decision. Both of these two effects work to provide agent Y with high incentives to invest. This tends to have \(I_{Y}^{*}>I_{Y}^{**}\).

All in all, from a theoretical viewpoint, the relative level of \(I_{Y}^{*}\) with respect of \(I_{Y}^{**}\) is undetermined.

By comparing (4) with (2), two remarks can be made:

  • \(- \frac{\gamma }{(\beta _{X}+\gamma )(\beta _{Y}+\gamma )} ln\left[ \frac{\gamma D_{Y}}{\delta _{Y}\beta _{Y}}\right] > - \frac{\gamma }{(\beta _{X}+\gamma )(\beta _{Y}+\gamma )} ln\left( \frac{\gamma H}{\delta _{Y}\beta _{Y}}\right)\) because \(D_{Y}<H\). Say this is effect 1.

  • \(\frac{1}{\beta _{X}+\gamma } ln\left[ \frac{\gamma (1-\frac{\gamma }{\beta _{Y}+\gamma })D_{X}}{\delta _{X}\beta _{X}}\right] < ln\left( \frac{\gamma H}{\delta _{X}\beta _{X}}\right) \frac{1}{\beta _{X}+\gamma }\) because \(D_{X} < H\) and \((1-\frac{\gamma }{\beta _{Y}+\gamma })<1\). Say this is effect 2.

Effect 1 is the effect from agent Y’s insolvency: knowing that the insolvency of agent Y provides her with less incentives to invest (because of partial risk internalization), agent X has thus incentives to provide a high investment in order to offset the effect of insolvency on Y. This tends to have: \(I_{X}^{*}>I_{X}^{**}\).

Effect 2 is the combination of two effects, which work together in the same way. \(D_{X} < H\) is the effect from agent X’s insolvency, and \((1-\frac{\gamma }{\beta _{Y}+\gamma }) < 1\) is the effect, on agent X, from being a leader in a Stackelberg setting. This latter effect comes from the fact that agent X anticipates agent Y’s reaction to agent X’s decision. Both of these two effects work to provide the agent X with low incentives to invest. This tends to have \(I_{X}^{*}<I_{X}^{**}\).

All in all, from a theoretical viewpoint, the relative level of \(I_{X}^{*}\) with respect of \(I_{X}^{**}\) is undetermined.

Note that the expression: \((1-\frac{\gamma }{\beta _{Y}+\gamma }) < 1\) in both the expressions of \(I_{X}^{*}\) and of \(I_{Y}^{*}\) highlights the Stackelberg effect, due to the sequential setting. Ceteris paribus, it provides the first-mover (Agent X) with less incentives to invest, and it provides the second-mover (Agent Y) with higher incentives to invest, in comparison with social optima \(I_{X}^{**}\) and of \(I_{Y}^{**}\). This is Prediction 1.

Then, we provide the following elements of comparative statics on private values.

By considering (10), the expression of \(I_{Y}^{*}\), and (9), the expression of \(I_{X}^{*}\), and recalling that \(D_{X} = \min \left\{ W_{X},(1-\lambda )H\right\}\) and \(D_{Y} = \min \left\{ W_{Y}, \lambda H\right\}\), we can easily check that:

$$\begin{aligned} \frac{\partial I_{Y}^{*}}{\partial W_{Y}}>0, \frac{\partial I_{Y}^{*}}{\partial W_{X}}<0, \frac{\partial I_{Y}^{*}}{\partial \lambda }>0 \end{aligned}$$

and

$$\begin{aligned} \frac{\partial I_{X}^{*}}{\partial W_{X}}>0, \frac{\partial I_{X}^{*}}{\partial W_{X}}<0, \frac{\partial I_{X}^{*}}{\partial \lambda }<0 \end{aligned}$$

when \(W_{X}<(1-\lambda )H\) and when \(W_{Y}<\lambda H\).

From these derivatives, we can state that: (i) the lower the solvency of one agent, the lower her investment; (ii) the lower the solvency of one agent, the higher the investment of the other agent ((i) and (ii) are Prediction 3); (iii) increasing (resp. decreasing) the share of liability apportioned to one agent increases (resp. decreases) her incentives to invest (Prediction 2).

1.7 Appendix C: Statistical tables

See Table 10.

Table 10 Overview of participants’ decisions, by matching groups

Table 11 reports statistics for a measure of relative social welfare \(SW_{rel}\) calculated as:

$$\begin{aligned} SW_{rel} = (SW_{max}-SW)/(SW_{max}-SW_{min}) \end{aligned}$$

with SW the social welfare as expressed in equation (1), \(SW_{max}\) the highest social welfare achievable in the treatment and \(SW_{min}\) the lowest social welfare achievable in the treatment.

Table 11 Measure of relative social welfare over treatments

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Jacob, J., Lambert, EA. & Peterle, E. Several liability with sequential care: an experiment. Eur J Law Econ 54, 283–326 (2022). https://doi.org/10.1007/s10657-022-09740-x

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