Other-regarding preferences and giving decision in a risky environment: experimental evidence

Abstract

We investigate whether and how an individual giving decision is affected in risky environments in which the recipient’s wealth is random. We demonstrate that, under risk neutrality, the donation of dictators with a purely ex post view of fairness should, in general, be affected by the riskiness of the recipient’s payoff, while dictators with a purely ex ante view should not be. Furthermore, we observe that some influential inequality aversion preferences functions yield opposite predictions when we consider ex post view of fairness. Hence, we report on dictator games laboratory experiments in which the recipient’s wealth is exposed to an actuarially neutral and additive background risk. Our experimental data show no statistically significant impact of the recipient’s risk exposure on dictators’ giving decisions. This result appears robust to both the experimental design (within subjects or between subjects) and the origin of the recipient’s risk exposure (chosen by the recipient or imposed on the recipient). Although we cannot sharply validate or invalidate alternative fairness theories, the whole pattern of our experimental data can be simply explained by assuming ex ante view of fairness and risk neutrality.

Appendix

1.1 Proof of Proposition 2

We first consider Fehr and Schmidt (1999)’s preferences function. Substituting $d(g)=15-g$ and $r(g)=5+g-\pi$ in (5), the dictator’s preferences function is

$$\begin{aligned} u_{3}\left( g\right) =15-g-\alpha \max \left\{ 2g-10-\pi ,0\right\} -\beta \max \left\{ 10-2g+\pi ,0\right\} \end{aligned}$$

(8)

or, equivalently,

$$\begin{aligned} u_{3}\left( g\right) =\left\{ \begin{array}{l} \left[ -2\alpha -1\right] g+10\alpha +15+\alpha \pi \quad \text {for}\quad g\ge 5+\frac{1}{2}\pi \quad \\ \left[ 2\beta -1\right] g-10\beta +15-\beta \pi \quad \text {for}\quad g\le 5+\frac{1}{2}\pi \end{array} \right. \end{aligned}$$

(9)

where $\alpha \ge \beta$ and $0\le \beta <1$. Observe that $u_{3}^{\prime }=u_{1}^{\prime }$. Therefore, if (i) $\frac{1}{2}<\beta <1$, $u_{3}$ is first increasing and then decreasing above the kink point at $5+\frac{1}{2} \pi$, where $u_{3}(5+\frac{1}{2}\pi )=10-\frac{1}{2}\pi$. The optimal donations is $g_{3}^{*}=5+\frac{1}{2}\pi$. If (ii) $\beta =\frac{1}{2}$ , $u_{3}$ is first flat, then decreasing above the kink point. The optimal donation is between 0 and $5+\frac{1}{2}\pi$ in this very special case. Let us assume that the optimal donation is $g_{1}^{*}=5+\frac{1}{2}\pi$ . If (iii) $0\le \beta <\frac{1}{2}$, $u_{3}$ is everywhere decreasing. The optimal donations is $g_{3}^{*}=0$.

Under risk aversion ($\pi >0$), we have shown that $g_{1}^{*}=5<5+\frac{1 }{2}\pi =g_{3}^{*}$ in case (i), $g_{1}^{*}=5<5+\frac{1}{2}\pi =g_{3}^{*}$ in case (ii), and $g_{1}^{*}=g_{3}^{*}=0$ in case (iii). Therefore we have shown that $g_{1}^{*}\le g_{3}^{*}$ everywhere, that is, for any parameter value of the Fehr and Schmidt (1999)’s preferences function. The opposite result obviously holds under risk seeking ($\pi <0$).

We now consider Bolton and Ockenfels (2000)’s preferences function. Substituting $d(g)=15-g$ and $r(g)=5+g-\pi$ in (6), the dictator’s preferences function is

$$\begin{aligned} v_{3}\left( g\right) =a\left[ 15-g\right] -\frac{1}{2}b\left[ \frac{5+\frac{1 }{2}\pi }{20-\pi }-\frac{g}{20-\pi }\right] ^{2} \end{aligned}$$

(10)

where $a\ge 0$ and $b>0$. The Kuhn-Tucker first order conditions are:

$$\begin{aligned} v_{3}^{\prime }\left( g_{3}^{*}\right) =\frac{\left[ 5+\frac{1}{2}\pi \right] }{\left[ 20-\pi \right] ^{2}}b-a-\frac{1}{\left[ 20-\pi \right] ^{2}} bg_{3}^{*}\left\{ \begin{array}{l} \le 0\quad \text {for}\quad g_{3}^{*}<10 \\ \ge 0\quad \text {for}\quad g_{3}^{*}>0 \end{array} \right. \end{aligned}$$

(11)

Because $v_{3}^{\prime }\left( 10\right) \ge 0$ is equivalent to $a\le \frac{\frac{1}{2}\pi -5}{\left[ 20-\pi \right] ^{2}}b$, where $\pi <5$, we conclude that $g_{1}=10$ cannot be an optimal choice. On the other hand, we could have $v_{3}^{\prime }\left( 0\right) \le 0$, which is equivalent to $\frac{a}{b}\ge \frac{5+\frac{1}{2}\pi }{\left[ 20-\pi \right] ^{2}}$. The optimal donation is

$$\begin{aligned} g_{3}^{*}=\left\{ \begin{array}{l} 0\quad \text {for}\quad \frac{a}{b}\ge \frac{5+\frac{1}{2}\pi }{\left[ 20-\pi \right] ^{2}} \\ 5+\frac{1}{2}\pi -\frac{a}{b}\left[ 20-\pi \right] ^{2}>0\quad \text {for} \quad 0\le \frac{a}{b}<\frac{5+\frac{1}{2}\pi }{\left[ 20-\pi \right] ^{2}} \end{array} \right. \end{aligned}$$

(12)

We observe that $g_{3}^{*}\le 5+\frac{1}{2}\pi$, and that the treshold $\frac{5+\frac{1}{2}\pi }{\left[ 20-\pi \right] ^{2}}$ increases with $\pi$.

Under risk aversion, with $\pi \in \left( 0,5\right)$, we have shown that: (i) $g_{1}^{*}=g_{3}^{*}=0$ if $\frac{a}{b}\ge \frac{5+\frac{1}{2} \pi }{\left[ 20-\pi \right] ^{2}}$, (ii) $g_{1}^{*}=0<g_{3}^{*}$ if $\frac{1}{80}\le \frac{a}{b}<\frac{5+\frac{1}{2}\pi }{\left[ 20-\pi \right] ^{2}}$, and (iii) $g_{1}^{*}<g_{3}^{*}$ if $0\le \frac{a}{b}<\frac{ 1}{80}$. Therefore we have shown that $g_{1}^{*}\le g_{3}^{*}$ everywhere, that is, for any parameter value of the Bolton and Ockenfels (2000)’s preferences function. The opposite result obviously holds under risk seeking , with $\pi \in \left( -5,0\right)$. This concludes the proof.

1.2 Proof of proposition 3

We first consider the case where the recipient’s endowment is certain, as in T1. Substituting $d(g)=15-g$ and $r(g)=5+g$ in (4), the dictator’s preferences function is

$$\begin{aligned} u_{1}\left( g\right) =15-g-\alpha \max \left\{ 2g-10,0\right\} -\beta \max \left\{ 10-2g,0\right\} \end{aligned}$$

(13)

or, equivalently,

$$\begin{aligned} u_{1}\left( g\right) =\left\{ \begin{array}{l} \left[ -2\alpha -1\right] g+10\alpha +15\quad \text {for}\quad g\ge 5 \\ \left[ 2\beta -1\right] g-10\beta +15\quad \text {for}\quad g\le 5 \end{array} \right. \end{aligned}$$

(14)

where $\alpha \ge \beta$ and $0\le \beta <1$. Observe that $u_{1}^{\prime }=2\beta -1$ for $g<5$ and $u_{1}^{\prime }=-2\alpha -1$ for $g>5$. Therefore, if (i) $\frac{1}{2}<\beta <1$, $u_{1}$ is first increasing, then decreasing above a kink point at $g=5$ (where $u_{1}(5)=10$). The optimal donation is $g_{1}^{*}=5$. If (ii) $\beta =\frac{1}{2}$, $u_{1}$ is first flat, then decreasing above the kink point at $g=5$. The optimal donation is between 0 and 5 in this very special case. Let us assume that the optimal donation is $g_{1}^{*}=5$. If (iii) $0\le \beta <\frac{ 1}{2}$, $u_{1}$ is everywhere decreasing. The optimal donation is $g_{1}^{*}=0$.

Now suppose that the recipient’s endowment is risky, as in T2. Substituting $d(g)=15-g$ and ${\widetilde{r}}(g)=(10+g,\frac{1}{2};g,\frac{1}{2})$ in (5), the dictator’s preferences function is

$$\begin{aligned} u_{2}\left( g\right) =15-g-\frac{1}{2}\alpha \left[ \begin{array}{c} \max \left\{ 2g-5,0\right\} \\ +\max \left\{ 2g-15,0\right\} \end{array} \right] -\frac{1}{2}\beta \left[ \begin{array}{c} \max \left\{ 5-2g,0\right\} \\ +\max \left\{ 15-2g,0\right\} \end{array} \right] \end{aligned}$$

(15)

or, equivalently,

$$\begin{aligned} u_{2}\left( g\right) =\left\{ \begin{array}{l} \left[ -2\alpha -1\right] g+10\alpha +15\quad \text {for}\quad g\ge \frac{15 }{2} \\ \left[ -\alpha +\beta -1\right] g+\frac{5}{2}\alpha -\frac{15}{2}\beta +15\quad \text {for}\quad \frac{5}{2}\le g\le \frac{15}{2} \\ \left[ 2\beta -1\right] g-10\beta +15\quad \text {for}\quad g\le \frac{5}{2} \end{array} \right. \end{aligned}$$

(16)

where $\alpha \ge \beta$ and $0\le \beta <1$. Observe that $u_{2}^{\prime }=2\beta -1$ for $g<\frac{5}{2}$, $u_{2}^{\prime }=-\alpha +\beta -1$ for $\frac{5}{2}<g<\frac{15}{2}$ and $u_{2}^{\prime }=-2\alpha -1$ for $g>\frac{15 }{2}$. Therefore, if (i) $\frac{1}{2}<\beta <1$, $u_{2}$ is first increasing and then decreasing above the kink point at $g=\frac{5}{2}$ (where $u_{2}\left( \frac{5}{2}\right) =-5\beta +\frac{25}{2}$). The optimal donation is $g_{2}^{*}=2$, with $u_{2}\left( 2\right) =-6\beta +13>- \frac{1}{2}\alpha -\frac{9}{2}\beta +12=u_{2}\left( 3\right)$). If (ii) $\beta =\frac{1}{2}$, $u_{2}$ is first flat and then decreases above the kink point. The optimal donation is between 0 and 2 in this very special case. Let us assume that the optimal donation is $g_{1}^{*}=2$. If (iii) $0\le \beta <\frac{1}{2}$, $u_{2}$ is everywhere decreasing. The optimal donation is $g_{2}^{*}=0$.

We have shown that $g_{1}^{*}=5>2=g_{2}^{*}$ in case (i), $g_{1}^{*}=5>2=g_{2}^{*}$ in case (ii), and $g_{1}^{*}=g_{2}^{*}=0$ in case (iii). Therefore we have shown that $g_{1}^{*}\ge g_{2}^{*}$ everywhere, that is, for any parameter value of the Fehr and Schmidt (1999)’s preferences function. This concludes the proof.

1.3 Proof of Proposition 4

We first consider the case where the recipient’s endowment is certain, as in T1. Substituting $d(g)=15-g$ and $r(g)=5+g$ in (6), the dictator’s preferences function is

$$\begin{aligned} v_{1}\left( g\right) =a\left[ 15-g\right] -\frac{1}{2}b\left[ \frac{1}{4}- \frac{g}{20}\right] ^{2} \end{aligned}$$

(17)

where $a\ge 0$ and $b>0$. The Kuhn and Tucker’s first-order conditions are

$$\begin{aligned} v_{1}^{\prime }\left( g_{1}^{*}\right) =\frac{1}{80}b-a-\frac{1}{400} bg_{1}^{*}\left\{ \begin{array}{l} \le 0\quad \text {for}\quad g_{1}^{*}<10 \\ \ge 0\quad \text {for}\quad g_{1}^{*}>0 \end{array} \right. \end{aligned}$$

(18)

Because $v_{1}^{\prime }\left( 10\right) \ge 0$ is equivalent to $80a\le -b$, we conclude that $g_{1}=10$ cannot be an optimal choice. On the other hand we could have $v_{1}^{\prime }\left( 0\right) \le 0$, which is equivalent to $b\le 80a$. The optimal donation is

$$\begin{aligned} g_{1}^{*}=\left\{ \begin{array}{l} 0\quad \text {for}\quad \frac{a}{b}\ge \frac{1}{80} \\ 5-400\frac{a}{b}>0\quad \text {for}\quad 0\le \frac{a}{b}<\frac{1}{80} \end{array} \right. \end{aligned}$$

(19)

We observe that $g_{1}^{*}\le 5$.

Now suppose that the recipient’s endowment is risky, as in T2. Substituting $d(g)=15-g$ and ${\widetilde{r}}(g)=(10+g,\frac{1}{2};g,\frac{1}{2})$ in (7), the dictator’s preferences function is

$$\begin{aligned} v_{2}\left( g\right) =a\left[ 15-g\right] -\frac{1}{4}b\left[ \left[ \frac{ 15-g}{25}-\frac{1}{2}\right] ^{2}+\left[ \frac{15-g}{15}-\frac{1}{2}\right] ^{2}\right] \end{aligned}$$

(20)

The Kuhn and Tucker’s first-order conditions are

$$\begin{aligned} v_{2}^{\prime }\left( g_{2}^{*}\right) =\frac{7}{375}b-a-\frac{17}{5625} bg_{2}^{*}\left\{ \begin{array}{l} \le 0\quad \text {for}\quad g_{2}^{*}<10 \\ \ge 0\quad \text {for}\quad g_{2}^{*}>0 \end{array} \right. \end{aligned}$$

(21)

Because $v_{2}^{\prime }\left( 10\right) \ge 0\$is equivalent to $1125a\le -13b$, we conclude that $g_{2}^{*}=10$ cannot be an optimal choice. On the other hand we could have $v_{2}^{\prime }\left( 0\right) \le 0$, which is equivalent to $7b\le 375a$. The optimal donation is

$$\begin{aligned} g_{2}^{*}=\left\{ \begin{array}{l} 0\quad \text {for}\quad \frac{a}{b}\ge \frac{7}{375} \\ \frac{105}{17}-\frac{5625}{17}\frac{a}{b}>0\quad \text {for}\quad 0\le \frac{ a}{b}<\frac{7}{375} \end{array} \right. \end{aligned}$$

(22)

We observe that $g_{2}^{*}\le \frac{105}{17}\simeq 6,2$.

We have shown that: (i) $g_{1}^{*}=g_{2}^{*}=0$ if $\frac{a}{b}\ge \frac{7}{375}$, (ii) $g_{1}^{*}=0<g_{2}^{*}$ if $\frac{1}{80}\le \frac{a}{b}<\frac{7}{375}$, and (iii) $g_{1}^{*}<g_{2}^{*}$ if $0\le \frac{a}{b}<\frac{1}{80}$. Therefore we have shown that $g_{1}^{*}\le g_{2}^{*}$ everywhere, that is, for any parameter value of the Bolton and Ockenfels (2000)’s preferences function. This concludes the proof.

1.4 Between-subject design: instructions for the benchmark riskless treatment T1

This is a translation of the original French version.

You are about to participate in an experiment to study decision making. Please carefully read the instructions, they should help you understand the experiment. All your decisions are anonymous. You will enter your choices in the computer in front of you.

The present experiment consists of two parts: “Part 1” and “Part 2”. The instructions for Part 1 are included hereafter. The instructions for Part 2 will be distributed once everybody has completed Part 1. At the end of the experiment, one of the two parts will be randomly drawn for real pay. You will then be paid in cash your earnings in euros.

During the experiment, you are not allowed to communicate. If you have questions, then please raise your hand, and one experimenter will come to you and answer your question in private.

In this experiment, there were two types of subjects (in equal number): player A and player B. You are randomly assigned a type for the entire experiment. It will be revealed privately before starting the experiment. You will be randomly paired to a player of another type than yours such that one player A is matched with one player B.

Part 1

At the beginning of the game, each player, whatever his/her type, receives an endowment of 5 euros.

Additionally, players A have to share 10 euros between them, and the players B they are paired with. Players A are free to send player B any amount between 0 and 10. The only compulsory limitation is that the amounts are integers (0, 1, 2, etc.). Playersï¿$\frac{1}{2}$ B have no decision to make.

The earnings for each type of player are as follows:

• Player A’s earnings: 5 euros + (10 euros - the amount sent to player B).

• Player B’s earnings: 5 euros + the amount received from player A.

Example 1: If player A sent 3 euros to the player B.

• Player A’s earnings: 5 + (10 - 3) = 12 euros.

• Player B’s earnings: 5 + 3 = 8 euros.

Example 2: If player A sent 5 euros to the player B.

• Player A’s earnings: 5 + (10 - 5) = 10 euros.

• Player B’s earnings: 5 + 5 = 10 euros.

Example 3: If player A sent 7 euros to the player B.

• Player A’s earnings: 5 + (10 - 7) = 8 euros.

• Player B’s earnings: 5 + 7 = 12 euros.

Part 2 (given at the end of Part 1, identical for all treatments)

Part 2 is independent from Part 1.

In this part of the experiment, you receive an endowment of 10 euros. You must decide which part of this amount you wish to invest in a risky option. You can choose any amount (in integer only) between 0 and 10.

The risky option consists of a coin toss. The risky option will bring you 3 times the invested amount if heads are drawn and 0 if tails are drawn.

Your final earnings are equal to the amount kept + the earnings of your investment.

Examples:

• 1. If you decide to invest 5 euros, your earnings will be 20 euros if Heads is drawn (5 euros + 3*5 euros invested in the risky option) and 5 euros if tails are drawn (5 euros + 0*5 euros from your investment in the risky option).

• 2. If you decide to invest 0 euro, your final earnings will be 10 euros whatever the result of the coin toss.

• 3. If you decide to invest 10 euros. Your final earnings will be 30 euros if Heads is drawn (0 euro + 3*10 euros invested in the risky option) and 0 euro if Tails is drawn (0 euro kept + 0*10 euros from your investment in the risky option).

To avoid calculations, the table below displays the earnings according to the amount invested in the risky option and the coin toss result.

Investment	Earnings
	If Heads	If Tails
0	10	10
1	12	9
2	14	8
3	16	7
4	18	6
5	20	5
6	22	4
7	24	3
8	26	2
9	28	1
10	30	0

1.5 Between-subject design: instructions of Part 1 for the risky treatment T2

Part 1

At the beginning of the game, player A receives an endowment of 5 euros.

The endowment of player B is risky and depends on coin toss result. If the result of the coin toss is Tails, player B has an endowment of 0 euro. If the result is Heads, player B receives an endowment of 10 euros. The coin toss result will only be known at the end of the experiment.

Additionally, players A have to share 10 euros between them, and the players B they are paired with. The players A are free to send player B any amount between 0 and 10. The only compulsory limitation is that the amounts are integers (0, 1, 2, etc.). Players B have no decision to make.

The earnings for each type of player are as follows:

• Player A’s earnings: 5 euros + (10 euros - the amount sent to player B).

• Player B’s earnings depend on the result of the coin toss:

  • If Tails, player B’s earnings: 0 euros + the amount received from the player A.

  • If Heads, player B’s earnings: 10 euros + the amount received from the player A.

Example 1: If player A sent 3 euros to the player B:

• Player A’s earnings: 5 + (10 - 3) = 12 euros.

• Player B’s earnings:

  • If Tails, player B’s earnings: 0 + 3 = 3 euros.

  • If Heads, player B’s earnings: 10 + 3 = 13 euros.

Example 2: If player A sent 5 euros to player B:

• Player A’s earnings: 5 + (10 - 5) = 10 euros.

• Player B’s earnings:

  • If Tails, player B’s earnings: 0 + 5 = 5 euros.

  • If Heads, player B’s earnings: 10 + 5 = 15 euros.

Example 3: If player A sent 7 euros to player B:

• Player A’s earnings: 5 + (10 - 7) = 8 euros.

• Player B’s earnings:

  • If Tails, player B’s earnings: 0 + 7 = 7 euros.

  • If Heads, player B’s earnings: 10 + 7 = 17 euros.

1.6 Between-subject design: instructions of part 1 for the choice treatment T3

Part 1

At the beginning of the game, player A receives an endowment of 5 euros.

Player B must make a choice between two alternatives:

• Choice 1: an endowment of 5 euros

• Choice 2: a risky endowment that depends on the result of a coin toss. If the result of the coin toss is Tails, player B has an endowment of 0 euro. If the result is Heads, player B receives an endowment of 10 euros. The coin toss results will only be known at the end of the experiment.

Player A and player B are informed about the choice (1 or 2) made by player B.

Additionally, players A have to share 10 euros between them, and the players B they are paired with. The players A are free to send player B any amount between 0 and 10. The only compulsory limitation is that the amounts are integers (0, 1, 2, etc. ).

The earnings for each type of player are as follows:

• Player A’s earnings: 5 euros + (10 euros - the amount sent to player B).

• Player B’s earnings depend on the choice:

  • If player B made choice 1, player B’s earnings: 5 euros + the amount received from the player A.

  • If player B made choice 2, player B’s earnings depend on the result of the coin toss:

    • *If Tails, player B’s earnings: 0 euros + the amount received from the player A.

    • *If Heads, player B’s earnings: 10 euros + the amount received from the player A.

Example 1: If player A sent 3 euros to the player B:

• Player A’s earnings: 5 + (10 - 3) = 12 euros.

• Player B’s earnings depend on the choice:

  • If player B made the choice 1: 5 + 3 = 8 euros.

  • If player B made choice 2, player B’s earnings depend on the result of the coin toss:

    • *If Tails, player B’s earnings: 0 + 3 = 3 euros.

    • *If Heads, player B’s earnings: 10 + 3 = 13 euros.

Example 2: If player A sent 5 euros to player B:

• Player A’s earnings: 5 + (10 - 5) = 10 euros.

• Player B’s earnings depend on the choice:

  • If player B made the choice 1: 5 + 5 = 10 euros.

  • If player B made choice 2, player B’s earnings depend on the result of the coin toss:

    • *If Tails, player B’s earnings: 0 + 5 = 5 euros.

    • *If Heads, player B’s earnings: 10 + 5 = 15 euros.

Example 3: If player A sent 7 euros to player B:

• Player A’s earnings: 5 + (10 - 7) = 8 euros.

• Player B’s earnings depend on the choice:

  • If player B made the choice 1: 5 + 7 = 12 euros

  • If player B made choice 2, player B’s earnings depend on the result of the coin toss:

    • *If Tails, player B’s earnings: 0 + 7 = 7 euros.

    • *If Heads, player B’s earnings: 10 + 7 = 17 euros.

1.7 Within-subject design: instructions for $T_{1-2}$

You are about to participate in an experiment to study decision making. Please carefully read the instructions, they should help you understand the experiment. All your decisions are anonymous. You will enter your choices on the computer in front of you.

The present experiment consists of three parts: “Part 1” “Part 2” and “Part 3”. The instructions for Part 1 are included hereafter. The instructions for Part 2 and Part 3 will be distributed once everybody has completed the previous part. At the end of the experiment, one of the three parts will be randomly drawn for real pay. You will then be paid in cash your earnings in euros.

During the experiment, you are not allowed to communicate. If you have questions, then please raise your hand, and an experimenter will come to you and answer your question in private.

In this experiment, there are two types of subjects (in equal number): player A and player B. You are randomly assigned a type for the entire experiment. It will be revealed privately before starting the experiment.

Part 1

At the beginning of the experiment, you are randomly paired with a player of type other than yours such that one player A is matched with one player B.

Each player, whatever his/her type, receives an endowment of 5 euros.

Additionally, players A have to share 10 euros between them, and the players B they are paired with. The players A are free to send player B any amount between 0 and 10. The only compulsory limitation is that the amounts are integers (0, 1, 2, etc. ).

Players B have an independent decision to take. They must give their opinion on the amount shared by player A of their pair. This prediction has no impact on their payment.

The earnings for each type of player are as follows:

• Player A’s earnings: 5 euros + (10 euros - the amount sent to player B).

• Player B’s earnings: 5 euros + the amount received from player A.

Example 1 : If player A sent 3 euros to the player B.

• Player A’s earnings: 5 + (10 - 3) = 12 euros.

• Player B’s earnings: 5 + 3 = 8 euros.

Example 2 : If player A sent 5 euros to the player B.

• Player A’s earnings: 5 + (10 - 5) = 10 euros.

• Player B’s earnings: 5 + 5 = 10 euros.

Example 3 : If player A sent 7 euros to the player B.

• Player A’s earnings: 5 + (10 - 7) = 8 euros.

• Player B’s earnings: 5 + 7 = 12 euros.

Part 2 (given at the end of Part 1)

Part 2 is independent of Part 1. At the beginning of this part, you are randomly paired to a player of another type than yours such that one player A is matched with one player B. The pair is different than the one you belonged to in Part 1.

Player A receives an endowment of 5 euros.

The endowment of player B is risky and depends on coin toss result. If the result of the coin toss is Tails, player B has an endowment of 0 euro. If the result is Heads, player B receives an endowment of 10 euros. The coin toss result will only be known at the end of the experiment.

Additionally, players A have to share 10 euros between them, and the players B they are paired with. The players A are free to send player B any amount between 0 and 10. The only compulsory limitation is that the amounts are integers (0, 1, 2, etc. ).

Players B have an additional independent decision to make. They must give their opinion on the amount shared by player A of their pair. This prediction has no impact on their payment.

The earnings for each type of player are as follows:

• Player A’s earnings: 5 euros + (10 euros - the amount sent to player B).

• Player B’s earnings depend on the result of the coin toss:

  • If Tails, player B’s earnings: 0 euros + the amount received from the player A.

  • If Heads, player B’s earnings: 10 euros + the amount received from the player A.

Example 1: If player A sent 3 euros to the player B:

• Player A’s earnings: 5 + (10 - 3) = 12 euros.

• Player B’s earnings:

  • If Tails, player B’s earnings: 0 + 3 = 3 euros.

  • If Heads, player B’s earnings: 10 + 3 = 13 euros.

Example 2: If player A sent 5 euros to player B:

• Player A’s earnings: 5 + (10 - 5) = 10 euros.

• Player B’s earnings:

  • If Tails, player B’s earnings: 0 + 5 = 5 euros.

  • If Heads, player B’s earnings: 10 + 5 = 15 euros.

Example 3: If player A sent 7 euros to player B:

• Player A’s earnings: 5 + (10 - 7) = 8 euros.

• Player B’s earnings:

  • If Tails, player B’s earnings: 0 + 7 = 7 euros.

  • If Heads, player B’s earnings: 10 + 7 = 17 euros.

Part 3 (given at the end of Part 2)

Part 3 is independent of parts 1 and 2.

In this part of the experiment, you receive an endowment of 10 euros. You must decide which part of this amount you wish to invest in a risky option. You can choose any amount (an integer only) between 0 and 10.

The risky option consists of a coin toss. The risky option will bring you 3 times the invested amount if heads are drawn and 0 if tails are drawn.

Your final earnings are equal to the amount kept + the earnings of your investment. Examples:

• 1. If you decide to invest 5 euros, your earnings will be 20 euros if Heads is drawn (5 euros + 3*5 euros invested in the risky option) and 5 euros if tails are drawn (5 euros + 0*5 euros from your investment in the risky option).

• 2. If you decide to invest 0 euro, your final earnings will be 10 euros whatever the result of the coin toss.

• 3. If you decide to invest 10 euros. Your final earnings will be 30 euros if Heads is drawn (0 euro + 3*10 euros invested in the risky option) and 0 euro if Tails is drawn (0 euro kept + 0*10 euros from your investment in the risky option).

To avoid calculations, the table below displays the earnings according to the amount invested in the risky option and the the coin toss result.

Investment	Earnings
	If Heads	If Tails
0	10	10
1	12	9
2	14	8
3	16	7
4	18	6
5	20	5
6	22	4
7	24	3
8	26	2
9	28	1
10	30	0

1.8 Within-subject design: instructions for $T_{2-1}$

The instructions are the same as for $T_{1-2}$ except for the order of Part 1 and Part 2.