The effect of the background risk in a simple chance improving decision model

Abstract

We experimentally investigate the effect of an independent and exogenous background risk to initial wealth on subjects’ risk attitudes and explore an appropriate incentive mechanism when identical or similar tasks are repeated in an experiment. Taking a simple chance improving decision model under risk where the winning probabilities are negatively related to the potential gain, we find that such a background risk tends to make risk-averse subjects behave more risk aversely. Furthermore, we find that risk-averse subjects tend to show decreasing absolute risk aversion (DARA), and that a random round payoff mechanism (RRPM) would control the possible wealth effect. This suggests that RRPM would be a better incentive mechanism for an experiment where repetition of a task is used.

Appendix

1.1 Appendix A: Proofs of propositions

(Proof of Proposition 1)

First we show that x _N  = M/2. Define $D{\left( x \right)} = u{\left( {w_{0} + M - x} \right)} - u{\left( {w_{0} } \right)} - x \cdot u\prime {\left( {w_{0} + M - x} \right)}$. For a risk-neutral person, ${{\left( {u{\left( {w_{0} + M - x} \right)} - u{\left( {w_{0} } \right)}} \right)}} \mathord{\left/ {\vphantom {{{\left( {u{\left( {w_{0} + M - x} \right)} - u{\left( {w_{0} } \right)}} \right)}} {{\left( {M - x} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {M - x} \right)}} = u\prime {\left( {w_{0} + M - x} \right)}$. Hence, $D_{N} {\left( {x_{N} } \right)} = u{\left( {w_{0} + M - x_{N} } \right)} - u{\left( {w_{0} } \right)} - x_{N} \cdot {{\left( {u{\left( {w_{0} + M - x_{N} } \right)} - u{\left( {w_{0} } \right)}} \right)}} \mathord{\left/ {\vphantom {{{\left( {u{\left( {w_{0} + M - x_{N} } \right)} - u{\left( {w_{0} } \right)}} \right)}} {{\left( {M - x_{N} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {M - x_{N} } \right)}} = 0$. Solving this gives x _N  = M/2. Now we show that x _A  > M/2. Since x _N  = M/2, we need to check D _A (x _N ). Note that for a risk-averse person, ${{\left( {u{\left( {w_{0} + M - x} \right)} - u{\left( {w_{0} } \right)}} \right)}} \mathord{\left/ {\vphantom {{{\left( {u{\left( {w_{0} + M - x} \right)} - u{\left( {w_{0} } \right)}} \right)}} {{\left( {M - x} \right)} > u\prime {\left( {w_{0} + M - x} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {M - x} \right)} > u\prime {\left( {w_{0} + M - x} \right)}}$. Hence, $D_{A} {\left( {x_{N} } \right)} - D_{N} {\left( {x_{N} } \right)} = D_{A} {\left( {M \mathord{\left/ {\vphantom {M 2}} \right. \kern-\nulldelimiterspace} 2} \right)} = u{\left( {w_{0} + M \mathord{\left/ {\vphantom {M 2}} \right. \kern-\nulldelimiterspace} 2} \right)} - u{\left( {w_{0} } \right)} - M \mathord{\left/ {\vphantom {M 2}} \right. \kern-\nulldelimiterspace} 2 \cdot u\prime {\left( {w_{0} + M \mathord{\left/ {\vphantom {M 2}} \right. \kern-\nulldelimiterspace} 2} \right)} > 0$. But, dD _A /dx < 0 by SOC. Hence, To satisfy D _A (x _A ) = 0, x _A should be higher than x _N  = M/2. Thus, x _A  > M/2. (Q.E.D.)

(Proof of Proposition 2)

Define v = k(u(.)) where k is a concave transformation of u(.), i.e., k(.) > 0, k′(.) > 0, and k″(.) < 0. Hence, k(u)/u > k′(u) ⬄ k(u) > u·k′(u). Then, v is more risk averse than u at every initial wealth level. If we show that v gives up more than u does, we are done. Solving v’s optimization problem generates FOC as following: $H{\left( x \right)} = k{\left( {u{\left( {w_{0} + M - x} \right)}} \right)} - x \cdot k\prime {\left( {u{\left( {w_{0} + M - x} \right)}} \right)} \cdot u\prime {\left( {w_{0} + M - x} \right)} - k{\left( {u{\left( {w_{0} } \right)}} \right)} = 0$. Suppose x = x _A which was the solution of D _A (x _A ) = 0 which implies $x_{A} \cdot u\prime {\left( {w_{0} + M - x_{A} } \right)} = u{\left( {w_{0} + M - x_{A} } \right)} - u{\left( {w_{0} } \right)}$. Denote $u{\left( {w_{0} + M - x_{A} } \right)} = u_{A}$, u(w ₀ ) = u ₀ and H(x _A ) = H _A where u _A  > u ₀  > 0. Then, by a simple calculation, we can show $H_{A} = k{\left( {u_{A} } \right)} - k{\left( {u_{0} } \right)} - {\left( {u_{A} - u_{0} } \right)}k\prime {\left( {u_{A} } \right)}$. Note that since k′(u) > 0, k″(u) < 0, and u _A  > u ₀  > 0, ${{\left( {k{\left( {u_{A} } \right)} - k{\left( {u_{0} } \right)}} \right)}} \mathord{\left/ {\vphantom {{{\left( {k{\left( {u_{A} } \right)} - k{\left( {u_{0} } \right)}} \right)}} {{\left( {u_{A} - u_{0} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {u_{A} - u_{0} } \right)}} > k\prime {\left( {u_{A} } \right)}$. That is, $k{\left( {u_{A} } \right)} - k{\left( {u_{0} } \right)} - {\left( {u_{A} - u_{0} } \right)}k\prime {\left( {u_{A} } \right)} > 0$. Thus, H _A  > 0 while D _A  = 0. Since it is easily seen that dH _A /dx < 0, we find that x _A ′ which satisfies H _A  = 0 should be higher than x _A . Thus, x _A ′ > x _A . In other words, a more risk-averse person v chooses a higher x than u does. (Q.E.D.)

(Proof of Proposition 3)

By totally differentiating FOC, we get ${dx_{A} } \mathord{\left/ {\vphantom {{dx_{A} } {dw_{0} }}} \right. \kern-\nulldelimiterspace} {dw_{0} } = {{\left( {u\prime {\left( {w_{0} + M - x_{A} } \right)} - u\prime {\left( {w_{0} } \right)} - x_{A} \cdot u\prime \prime {\left( {w_{0} + M - x_{A} } \right)}} \right)}} \mathord{\left/ {\vphantom {{{\left( {u\prime {\left( {w_{0} + M - x_{A} } \right)} - u\prime {\left( {w_{0} } \right)} - x_{A} \cdot u\prime \prime {\left( {w_{0} + M - x_{A} } \right)}} \right)}} {{\left( {2u\prime {\left( {w_{0} + M - x_{A} } \right)} - x_{A} \,u\prime \prime {\left( {w_{0} + M - x_{A} } \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {2u\prime {\left( {w_{0} + M - x_{A} } \right)} - x_{A} \,u\prime \prime {\left( {w_{0} + M - x_{A} } \right)}} \right)}}$. Since u′ > 0, u″ < 0, $\operatorname{sgn} {\left( {{dx_{A} } \mathord{\left/ {\vphantom {{dx_{A} } {dw_{0} }}} \right. \kern-\nulldelimiterspace} {dw_{0} }} \right)} = \operatorname{sgn} {\left( {u\prime {\left( {w_{0} + M - x_{A} } \right)} - u\prime {\left( {w_{0} } \right)} - x_{A} \cdot \,u\prime \prime {\left( {w_{0} + M - x_{A} } \right)}} \right)}$. Thus $u\prime {\left( {w_{0} + M - x_{A} } \right)} - u\prime {\left( {w_{0} } \right)} - x_{A} \,u\prime \prime {\left( {w_{0} + M - x_{A} } \right)} < 0$ would imply dx _A /dw ₀  < 0. But note that DARA preference implies that v = −u′ is a concave transformation of u where v′ > 0 and v″ < 0 (Eeckhoudt, Gollier and Schlesinger 2005). That is, DARA implies that v is more risk averse than u. If we solve the maximisation problem of v, then we get the FOC, ${\text{I}}{\left( {x_{A} \prime } \right)} = - {\left\{ {u\prime {\left( {w_{0} + M - x_{A} \prime } \right)} - u\prime {\left( {w_{0} } \right)} - x_{A} \prime \cdot \,u\prime \prime {\left( {w_{0} + M - x_{A} \prime } \right)}} \right\}} = 0$ where x _A ′ is a giving-up amount satisfying FOC, I(x _A ′) = 0. From Proposition 2, we know x _A ′ > x _A since v is more risk averse than u. This implies that I(x _A ) > 0 since x _A ′, which is higher than x _A , should satisfy I(x _A ′) = 0 and by SOC, dI(x _A )/dx _A  < 0 if and only if v is concave.

Note that ${\text{I}}{\left( {x_{A} } \right)} = - {\left( {u\prime {\left( {w_{0} + M - x_{A} } \right)} - u\prime {\left( {w_{0} } \right)} - x_{A} \cdot \,u\prime \prime {\left( {w_{0} + M - x_{A} } \right)}} \right)} = { - dx_{A} } \mathord{\left/ {\vphantom {{ - dx_{A} } {dw_{0} }}} \right. \kern-\nulldelimiterspace} {dw_{0} }$. Thus, I(x _A ) > 0 implies dx _A /dw ₀  < 0. (Q.E.D.)

(Proof of Proposition 4)

We here mainly follow Gollier and Pratt (1996). Note from Proposition 2 that if a person is more risk averse at any wealth level, then he or she would choose a higher x. Hence, if we show that a risk-averse person who faces an independent and exogenous background risk to initial wealth would be more risk averse than one who does not face the background risk to initial wealth, then we are done. Denote $v{\left( {w_{0} + M - x} \right)} = Eu{\left( {w_{\varepsilon } + M - x} \right)}$. If we denote $z = w_{0} + M - x$, then $v{\left( z \right)} = Eu{\left( {z + \varepsilon } \right)}$. Since we know that ${ - v\prime \prime {\left( z \right)}} \mathord{\left/ {\vphantom {{ - v\prime \prime {\left( z \right)}} {v\prime {\left( z \right)}}}} \right. \kern-\nulldelimiterspace} {v\prime {\left( z \right)}} > { - u\prime \prime {\left( z \right)}} \mathord{\left/ {\vphantom {{ - u\prime \prime {\left( z \right)}} {u\prime {\left( z \right)}}}} \right. \kern-\nulldelimiterspace} {u\prime {\left( z \right)}}$ implies x _v  > x _u at any z, we only need to show ${ - v\prime \prime {\left( z \right)}} \mathord{\left/ {\vphantom {{ - v\prime \prime {\left( z \right)}} {v\prime {\left( z \right)}}}} \right. \kern-\nulldelimiterspace} {v\prime {\left( z \right)}} > { - u\prime \prime {\left( z \right)}} \mathord{\left/ {\vphantom {{ - u\prime \prime {\left( z \right)}} {u\prime {\left( z \right)}}}} \right. \kern-\nulldelimiterspace} {u\prime {\left( z \right)}}$. Rearranging ${ - v\prime \prime {\left( z \right)}} \mathord{\left/ {\vphantom {{ - v\prime \prime {\left( z \right)}} {v\prime {\left( z \right)}}}} \right. \kern-\nulldelimiterspace} {v\prime {\left( z \right)}} = { - Eu\prime \prime {\left( {z + \varepsilon } \right)}} \mathord{\left/ {\vphantom {{ - Eu\prime \prime {\left( {z + \varepsilon } \right)}} {Eu\prime {\left( {z + \varepsilon } \right)}}}} \right. \kern-\nulldelimiterspace} {Eu\prime {\left( {z + \varepsilon } \right)}} > { - u\prime \prime {\left( z \right)}} \mathord{\left/ {\vphantom {{ - u\prime \prime {\left( z \right)}} {u\prime {\left( z \right)}}}} \right. \kern-\nulldelimiterspace} {u\prime {\left( z \right)}}$, the condition implies $Eg{\left( {z,\varepsilon } \right)} = Eu\prime \prime {\left( {z + \varepsilon } \right)} \cdot u\prime {\left( z \right)} - Eu\prime {\left( {z + \varepsilon } \right)} \cdot u\prime \prime {\left( z \right)} < 0$. By noting g(z,E(ɛ)) = 0 since E(ɛ) = 0, we find the condition implies that g(z,ɛ) should be concave in ɛ at any z, that is, g ₂₂ (z,ɛ) < 0. This implies −u″(z)/u′(z) < −u″″(z)/u‴(z) when evaluated at ɛ = 0. By our assumptions of u‴ > 0 and u″″ < 0, this could be satisfied. So this is the necessary condition. In fact, it is easily shown that u‴ > 0 and u″″ < 0 are sufficient for ${ - v\prime \prime {\left( z \right)}} \mathord{\left/ {\vphantom {{ - v\prime \prime {\left( z \right)}} {v\prime {\left( z \right)}}}} \right. \kern-\nulldelimiterspace} {v\prime {\left( z \right)}} > { - u\prime \prime {\left( z \right)}} \mathord{\left/ {\vphantom {{ - u\prime \prime {\left( z \right)}} {u\prime {\left( z \right)}}}} \right. \kern-\nulldelimiterspace} {u\prime {\left( z \right)}}$ by using the implied relationship of $- Eu\prime \prime {\left( {z + \varepsilon } \right)} = E{\left[ {A{\left( {z + \varepsilon } \right)} \cdot u\prime {\left( {z + \varepsilon } \right)}} \right]} > EA{\left( {z + \varepsilon } \right)} \cdot Eu\prime {\left( {z + \varepsilon } \right)} > A{\left( z \right)} \cdot Eu\prime {\left( {z + \varepsilon } \right)}$ where A(.) denotes the absolute risk aversion coefficient. (Q.E.D.)

1.2 Appendix B: Instructions

These instructions have been used for the RRPM&Stranger group. The instructions for the other treatments are almost identical to this except only a few sentences to explain Partner treatment and/or APM treatment.

1.2.1 Instructions

This is an experiment in the economics of decision-making. If you follow the instructions carefully and make good decisions you may earn a considerable amount of money. You will be paid in private and in cash at the end of the whole session. It is important that you remain silent and do not communicate with the other participants. If you have any questions, please raise your hand and an experimenter will answer your question in private.

1.2.2 Overview

This experiment consists of 10 Rounds. Each Round consists of two Tasks, Task 1 and Task 2 in that order. In each task, your decision problem is to choose your number, which we call x. In each task, another number, which we call y, will be determined. Your potential earnings for each task and for each round will depend on x and y. Your final cash earnings from this experiment will depend on your potential earnings in each round and in each task.

Choosing your number x

In each Task of each Round, you have to choose one number from the set of numbers 2, 4, 5, 6, 7, 8, 16, 21, 24, and 27.

How the other number y will be determined

In Task 1, the other number y will be chosen at random from the same set of numbers 2, 4, 5, 6, 7, 8, 16, 21, 24, and 27.

In Task 2, the other number y will be chosen from the same set of numbers by some other participant in the experiment, who we call your partner for that Round. Your partner will be changed in each Round. Hence, you will never meet the same partner more than once through the 10 Rounds. Your partner will face exactly the same decision task as you, and you and your partner will not know the identity of each other.

How your potential earnings are determined

In every Round and in both Tasks, your potential earnings will depend on x and y in the following way:

• If x is greater than y then your potential earnings will be (35 − x) times 4 pence.

• If x is equal to y then your potential earnings will be (35 − x) times 2 pence.

• If x is less than y then your potential earnings will be zero.

Tables 1 and 2, which are provided on separate sheets, show you the calculated potential earnings for Task 1 and for Task 2 corresponding to your various choices, respectively. Please see the tables and read the examples on those sheets now.

What feedback is given to you after each Task in each Round

For Task 1, a history table for Task 1 on the screen will present you with only the number x you chose for Task 1.

For Task 2, a history table for Task 2 on the screen will present you with the following: the number x you chose in Task 2, the number y your partner chose in Task 2, and your potential earnings for Task 2.

1.2.3 How your cash earnings will be determined

Your cash earnings for Task 1 will be 10 times your potential earnings for Task 1 from a randomly chosen Round: that is, after you completed all the ten Rounds, one of the ten Rounds will be selected at random and the number x , which you choose for Task 1 in that Round, will be recalled. Then, y will be chosen at random. Your potential earnings for Task 1 in that Round are calculated by Table 1 with the x and y, and your cash earnings for Task 1 will be 10 times your potential earnings for Task 1 in that Round.

Your cash earnings for Task 2 will be 10 times your potential earnings for Task 2 from a randomly chosen Round: that is, after you completed all the ten Rounds, one of the ten Rounds will be selected at random, and your potential earnings for Task 2 in that Round will be recalled. Your cash earnings for Task 2 will be 10 times your potential earnings for Task 2 of that randomly selected Round.

Your cash earnings for the experiment as a whole will be the sum of your cash earnings for Task 1 and your cash earnings for Task 2.

1.2.4 An Example

Your cash earnings from Task 1

Suppose that Round 2 is randomly selected after you completed all the ten Rounds and that you chose 4 in Task 1 in that Round. That number will be recalled from your history table for Task 1. Then, the other number y is randomly chosen. For example, if the randomly chosen y is 2, then your potential earnings from Task 1 will be 124p as shown in Table 1 (with x = 4, y = 2). Hence, your cash earnings from Task 1 will be £12.40 (= 10 times 124p). Similarly, if the randomly chosen y is 4, then your potential earnings are 62p as shown in Table 1 (with x = 4 and y = 4) and so your cash earnings from Task 1 will be £6.20 (= 10 times 62p). If the randomly chosen y is 7, then your potential earnings will be 0p as shown in Table 1 (with x = 4 and y = 7) and so your cash earnings from Task 1 will be £0 (10 times 0p).

Your cash earnings from Task 2

Suppose that Round 5 is randomly selected for Task 2 after you completed all the ten Rounds. Your potential earnings in Task 2 in that Round will be recalled from your history table for Task 2. For example, if your potential earnings for Task 2 in that Round was 124p, then your cash earnings from Task 2 will be £12.40 (=10 times 124p). Similarly, if your potential earnings for Task 2 in that Round was 44p, then your cash earnings from Task 2 will be £4.40 (=10 times 44p), and if your potential earnings for Task 2 in that Round was 0p, then your cash earnings from Task 2 will be £0 (=10 times 0p).

Your final cash earnings for the experiment as a whole

For example, if your cash earnings from Task 1 and Task 2 are £5.00 and £3.50 respectively, then your final cash earnings for this experiment will be £8.50 (=£5.00+£3.50).

1.2.5 Some details

Throughout this experiment, the Round number and the Remaining time for the Task will appear on the upper-left part and the upper right part of the screen, respectively. The Task number will be on the center of the screen. Each Round begins with Task 1 and ends with Task 2.

In each Task, please put the number that you choose in the box on the screen, and then click the OK button.

After each Task of each Round, you will have feedback about your choice in that task of that Round, and your history tables for Task 1 and Task 2 will be in the bottom of the feedback screen.

After you finish the 10th Round, an experimenter will come over to you. You will pick one coin at random from an opaque bag (labeled with ‘Round for Task 1’) containing 10 coins numbered from 1 to 10: the number on it determines the Round in which your chosen number x for Task 1 will be recalled. Then, you will pick one coin at random from an opaque bag (labeled with ‘y for Task 1’) containing 10 coins numbered from the set 2, 4, 5, 6, 7, 8, 16, 21, 24 and 27 to decide y for Task 1. As explained above, your cash earnings from Task 1 will be 10 times the potential earnings determined by these x and y. After doing this, you will pick one coin at random again from an opaque bag (labeled with ‘Round for Task 2’) containing 10 coins numbered from 1 to 10 to decide the Round in which your potential earnings for Task 2 will be recalled, and as explained above, your cash earnings from Task 2 will be 10 times the potential earnings for Task 2 of that Round. Your final cash earnings from this experiment will be the sum of your cash earnings from Task 1 and Task 2 as already explained.

The experiment will be finished after you have answered a brief questionnaire and you have been paid your final cash earnings.

Many thanks for taking part.