Prospect theory in multiple price list experiments: further insights on behaviour in the loss domain

Abstract

In the theoretical description of prospect theory, distinct sets of parameters can control the curvature of the value function and the shape of the probability weighting function. There is one for the gain domain and one for the loss domain. However, in most estimations, behaviour over losses is assumed to perfectly reflect behaviour over gains, through a unique set of parameters. We examine the consequences of relaxing this simplifying assumption in the context of Tanaka et al.’s (Am Econ Rev 100(1):557–571, 2010) risk-elicitation procedure based on multiple price lists. We show that subjects’ behaviour for gains is mostly reflected for losses at the aggregate and individual levels, and is consistent with the distinctive prospect theory fourfold pattern. Reflection is only partial as the mean curvature of the value function is slightly less convex for losses than it is concave for gains. These results are robust to a high-stake context. However, we demonstrate that assuming reflection can have huge consequences on loss-aversion measures. Incidentally, we also highlight the existence of a strong, negative and persistent pure loss-frame effect on elicited loss aversion. We recommend that future practitioners and modellers are particularly cautious about the loss-aversion values they obtain or use because these are especially sensitive to parametric assumptions and framing.

Appendices

Appendix A: Lotteries of the LLo treatment

See Table 13.

Table 13 Lottery options corresponding to treatment LLo

Appendix B: Calculation of the individual PT parameters according to frame

We distinguish parameters and functions between the gain and loss frames through index d (d is $+$ in the gain frame, d is − in the loss frame).

1.1 B.1 Calculation of $\sigma ^{d}$ and $\gamma ^{d}$

The following calculations are valid for Series 1 and 2.

1.1.1 B.1.1 Gain frame

Let $A(x_A,p_A;y_A)$ be the LHS lottery and $B^l(x_{B_{l}},p_B;y_B)$ the RHS lottery of row l ($x_A,y_A,x_{B_{l}}$ and $y_B$ are strictly positive $\forall l$, and $0<p_A<1$ and $0<p_B<1$). The lottery structure is such that only $x_{B_{l}}$ varies over rows l, while the other lottery attributes remain similar. For a given subject, switching at row s means the following inequalities in terms of prospect value:

$$\begin{aligned} \left\{ \begin{array}{ll} PV(A)>PV(B_{s-1}) \\ PV(A)<PV(B_{s}). \end{array} \right. \end{aligned}$$

(4)

From Eq. (2), we know that the prospect value of any lottery (x, p; y) is $v^d(y)+ \omega ^d(p) \cdot (v^d(x)-v^d(y))$ when $xy \ge 0$ and $|x| > |y|$. Equations (1) and (3) give the functional forms for the value and probability weighting functions, respectively. Thus, we obtain

$$\begin{aligned}&(4) \Leftrightarrow \left\{ \begin{array}{lll} y_{A}^{\sigma ^+} + \exp { [-(-\ln {p_A})^{\gamma ^+} ] } ( x_{A}^{\sigma ^+} - y_{A}^{\sigma ^+} ) &{}> &{}y_{B}^{\sigma ^+} + \exp { [-(-\ln {p_B})^{\gamma ^+} ] } ( x_{B_{s-1}}^{\sigma ^+} - y_{B}^{\sigma ^+} ) \\ y_{A}^{\sigma ^+} + \exp { [-(-\ln {p_A})^{\gamma ^+}] } ( x_{A}^{\sigma ^+} - y_{A}^{\sigma ^+} ) &{}< &{}y_{B}^{\sigma ^+} + \exp { [-(-\ln {p_B})^{\gamma ^+} ] } ( x_{B_{s}}^{\sigma ^+} - y_{B}^{\sigma ^+} ) \end{array} \right. \nonumber \\&\quad \Leftrightarrow y_{B}^{\sigma ^+} + \exp { [-(-\ln {p_B})^{\gamma ^+} ] } ( x_{B_{s-1}}^{\sigma ^+} - y_{B}^{\sigma ^+} )< y_{A}^{\sigma ^+} + \exp { [-(-\ln {p_A})^{\gamma ^+} ] } ( x_{A}^{\sigma ^+} - y_{A}^{\sigma ^+} ) \nonumber \\&\quad < y_{B}^{\sigma ^+} + \exp { [-(-\ln {p_B})^{\gamma ^+} ] } ( x_{B_{s}}^{\sigma ^+} - y_{B}^{\sigma ^+} ). \end{aligned}$$

(5)

1.1.2 B.1.2 Loss frame

Let $C^l(x_{C_{l}},p_C;y_C)$ be the LHS lottery of row l and $D(x_D,p_D;y_D)$ the RHS lottery ($x_{C_{l}},y_C,x_D,$ and $y_D$ are strictly negative $\forall l$, and $0<p_C<1$ and $0<p_D<1$). The lottery structure is such that only $x_{C_{l}}$ varies over rows l. This time, switching at row s means

$$\begin{aligned} \left\{ \begin{array}{ll} PV(C)>PV(D_{s-1}) \\ PV(C)<PV(D_{s}). \end{array} \right. \end{aligned}$$

(6)

As $xy \ge 0$ still, any lottery (x, p; y) has the same prospect value than in the previous section, i.e. $v^d(y)+ \omega ^d(p) \cdot (v^d(x)-v^d(y))$ when $|x| > |y|$. Equation (1) gives the specific value function for the loss domain $v^d(x)=-\lambda (-x)^{\sigma ^d}$ $\forall x<0$. Thus,

$$\begin{aligned} (6) \Leftrightarrow \left\{ \begin{array}{l} -\lambda (-y_{C})^{\sigma ^-} + \exp { [-(-\ln {p_C})^{\gamma ^-} ] } (-\lambda ) ( (-x_{C_{s-1}})^{\sigma ^-} - (-y_{C})^{\sigma ^-} ) \\ > -\lambda (-y_{D})^{\sigma ^-} + \exp { [-(-\ln {p_D})^{\gamma ^-} ] } (-\lambda ) ( (-x_{D})^{\sigma ^-} - (-y_{D})^{\sigma ^-} ) \\ -\lambda (-y_{C})^{\sigma ^-} + \exp { [-(-\ln {p_C})^{\gamma ^-} ] } (-\lambda ) ( (-x_{C_{s}})^{\sigma ^-} (-y_{C})^{\sigma ^-} ) \\ < -\lambda (-y_{D})^{\sigma ^-} + \exp { [-(-\ln {p_D})^{\gamma ^-} ] } (-\lambda ) ( (-x_{D})^{\sigma ^-} - (-y_{D})^{\sigma ^-} ). \end{array} \right. \end{aligned}$$

Simplifying by $-\lambda$, we obtain

$$\begin{aligned} (6) \Leftrightarrow \left\{ \begin{array}{l} (-y_{C})^{\sigma ^-} + \exp { [-(-\ln {p_C})^{\gamma ^-} ] } ( (-x_{C_{s-1}})^{\sigma ^-} - (-y_{C})^{\sigma ^-} ) \\ < (-y_{D})^{\sigma ^-} + \exp { [-(-\ln {p_D})^{\gamma ^-} ] } ( (-x_{D})^{\sigma ^-} - (-y_{D})^{\sigma ^-} ) \\ (-y_{C})^{\sigma ^-} + \exp { [-(-\ln {p_C})^{\gamma ^-} ] } ( (-x_{C_{s}})^{\sigma ^-} (-y_{C})^{\sigma ^-} ) \\ > (-y_{D})^{\sigma ^-} + \exp { [-(-\ln {p_D})^{\gamma ^-} ] } ( (-x_{D})^{\sigma ^-} - (-y_{D})^{\sigma ^-} ). \end{array} \right. \end{aligned}$$

As $x_{C_{l}}= -x_{B_{l}}$ , $y_C= -y_B$, $x_D= -x_A$, $y_D= -y_A$, and $p_D= p_A$, $p_C= p_B$, we further obtain

$$\begin{aligned}&(6) \Leftrightarrow \left\{ \begin{array}{lll} y_{B}^{\sigma ^-} + \exp { [-(-\ln {p_B})^{\gamma ^-} ] } (x_{B_{s-1}}^{\sigma ^-} - y_{B}^{\sigma ^-} ) &{}< &{}y_{A}^{\sigma ^-} + \exp { [-(-\ln {p_A})^{\gamma ^-} ] } ( x_{A}^{\sigma ^-} - y_{A}^{\sigma ^-} ) \\ y_{B}^{\sigma ^-} + \exp { [-(-\ln {p_B})^{\gamma ^-} ] } (x_{B_{s}}^{\sigma ^-} - y_{B}^{\sigma ^-} ) &{}> &{}y_{A}^{\sigma ^-} + \exp { [-(-\ln {p_A})^{\gamma ^-} ] } ( x_{A}^{\sigma ^-} - y_{A}^{\sigma ^-} ) \end{array} \right. \nonumber \\&y_{B}^{\sigma ^-} + \exp { [-(-\ln {p_B})^{\gamma ^-} ] } (x_{B_{s-1}}^{\sigma ^-} - y_{B}^{\sigma ^-} )< y_{A}^{\sigma ^-} + \exp { [-(-\ln {p_A})^{\gamma ^-} ] } ( x_{A}^{\sigma ^-} - y_{A}^{\sigma ^-} ) \nonumber \\&\quad < y_{B}^{\sigma ^-} + \exp { [-(-\ln {p_B})^{\gamma ^-} ] } (x_{B_{s}}^{\sigma ^-} - y_{B}^{\sigma ^-} ). \end{aligned}$$

(7)

We can see that (7) is equivalent to (5), meaning that a similar couple of switching points in Series 1 and 2 $(s_1,s_2)$ in the gain domain and in the loss domain leads to a similar couple of parameters$(\sigma ^d,\gamma ^d)$.

1.2 B.2 Calculation of $\lambda$, $\lambda _\mathrm{oppos}$, and $\lambda _\mathrm{gen}$ conditionally to $\sigma ^{d}$

The following calculations are valid for Series 3 only, where lotteries mix positive and negative payoffs. We distinguish $\sigma ^{d1}$ and $\sigma ^{d2}$, depending on which type of payoff the parameter applies to: $\sigma ^{d1}$ applies to gains while $\sigma ^{d2}$ applies to losses.

1.2.1 B.2.1 Gain frame

Let $A^l(x_{A_{l}},p;y_{A_{l}})$ be the LHS lottery and $B^l(x_B,p;y_{B_{l}})$ the RHS lottery of row l ($x_{A_{l}}$ and $x_B$ are strictly positive $\forall l$, while $y_{A_{l}}$ and $y_{B_{l}}$ are strictly negative $\forall l$, and $p=\frac{1}{2}$ ). The lottery structure is such that only $x_B$ does not vary over rows. A subject switching at row s means

$$\begin{aligned} \left\{ \begin{array}{ll} PV(A_{s-1})>PV(B_{s-1}) \\ PV(A_{s})<PV(B_{s}). \end{array} \right. \end{aligned}$$

(8)

From Eq. (2), we know that the prospect value of any binary lottery (x, p; y) is $\omega ^d(p) \cdot v^d(x) + \omega ^d(p)(1-p) \cdot v^d(y)$ when $xy \le 0$. As $p = \frac{1}{2}$ it simplifies to $\omega ^d(\frac{1}{2}) \cdot (v^d(x)+v^d(y))$. We designate as $\lambda _{12}$ the loss-aversion parameter, which is equivalent to $\lambda$, $\lambda _\mathrm{oppos}$ or $\lambda _\mathrm{gen}$ depending on the hypotheses relative to $\sigma ^{d1}$ and $\sigma ^{d2}$. We obtain

$$\begin{aligned} (8) \Leftrightarrow \left\{ \begin{array}{lll} \exp { [-(-\ln {\frac{1}{2}})^{\gamma ^{d}} ] } [( x_{A_{s-1}})^{\sigma ^{d1}} + (-\lambda _{12})(-y_{A_{s-1}})^{\sigma ^{d2}}] \\> \exp { [-(-\ln {\frac{1}{2}})^{\gamma ^d} ] } [(x_{B_{s-1}})^{\sigma ^{d1}} + (-\lambda _{12})(-y_{B_{s-1}})^{\sigma ^d}] \\ \exp { [-(-\ln {\frac{1}{2}})^{\gamma ^{d}} ] } [(x_{A_{s}})^{\sigma ^{d1}} + (-\lambda _{12})(-y_{A_{s}})^{\sigma ^{d2}}] \\< \exp { [-(-\ln {\frac{1}{2}})^{\gamma ^d} ] } [( x_{B_{s}})^{\sigma ^{d1}} + (-\lambda _{12})(-y_{B_{s}})^{\sigma ^{d2}}] \end{array} \right. \\ \Leftrightarrow \left\{ \begin{array}{lll} ( x_{A_{s-1}})^{\sigma ^{d1}} -\lambda _{12}(-y_{A_{s-1}})^{\sigma ^{d2}} &{}> &{} (x_{B_{s-1}})^{\sigma ^{d1}} -\lambda _{12}(-y_{B_{s-1}})^{\sigma ^{d2}} \\ (x_{A_{s}})^{\sigma ^{d1}} -\lambda _{12}(-y_{A_{s}})^{\sigma ^{d2}} &{}< &{} ( x_{B_{s}})^{\sigma ^{d1}} -\lambda _{12}(-y_{B_{s}})^{\sigma ^{d2}} \end{array} \right. \\ \Leftrightarrow \left\{ \begin{array}{lll} \lambda _{12}[(-y_{B_{s-1}})^{\sigma ^{d2}} - (-y_{A_{s-1}})^{\sigma ^{d2}}] &{}> &{}(x_{B_{s-1}})^{\sigma ^{d1}} - ( x_{A_{s-1}})^{\sigma ^{d1}} ) \\ \lambda _{12}[(-y_{B_{s}})^{\sigma ^{d2}} - (-y_{A_{s}})^{\sigma ^{d2}}] &{}< &{}(x_{B_{s}})^{\sigma ^{d1}} - ( x_{A_{s}})^{\sigma ^{d1}} ). \end{array} \right. \end{aligned}$$

Last, as $y_{A_{l}}$ and $y_{B_{l}}$ are negative payoffs such as $|y_{A_{l}}|<|y_{B_{l}}|$, $\sigma ^{d1} >0$, and $\sigma ^{d2} >0$, we can write

$$\begin{aligned} (8) \Leftrightarrow \left\{ \begin{array}{lll} \lambda _{12} &{}> &{}\frac{(x_{B_{s-1}})^{\sigma ^{d1}} -( x_{A_{s-1}})^{\sigma ^{d1}} }{(-y_{B_{s-1}})^{\sigma ^{d2}} -(-y_{A_{s-1}})^{\sigma ^{d2}} }\\ \lambda _{12} &{}< &{}\frac{(x_{B_{s}})^{\sigma ^{d1}} -( x_{A_{s}})^{\sigma ^{d1}} }{(-y_{B_{s}})^{\sigma ^{d2}} -(-y_{A_{s}})^{\sigma ^{d2}} }. \end{array} \right. \end{aligned}$$

(9)

Inequations (9) define the bound values of $\lambda$ when $\sigma ^{d1}=\sigma ^{d2}=\sigma ^{+}$, $\lambda _\mathrm{oppos}$ when $\sigma ^{d1}=\sigma ^{d2}=\sigma ^{-}$, and $\lambda _\mathrm{gen}$ when $\sigma ^{d1}=\sigma ^{+}$ and $\sigma ^{d2}=\sigma ^{-}$.

1.2.2 B.2.2 Loss frame

Let $C^l(x_C,p;y_{C_{l}})$ be the LHS lottery and $D^l(x_{D_{l}},p;y_{D_{l}})$ the RHS lottery of row l ($x_C$ and $x_{D_{l}}$ are strictly negative $\forall l$, while $y_{C_{l}}$ and $y_{D_{l}}$ are strictly positive $\forall l$, and $p=\frac{1}{2}$ ). The lottery structure is such that only $x_C$ does not vary over rows. A subject switching at row s means

$$\begin{aligned} \left\{ \begin{array}{ll} PV(C_{s-1})>PV(D_{s-1}) \\ PV(C_{s})<PV(D_{s}). \end{array} \right. \end{aligned}$$

(10)

As $xy \le 0$ still, any lottery (x, p; y) has the same prospect value than in the previous section, i.e. $\omega ^d(\frac{1}{2}) \cdot (v^d(x)+v^d(y))$. Equation (1) gives the specific value function for the loss domain $v^d(x)=-\lambda _{12}(-x)^{\sigma ^d}$ $\forall x<0$. We obtain

$$\begin{aligned}&(10) \Leftrightarrow \left\{ \begin{array}{ll} \exp { [-(-\ln {\frac{1}{2}})^\gamma ] } [(-\lambda )( -x_{C_{s-1}})^{\sigma ^{d2}} + (y_{C_{s-1}})^{\sigma ^{d1}}] \\> \exp { [-(-\ln {\frac{1}{2}})^\gamma ] } [(-\lambda _{12})(-x_{D_{s-1}})^{\sigma ^{d2}} + (y_{D_{s-1}})^{\sigma ^{d1}}] \\ \exp { [-(-\ln {\frac{1}{2}})^\gamma ] } [(-\lambda )(-x_{C_{s}})^{\sigma ^{d2}} + (y_{C_{s}})^{\sigma ^{d1}}]&{} \\< \exp { [-(-\ln {\frac{1}{2}})^\gamma ] } [ (-\lambda _{12})( -x_{D_{s}})^{\sigma ^{d2}} + (y_{D_{s}})^{\sigma ^{d1}}] \end{array} \right. \\&\quad \Leftrightarrow \left\{ \begin{array}{lll} (y_{C_{s-1}})^{\sigma ^{d1}} -\lambda _{12}( -x_{C_{s-1}})^{\sigma ^{d2}} &{}> &{}(y_{D_{s-1}})^{\sigma ^{d1}} -\lambda _{12}(-x_{D_{s-1}})^{\sigma ^{d2}} \\ (y_{C_{s}})^{\sigma ^{d1}} -\lambda _{12}(-x_{C_{s}})^{\sigma ^{d2}} &{}< &{}(y_{D_{s}})^{\sigma ^{d1}} -\lambda _{12}( -x_{D_{s}})^{\sigma ^{d2}} \end{array} \right. \\&\quad \Leftrightarrow \left\{ \begin{array}{lll} \lambda [(-x_{D_{s-1}})^{\sigma ^{d2}} - (-x_{C_{s-1}})^{\sigma ^{d2}}] &{}> &{}(y_{D_{s-1}})^{\sigma ^{d1}} - ( y_{C_{s-1}})^{\sigma ^{d1}} \\ \lambda [(-x_{D_{s}})^{\sigma ^{d2}} - (-x_{C_{s}})^{\sigma ^{d2}}] &{}< &{}(y_{D_{s}})^{\sigma ^{d1}} - ( y_{C_{s}})^{\sigma ^{d1}}. \end{array} \right. \end{aligned}$$

Last, as $x_{C}$ and $x_{D_{l}}$ are negative payoffs such as $|x_{C}|>|x_{D_{l}}|$, $\sigma ^{d1} >0$, and $\sigma ^{d2} >0$, we can write:

$$\begin{aligned} (10) \Leftrightarrow \left\{ \begin{array}{lll} \lambda _{12} &{}< &{}\frac{ ( y_{D_{s-1}})^{\sigma ^{d1}} -( y_{C_{s-1}})^{\sigma ^{d1}} }{ (-x_{D_{s-1}})^{\sigma ^{d2}} -(-x_{C_{s-1}})^{\sigma ^{d2}} } \\ \lambda _{12} &{}> &{}\frac{ ( y_{D_{s}})^{\sigma ^{d1}} -( y_{C_{s}})^{\sigma ^{d1}} }{ (-x_{D_{s}})^{\sigma ^{d2}} -(-x_{C_{s}})^{\sigma ^{d2}} }. \end{array} \right. \end{aligned}$$

(11)

As $x_C = -x_B$ , $y_C = -y_B$, $x_D = -x_A$ and $y_D = -y_A$, then Eq. (11) can be written as

$$\begin{aligned}&\left\{ \begin{array}{lll} \lambda _{12} &{}< &{}\frac{(-y_{A_{s-1}})^{\sigma ^{d1}} -(-y_{B_{s-1}})^{\sigma ^{d1}} }{(x_{A_{s-1}})^{\sigma ^{d2}} -(x_{B_{s-1}})^{\sigma ^{d2}} } \\ \lambda _{12} &{}> &{}\frac{(-y_{A_{s}})^{\sigma ^{d1}} -(-y_{B_{s}})^{\sigma ^{d1}} }{(x_{A_{s}})^{\sigma ^{d2}} -(x_{B_{s}})^{\sigma ^{d2}} } \end{array} \right. \nonumber \\&\quad \Leftrightarrow \left\{ \begin{array}{lll} \lambda _{12} &{}< &{}\frac{ (-y_{B_{s-1}})^{\sigma ^{d1}} - (-y_{A_{s-1}})^{\sigma ^{d1}} }{(x_{B_{s-1}})^{\sigma ^{d2}} - (x_{A_{s-1}})^{\sigma ^{d2}}} \\ \lambda _{12} &{}> &{}\frac{ (-y_{B_{s}})^{\sigma ^{d1}} - (-y_{A_{s}})^{\sigma ^{d1}} }{ (x_{B_{s}})^{\sigma ^{d2}} -(x_{A_{s}})^{\sigma ^{d2}} } . \end{array} \right. \end{aligned}$$

(12)

Inequations (12) define the bound values of $\lambda$ when $\sigma ^{d1}=\sigma ^{d2}=\sigma ^{-}$, $\lambda _\mathrm{oppos}$ when $\sigma ^{d1}=\sigma ^{d2}=\sigma ^{+}$, and $\lambda _\mathrm{gen}$ when $\sigma ^{d1}=\sigma ^{+}$ and $\sigma ^{d2}=\sigma ^{-}$.

As a last remark, in the special case when $\sigma ^{d1}=\sigma ^{d2}$, note that the bound values of the loss-aversion parameter in the gain and in the loss frames are linked. Recall that, in the gain frame, the bounds of the loss-aversion parameter are (Eq. (9)):

$$\begin{aligned} \left\{ \begin{array}{lll} \lambda ^{G}_{12,\min } &{}= &{}\frac{(x_{B_{s-1}})^{\sigma ^{d1}} -( x_{A_{s-1}})^{\sigma ^{d1}} }{(-y_{B_{s-1}})^{\sigma ^{d2}} -(-y_{A_{s-1}})^{\sigma ^{d2}} }\\ \lambda ^{G}_{12,\max } &{}= &{}\frac{(x_{B_{s}})^{\sigma ^{d1}} -( x_{A_{s}})^{\sigma ^{d1}} }{(-y_{B_{s}})^{\sigma ^{d2}} -(-y_{A_{s}})^{\sigma ^{d2}} }. \end{array} \right. \end{aligned}$$

Comparing with Eq. (12) provides the following equivalencies for $\lambda$ bounds, which also apply to $\lambda _\mathrm{oppos}$:

$$\begin{aligned} \left\{ \begin{array}{lll} \lambda ^{L}_{\max } &{}= &{}1/ \lambda ^{G}_{\min } \\ \lambda ^{L}_{\min } &{}= &{}1/ \lambda ^{G}_{\max } . \end{array} \right. \end{aligned}$$

Appendix C: Raw results

See Tables 14, 15, 16 and 17.

Table 14 Distribution of switching points for treatment GLo

Table 15 Distribution of switching points for treatment GHi

Table 16 Distribution of switching points for treatment LLo

Table 17 Distribution of switching points for treatment LHi

Appendix D: Comparison of PT parameter values from various studies

See Fig. 1.

Fig. 1

Mean parameter values from studies using the TCN methodology

Appendix E: Alternative regressions

See Tables 18, 19, 20, and 21

Table 18 Interval regression of $\sigma ^d$ on frame characteristics and socio-demographics

Table 19 Interval regression of $\gamma ^d$ on frame characteristics and socio-demographics

Table 20 Interval regression of $\lambda$ on frame characteristics and socio-demographics

Table 21 Log-interval regression of $\lambda$ on frame characteristics and socio-demographics

Appendix F: Distribution of individual PT parameter values

See Figs. 2, 3, 4, and 5.

Fig. 2

Distribution of individual PT parameters for baseline treatment GLo

Fig. 3

Distribution of individual $\sigma ^d$ parameters over treatments

Fig. 4

Distribution of individual $\gamma ^d$ parameters over treatments

Fig. 5

Distribution of individual $\lambda$ parameters over treatments

Appendix G: Distribution of selected individual loss-aversion values

See Figs. 6 and 7.

Fig. 6

Distribution of individual $\lambda$ values in baseline GLo and $\lambda _\mathrm{oppos}$ values in treatment LLo

Fig. 7

Distribution of individual $\lambda$ values in treatment GHi and $\lambda _\mathrm{oppos}$ values in treatment LHi