When does less information translate into more giving to public goods?

Abstract

This paper theoretically and experimentally investigates the impact of information provision on voluntary contributions to a linear public good with an uncertain marginal per-capita return (MPCR). Uninformed donors make contribution decisions based only on the expected MPCR (i.e. the prior distribution), while informed donors observe the realized MPCR before contributing. The theoretical analysis predicts that the impact of information on average contributions crucially depends on the generosity level of the population, modeled as a stochastic change in the pro-social preferences. In particular, a less generous population increases contributions substantially in response to good news of higher than expected MPCR and reduces contributions relatively little in response to bad news of lower than expected MPCR. The opposite is true for a more generous population. Thus, the theory predicts that information provision increases (reduces) average contributions when the population is less (more) generous. This prediction finds strong support in a two-stage lab experiment. The first stage measures subjects’ generosity in the public good game using an online experiment. The resulting measure is used to create more and less generous groups in the public good lab experiment, which varies the information provided to these groups in the lab. The findings are in line with the theoretical predictions, suggesting that targeted information provision to less generous groups may be more beneficial for public good contributions than uniform information provision.

Appendices

Appendix 1

Proof of Lemma 1

For the purpose of this proof, note that

$$\begin{aligned} \frac{\partial \beta _1(v)}{\partial v}=-\frac{N(N-1)}{(Nv-1)^{2}}<0 \end{aligned}$$

(8)

and

$$\begin{aligned} \beta _2(v,\gamma )=(1+\gamma )\beta _1(v)+\gamma , \end{aligned}$$

(9)

implying that $\frac{\partial \beta _2(v,\gamma )}{\partial v}=(1+\gamma )\frac{\partial \beta _1(v)}{\partial v}$.

To establish that ${\overline{g}}(v)$ is increasing in v, note by Eq. (5) that

$$\begin{aligned} {\overline{g}}^{\prime }(v)=-{\overline{g}}(v)\frac{R^{\prime }(v)}{1+R(v)} \end{aligned}$$

(10)

Moreover, since $\beta _i\sim Exp(1/\lambda )$, $R(v)=\frac{1-e^{-\beta _1(v)/\lambda }}{e^{-\beta _2(v,\gamma )/\lambda }}$. Therefore, differentiating R(v) with respect to v yields

$$\begin{aligned} R^{\prime }(v)= & {} \frac{1}{\lambda }\frac{\partial \beta _2(v,\gamma )}{\partial v}e^{\beta _2(v,\gamma )/\lambda }-\frac{1}{\lambda }\left[ \frac{\partial \beta _2(v,\gamma )}{\partial v}-\frac{\partial \beta _1(v)}{\partial v}\right] e^{(\beta _2(v,\gamma )-\beta _1(v))/\lambda } \nonumber \\= & {} -\frac{N(N-1)}{(Nv-1)^2}\frac{1}{\lambda }\left[ e^{\beta _2(v,\gamma )/\lambda }+\gamma R(v)\right] <0, \end{aligned}$$

(11)

where the last equality follows from Eqs. (8) and (9). Then, $R'(v)<0$ and Eq. (10) immediately implies that ${\overline{g}}^{\prime }(v)>0$.

To show that $\lim _{v\rightarrow \frac{1}{N}}{\overline{g}}(v)=0$, we need to show that $\lim _{v\rightarrow \frac{1}{N}}R(v)=\infty$. Note that $\lim _{v\rightarrow \frac{1}{N}}\beta _{1}(v)=\lim _{v\rightarrow \frac{1}{N} }\beta _{2}(v,\gamma )=\infty$. Therefore, $\lim _{v\rightarrow \frac{1}{N} }e^{-\beta _2(v,\gamma )/\lambda }=\lim _{v\rightarrow \frac{1}{N}}e^{-\beta _1(v)/\lambda }=0$, resulting in $\lim _{v\rightarrow \frac{1}{N}}R(v)=\infty$. To see that $\lim _{v\rightarrow 1}{\overline{g}}(v)=W$ note that $\lim _{v\rightarrow 1} \beta _{1}(v)=0$ and $\lim _{v\rightarrow 1}\beta _2(v,\gamma )=\gamma$. This implies that $\lim _{v\rightarrow 1} R(v)=0$ and $\lim _{v\rightarrow 1}{\overline{g}}(v)=W$.

To establish the existence and uniqueness of ${\tilde{v}}(\lambda )$ and its corresponding properties, we first derive ${\overline{g}}^{\prime \prime }(v)$ by differentiating ${\overline{g}}^{\prime }(v)$ with respect to v, yielding

$$\begin{aligned} {\overline{g}}^{\prime \prime }(v)=\frac{{\overline{g}}(v)}{(1+R(v))}\left[ 2\frac{\left[ R^{\prime }(v)\right] ^{2}}{1+R(v)}-R^{\prime \prime }(v)\right] . \end{aligned}$$

(12)

Differentiating Eq. (11) with respect to v and simplifying yields

$$\begin{aligned} R^{\prime \prime }(v)= & {} \frac{N^2(N-1)^2}{\lambda ^2(Nv-1)^4}\left[ (e^{\beta _2(v,\gamma )/\lambda }+\gamma R(v))\left( \frac{2\lambda (Nv-1)}{(N-1)}+\gamma \right) \right. \nonumber \\&\left. +(1+\gamma )e^{\beta _2(v,\gamma )/\lambda }\right] >0. \end{aligned}$$

(13)

Substituting for $R^{\prime }(v)$ and $R^{\prime \prime }(v)$ in Eq. (12) and simplifying results in

$$\begin{aligned} {\overline{g}}^{\prime \prime }(v)= & {} \frac{{\overline{g}}(v)}{(1+R(v))}\frac{N^2(N-1)^2}{\lambda ^2(Nv-1)^4}[e^{\beta _2(v,\gamma )/\lambda }+\gamma R(v)]\\&\times \left[ 2\frac{e^{\beta _2(v,\gamma )/\lambda }+\gamma R(v)}{1+R(v)}-\frac{(1+\gamma )e^{\beta _2(v,\gamma )/\lambda }}{e^{\beta _2(v,\gamma )/\lambda }+\gamma R(v)}-\frac{2\lambda (Nv-1)}{(N-1)}-\gamma \right] . \end{aligned}$$

Note that

$$\begin{aligned}&g''(v)\overset{sign}{=} \frac{1}{\lambda }\left[ 2\frac{e^{\beta _2(v,\gamma )/\lambda }+\gamma R(v)}{1+R(v)}-\frac{(1+\gamma )e^{\beta _2(v,\gamma )/\lambda }}{e^{\beta _2(v,\gamma )/\lambda }+\gamma R(v)}-\frac{2\lambda (Nv-1)}{(N-1)}-\gamma \right] \nonumber \\&\quad =\Omega (v,\lambda ). \end{aligned}$$

(14)

Substituting for R(v) in the above expression and further simplifying yields

$$\begin{aligned} \Omega (v,\lambda )= & {} \frac{1}{\lambda }\left[ 2\frac{1+\gamma (1-e^{-\beta _1(v)/\lambda })}{e^{-\beta _2(v,\gamma )/\lambda }+1-e^{-\beta _1(v)/\lambda }}-\frac{1+\gamma }{1+\gamma (1-e^{-\beta _1(v)/\lambda })}\right. \nonumber \\&\left. -\frac{2\lambda (Nv-1)}{(N-1)}-\gamma \right] . \end{aligned}$$

(15)

Note that

$$\begin{aligned} \lim _{v\rightarrow \frac{1}{N}} \Omega (v,\lambda )=\frac{1+\gamma }{\lambda }>0, \end{aligned}$$

(16)

since $\lim _{v\rightarrow \frac{1}{N}} \beta _1(v)=\lim _{v\rightarrow \frac{1}{N}} \beta _2(v,\gamma )=\infty$, and

$$\begin{aligned} \lim _{v\rightarrow 1} \Omega (v,\lambda )=\frac{1}{\lambda }\left[ 2e^{\gamma /\lambda }-2\lambda -2\gamma -1\right] =\Omega ^1(\lambda ), \end{aligned}$$

(17)

since $\lim _{v\rightarrow 1} \beta _1(v)=0$ and $\lim _{v\rightarrow 1} \beta _2(v,\gamma )=\gamma$. Note that $\lim _{\lambda \rightarrow 0}\Omega ^1(\lambda )=\infty$ and $\lim _{\lambda \rightarrow \infty }\Omega ^1(\lambda )=-2$. Moreover, the term in brackets in Eq. (17) is strictly decreasing in $\lambda$, implying the existence a unique ${\tilde{\lambda }}>0$ such that $\Omega ^1({\tilde{\lambda }})=0$. For $\lambda >{\tilde{\lambda }}$, $\Omega ^1(\lambda )<0$ and thus $\lim _{v\rightarrow 1} \Omega (v,\lambda )<0$. Note that $\lim _{v\rightarrow \frac{1}{N}} \Omega (v,\lambda )>0$ and $\lim _{v\rightarrow 1} \Omega (v,\lambda )<0$ together imply the existence of ${\tilde{v}}(\lambda )\in (\frac{1}{N},1)$ that solves $\Omega ({\tilde{v}}(\lambda ),\lambda )=0$.

To establish the uniqueness of ${\tilde{v}}(\lambda )$, it suffices to show that $\frac{\partial \Omega ({\tilde{v}}(\lambda ),\lambda )}{\partial v}<0$. Note that by Eq. (12), $\Omega ({\tilde{v}}(\lambda ),\lambda )=0$ can be re-written as

$$\begin{aligned} \Phi ({\tilde{v}}(\lambda ))=2[R'({\tilde{v}})]^2-R''({\tilde{v}})(1+R({\tilde{v}}))=0. \end{aligned}$$

Thus, $\frac{\partial \Phi ({\tilde{v}}(\lambda ))}{\partial v}<0$ implies $\frac{\partial \Omega ({\tilde{v}}(\lambda ),\lambda )}{\partial v}<0$. Differentiating $\Phi (v)$ and evaluating at ${\tilde{v}}(\lambda )$ results in

$$\begin{aligned} \frac{\partial \Phi ({\tilde{v}}(\lambda ))}{\partial v}=\frac{R'({\tilde{v}})}{R''({\tilde{v}})}\left[ 3[R''({\tilde{v}})]^2-2R'''({\tilde{v}})R'({\tilde{v}})\right] , \end{aligned}$$

where we have taken into account that $1+R({\tilde{v}})=\frac{2[R'({\tilde{v}})]^2}{R''({\tilde{v}})}$. By Eq. (11), $R'({\tilde{v}})<0$ and by Eq. (13), $R''({\tilde{v}})>0$. Therefore,$\frac{\partial \Phi ({\tilde{v}}(\lambda ))}{\partial v}<0$ requires $3[R''({\tilde{v}})]^2-2R'''({\tilde{v}})R'({\tilde{v}})=\zeta ({\tilde{v}})>0$. Somewhat tedious, but straightforward algebra yields:²⁴

$$\begin{aligned} \zeta ({\tilde{v}})= & {} \frac{N^2(N-1)^2}{(N{\tilde{v}}-1)^4}\\&\times \left[ \frac{(1+\gamma )^2[R'({\tilde{v}})]^2}{{\lambda }^2}-R'({\tilde{v}})e^{\frac{\beta _2({\tilde{v}})-\beta _1({\tilde{v}})}{\lambda }}\frac{N(N-1)\gamma (4+2\gamma )}{\lambda ^3(N{\tilde{v}}-1)^2}\right. \\&\left. +\frac{3\gamma ^2N^2(N-1)^2}{(N{\tilde{v}}-1)^4\lambda ^4} e^{\frac{2(\beta _2({\tilde{v}})-\beta _1({\tilde{v}}))}{\lambda }}\right] \end{aligned}$$

Since $R'(v)<0$, it follows immediately that $\zeta ({\tilde{v}})>0$ and thus $\frac{\partial \Phi ({\tilde{v}}(\lambda ))}{\partial v}<0$. This establishes the uniqueness of ${\tilde{v}}(\lambda )$. To establish property 1), note that by the continuity of $\Omega (v,\lambda )$ in v and Eq. (16), it follows that $\Omega (v,\lambda )>0$ for all $v<{\tilde{v}}(\lambda )$, implying that ${\overline{g}}''(v)>0$ for all $v<{\tilde{v}}(\lambda )$. The uniqness of ${\tilde{v}}(\lambda )$ also implies that for all $\lambda >{\tilde{\lambda }}$ (i.e. $\Omega ^1(\lambda )<0$), $\Omega (v,\lambda )<0$ for all $v>{\tilde{v}}(\lambda )$. This, in turn, implies that ${\overline{g}}''(v)<0$ for all $v>{\tilde{v}}(\lambda )$.

To establish property (2), note first that the uniqueness of ${\tilde{v}}(\lambda )$ implies that for $\lambda <{\tilde{\lambda }}$, $\Omega (v,\lambda )>0$ for any $v\in \left( \frac{1}{N},1\right)$. Therefore, for $\lambda <{\tilde{\lambda }}$, ${\tilde{v}}(\lambda )=1$. For $\lambda >{\tilde{\lambda }}$, implicit differentiation of $\Omega ({\tilde{v}}(\lambda ),\lambda )=0$ results in

$$\begin{aligned} {\tilde{v}}^{\prime }(\lambda )=-\frac{\partial \Omega ({\tilde{v}}(\lambda ),\lambda )/\partial \lambda }{\partial \Omega ({\tilde{v}}(\lambda ),\lambda )/\partial v} \end{aligned}$$

(18)

Recall that $\frac{\partial \Phi ({\tilde{v}}(\lambda ))}{\partial v}<0$, implies $\frac{\partial \Omega ({\tilde{v}}(\lambda ),\lambda )}{\partial v}<0$. Thus, it suffices to show that $\frac{\partial \Omega ({\tilde{v}}(\lambda ),\lambda )}{\partial \lambda }<0$. For this purpose, let ${\tilde{\beta }}_{1}({\tilde{v}},\lambda )=\frac{\beta _1({\tilde{v}})}{\lambda }$, ${\tilde{\beta }}_{2}({\tilde{v}},\lambda ,\gamma )=\frac{\beta _{2}({\tilde{v}},\gamma )}{\lambda }$, and $\omega ({\tilde{\beta }}_1,{\tilde{\beta }}_2)=2\frac{1+\gamma (1-e^{-{\tilde{\beta }}_{1}})}{e^{-{\tilde{\beta }}_{2}}+1-e^{-{\tilde{\beta }}_{1}}}-\frac{1+\gamma }{1+\gamma (1-e^{-{\tilde{\beta }}_{1}})}$. Then, $\Omega ({\tilde{v}},\lambda )=0$ can be re-written as

$$\begin{aligned} \Omega ({\tilde{v}},\lambda )=\frac{1}{\lambda }\left[ \omega ({\tilde{\beta }}_1,\tilde{\beta _2})-2\lambda \frac{N{\tilde{v}}-1}{N-1}-\gamma \right] =0. \end{aligned}$$

(19)

Differentiating the above expression w.r.t. v yields

$$\begin{aligned} \frac{\partial \Omega ({\tilde{v}},\lambda )}{\partial v}= & {} \frac{1}{\lambda ^2}\left[ \frac{\partial \omega ({\tilde{\beta }}_1,\tilde{\beta _2})}{\partial {\tilde{\beta }}_1}\frac{\partial \beta _1({\tilde{v}})}{\partial v}+\frac{\partial \omega ({\tilde{\beta }}_1,\tilde{\beta _2})}{\partial {\tilde{\beta }}_2}\frac{\partial \beta _2({\tilde{v}},\gamma )}{\partial v}\right] - \frac{2 N}{N-1}\\= & {} -\frac{1}{\lambda ^2}\left[ \frac{\partial \omega ({\tilde{\beta }}_1,\tilde{\beta _2})}{\partial {\tilde{\beta }}_1}+\frac{\partial \omega ({\tilde{\beta }}_1,\tilde{\beta _2})}{\partial {\tilde{\beta }}_2}(1+\gamma )\right] \frac{N(N-1)}{(N{\tilde{v}}-1)^2}- \frac{2 N}{N-1}<0 \end{aligned}$$

where we have substituted for $\frac{\partial \beta _1({\tilde{v}})}{\partial v}$ and $\frac{\partial \beta _2({\tilde{v}},\gamma )}{\partial v}$ in the second equation using Eqs. (8) and (9). The strict inequality follows by our earlier result. Similarly, differentiating $\Omega ({\tilde{v}},\lambda )$ w.r.t. $\lambda$ results in

$$\begin{aligned} \frac{\partial \Omega ({\tilde{v}},\lambda )}{\partial \lambda }= & {} \frac{1}{\lambda }\left\{ \left[ -\frac{\partial \omega ({\tilde{\beta }}_1,\tilde{\beta _2})}{\partial {\tilde{\beta }}_1}\frac{\beta _1({\tilde{v}})}{\lambda ^2}-\frac{\partial \omega ({\tilde{\beta }}_1,\tilde{\beta _2})}{\partial {\tilde{\beta }}_2}\frac{\beta _2({\tilde{v}},\gamma )}{\lambda ^2}\right] -2\frac{N{\tilde{v}}-1}{N-1}\right\} \\= & {} \frac{1}{\lambda }\left\{ -\left[ \frac{\partial \omega ({\tilde{\beta }}_1,\tilde{\beta _2})}{\partial {\tilde{\beta }}_1}+\frac{\partial \omega ({\tilde{\beta }}_1,\tilde{\beta _2})}{\partial {\tilde{\beta }}_2}(1+\gamma )\right] \frac{\beta _1({\tilde{v}})}{\lambda ^2}\right. \\&\left. -\frac{\gamma }{\lambda ^2}\frac{\partial \omega ({\tilde{\beta }}_1,\tilde{\beta _2})}{\partial {\tilde{\beta }}_2}-2\frac{N{\tilde{v}}-1}{N-1}\right\} \\= & {} \frac{1}{\lambda }\left\{ \frac{(N{\tilde{v}}-1)(1-{\tilde{v}})}{N-1}\frac{d \Omega ({\tilde{v}},\lambda )}{d v}-2\frac{(N{\tilde{v}}-1)^2}{(N-1)^2}-\frac{\gamma }{\lambda ^2}\frac{\partial \omega ({\tilde{\beta }}_1,\tilde{\beta _2})}{\partial {\tilde{\beta }}_2}\right\} \end{aligned}$$

The first equality follows from $\Omega ({\tilde{v}},\lambda )=0$, the second equality follows from substituting for $\beta _2({\tilde{v}},\gamma )$ from Eq. (9) in the first one, and the last equality is derived by substituting for $\beta _1({\tilde{v}})$ and re-arranging the terms. It is straightforward to verify that $\frac{\partial \omega ({\tilde{\beta }}_1,\tilde{\beta _2})}{\partial {\tilde{\beta }}_2}>0$ and thus $\frac{\partial \Omega ({\tilde{v}},\lambda )}{\partial v}<0$ implies that $\frac{\partial \Omega ({\tilde{v}},\lambda )}{\partial \lambda }<0$. Therefore, by Eq. (18), ${\tilde{v}}^{\prime }(\lambda )<0$ for $\lambda >{\tilde{\lambda }}$.

Finally, to establish that $\lim _{\lambda \rightarrow \infty } {\tilde{v}}(\lambda )=\frac{1}{N}$, note that Eq. (15) implies that $\lim _{\lambda \rightarrow \infty } \Omega (v,\lambda )=-2\frac{Nv-1}{N-1}$. By definition, $\Omega ({\tilde{v}}(\lambda ),\lambda )=0$ for $\lambda >{\tilde{\lambda }}$. Therefore,

$$\begin{aligned} \lim _{\lambda \rightarrow \infty } \Omega ({\tilde{v}}(\lambda ),\lambda )=\lim _{\lambda \rightarrow \infty } -2\frac{N{\tilde{v}}(\lambda )-1}{N-1}=0\Longrightarrow \lim _{\lambda \rightarrow \infty } {\tilde{v}}(\lambda )=\frac{1}{N}. \end{aligned}$$

This completes the proof. $\square$

Proof of Proposition 1

Given $\frac{1}{N}<v_L<v_H<1$, by Lemma 1, there exist $\lambda _1>0$ such that ${\tilde{v}}(\lambda _1)=v_H$ and $\lambda _2>\lambda _1$ such that ${\tilde{v}}(\lambda _2)=v_L$. Furthermore, by Lemma 1, ${\overline{g}}(v)$ is convex for all $v<v_H$ if $\lambda \le \lambda _1$. Thus, by definition of convexity,

$$\begin{aligned} p_L{\overline{g}}(v_L)+p_H{\overline{g}}(v_H)>{\overline{g}}(p_Lv_L+p_Hv_H) \end{aligned}$$

Analogously, for $\lambda \ge \lambda _2$, ${\overline{g}}(v)$ is concave for all $v\ge v_L$, implying the reverse inequality. $\square$

Appendix 2: Behavioral types in the conditional contribution game

A total of 360 subjects participated in the first stage of this experiment and played the Fischbacher et al. (2001) conditional contribution game. Using this data, we classify subjects’ contribution behavior into the following categories:

• Conditional Cooperators About 73.6% of our subjects (265 subjects) are classified as the conditional cooperative types. Based on their contribution schedule, we further classify them into the following three distinct categories:

  • Pro-Social Conditional Cooperators 11.7% of our subjects (42 subjects) are classified as pro-social conditional cooperators. Their contributions are either equal to or more than the average contributions of others. On average, their contributions lie above the 45 degree line. The Spearman rank correlation coefficients of pro-social conditional cooperators are always positive and highly significant.

  • Perfect Conditional Cooperators 16.9% (61 subjects) are classified as perfect conditional cooperators. These subjects always perfectly match the contributions of others and thus their contributions on average lie exactly on the 45 degree line. Their Spearman rank correlation coefficients are equal to 1 and highly significant.

  • Weak Conditional Cooperators: 45% (162 subjects) are classified as weak conditional cooperators. The average contributions made by these subjects lie below the 45 degree line. While these subjects mostly exhibit weakly monotonic and increasing contribution behavior, 6.7% of our subjects (24 subjects) show some slight deviations from the increasing trend. The Spearman rank correlation coefficients of weak conditional cooperators are always positive and highly significant except for one subject whose coefficient is not significant.

• Free Riders 10% (36 subjects) are classified as free-riders or selfish. Independent of the others’ average contributions, these subjects never contribute anything.

• Hump-Shaped Contributors About 6.7% (24 subjects) are classified as hump-shaped contributors. These subjects’ contributions are increasing in others’ average contributions but up to a level (about 10 tokens on average). Beyond this level, their contributions start decreasing as others’ average contributions increase.

• Unconditional Full Cooperators 1.7% of our subjects (6 subjects) are classified as unconditional full cooperators. They always contribute all of their endowment independent of others’ average contribution.

• Other We classify the remaining 8% (29 subjects) as other. While, 3.9% (14 subjects) exhibits random behavior, about 3.6% (13 subjects) contributed a fixed amount such as 1 or 5 tokens with sometimes slight deviations. Interestingly, the contribution behavior for the 0.5% of subjects (2 subjects) exhibit a reserve hump-shape.