The joy of ruling: an experimental investigation on collective giving

Abstract

We analyse team dictator games with different voting mechanisms in the laboratory. Individuals vote to select a donation for all group members. Standard Bayesian analysis makes the same prediction for all three mechanisms: participants should cast the same vote regardless of the voting mechanism used to determine the common donation level. Our experimental results show that subjects fail to choose the same vote. We show that their behaviour is consistent with a joy of ruling: individuals get an extra utility when they determine the voting outcome.

Appendix: Proofs

Proof of Theorem 1

• (a) MAX Mechanism. Focus on the best response function of individual i to any profile of reported types $\hat{\theta }_{-i}$. By substituting the restrictions of the maximization problem in the utility function, the best response function is obtained from the solution of the following maximization problem

  $$\begin{aligned} max_{\left\{ {\hat{\theta }_i } \right\} } U_i \left( {w-max\left\{ {\hat{\theta }_i ,\hat{\theta }_{-i} } \right\} ,n \, max\left\{ {\hat{\theta }_i ,\hat{\theta }_{-i} } \right\} } \right) \end{aligned}$$

  • Step a.1. Consider first the profile $\hat{\theta }_{-i} =\{ {0,\ldots ,0}\}$. In this case, $\max \{{\hat{\theta }_{-i} }\}=0$ and therefore player i’s best response solves $\max _{\{ {\hat{\theta }_i }\}} U_i ({w-\hat{\theta }_i ,n\hat{\theta }_i})$. Given that the utility function is strictly quasi-concave, this maximisation problem has a unique solution $g_i^{\mathrm{SD}}$.

  • Step a.2. Focus now on those vote profiles $\hat{\theta }_{-i}$ for which $\max \{ {\hat{\theta }_{-i} }\}\le g_i^{\mathrm{SD}}$. In this case, player i will report $g_i^{\mathrm{SD}}$. Hence, $g_i^{\mathrm{SD}}$ is best response to those strategy profiles with $\max \{ {\hat{\theta }_{-i} }\}\le g_i^{\mathrm{SD}}$.

  • Step a.3. We finally consider those vote profiles such that $\max \{ {\hat{\theta }_{-i} } \}>g_i^{\mathrm{SD}}$. In this case, the strict quasi-concavity assures that all reported types larger than $\max \{ {\hat{\theta }_{-i} } \}$ yield a strictly lower utility level than that associated with voting $g_i^{\mathrm{SD}}$. This completes the proof. $\square$

• (b) MIN Mechanism. The proof follows the same lines as those of Proposition 1. The best response function comes from the solution of the following maximization problem

$$\begin{aligned} \max _{\left\{ {\hat{\theta }_i } \right\} } U_i \left( {w-\mathrm{min}\left\{ {\hat{\theta }_i ,\hat{\theta }_{-i} } \right\} ,n\, \mathrm{min}\left\{ {\hat{\theta }_i ,\hat{\theta }_{-i} } \right\} } \right) \end{aligned}$$

For all profiles such that $\mathrm{min}\{ {\hat{\theta }_i ,\hat{\theta }_{-i} } \}=\hat{\theta }_i$, player i will solve the problem $\max _{\{ {\hat{\theta }_i }\}} U_i ({w-\hat{\theta }_i ,n\hat{\theta }_i })$ whose solution is $g_i^{\mathrm{SD}}$. For all profiles such that $\hbox {min}\{{\hat{\theta }_{-i}}\}<g_i^{\mathrm{SD}}$, player i is indifferent between reporting $\hbox {min}\{ {\hat{\theta }_{-i} } \}$ or reporting $g_i^{\mathrm{SD}}$. This completes the proof. $\square$

Proof of Proposition 1

Firstly, we consider the MAX Institution. Let b denote the benefit from setting the group contribution level. Let $F_i ({M|\theta _i })$ be player i’s belief about the highest message M sent by the other players given his own type. Then, player i’s problem is

$$\begin{aligned} Max_{\left\{ {\hat{\theta }_i } \right\} }\mathop {\underbrace{\mathop {\int }\limits _0^{\hat{\theta }_i} \left[ {u_i \left( {w-\hat{\theta }_i ,n\hat{\theta }_i } \right) +b} \right] \mathrm{d}F_i \left( {M |\theta _i} \right) }} \limits _{\begin{array}{c}\hbox {Player }{i}\hbox {'s message is selected} \\ \hbox {by the mechanism} \end{array}} +\mathop {\underbrace{\mathop {\int }\limits _{\hat{\theta }_i}^w u_i \left( {w-M,nM} \right) \mathrm{d}F_i \left( {M |\theta _i}\right) }} \limits _{\begin{array}{c}\hbox {Player }{i}\hbox {'s message is not selected} \\ \hbox {by the mechanism} \end{array}} \end{aligned}$$

The first-order condition is

$$\begin{aligned}&\frac{\partial \left( {\int \nolimits _0^{\hat{\theta }_i } \left[ {u_i \left( {w-\hat{\theta }_i ,n\hat{\theta }_i } \right) +b} \right] \mathrm{d}F_i \left( {M |\theta _i } \right) +\mathop \int \nolimits _{\hat{\theta }_i }^w u_i \left( {w-M,nM} \right) \mathrm{d}F_i \left( {M |\theta _i } \right) } \right) }{\partial \hat{\theta }_i} \\&\quad +b\frac{\partial \left( {\int \nolimits _0^{\hat{\theta }_i } \mathrm{d}F_i \left( {M |\theta _i } \right) } \right) }{\partial \hat{\theta }_i }=0 \end{aligned}$$

We can actually prove that the equilibrium message will not be the super-dictatorial decision $g_i^{\mathrm{SD}}$. To prove it, we evaluate this first-order condition at the dominant strategy $g_i^{\mathrm{SD}}$. By definition, the first term on the left-hand side is zero, because $g_i^{\mathrm{SD}}$ is the optimal behaviour in the absence of joy of ruling. This implies that the value of the first-order condition evaluated at $g_i^{\mathrm{SD}}$ is $bf_i ({g_i^{\mathrm{SD}} |\theta _i })$, which is different from zero, where $f_i ({g_i^{\mathrm{SD}} |\theta _i })$ is the derivative of $F ({g_i^{\mathrm{SD}} |\theta _i })$. This means that if $b>(<)0$, then player i improves by sending a message larger (smaller) than $g_i^{\mathrm{SD}}$.

The analysis of the MIN Institution is analogous. $\square$