The pivotal mechanism versus the voluntary contribution mechanism: an experimental comparison

Abstract

We conduct an experimental comparison of two well-known mechanisms for undertaking a binary public project: the pivotal mechanism and the voluntary contribution mechanism. We compare the two mechanisms under complete information in which each subject knows the other subjects’ payoffs. We then observe that the voluntary contribution mechanism works better than the pivotal mechanism from the perspectives of Pareto efficiency, individual rationality, and surplus maximization. On the other hand, there is no significant difference between the two mechanisms in terms of decision efficiency. These results suggest that the voluntary contribution mechanism is more favorable than the pivotal mechanism in practical applications.

Appendices

Appendices

A Proofs

1.1 A.1 Proof of Fact 1

The “if” part. Suppose, by contradiction, that $(y, t) \notin P(v)$. Then, there is $(y^{\prime }, t^{\prime }) \in A$ such that $u_i(y^{\prime }, t^{\prime }_i; v_i) \ge u_i(y, t_i; v_i)$ for each $i \in \{1, 2\}$, with strict inequality for some $i \in \{1, 2\}$. There are two cases.

$\bullet$ Case 1: ${\varvec{{v}_{1} + {v}_{2} \ge {c}}}$. Then,

$$\begin{aligned} u_1(y^{\prime }, t^{\prime }_1; v_1) + u_2(y^{\prime }, t^{\prime }_2; v_2) > u_1(y, t_1; v_1) + u_2(y, t_2; v_2) = v_1 + v_2 - c + \omega _1 + \omega _2. \end{aligned}$$

If $y^{\prime }= 1$, then

$$\begin{aligned} v_1 + v_2 + \omega _1 + \omega _2 + t^{\prime }_1 + t^{\prime }_2&= u_1(y^{\prime }, t^{\prime }_1; v_1) + u_2(y^{\prime }, t^{\prime }_2; v_2) \\&> u_1(y, t_1; v_1) + u_2(y, t_2; v_2) \\&= v_1 + v_2 - c + \omega _1 + \omega _2, \end{aligned}$$

which implies $-(t^{\prime }_1 + t^{\prime }_2) < c$, a contradiction. If $y^{\prime }= 0$, then

$$\begin{aligned} \omega _1 + \omega _2 + t^{\prime }_1 + t^{\prime }_2&= u_1(y^{\prime }, t^{\prime }_1; v_1) + u_2(y^{\prime }, t^{\prime }_2; v_2) \\&> u_1(y, t_1; v_1) + u_2(y, t_2; v_2) \\&= v_1 + v_2 - c + \omega _1 + \omega _2, \end{aligned}$$

which implies $-(t^{\prime }_1 + t^{\prime }_2) < c - (v_1 + v_2 ) \le 0$, a contradiction.

$\bullet$ Case 2: ${\varvec{{v}_{1} + {v}_{2} < {c}}}$. Then,

$$\begin{aligned} u_1(y^{\prime }, t^{\prime }_1; v_1) + u_2(y^{\prime }, t^{\prime }_2; v_2) > u_1(y, t_1; v_1) + u_2(y, t_2; v_2) = \omega _1 + \omega _2. \end{aligned}$$

If $y^{\prime }= 1$, then

$$\begin{aligned} v_1 + v_2 + \omega _1 + \omega _2 + t^{\prime }_1 + t^{\prime }_2&= u_1(y^{\prime }, t^{\prime }_1; v_1) + u_2(y^{\prime }, t^{\prime }_2; v_2) \\&> u_1(y, t_1; v_1) + u_2(y, t_2; v_2) \\&= \omega _1 + \omega _2, \end{aligned}$$

which implies $-(t^{\prime }_1 + t^{\prime }_2)< v_1 + v_2 < c$, a contradiction. If $y^{\prime }= 0$, then

$$\begin{aligned} \omega _1 + \omega _2 + t^{\prime }_1 + t^{\prime }_2&= u_1(y^{\prime }, t^{\prime }_1; v_1) + u_2(y^{\prime }, t^{\prime }_2; v_2) \\&> u_1(y, t_1; v_1) + u_2(y, t_2; v_2) \\&= \omega _1 + \omega _2, \end{aligned}$$

which implies $-(t^{\prime }_1 + t^{\prime }_2) < 0$, a contradiction.

The “only if” part. Let $(y, t) \in P(v)$. There are three cases.

$\bullet$ Case 1: ${\varvec{{v}_{1} + {v}_{2} > {c}}}$. In this case, it suffices to show that $y = 1$ because this together with $(y, t) \in P(v)$ implies that $u_1(y, t_1; v_1) + u_2(y, t_2; v_2) = v_1 + v_2 - c + \omega _1 + \omega _2$. To show this, suppose by contradiction that $y=0$. Then, $u_1(y, t_1; v_1) = \omega _1 + t_1$, $u_2(y, t_2; v_2) = \omega _2 + t_2$, and $-(t_1 + t_2) \ge 0$. Without loss of generality, assume $t_1 \ge t_2$. Let $(y^{\prime }, t^{\prime }) \in A$ be such that $y^{\prime }= 1$, $-(t^{\prime }_1 + t^{\prime }_2) = c$, and $t'_1 = t_1 - v_1$. Since $v_1 + v_2 > c$, $v_1 + v_2 - c = v_1 + v_2 + (t'_1 + t'_2) > 0 \ge t_1 + t_2$, which implies that $v_2 + t'_2 > t_2$. It then follows that

$$\begin{aligned} u_1(y^{\prime }, t^{\prime }_1; v_1) = v_1 + t'_1 + \omega _1&= t_1 + \omega _1 = u_1(y, t_1; v_1),\\ u_2(y^{\prime }, t^{\prime }_2; v_2) = v_2 + t'_2 + \omega _1&> t_2 + \omega _2 = u_2(y, t_2; v_2), \end{aligned}$$

which contradict $(y, t) \in P(v)$.

$\bullet$ Case 2: ${\varvec{{v}_{1} + {v}_{2} < {c}}}$. In this case, it suffices to show that $y = 0$, because this together with $(y, t) \in P(v)$ implies that $u_1(y, t_1; v_1) + u_2(y, t_2; v_2) = \omega _1 + \omega _2$. To show this, suppose by contradiction that $y=1$. Then, $u_1(y, t_1; v_1) = v_1 + \omega _1 + t_1$, $u_2(y, t_2; v_2) = v_2 + \omega _2 + t_2$, and $-(t_1 + t_2) \ge c > 0$. Without loss of generality, assume $v_1 + t_1 \ge v_2 + t_2$. Let $(y^{\prime }, t^{\prime }) \in A$ be such that $y^{\prime }= 0$, $t^{\prime }_1 + t^{\prime }_2 = 0$, and $t'_1 = v_1 + t_1$. Since $v_1 + v_2 < c$, $t'_1 + t'_2 = 0 > v_1 + v_2 - c \ge v_1 + v_2 + (t_1 + t_2)$, which implies that $t'_2 > v_2 + t_2$. It then follows that

$$\begin{aligned} u_1(y^{\prime }, t^{\prime }_1; v_1) = \omega _1 + t'_1&= v_1 + t_1 + \omega _1 = u_1(y, t_1; v_1),\\ u_2(y^{\prime }, t^{\prime }_2; v_2) = \omega _1 + t'_2&> v_2 + t_2 + \omega _2 = u_2(y, t_2; v_2), \end{aligned}$$

which contradict $(y, t) \in P(v)$.

$\bullet$ Case 3: ${\varvec{{v}_{1} + {v}_{2} = {c}}}$. Then, either $y = 1$ or $y = 0$. If $y = 1$, then it follows from $(y, t) \in P(v)$ that $u_1(y, t_1; v_1) + u_2(y, t_2; v_2) = v_1 + v_2 - c + \omega _1 + \omega _2$. If $y = 0$, then it follows from $(y, t) \in P(v)$ that $v_1+ v_2 = c$, that $u_1(y, t_1; v_1) + u_2(y, t_2; v_2) = \omega _1 + \omega _2 = v_1 + v_2 - c + \omega _1 + \omega _2$. $\square$

1.2 A.2 Proof of Fact 2

We prove this fact by contraposition. Suppose that an allocation $(y, t) \in A$ is not decision efficient for v. Then, $y \notin \arg \max _{y^{\prime }\in \{0, 1\}} (v_1 + v_2) \cdot y^{\prime }- c \cdot y^{\prime }$. There are two cases.

$\bullet$ Case 1: ${\varvec{{y} ={1}}}$. Then,

$$\begin{aligned} 0 > v_1 + v_2 - c. \end{aligned}$$

(2)

Let $(y^{\prime }, t^{\prime }) \in A$ be such that $y^{\prime }= 0$ and for each $i \in \{1, 2\}$,

$$\begin{aligned} t^{\prime }_i \equiv t_i + v_i + \frac{c - (v_1 + v_2)}{2}. \end{aligned}$$

Note that by $-(t_1 + t_2) \ge c$,

$$\begin{aligned} t^{\prime }_1 + t^{\prime }_2&= t_1 + t_2 + v_1 + v_2+ c - (v_1 + v_2) \\&= t_1 + t_2 + c \\&\le 0. \end{aligned}$$

By (2), we have

$$\begin{aligned} u_1(y^{\prime }, t^{\prime }_1; v_1)&= \omega _1 + t^{\prime }_1 = \omega _1 + t_1 + v_1 + \frac{c - (v_1 + v_2)}{2}> \omega _1 + t_1 + v_1 = u_1(y, t_1; v_1),\\ u_2(y^{\prime }, t^{\prime }_2; v_2)&= \omega _2 + t^{\prime }_2 = \omega _2 + t_2 + v_2 + \frac{c - (v_1 + v_2)}{2} > \omega _2 + t_2 + v_2 = u_2(y, t_2; v_2). \end{aligned}$$

These imply that $(y, t) \notin P(v)$.

$\bullet$ Case 2: ${\varvec{y=0}}$. Then,

$$\begin{aligned} v_1+ v_2 - c > 0. \end{aligned}$$

(3)

Let $(y^{\prime }, t^{\prime }) \in A$ be such that $y^{\prime }= 1$ and for each $i \in \{1, 2\}$,

$$\begin{aligned} t^{\prime }_i \equiv t_i - v_i + \frac{v_1 + v_2 -c}{2}. \end{aligned}$$

Note that by $-(t_1 + t_2) \ge 0$,

$$\begin{aligned} t^{\prime }_1 + t^{\prime }_2&= t_1 + t_2 - v_1 - v_2+ v_1 + v_2 -c \\&= t_1 + t_2 - c \\&\le -c. \end{aligned}$$

By (3), we have

$$\begin{aligned} u_1(y^{\prime }, t^{\prime }_1; v_1)&= v_1 + \omega _1 + t^{\prime }_1 = \omega _1 + t_1 + \frac{v_1 + v_2 -c}{2}> \omega _1 + t_1 = u_1(y, t_1; v_1), \\ u_2(y^{\prime }, t^{\prime }_2; v_2)&= v_2 + \omega _2 + t^{\prime }_2 = \omega _2 + t_2 + \frac{v_1 + v_2 -c}{2} > \omega _2 + t_2 = u_2(y, t_2; v_2). \end{aligned}$$

These imply that $(y, t) \notin P(v)$. $\square$

1.3 A.3 Proof of Fact 3

The “if” part. Suppose, by contradiction, that $(y, t) \in A$ is not individually rational for v. Then, there are $i \in \{1, 2\}$ and $(y^{\prime }, t^{\prime }_i) \in A_{i}$ such that $u_i(y^{\prime }, t^{\prime }_i; v_i) > u_i(y, t_i; v_i)$. There are two cases.

$\bullet$ Case 1: ${\varvec{{v}_{i} \ge {c}}}$. Then, $u_i(y^{\prime }, t^{\prime }_i; v_i) > u_i(y, t_i; v_i) \ge v_i - c + \omega _i$. If $y^{\prime }= 1$, then $v_i + \omega _i + t^{\prime }_i > v_i - c + \omega _i$, which implies $-t^{\prime }_i < c$, a contradiction. If $y^{\prime }= 0$, then $\omega _i + t^{\prime }_i > v_i - c + \omega _i$, which implies $-t^{\prime }_i< c - v_i < 0$, a contradiction.

$\bullet$ Case 2: ${\varvec{{v}_{{i}} < {c}}}$. Then, $u_i(y^{\prime }, t^{\prime }_i; v_i) > u_i(y, t_i; v_i) \ge \omega _i$. If $y^{\prime }= 1$, then $v_i + \omega _i + t^{\prime }_i > \omega _i$, which implies $- t^{\prime }_i< v_i < c$, a contradiction. If $y^{\prime }= 0$, then $\omega _i + t^{\prime }_i > \omega _i$, which implies $- t^{\prime }_i < 0$, a contradiction.

The “only if” part. We prove this by contraposition. Suppose that there are $(y, t) \in A$ and $i \in \{1, 2\}$ such that (1) does not hold. If $v_{i} \ge c$, then $u_{i}(y, t_{i}; v_{i}) < v_{i} - c + \omega _{i}$. Let $(y^{\prime }, t^{\prime }_{i}) \in A_{i}$ be such that $y^{\prime }= 1$ and $t^{\prime }_{i} = -c$. Then, $u_{i}(y, t_{i}; v_{i}) < u_{i}(y^{\prime }, t^{\prime }_{i}; v_{i})$, which implies $(y, t) \notin I(v)$. If $v_{i} < c$, then $u_{i}(y, t_{i}; v_{i}) < \omega _{i}$. Let $(y^{\prime }, t^{\prime }_{i}) \in A_{i}$ be such that $y^{\prime }= 0$ and $t^{\prime }_{i} = 0$. Then, $u_{i}(y, t_{i}; v_{i}) < u_{i}(y^{\prime }, t^{\prime }_{i}; v_{i})$, which implies $(y, t) \notin I(v)$. $\square$

1.4 A.4 Proof of Proposition 1

Let $v \in V$. We proceed in two steps.³³

Step 1: For each ${\varvec{{i } \in \{{1, 2 }\}}}$, ${\varvec{{U} ({S} ^{\bf{VC} }_{i} ; ({\Gamma }^{\bf{VC} }, {v} )) = [{0}, v _{i }[}}$. Let $i \in \{1, 2\}$. Without loss of generality, assume $i = 1$.

$\circ$ Substep 1-1: Any $s_{1} \in [v_{1}, \omega _{1}]$ is weakly dominated. Let $s_1 \in [0, v_1[$. Since $\omega _{2} \ge c$ by A1, there is $s^{*}_{2} \in S^{\text {VC}}_{2}$ such that $s_{1} + s^{*}_{2} = c$. Let $s^{**}_{2} \in S^{\text {VC}}_{2}$ be such that $s^{\prime }_{1} + s^{**}_{2} = c$. Note that $s^{**}_{2} < s^{*}_{2}$. Let $s^{\prime }_{2} \in S^{\text {VC}}_{2}$. There are three cases.

$\bullet$ Case 1: ${\varvec{{s}^{\prime }_{{2}} \in [{0}, {s}^{**}_{{2}}[}}$. Then, $g^{\text {VC}}(s_{1}, s^{\prime }_{2}) = g^{\text {VC}}(s^{\prime }_{1}, s^{\prime }_{2}) =(0, (0, 0))$, which implies

$$\begin{aligned} u_{1}(g^{\text {VC}}_{1}(s_{2}, s^{\prime }_{2} ); v_{1}) = \omega _{1} = u_{1}(g^{\text {VC}}_{1}(s^{\prime }_{2}, s^{\prime }_{2} ); v_{1}). \end{aligned}$$

$\bullet$ Case 2: ${\varvec{{s}^{\prime }_{{2}} \in [{s}^{**}_{{2}}, {s}^{*}_{{2}}[}}$. Then, $g^{\text {VC} }(s_{1}, s^{\prime }_{2} ) = (0, (0, 0))$ and $g^{\text {VC}}(s^{\prime }_{1}, s^{\prime }_{2} ) =(1, (-s^{\prime }_{1}, -s^{\prime }_{2}))$. Since $s^{\prime }_{1} \ge v_{1}$,

$$\begin{aligned} u_{1}(g^{\text {VC}}_{1}(s_{1}, s^{\prime }_{2} ); v_{1}) = \omega _{1} \ge v_{1} + \omega _{1} - s^{\prime }_{1} = u_{1}(g^{\text {VC}}_{1}(s^{\prime }_{1}, s^{\prime }_{2} ); v_{1}). \end{aligned}$$

$\bullet$ Case 3: ${\varvec{{s}^{\prime }_{{2}} \in [{s}^{*}_{{2}}, \omega _{{2}}]}}$. Then, $g^{\text {VC}}(s_{1}, s^{\prime }_{2}) = (1, (-s_{1}, -s^{\prime }_{2}))$ and $g^{\text {VC}}(s^{\prime }_{1}, s^{\prime }_{2}) =(1, (-s^{\prime }_{1}, -s^{\prime }_{2}))$. Since $s^{\prime }_{1} > s_{1}$,

$$\begin{aligned} u_{1}(g^{\text {VC}}_{1}(s_{1}, s^{\prime }_{2} ); v_{1}) = v_{1} + \omega _{1} - s_{1} > v_{1} + \omega _{1} - s^{\prime }_{1} = u_{1}(g^{\text {VC}}_{1}(s^{\prime }_{1}, s^{\prime }_{2}); v_{1}). \end{aligned}$$

By Cases 1–3, we can conclude that $s^{\prime }_{1}$ is weakly dominated.

$\circ$ Substep 1-2: Any $s_{1} \in [0, v_{1}[$ is not weakly dominated. Note that there is $s^{*}_{2} \in S^{\text {VC}}_{2}$ be such that $s_{1} + s^{*}_{2} = c$. Let $s^{\prime }_{1} \in S^{\text {VC}}_{1} \setminus \{s_{1}\}$. There are two cases.

$\bullet$ Case 1: ${\varvec{{s}_{{1}} > {s}^{\prime }_{{1}}}}$.Then $g^{\text {VC}}(s_{1}, s^{*}_{2} ) = (1, (-s_{1}, -s^{*}_{2}))$ and $g^{\text {VC}}(s^{\prime }_{1}, s^{*}_{2} ) =(0, (0, 0))$. Since $v_{1} > s_{1}$,

$$\begin{aligned} u_{1}(g^{\text {VC}}_{1}(s_{1}, s^{*}_{2}); v_{1}) =v_{1} + \omega _{1} - s_{1} > \omega _{1} = u_{1}(g^{\text {VC}}_{1}(s^{\prime }_{1}, s^{*}_{2}); v_{1}). \end{aligned}$$

$\bullet$ Case 2: ${\varvec{{s}_{{1}} < {s}^{\prime }_{{1}}}}$. Then, $g^{\text {VC}}(s_{1}, s^{*}_{2} ) = (1, (-s_{1}, -s^{*}_{2}))$ and $g^{\text {VC}}(s^{\prime }_{1}, s^{*}_{2} ) = (1, (-s^{\prime }_{1}, -s^{*}_{2}))$. Since $s^{\prime }_{1} > s_{1}$,

$$\begin{aligned} u_{1}(g^{\text {VC}}_{1}(s_{1}, s^{*}_{2}); v_{1}) =v_{1} + \omega _{1} - s_{1} > v_{1} + \omega _{1} - s^{\prime }_{1} = u_{1}(g^{\text {VC}}_{1}(s^{\prime }_{1}, s^{*}_{2}); v_{1}). \end{aligned}$$

By Cases 1–2, $s_1$ is not weakly dominated.

Step 2: ${\varvec{{{{\mathcal {A}}}}({{\Gamma }}^{\bf{VC} }, {v}) {\subseteq } {{C(v)}}}}$. There are two cases.

$\bullet$ Case 1: ${\varvec{{v}_{{1}} + {v}_{{2}} > {c}}}$. For each $i \in \{1, 2\}$, let

$$\begin{aligned} {\underline{s}}_{i} \equiv \inf \left\{ s_i \in [0, v_{i}[ :\ s_i + s_j = c\ \text{ for } \text{ some } s_j \in [0, v_{j}[ \right\} . \end{aligned}$$

Note that under A1, for each $s_{i} \in [0, {\underline{s}}_{i}]$ and each $s_j \in [0, v_{j}[$, $g^{\text {VC}}(s_i, s_{j}) =(0, (0, 0))$.

$\circ$ Substep 2-1: For each $i \in \{1, 2\}$ and each $s^{\prime }_{i} \in [0, {\underline{s}}_{i}]$, $s^{\prime }_{i}$ is weakly dominated in the game $(\Gamma ^{\text {{VC}}}(U), v)$. Let $i \in \{1, 2\}$. Without loss of generality, assume $i = 1$. Let $s_{1}, s^{\prime }_{1} \in [0, v_{1}[$ be such that $s^{\prime }_{1} \le {\underline{s}}_{1} < s_{1}$. Then, there is $s^{*}_{2} \in [0, v_{2}[$ such that $s_{1} + s^{*}_{2} = c$. It follows that $g^{\text {VC}}(s_{1}, s_{2}) =(0, (0, 0))$ if $s_{2} < s^{*}_{2}$ and $g^{\text {VC}}(s_{1}, s_{2}) =(1, (-s_{1}, -s_{2}))$ otherwise. Let $s_2 \in [0, v_{2}[$. We also distinguish two subcases.

• 1. If $s_{2} < s_{2}^{*}$, then $g^{\text {VC}}(s_{1}, s_{2}) = g^{\text {VC}}(s^{\prime }_{1}, s_{2}) = (0, (0, 0))$, which implies

  $$\begin{aligned} u_{1}(g^{\text {VC}}_{1}(s_{1}, s_{2}); v_{1}) = \omega _{1} = u_{1}(g^{\text {VC}}(s^{\prime }_{1}, s_{2}); v_{1}). \end{aligned}$$

• 2. If $s_{2} \ge s_{2}^{*}$, then $g^{\text {VC}}(s_{1}, s_{2}) = (1, (-s_{1}, -s_{2}))$ and $g^{\text {VC}}(s^{\prime }, s_{2}) = (0, (0, 0))$. Since $v_{1} > s_{1}$,

  $$\begin{aligned} u_{1}(g^{\text {VC}}_{1}(s_{1}, s_{2}); v_{1}) = v_{1} + \omega _{1} - s_{1} > \omega _{1} = u_{1}(g^{\text {VC}}_{1}(s^{\prime }_{1}, s_{2}); v_{1}). \end{aligned}$$

Hence $s^{\prime }_{1}$ is weakly dominated.

$\circ$ Substep 2-2: For each $i \in \{1, 2\}$ and each $s_{i} \in \ ]{\underline{s}}_{i}, v_{i}[$, $s_{i}$ is not weakly dominated in the game $(\Gamma ^{\text {VC }}(U), v)$. Let $i \in \{1,2\}$. Without loss of generality, assume $i = 1$. Let $s_{1} \in \ ]{\underline{s}}_{1}, v_{1}[$. Note that there is $s^{*}_{2} \in [0, v_{2}[$ such that $s_{1} + s^{*}_{2} = c$. Let $s^{\prime }_{1} \in [0, v_{1}[\ \setminus \{s_{1}\}$. We also distinguish two subcases.

• 1. If $s_{1} > s^{\prime }_{1}$, then $g^{\text {VC}}(s_{1}, s^{*}_{2}) = (1, (-s_{1}, -s^{*}_{2}))$ and $g^{\text {VC}}(s^{\prime }_{1}, s^{*}_{2}) = (0, (0, 0))$. Since $v_{1} > s_{1}$,

  $$\begin{aligned} u_{1}(g^{\text {VC}}_{1}(s_{1}, s^{*}_{2}); v_{1}) =v_{1} + \omega _{1} - s_{1} > \omega _{1} = u_{1}(g^{\text {VC}}_{1}(s^{\prime }_{1}, s^{*}_{2}); v_{1}). \end{aligned}$$

• 2. If $s_{1} < s^{\prime }_{1}$, then $g^{\text {VC}}(s_{1}, s^{*}_{2}) = (1, (-s_{1}, -s^{*}_{2}))$ and $g^{\text {VC}}(s^{\prime }_{1}, s^{*}_{2}) = (1, (-s^{\prime }_{1}, -s^{*}_{2}))$. Since $s_{1} < s^{\prime }_{1}$,

  $$\begin{aligned} u_{1}(g^{\text {VC}}_{1}(s_{1}, s^{*}_{2}); v_{1}) =v_{1} + \omega _{1} - s_{1} > v_{1} + \omega _{1} - s^{\prime }_{1} = u_{1}(g^{\text {VC}}_{1}(s^{\prime }_{1}, s^{*}_{2}); v_{1}). \end{aligned}$$

Hence $s_{1}$ is not weakly dominated.

$\circ$ Substep 2-3: ${\mathcal {E}}(\Gamma ^{\text {VC }}, v) = \{(s_{1}, s_{2}) \in [0, v_{1}[\ \times \ [0, v_{2}[ :s_1 + s_2 = c\}$. We first show that ${\mathcal {E}}(\Gamma ^{\text {VC}}, v) \subseteq \{(s_{1}, s_{2}) \in [0, v_{1}[\ \times \ [0, v_{2}[ :s_1 + s_2 = c\}$. Let $s \in {\mathcal {E}}(\Gamma ^{\text {VC}}, v)$. Suppose, by contradiction, that $s_1 + s_2 \ne c$. If $s_1 + s_2 > c$, there is $i \in \{1, 2\}$ with $s_i > 0$. Let $\varepsilon > 0$ be such that $s_i + s_j > c + \varepsilon$. Let $s^{\prime }_i = s_i - \varepsilon$. Then,

$$\begin{aligned} u_i(g^{\text {VC}}_{i}(s^{\prime }_i, s_{j} ); v_i) =v_i + \omega _i - s_i + \varepsilon > v_i + \omega _i - s_i = u_i(g^{\text {VC}}_{i}(s_i, s_{j} ); v_i), \end{aligned}$$

a contradiction. If $s_1 + s_2 < c$, by Substeps 2.1 and 2.2, then $s_1 > {\underline{s}}_{1}$ and $s_2 > {\underline{s}}_{2}$. Let $i \in \{1, 2\}$. Then, there is $s^{\prime }_i$ such that $s^{\prime }_i + s_j = c$. Since $v_{i} > s_{i}$,

$$\begin{aligned} u_i(g^{\text {VC}}_{i}(s^{\prime }_i, s_{j} ); v_i) =v_i + \omega _i - s_i > \omega _i = u_i(g^{\text {VC}}_{i}(s_i, s_{j} ); v_i), \end{aligned}$$

a contradiction.

We next show that $\{(s_{1}, s_{2}) \in [0, v_{1}[\ \times \ [0, v_{2}[ :s_1 + s_2 = c\} \subseteq {\mathcal {E}}(\Gamma ^{\text {VC}}, v)$. Let $s^{\prime } = (s^{\prime }_{1}, s^{\prime }_{2}) \in \{(s_{1}, s_{2}) \in [0, v_{1}[\ \times \ [0, v_{2}[ :s_1 + s_2 = c\}$. Then, for each $i, j \in \{1, 2\}$ and each $s^{\prime \prime }_i (> {\underline{s}}_{i})$,

$$\begin{aligned} u_i(g^{\text {VC}}_{i}(s^{\prime }_i, s^{\prime }_{j} ); v_i) = v_i + \omega _i - s_i \ge u_i(g^{\text {VC}}_{i}(s^{\prime \prime }_i, s^{\prime }_{j} ); v_i). \end{aligned}$$

Moreover, since $s^{\prime }_1 + s^{\prime }_2 = c$, both $s^{\prime }_1$ and $s^{\prime }_2$ are not weakly dominated. Thus, $s^{\prime }\in {\mathcal {E}}(\Gamma ^{\text {VC}}, v)$.

$\circ$ Substep 2-4: Concluding. It follows from Substep 2-3 that ${\mathcal {A}}(\Gamma ^{\text {VC}}, v) \subseteq C(v)$.

$\bullet$ Case 2: ${\varvec{{v}_{{1}}+{{v}_{2}}\le {c}}}$. To show ${\mathcal {A}}(\Gamma ^{\text {VC}},v)\subseteq C(v)$, we first show that ${\mathcal {E}}(\Gamma ^{\text {VC}},v)=[0,v_{1}[\ \times \ [0,v_{2}[$. It is obvious that ${\mathcal {E}}(\Gamma ^{\text {VC}},v)\subseteq [0,v_{1}[\ \times \ [0,v_{2}[$. Therefore, it suffices to show that $[0,v_{1}[\ \times \ [0,v_{2}[\ \subseteq {\mathcal {E}}(\Gamma ^{\text {VC}},v)$. Let $(s_{1},s_{2}) \in [0,v_{1}[\ \times \ [0,v_{2}[$. Since $s_{1}+s_{2}<c$, $g^{\text {VC}}(s)=(0,(0,0))$. It follows that for each $i\in \{1,2\}$, $s_{i}$ is not weakly dominated in the game $(\Gamma ^{\text {VC}}(U),v)$ and, moreover, $(s_{1},s_{2})$ is a Nash equilibrium of the game $(\Gamma ^{\text {VC}}(U),v)$. Hence $(s_{1},s_{2})\in {\mathcal {E}}(\Gamma ^{\text {VC}},v)$. It then follows that ${\mathcal {A}}(\Gamma ^{\text {VC}},v)=\{(0,(0,0))\}\subseteq C(v)$. $\square$

B Implementation in iterated elimination of weakly dominated strategies

This section shows that the VCM cannot implement any sub-correspondence of the core in iterated elimination of weakly dominated strategies. To see this, suppose that $\omega _1 = \omega _2 = c = 8$.

We now consider the case where $v = (7, 7)$. Let $U^{1}_{i} \equiv U(S^{\text {VC}}_i; (\Gamma ^{\text {VC}}, v))$ and $U^{1} \equiv U^{1}_{1} \times U^{1}_{2}$. From the proof of Proposition 1 (Step 2), we obtain the following facts:

• For each $i \in \{1, 2\}$, $U^1_i = [0, 7[$;

• For each $i \in \{1, 2\}$ and each $s_i \in [0, 1[$, $s_i$ is weakly dominated in the game $(\Gamma ^{\text {VC}}(U^1), v)$; and

• For each $i \in \{1, 2\}$ and each $s_i \in U^{1}_{i} \setminus [0,1[\ =\ ]1, 7[$, $s_i$ is not weakly dominated in the game $(\Gamma ^{\text {VC}}(U^1), v)$.

Therefore, for each $i \in \{1,2\}$, $U^2_1 \equiv U(U^1_i; (\Gamma ^{\text {VC}}(U^1), v)) =\ ]1, 7[$. We now show that for each $i \in \{1, 2\}$ and each $s_i \in U^2_i$, $s_i$ is not weakly dominated in the game $(\Gamma ^{\text {VC}}(U^2), v)$. Let $i \in \{1,2\}$. Without loss of generality, assume $i = 1$. Let $s_{1} \in U^2_1$. Note that there is $s^{*}_{2} \in U^{2}_{2}$ such that $s_{1} + s^{*}_{2} = c = 8$. Let $s'_{1} \in U^2_1 \setminus \{s_1\}$. There are two cases.

$\bullet$ Case 1: ${\varvec{s_{1} > s^{\prime }_{1}}}$. Then, $g^{\text {VC}}(s_{1}, s^{*}_{2}) = (1, (-s_{1}, -s^{*}_{2}))$ and $g^{\text {VC}}(s^{\prime }_{1}, s^{*}_{2}) = (0, (0, 0))$. Since $v_{1} = 7 > s_{1}$,

$$\begin{aligned} u_{1}(g^{\text {VC}}_{1}(s_{1}, s^{*}_{2}); v_{1}) =v_{1} + \omega _{1} - s_{1} = 8 + (7- s_{1}) > 8 = \omega _{1} = u_{1}(g^{\text {VC}}_{1}(s^{\prime }_{1}, s^{*}_{2}); v_{1}). \end{aligned}$$

$\bullet$ Case 2: ${\varvec{{s}_{{1}} < {s}^{\prime }_{{1}}}}$. Then, $g^{\text {VC}}(s_{1}, s^{*}_{2}) = (1, (-s_{1}, -s^{*}_{2}))$ and $g^{\text {VC}}(s^{\prime }_{1}, s^{*}_{2}) = (1, (-s^{\prime }_{1}, -s^{*}_{2}))$. Since $s_{1} < s^{\prime }_{1}$,

$$\begin{aligned} u_{1}(g^{\text {VC}}_{1}(s_{1}, s^{*}_{2}); v_{1}) =v_{1} + \omega _{1} - s_{1} > v_{1} + \omega _{1} - s^{\prime }_{1} = u_{1}(g^{\text {VC} }_{1}(s^{\prime }_{1}, s^{*}_{2}); v_{1}). \end{aligned}$$

From Cases 1–2, we can conclude that $s_{1}$ is not weakly dominated. This fact implies that the set of strategies that survive iterated elimination of weakly dominated strategies is $]1,7[\ \times \ ]1,7[$. We now consider the strategy profile (2, 2). Then, $(2, 2) \in \ ]1,7[\ \times \ ]1,7[$ but $g^{\text {VC}}(2,2) = (0, (0, 0)) \notin C(v)$. This implies that the VCM fails to implement any sub-correspondence of the core in iterated elimination of weakly dominated strategies.

C Individual-oriented analysis

1.1 C.1 Treatment V

We first investigate the data in Treatment V more individually. To do this, we plot the strategies chosen by each of Type 1 and Type 2 subjects in all periods in Fig. 4. From Fig. 4a, we classify the decisions of Type 1 subjects into the following two categories:

• Equilibrium Strategy. The strategy choice patterns of 18 subjects (i.e., $90\% = \frac{18}{20}$) fall into this category. That is, either (i) each of these subjects chose $s_{1}=8$ at least in the last 10 periods; or (ii) the frequency at which each of these subjects chose $s_{1}=8$ was at least 80% across all periods. Thirteen subjects (ID Numbers 41, 44, 46, 47, 48, 49, 50, 62, 63, 65, 67, 68, and 69) consistently chose $s_{1}=8$ in almost all the periods. Four subjects (ID numbers 42, 64, 66, and 70) consistently chose $s_{1}=8$ in most of the remaining periods after experiencing other strategies in the first few periods. One subject (ID number 61) switched his or her strategies to $s_{1}=8$ a little later (from period 9).

• Other patterns. One subject (ID number 45) chose $s_{1}=8$ in the first 3 periods, then switched to other strategies (9 periods for $s_{1}=9$ and one period for $s_{1}=10$), and finally returned to his or her original strategy in the last 7 periods. One subject (ID number 43) had never chosen $s_{1}=8$ in all the periods.

Fig. 4

Strategies chosen by each subject in Treatment V. The red lines indicate the strategies $s_{1}=8$ for Type 1 subjects and $s_{2}=12$ for Type 2 subjects, respectively

From Fig. 4b, we classify the decisions of Type 2 subjects into the following two categories:

• Equilibrium Strategy. The strategy choice patterns of 17 subjects (i.e., $85\% = \frac{17}{20}$) fall into this category. That is, either (i) each of these subjects chose $s_{2}=12$ at least in the last 10 periods; or (ii) the frequency at which each of these subjects chose $s_{2}=12$ was at least 80% across all periods. Twelve subjects (ID numbers 51, 56, 59, 60, 71, 73, 74, 75, 76, 77, 78, and 80) consistently chose $s_{2}=12$ in almost all the periods. Two subjects (ID numbers 53 and 55) consistently chose $s_{2}=12$ from period 4. Three subjects (ID numbers 52, 58, and 79) started from $s_{2}=12$, then fluctuated their strategies between $s_{2}=12$ and others, but finally returned to their original strategy at least in the last 10 periods.

• Other patterns. One subject (ID number 57) also started from $s_{2}=12$, then fluctuated his or her strategies between $s_{2}=12$ and neighbors, and returned to $s_{2}=12$ only in the last 3 periods. One subject (ID number 54) had tried other strategies for 3 periods after consistently choosing $s_{2}=12$ in the first 10 periods, and then returned to $s_{2}=12$ for 6 periods, but finally chose $s_{2}=11$ in the last period. In contrast, another subject (ID number 72) started from other strategies, and then fluctuated between $s_{2}=12$ and others, but later consistently chose $s_{2}=12$ in the last 9 periods.

1.2 C.2 Treatment P

We next investigate the data in Treatment P more individually. To do this, we plot the strategies chosen by each of Type 1 and Type 2 subjects in all periods in Fig. 5. From Fig. 5a, we classify the decisions of Type 1 subjects into three categories:

• Dominant strategies. The strategy choice patterns of 12 subjects (i.e., $60\% = \frac{12}{20}$) fall into this category. That is, either (i) each of these subjects chose $s_{1}=8$ or 9 at least in the last 10 periods; or (ii) the frequency at which each of these subjects chose $s_{1}=8$ or $s_{1}=9$ was at least 80% across all periods. Six subjects (ID numbers 4, 10, 21, 26, 29, and 30) consistently chose either $s_{1}=8$ or $s_{1}=9$ in almost all the periods. Three subjects (ID numbers 1, 2, and 3) consistently chose either $s_{1}=8$ or $s_{1}=9$ in most of the remaining periods after experiencing other strategies in the first few periods. Three subjects (ID numbers 7, 22, and 24) switched to consistently choosing $s_{1}=8$ or $s_{1}=9$ a little later.

• “Overbidding”: choosing a number above dominant strategies. Three subjects (i.e., $15\% = \frac{3}{20}$) displayed such a behavior. Two subjects (ID numbers 8 and 9) had chosen numbers above $s_{1}=9$ since period 5. One subject (ID number 23) chose $s_{1}=10$ in all periods.³⁴

• Other patterns. Four subjects (ID numbers 5, 6, 25, and 27) fluctuated between other strategies and $s_{1}=8$ or $s_{1}=9$. One subject (ID number 28) chose $s_{1}=8$ and $s_{1}=9$ once each, then tried $s_{1}=10$ for 5 periods, and consistently chose $s_{1}=8$ in the remaining periods.

Fig. 5

Strategies chosen by each subject in Treatment P. The red lines indicate the strategies $s_{1}=8$ and $s_{1}=9$ for Type 1 subjects and $s_{2}=12$ and $s_{2}=13$ for Type 2 subjects, respectively

From Fig. 5b, we classify the decisions of Type 2 subjects into three categories:

• Dominant strategies. The strategy choice patterns of 15 subjects (i.e., $75\% = \frac{15}{20}$) fall into this category. That is, either (i) each of these subjects chose $s_{2}=12$ or $s_{2}=13$ at least in the last 10 periods; or (ii) the frequency at which each of these subjects chose $s_{2}=12$ or 13 was at least 80% across all periods. Twelve subjects (ID numbers 11, 12, 17, 18, 19, 31, 33, 34, 35, 38, 39, and 40) consistently chose either $s_{2}=12$ or $s_{2}=13$ in almost all the periods. Two subjects (ID numbers 14 and 16) consistently chose either $s_{2}=12$ or $s_{2}=13$ from period 5 and period 6, respectively. One subject (ID number 32) switched to consistently choosing either $s_{2}=12$ or $s_{2}=13$ a little later (from period 9).

• “Underbidding”: choosing a number below dominant strategies. Three subjects (i.e., $15\% = \frac{3}{20}$) displayed such a behavior. Two subjects (ID numbers 13 and 15) chose numbers below $s_{2}=12$ at most periods. One subject (ID number 20) started from $s_{2}=10$, and then switched to dominant strategies, $s_{2}=12$ or $s_{2}=13$, for 7 periods, but consistently chose $s_{2}=10$ until period 20.³⁵

• Other patterns. One subject (ID number 36) had never chosen either $s_{2}=12$ or $s_{2}=13$ in all the periods. One subject (ID number 37) chose $s_{2}=13$ only once in period 2, but chose numbers above $s_{2}=13$ (“overbidding”) in most periods.³⁶

D Behavioral differences between the two treatments

We observed that most of the subjects chose their own equilibrium strategy in Treatment V. On the other hand, in Treatment P, the majority of subjects chose dominant strategies, but many chose strategies other than dominant strategies. In this section, we explain why the behaviors in Treatment V are consistent with our theoretical prediction, whereas those in Treatment P are not.³⁷

As we mentioned in the main text, in Treatment V, a twice iteratively undominated Nash equilibrium uniquely exists, whereas there are many Nash equilibria. Moreover, the twice iteratively undominated Nash equilibrium is also the unique Nash equilibrium in which the sum of two agents’ payoffs is maximized and both agents’ payoffs are equal.³⁸ Thus, there is a possibility that the twice iteratively undominated Nash equilibrium is salient in Treatment V. We, in fact, observe that the twice iteratively undominated Nash equilibrium is the highest frequency strategy profile. On the other hand, in Treatment P, there seems to be no such salient equilibrium owing to the multiplicity of dominant strategy equilibria as well as asymmetry of all good Nash equilibria. As can be seen from Fig. 1, the frequency distribution in Treatment P is more spread out than that in Treatment V.

Our observations suggest that some subjects do not seek simply to maximize their own monetary payoffs. Recently, several studies have developed models that incorporate non-monetary factors into players’ objective functions (e.g., Fehr and Schmidt (1999), Bolton and Ockenfels (2000), and Charness and Rabin (2002)). Our experiment was not designed to differentiate between these alternative models and hence, we do not wish to overstate what our experiment can say about them. However, it is worthwhile to briefly evaluate our observations in the context of social preferences.

In the “inequality aversion” model proposed by Fehr and Schmidt (1999), subjects prefer higher, but more equal, earnings among participants in their group. In this model, a subject’s utility payoff is equal to his or her own monetary payoff minus the “inequality costs” that rise as the difference between his or her own and the other agent’s monetary payoff increases. To formally explain this, let $\pi _{i}(s_{1},s_{2})$ be the monetary payoff that Type $i \in \{1, 2\}$ receives when Type 1 chooses $s_{1}$ and Type 2 selects $s_{2}$. In the inequality aversion model, for each $i \in \{1,2\}$ and each $(s_1, s_2) \in S$, Type i’s utility payoff $U_i(s_1, s_2)$ is given by:

$$\begin{aligned} U_{i}(s_{1},s_{2})&=\pi _{i}(s_{1},s_{2})-\alpha _{i} \cdot \max \{\pi _{j}(s_{1},s_{2})-\pi _{i}(s_{1},s_{2}),0\}\\&\quad -\beta _{i} \cdot \max \{\pi _{i}(s_{1},s_{2})-\pi _{j}(s_{1},s_{2}),0\}, \end{aligned}$$

where $\alpha _{i}\ge \beta _{i}\ge 0$, and $\beta _{i}<1$, and $j\ne i$. The parameter $\alpha _{i}$ measures aversion against disadvantageous inequity (envy) and $\beta _{i}$ measures aversion against advantageous inequity (guilt).

We can easily confirm that the twice iteratively undominated Nash equilibrium in Treatment V remains the only twice iteratively undominated Nash equilibrium for any parameters, even under the inequality aversion model. On the other hand, each dominant strategy equilibrium in Treatment P is no longer Nash equilibrium for some parameters when we consider the inequality aversion model.³⁹ These theoretical differences between the two treatments correspond to the differences in observed behaviors between the two treatments in our experiment.

E Further analysis

• Figure 6a displays the frequency distribution of the last 10 periods in Treatment V and Fig. 6b displays that of the last 10 periods in Treatment P.

• Table 10 shows the frequency of distribution of all data in Treatment V.

• Table 11 shows the frequency of distribution of all data in Treatment P.

Fig. 6

All pairs choices in the last 10 periods

Table 10 Data frequency in Treatment V

Table 11 Data frequency in Treatment P

F Experiments with payoff tables

1.1 F.1 Experimental design and procedure

As mentioned in Sect. 6, we also conducted experiments presenting the payoff tables. There are two types of treatments: Treatment VT (resp. Treatment PT) implements the VCM (resp. PM) for a two-agent group with the use of payoff tables. The settings for each treatment are identical to the setting for Treatments V and P. The payoff tables that we employ in Treatment VT and PT are the same as the original payoff tables with the exception that a linear transformation of the valuation functions is employed: $10u_{i}-40$ for each Type $i\in \{1,2\}$, where $u_i$ is Type i’s payoff in the original payoff table.

We conducted two sessions in each of the two treatments at Tokyo Institute of Technology during January 2013, June 2013, and July 2014. Twenty subjects participated in each session (eighty separate subjects in total). We recruited the student subjects by campus-wide advertisement. These students were told that there would be an opportunity to earn money in a research experiment. None of them had prior experience in a public project experiment. No subject attended more than one session. Each session took approximately two hours to complete. The mean payoff per subject was $33.56 in Treatment VT and $32.57 in Treatment PT.

The experimental procedure is exactly the same as that described in Sect. 4.3 except for providing payoff tables to the subjects instead of explaining the rule of mechanism.⁴⁰ Each subject received written instructions, a record sheet, and a payoff table.⁴¹ Each subject chose an integer number between 0 and 20 by looking at their own payoff table only. In each subject’s payoff table, both their own payoffs and the payoffs of the other type were shown in Treatments VT and PT. Before the real 20 periods, the subjects had an opportunity to practice in two periods using a payoff table that differed from the table employed in the actual experiment. In these non-monetary periods, the numbers to be chosen were decided in advance by the experimenters. We allowed the subjects 10 minutes to examine the payoff table before the real periods started. No communication among the subjects was allowed, and we declared that the experiment would be stopped if we observed any communication among the subjects. This did not happen.

1.2 F.2 Experimental results

1.2.1 F.2.1 Treatment VT

Figure 7a shows the frequency of distribution of all data in Treatment VT. The maximum frequency pair was the unique twice iteratively undominated Nash equilibrium (8, 12) with 313 pairs out of 400, the second frequency pair was (8, 10) with 31 pairs, the third frequency pair was (8, 11) with 11 pairs, and the fourth frequency pair was (7, 12) with 6 pairs. The total frequency of Nash equilibria other than (8, 12) was 8. All of them were bad Nash equilibria that failed to achieve decision efficiency. The total frequency of strategy profiles that achieved decision efficiency was 316. The maximum frequency strategy chosen by Type 1 subjects was $s_{1}=8$ with 377 choices out of 400, the second maximum frequency strategy was $s_{1}=7$ with 8 choices, and the third maximum frequency strategy was $s_{1}=4$ with 6 choices. The maximum frequency strategy chosen by Type 2 subjects was $s_{2}=12$ with 328 choices out of 400, the second was $s_{2}=10$ with 32 choices, and the third was $s_{1}=11$ with 14 choices.

Fig. 7

All pairs choices

Figure 8a displays the rates that Type 1 and Type 2 subjects chose $s_{1}=8$ and $s_{2}=12$, respectively. After period 3, the ratios of Type 1 subjects who chose $s_{1}=8$ were at least $90\%$. The ratios of Type 2 subjects who chose $s_{2}=12$ were at least $80\%$ after period 4. Figure 8a also demonstrates the rates that pairs of subjects played the unique twice iteratively undominated Nash equilibrium (8, 12) for all periods. These equilibrium rates were at least $70\%$ after period 4.

Fig. 8

Rates that Types 1 and 2 subjects equilibrium strategies. The acronym DSE in the legend stands for dominant strategy equilibrium

These results are summarized by the following observations.

Observation F.1

• (a) The frequency of the unique twice iteratively undominated Nash equilibrium was $78\%$ $(= \frac{313}{400})$ across all periods in Treatment VT.

• (b) The frequency that Type 1 subjects chose $s_{1}=8$ was $94\%$ $(= \frac{377}{400})$ across all periods in Treatment VT.

• (c) The frequency that Type 2 subjects chose $s_{2}=12$ was $82\%$ $(= \frac{328}{400})$ across all periods in Treatment VT.

To investigate the data more individually, we plot the strategies chosen by each of Type 1 and Type 2 subjects in Fig. 9. From Figure 9a, we classify the decisions of Type 1 subjects into the following two categories:

• Equilibrium Strategy. The strategy choice patterns of 20 subjects (i.e., 100%) fall into this category. That is, either (i) each of these subjects chose $s_1= 8$ at least in the last 10 periods; or (ii) the frequency at which each of these subjects chose $s_1= 8$ was at least 80% across all periods. Fifteen subjects (ID numbers 121, 122, 123, 125, 126, 127, 130, 141, 143, 144, 146, 147, 148, 149, and 150) consistently chose $s_1= 8$ in almost all the periods. Four subjects (ID numbers 124, 128, 129, and 142) consistently chose $s_1= 8$ in most of the remaining periods after experiencing other strategies in the first few periods. One subject (ID number 145) switched his or her strategies to consistently choose $s_1= 8$ a little later (from period 9).

• Other patterns. No one falls into this category.

From Fig. 9b, we classify the decisions of Type 2 subjects into the following two categories:

• Equilibrium Strategy. The strategy choice patterns of 16 subjects (i.e., 80%) fall into this category. That is, either (i) each of these subjects chose $s_2= 12$ at least in the last 10 periods; or (ii) the frequency at which each of these subjects chose $s_2= 12$ was at least 80% across all periods. Twelve subjects (ID numbers 131, 132, 133, 135, 136, 137, 138, 139, 151, 157, 158, and 160) consistently chose $s_2= 12$ in almost all the periods. Three subjects (ID numbers 140, 153, and 155) consistently chose $s_2= 12$ in most of the remaining periods after experiencing other strategies in the first few periods. One subject (ID number 152) switched his or her strategies to consistently choose $s_2= 12$ a little later (from period 8).

• Other patterns. The strategy choice patterns of 4 subjects (i.e., 20%) fall into this category. One subject (ID number 134) started from $s_2= 12$, then fluctuated his or her strategies between $s_2= 12$ and neighbors, and consistently chose $s_2= 10$ in the last 7 periods. One subject (ID number 156) chose $s_2= 11$ in the first 2 periods, then almost consistently chose $s_2= 12$ in the following 15 periods, but finally chose other strategies in the last 3 periods. One subject (ID number 159) chose other strategies in the first 5 periods, then fluctuated his or her strategies between $s_2= 12$ and others, but finally chose the strategy far away from $s_2= 12$ in the last period. One subject (ID number 154) never chose $s_2= 12$.

Table 12 summarizes the classification of strategies into two categories by types in Treatment VT.

Fig. 9

Strategies chosen by each subject in Treatment VT. The red lines indicate the strategies $s_{1}= 8$ for Type 1 subjects and $s_{2}= 12$ for Type 2 subjects, respectively

Table 12 Strategy category by types in Treatment VT

1.2.2 F.2.2 Treatment PT

Figure 7b shows the frequency of distribution of all data in Treatment PT. The maximum frequency pair was (8, 13) with 95 pairs out of 400, the second was (8, 12) with 91 pairs, the third was (9, 12) with 30 pairs, the fourth was (10, 12) with 29 pairs, and the fifth was (9, 13) with 26 pairs. The total frequency of dominant strategy equilibria was 242. The total frequency of Nash equilibria was 370. All Nash equilibria were good. The maximum frequency strategy chosen by Type 1 subjects was $s_{1}=8$ with 224 choices out of 400, the second was $s_{1}=9$ with 72 choices, and the third was $s_{1}=10$ with 70 choices. The total frequency of dominant strategies selected by Type 1 subjects was 296. The maximum frequency strategy chosen by Type 2 was $s_{2}=12$ with 167 choices out of 400, the second was $s_{2}=13$ with 160 choices, and the third was $s_{1}=20$ with 15 choices. Hence, the total frequency of dominant strategies selected by Type 2 subjects was 327.

Figure 8b displays the rates that each type of subjects chose dominant strategies separately as well as the rates that pairs of subjects played dominant strategy equilibria for all periods.

These results are summarized by the following observations.

Observation F.2

• (a) The frequency of dominant strategy equilibria was $61\%$ $(= \frac{242}{400})$ across all periods in Treatment PT.

• (b) The frequency of good Nash equilibria was $93\%$ $(= \frac{370}{400})$ across all periods in Treatment PT.

• (c) The frequency that Type 1 subjects chose dominant strategies was $74\%$ $(= \frac{296}{400})$ across all periods in Treatment PT.

• (d) The frequency that Type 2 subjects chose dominant strategies was $82\%$ $(= \frac{327}{400})$ across all periods in Treatment PT.

To investigate the data more individually, we plot the strategies chosen by each of Type 1 and Type 2 subjects in Fig. 10. From Figure 10a, we classify the decisions of Type 1 subjects into three categories:

• Dominant strategies. The strategy choice patterns of 11 subjects (i.e., 55%) fall into this category. That is, either (i) each of these subjects chose either $s_1= 8$ or $s_1= 9$ at least in the last 10 periods; or (ii) the frequency at which each of these subjects chose either $s_1= 8$ or $s_1= 9$ was at least 80% across all periods. These subjects (ID numbers 83, 86, 88, 89, 101, 103, 104, 105, 106, 108, and 109) consistently chose either $s_1= 8$ or $s_1= 9$ in almost all the periods.

• “Overbidding”: choosing a number above dominant strategies. Six subjects (i.e., 30%) displayed such a behavior. Three subjects (ID numbers 81, 90, and 102) chose between the dominant strategies (i.e., $s_1= 8$ or $s_1= 9$) and $s_1= 10$, and consistently chose the latter at least in the last 6 periods. Three subjects (ID numbers 82, 85, and 107) chose numbers above the dominant strategies at least in 6 periods and finally ended by choosing the dominant strategies.

• Other patterns. Three subjects (ID numbers 84, 87, and 110) fluctuated among the dominant strategies, numbers above $s_1= 8$ or $s_2=9$, and numbers below $s_1= 8$ or $s_2=9$.

Fig. 10

Strategies chosen by each subject in Treatment PT. The red lines indicate the strategies $s_{1} = 8$ and $s_{1} = 9$ for Type 1 subjects and $s_{2} = 12$ and $s_{2} = 13$ for Type 2 subjects, respectively

From Figure 10b, we classify the decisions of Type 2 subjects into the following three categories:

• Dominant strategies. The strategy choice patterns of 14 subjects (i.e., 70%) fall into this category. That is, either (i) each of these subjects chose either $s_2= 12$ or $s_2= 13$ at least in the last 10 periods; or (ii) the frequency at which each of these subjects chose either $s_2= 12$ or $s_2 = 13$ was at least 80% across all periods. Twelve subjects (ID numbers 91, 93, 94, 95, 97, 98, 99, 100, 112, 113, 114, and 115) consistently chose either $s_2= 12$ or $s_2= 13$ in almost all the periods. One subject (ID number 92) started from choosing other strategies in the first 3 periods and then consistently chose either $s_2= 12$ or $s_2= 13$ in the remaining 17 periods. One subject (ID number 119) chose either $s_2= 12$ or $s_2= 13$ almost consistently, except for choosing other numbers above the dominant strategies in the periods 9, 13, and 19.

• “Underbidding”: choosing a number below dominant strategies. Two subjects (i.e., 10%) displayed such a behavior. One subject (ID number 96) chose numbers under the dominant strategies (i.e., $s_2= 12$ or $s_2=13$) in more than half of the periods. One subject (ID number 120) chose either $s_2= 11$ or $s_2=10$ in 5 periods including the last period.

• Other patterns. Two subjects (ID numbers 116 and 117) fluctuated among the dominant strategies, numbers above $s_2= 12$ or $s_2=13$, and numbers below $s_2= 12$ or $s_2=13$. Two subjects (ID numbers 111 and 118) chose numbers above the dominant strategies in 10 and 7 periods, respectively. They never chose numbers below the dominant strategies.

Table 13 summarizes the classification of strategies into two categories by types in Treatment PT.

Table 13 Strategy category by types in Treatment VT

1.2.3 F.2.3 Comparing the two mechanisms

We compare the performances of the VCM with those of the PM based on the following four indices that are introduced in Sect. 5.3: the Pareto efficiency index, the individual rationality index, the surplus index, and the decision efficiency index.

Figure 11 displays the mean Pareto efficiency index, the mean individual rationality index, the mean surplus index, and the mean decision efficiency index at each period in Treatments VT and PT. As shown in this figure, we observed that the mean Pareto efficiency index, the mean individual rationality index, and the surplus index for Treatment VT were higher than those for Treatment PT in each period (except for the first period for the surplus index). We also observed that the mean decision efficiency index for Treatment PT was higher than that for Treatment VT in 18 out of 20 periods (except for periods 8 and 11), although the mean decision efficiency index for Treatment VT attained at least 80% in 16 out of 20 periods (except for periods 1–3 and 20).

Fig. 11

Comparing the voluntary contribution mechanism and the pivotal mechanism based on four indices. The blue (resp. red) dot lines indicate the theoretical benchmarks for Treatment VT (resp. Treatment PT)

To formally test these graphical observations, we ran the panel regressions. Table 14 reports the panel regression results of the pooled data. Among the four regressions, the random effects panel probit model was applied in the regressions of the Pareto efficiency index, the individual rationality index, and the decision efficiency index, while the random effects panel model was conducted for the regression of the surplus index. Besides the continuous variable of Period controlling for round effects, the dummy variable of Treatment PT was included in the four regressions as the independent variable, leaving Treatment VT as the base of comparison.

Table 14 Panel regressions on the four indices

As presented in Table 14, the dummy variable of Treatment PT was estimated with significantly negative sign in the regressions of the Pareto efficiency index, the individual rationality index, and the surplus index, implying that the VCM is superior to the PM based on these criteria. However, this variable was significantly and positively estimated in the regression of the decision efficiency index, which exhibits a better performance of the PM than the VCM in the sense of decision efficiency.

These results are summarized by the following observations.

Observation F.3

• (a) The Pareto efficiency index for Treatment VT was significantly higher than the Pareto efficiency index for Treatment PT.

• (b) The individual rationality index for Treatment VT was significantly higher than the individual rationality index for Treatment PT.

• (c) The surplus index for Treatment VT was significantly higher than the surplus index for Treatment PT.

• (d) The decision efficiency index for Treatment VT was significantly lower than the decision efficiency index for Treatment PT.

G Experimental instructions (Treatment V)

Note: In this experiment, please remember that you cannot talk or communicate with other participants. If there is communication, the experiment will be stopped at that point.

The reward you receive after the experiment is decided according to the points you earn through the experiment. A detailed explanation of the reward you receive will be provided later in Sect. G.5 (Reward).

First, please confirm that the following items are on your desk. If any of the items are missing, please contact an experimenter.

• Instructions (this handout)

• Table 15

• Record sheet

• Calculation sheet

• Two pencils and one eraser

1.1 G.1 Overview

In this experiment, two participants (Individuals 1 and 2) are about to decide whether to invest in a certain project. Your role in this experiment will be either Individual 1 or Individual 2. At the beginning of each period, an experimenter will choose the person you are paired with from the other subjects at random. For each round, each participant has 20 units of money, and reports how much you may invest in the project (willingness to pay) to the planner of the project. All participants report their willingness to pay simultaneously. This experiment consists of 20 periods. The person you are paired with will change each round. No subject knows who they have been paired with either during or after the experiment.

1.2 G.2 Details

Two individuals (Individuals 1 and 2) are about to decide whether to invest in a certain project. Individual 1’s benefit gained when the project is undertaken is different from Individual 2’s one. The following table (Table 15) summarizes each individual’s benefits.

Table 15 Each individual’s benefits

In this experiment your role is:

Individual 1

The person you are paired with (Individual 2) is distributed similar instructions. The roles of all participants are invariant throughout the experiment.

To undertake this project, 20 units of money are required. Each individual has 20 units of money.

The planner of the project does not know each individual’s benefits gained from the project. Each individual will report how much he or she may invest in the project (willingness to pay) to the planner. You may report any integer number from 0 to 20. When you report your willingness to pay, you cannot observe how much the person you are paired with reports.

Based on the willingness to pay reported by each individual, (1) whether to undertake the project, (2) payment for each individual, and (3) score for each individual are determined.

• 1) Whether to undertake the project

  • If the sum of the willingness to pay reported by each individual is not less than 20, then the project is undertaken.

  • If the sum of the willingness to pay reported by each individual is less than 20, then the project is not undertaken.

• 2) Payment for each individual

  • If the project is undertaken, each individual pays the willingness to pay reported by him or her.

  • If the project is not undertaken, then the payment for each individual is 0.

• 3) Score for each individual

  • If the project is undertaken, then your score is equal to the sum of “the benefit from the project” and “the amount of money left.” Here “the amount of money left” is equal to $20-($your payment).

  • If the project is not undertaken, your score is equal to “the amount of money left.” Since your payment is 0, “the amount of money left” is equal to 20. Hence, your score is 20.

1.3 G.3 Examples

Here, we illustrate how to calculate each individual’s payment and score based on several numeral examples. Suppose that Individuals 1 and 2’s benefits are shown in the following table (Table 16). Note that these benefits are different from the benefits in the actual experiment.

Table 16 Each individual’s benefits

To undertake the project, 20 units of money are required. Each individual has 20 units of money.

In what follows, we consider four numerical examples. Then, for each example, we calculate 1) whether to undertake the project, 2) payment for each individual, and 3) score for each individual.

Example 1

Suppose that the willingness to pay reported by each individual is as follows:

• Individual 1: 7 units of money

• Individual 2: 9 units of money

• 1) Whether to undertake the project. The sum of the willingness to pay reported by each individual is $7+9=16$. Since the sum of the willingness to pay is less than 20, the project is not undertaken.

• 2) Payments

  • Individual 1: Since the project is not undertaken, his or her payment is 0.

  • Individual 2: Since the project is not undertaken, his or her payment is 0.

• 3) Scores

  • Individual 1: Since his or her payment is 0, the amount of money left is $20 - 0 = 20$. Hence, his or her score is 20.

  • Individual 2: Since his or her payment is 0, the amount of money left is $20 - 0 = 20$. Hence, his or her score is 20.

Example 2

Suppose that the willingness to pay reported by each individual is as follows:

• Individual 1: 7 units of money

• Individual 2: 12 units of money

• 1) Whether to undertake the project. The sum of the willingness to pay reported by each individual is $7+12=19$. Since the sum of the willingness to pay is less than 20, the project is not undertaken.

• 2) Payments

  • Individual 1: Since the project is not undertaken, his or her payment is 0.

  • Individual 2: Since the project is not undertaken, his or her payment is 0.

• 3) Scores

  • Individual 1: Since his or her payment is 0, the amount of money left is $20 - 0 = 20$. Hence, his or her score is 20.

  • Individual 2: Since his or her payment is 0, the amount of money left is $20 - 0 = 20$. Hence, his or her score is 20.

Example 3

Suppose that the willingness to pay reported by each individual is as follows:

• Individual 1: 11 units of money

• Individual 2: 12 units of money

• 1) Whether to undertake the project. The sum of the willingness to pay reported by each individual is $11+12=23$. Since the sum of the willingness to pay is not less than 20, the project is undertaken.

• 2) Payments

  • Individual 1: Since the project is undertaken, his or her payment is equal to the willingness to pay reported by him or her. Hence, his or her payment is 11.

  • Individual 2: Since the project is undertaken, his or her payment is equal to the willingness to pay reported by him or her. Hence, his or her payment is 12.

• 3) Scores

  • Individual 1: Since his or her payment is 11, the amount of money left is $20-11=9$. Hence, his or her score is his or her benefit $+$ the amount of money left $=$ $14+9=23$.

  • Individual 2: Since his or her payment is 12, the amount of money left is $20-12=8$. Hence, his or her score is his or her benefit $+$ the amount of money left $=$ $7+8=15$.

Example 4

Suppose that the willingness to pay reported by each individual is as follows:

• Individual 1: 11 units of money

• Individual 2: 9 units of money

• 1) Whether to undertake the project. The sum of the willingness to pay reported by each individual is $11+9=20$. Since the sum of the willingness to pay is not less than 20, the project is undertaken.

• 2) Payments

  • Individual 1: Since the project is undertaken, his or her payment is equal to the willingness to pay reported by him or her. Hence, his or her payment is 11.

  • Individual 2: Since the project is undertaken, his or her payment is equal to the willingness to pay reported by him or her. Hence, his or her payment is 9.

• 3) Scores

  • Individual 1: Since his or her payment is 11, the amount of money left is $20-11=9$. Hence, his or her score is his or her benefit $+$ the amount of money left $=$ $14+9=23$.

  • Individual 2: Since his or her payment is 12, the amount of money left is $20-9=11$. Hence, his or her score is his or her benefit $+$ the amount of money left $=$ $7+11=18$.

1.4 G.4 Operation

This experiment consists of 20 periods. Here, we will explain how to operate the computer you will use in period 1 of the experiment. The operation after period 1 is the same as the operation in period 1.

• 1. At the beginning of period 1, you will be paired with the person an experimenter will choose from the other subjects at random. Then, “Input one integer number from 0 to 20” is displayed on the screen of your computer. The following figure is an example of the screen.

• 2. After deciding how much you report your willingness to pay, you input that number in the cell labeled “The number you chose.” Then, click the “OK” button at the bottom of the screen. The time limit is 5 min.

• 3. After inputting the number you chose on your computer, fill out the number you chose in the column “Willingness to pay you chose” in the record sheet. Moreover, please fill out why you chose that number in the column “Reason for your decision” in the record sheet.

• 4. After all subjects click the “OK” button, period 1 is complete. Then, “Willingness to pay you chose,” “Willingness to pay the other person chose,” “Your payment,” “The other person’s payment,” “Your score,” and “The other person’s score” are displayed on the screen. We ask that you transcribe this information in the record sheet.

• 5. After the transcription, click the “NEXT” button at the bottom of the screen.

Once all subjects click the “NEXT” button, period 2 will start. At the beginning of period 2, you will be paired with the person an experimenter will choose from the other subjects at random. The operation after period 1 is the same as the operation in period 1. The experiment is complete when period 20 is complete.

1.5 G.5 Reward

After completing 20 periods, the experimenter selects one participant to draw a card from 20 cards. A number from 1 to 20 is shown on each card. The reward for each participant is determined based on his or her score on the period shown on the drawn card.

The method to calculate your reward is described by the following equation.

$${\mathbf{Your}} \, {\mathbf{Reward }} = {\mathbf{Your}}\,{\mathbf{Score}}\times {\mathbf{185}}\,{\mathbf{(yen) }}$$

1.6 G.6 Comprehension test

To check the level of understanding, we will now conduct a test. After you finish the test, please raise your hand. Then, the experimenter will check your answers. Please feel free to ask any questions you may have during or after the test. However, the experimenter may not able to answer your question depending on the content of your question.

Question. Individuals 1 and 2’s benefits are shown in Table 17 in Sect. G.1 (Details). To undertake this project, 20 units of money are required. Each individual has 20 units of money.

• 1) The willingness to pay reported by each individual is as follows:

  • Individual 1: 4 units of money

  • Individual 2: 15 units of money

  Please answer Individuals 1 and 2’s payments and scores.

• 2) The willingness to pay reported by each individual is as follows:

  • Individual 1: 18 units of money

  • Individual 2: 6 units of money

  Please answer Individuals 1 and 2’s payments and scores.

• 1) Individual 1’s payment:_—

  Individual 2’s payment:_—

  Individual 1’s score:_—

  Individual 2’s score:_—

• 2) Individual 1’s payment:_—

  Individual 2’s payment:_—

  Individual 1’s score:_—

  Individual 2’s score:_—

Table 17 Each individual’s benefits

After the test, you will now have 3 minutes to confirm the experimental instructions before the actual experiment. If you have any questions, please raise your hand.

H Experimental instructions (Treatment P)

Note: In this experiment, please remember that you cannot talk or communicate with other participants. If there is communication, the experiment will be stopped at that point.

The reward you receive after the experiment is decided according to the points you earn through the experiment. A detailed explanation of the reward you receive will be provided later in Sect. H.5 (Reward).

First, please confirm that the following items are on your desk. If any of the items are missing, please contact an experimenter.

• Instructions (this handout)

• Table 18

• Record sheet

• Calculation sheet

• Two pencils and one eraser

1.1 H.1 Overview

In this experiment, two participants (Individuals 1 and 2) are about to decide whether to invest in a certain project. Your role in this experiment will be either Individual 1 or Individual 2. At the beginning of each period, an experimenter will choose the person you are paired with from the other subjects at random. For each round, each participant has 20 units of money, and reports how much you may invest in the project (willingness to pay) to the planner of the project. All participants report their willingness to pay simultaneously. This experiment consists of 20 periods. The person you are paired with will change each round. No subject knows who they have been paired with either during or after the experiment.

1.2 H.2 Details

Two individuals (Individuals 1 and 2) are about to decide whether to invest in a certain project. Individual 1’s benefit gained when the project is undertaken is different from Individual 2’s one. The following table (Table 18) summarizes each individual’s benefits.

Table 18 Each individual’s benefits

In this experiment your role is:

Individual 1

The person you are paired with (Individual 2) is distributed similar instructions. The roles of all participants are invariant throughout the experiment.

To undertake this project, 20 units of money are required. Each individual has 20 units of money.

The planner of the project does not know each individual’s benefits gained from the project. Each individual will report how much he or she may invest in the project (willingness to pay) to the planner. You may report any integer number from 0 to 20. When you report your willingness to pay, you cannot observe how much the person you are paired with reports.

Based on the willingness to pay reported by each individual, (1) whether to undertake the project, 2) payment for each individual, and 3) score for each individual are determined.

• 1) Whether to undertake the project

  • If the sum of the willingness to pay reported by each individual is not less than 20, then the project is undertaken.

  • If the sum of the willingness to pay reported by each individual is less than 20, then the project is not undertaken.

• 2) Payment for each individual

  • If the project is undertaken, you (Individual 1) pay half of the project’s cost (10 units) and the “tax” described as follows.

    • A) If the willingness to pay reported by Individual 2 is not less than 10, then the amount of Individual 1’s tax is 0.

    • B) If the willingness to pay reported by Individual 2 is less than 10, then the amount of Individual 1’s tax is 10−(willingness to pay reported by Individual 2).

  • If the project is not undertaken, then you (Individual 1) pay the “tax” described as follows.

    • C) If the willingness to pay reported by Individual 2 is not less than 10, then the amount of Individual 1’s tax is (willingness to pay reported by Individual 2)−10.

    • D) If the willingness to pay reported by Individual 2 is less than 10, then the amount of Individual 1’s tax is 0.

• 3) Score for each individual

  • If the project is undertaken, your score is equal to the sum of “the benefit from the project” and “the amount of money left.” Here “the amount of money left is equal to 20−(your payment).

  • If the project is not undertaken, your score is equal to “the amount of money left.” Here “the amount of money left” is equal to 20−(your payment ).

1.3 H.3 Examples

Here, we illustrate how to calculate each individual’s payment and score based on several numeral examples. Suppose that Individuals 1 and 2’s benefits are shown in the following table (Table 19). Note that these benefits are different from the benefits in the actual experiment.

Table 19 Each individual’s benefits

To undertake the project, 20 units of money are required. Each participant has 20 units of money.

In what follows, we consider four numerical examples. Then, for each example, we calculate (1) whether to undertake the project, (2) payment for each individual, and (3) score for each individual.

Example 1

Suppose that the willingness to pay reported by each individual is as follows:

• Individual 1: 7 units of money

• Individual 2: 9 units of money

• 1) Whether to undertake the project. The sum of the willingness to pay reported by each individual is $7+9=16.$ Since the sum of the willingness to pay is less than 20, the project is not undertaken.

• 2) Payments

  • Individual 1: Since the project is not undertaken, he or she pays only the tax. Since the willingness to pay reported by Individual 2 is 9, which is less than 10, the amount of Individual 1’s tax is 0. Hence, Individual 1’s payment is 0.

  • Individual 2: Since the project is not undertaken, he or she pays only the tax. Since the willingness to pay reported by Individual 1 is 7, which is less than 10, the amount of Individual 2’s tax is 0. Hence, Individual 2’s payment is 0.

• 3) Scores

  • Individual 1: Since his or her payment is 0, the amount of money left is $20 - 0 = 20$. Hence, Individual 1’s score is 20.

  • Individual 2: Since his or her payment is 0, the amount of money left is $20 - 0 =20$. Hence, Individual 2’s score is 20.

Example 2

Suppose that the willingness to pay reported by each individual is as follows:

• Individual 1: 7 units of money

• Individual 2: 12 units of money

• 1) Whether to undertake the project. The sum of the willingness to pay reported by each individual is $7+12=19.$ Since the sum of the willingness to pay is less than 20, the project is not undertaken.

• 2) Payments

  • Individual 1: Since the project is not undertaken, he or she pays only the tax. Since the willingness to pay reported by Individual 2 is 12, which is not less than 10, the amount of Individual 1’s tax is $12-10=2$. Hence, Individual 1’s payment is 2.

  • Individual 2: Since the project is not undertaken, he or she pays only the tax. Since the willingness to pay reported by Individual 1 is 7, which is less than 10, the amount of Individual 2’s tax is 0. Hence, Individual 2’s payment is 0.

• 3) Scores

  • Individual 1: Since his or her payment is 2,  the amount of money left is $20-2=18$. Hence, Individual 1’s score is 18.

  • Individual 2: Since his or her payment is 0,  the amount of money left is $20-0=20$. Hence, Individual 2’s score is 20.

Example 3

Suppose that the willingness to pay reported by each individual is as follows:

• Individual 1: 11 units of money

• Individual 2: 12 units of money

• 1) Whether to undertake the project. The sum of the willingness to pay reported by each individual is $11+12=23$. Since the sum of the willingness to pay is not less than 20, the project is undertaken.

• 2) Payments

  • Individual 1: Since the project is undertaken, he or she pays half of the project’s cost (10 units) and the tax. Since the willingness to pay reported by the individual 2 is 12, which is not less than 10, the amount of Individual 1’s tax is 0. Hence, Individual 1’s payment is $10+0=10$.

  • Since the project is undertaken, he or she pays half of the project’s cost (10 units) and the tax. Since the willingness to pay reported by Individual 1 is 11, which is not less than 10, the amount of Individual 2’s tax is 0. Hence, Individual 2’s payment is $10+0=10$.

• 3) Scores

  • Individual 1: Since his or her payment is 10,  the amount of money left is $20-10=10$. Hence, Individual 1’s score is his or her benefit $+$ the amount of money left $=$ $14+10=24$.

  • Individual 2: Since his or her payment is 10,  the amount of money left is $20-10=10$. Hence, Individual 2’s score is his or her benefit $+$ the amount of money left $=$ $7+10=17$.

Example 4

Suppose that the willingness to pay reported by each individual is as follows:

• Individual 1: 11 units of money

• Individual 2: 9 units of money

• 1) Whether to undertake the project. The sum of the willingness to pay reported by each individual is $11+9=20.$ Since the sum of the willingness to pay reported is not less than 20, the project is undertaken.

• 2) Payments

  • Individual 1: Since the project is undertaken, he or she pays half of the project’s cost (10 units) and the tax. Since the willingness to pay reported by Individual 2 is 9, which is not less than 9, the amount of Individual 1’s tax is $10-9=1.$ Hence, Individual 1’s payment is $10+1=11.$

  • Since the project is undertaken, he or she pays half of the project’s cost (10 units) and the tax. Since the willingness to pay reported by Individual 1 is 11, which is not less than 10, the amount of Individual 2’s tax is 0. Hence, Individual 2’s payment is $10+0=10.$

• 3) Scores

  • Individual 1: Since his or her payment is 11,  the amount of money left is $20-11=9$. Hence, Individual 1’s score is his or her benefit $+$ the amount of money left $=$ $14+9=23$.

  • Individual 2: Since his or her payment is 10,  the amount of money left is $20-10=10$. Hence, Individual 2’s score is his or her benefit $+$ the amount of money left $=$ $7+10=17$.

1.4 H.4 Operation

This experiment consists of 20 periods. Here, we will explain how to operate the computer you will use in period 1 of the experiment. The operation after period 1 is the same as the operation in period 1.

• 1. At the beginning of period 1, you will be paired with the person an experimenter will choose from the other subjects at random. Then, “Input one integer number from 1 to 20” is displayed on the screen of your computer. The following figure is an example of the screen.

• 2. After deciding how much you report your willingness to pay, you input that number in the cell labeled “The number you chose.” Then, click the “OK” button at the bottom of the screen. The time limit is 5 minutes.

• 3. After inputting the number you chose on your computer, fill out the number you chose in the column “Willingness to pay you chose” in the record sheet. Moreover, please fill out why you chose that number in the column “Reason for your decision” in the record sheet.

• 4. After all subjects click the “OK” button, period 1 is complete. Then, “Willingness to pay you chose,” “Willingness to pay the other person chose,” “Your payment,” “The other person’s payment,” “Your score,” and “The other person’s score” are displayed on the screen. We ask that you transcribe this information in the record sheet.

• 5. After the transcription, click the “NEXT” button at the bottom of the screen.

Once all subjects click the “NEXT” button, period 2 will start. At the beginning of period 2, you will be paired with the person an experimenter will choose from the other subjects at random. The operation after period 1 is the same as the operation in period 1. The experiment is complete when period 20 is complete.

1.5 H.5 Reward

After completing 20 periods, the experimenter selects one participant to draw a card from 20 cards. A number from 1 to 20 is shown on each card. Reward for each participant is determined based on his or her score on the period shown on the drawn card.

Your reward is determined as follows.

$$ \mathbf{Your\, Reward} \,{=}\, \mathbf{Your\,Score} \, {\times }\, \mathbf{177} {+} \, \mathbf{531}\, (\bf{yen}) $$

1.6 H.6 Comprehension test

To check the level of understanding, we will now conduct a test. After you finish the test, please raise your hand. Then, the experimenter will check your answers. Please feel free to ask any questions you may have during or after the test. However, the experimenter may not able to answer your question depending on the content of your question.

Question. Individuals 1 and 2’s benefits are shown in Table 20 in Sect. H.1 (Details). To undertake this project, 20 units of money are required. Each individual has 20 units of money.

• 1) The willingness to pay reported by each individual is as follows:

  • Individual 1: 4 units of money

  • Individual 2: 15 units of money

  Please answer Individuals 1 and 2’s payments and scores.

• 2) The willingness to pay reported by each individual is as follows:

  • Individual 1: 18 units of money

  • Individual 2: 6 units of money

  Please answer Individuals 1 and 2’s payments and scores.

• 1) Individual 1’s payment:_—

  Individual 2’s payment:_—

  Individual 1’s score:_—

  Individual 2’s score:_—

• 2) Individual 1’s payment:_—

  Individual 2’s payment:_—

  Individual 1’s score:_—

  Individual 2’s score:_—

Table 20 Each individual’s benefits

After the test, you will now have 3 minutes to confirm the experimental instructions before the actual experiment. If you have any questions, please raise your hand.

I Experimental instructions (Treatments VT and PT)

Note: In this experiment, please remember that you cannot talk or communicate with other participants. If there is communication, the experiment will be stopped at that point.

First, please confirm that the following items are on your desk. If any of the items are missing, please contact an experimenter.

• Instructions (this handout)

• Payoff table for practice

• Record sheet for practice

• Pencil and eraser

1.1 I.1 Overview

In this experiment, at the beginning of each period, an experimenter will choose the person you are paired with from the other subjects at random. The person you are paired with will change each period. This experiment consists of 20 periods. As explained in more detail below, at the beginning of each period, you will make decisions using a computer assigned to you based on the “Payoff table.” You will choose one integer number from 0 to 20 in each period. No subject knows who they have been paired with either during or after the experiment.

The rewards you receive after the experiment is complete are proportional to the total payoffs you earn throughout the 20 periods of the experiment. A detailed explanation of the rewards you receive will be provided later in Sect. I.4 (Reward).

1.2 I.2 Details

You will choose one integer number from 0 to 20 in each period. Your payoff in each period is determined by the “Outcome” of the period. The outcome is determined by the number you choose as well as the number the other person chooses. Your “Payoff table” describes the relation between outcomes and the payoffs you earn.

Your “Payoff table” displays both your payoff and the other person’s payoff when the number you choose and the number the other person chooses are determined. The following table is a part of the “Payoff table for practice.”

The vertical line of the above table shows the number which you choose, and the horizontal line is the number which the other person chooses. The lower left-hand red number is your payoff, and the upper right-hand blue number is the other person’s payoff in each cell. The numbers in each cell are displayed in yen.

For example, suppose that the number you choose is “3,” and the number the other person chooses is “1.” In this case, your payoff is “140” when the vertical line is 3 and the horizontal line is 1. Then, the other person’s payoff is “100.”

The person you are paired with is chosen from the other subjects at random by an experimenter. The person you are paired with has a payoff table in which the vertical line and the horizontal line of your payoff table are reversed.

1.3 I.3 Operation

This experiment consists of 20 periods. Here, we will explain how to operate the computer you will use in period 1 of the experiment. The operation after period 1 is the same as the operation in period 1.

• 1. At the beginning of period 1, you will be paired with the person an experimenter will choose from the other subjects at random. Then, “Input one integer number from 0 to 20” is displayed on the screen of your computer. The following figure is an example of the screen.

• 2. Please look at the “Payoff table” and confirm the relation between the numbers you and the other person choose and the payoffs.

• 3. After deciding which number to choose, you input that number in the cell labeled “The number you chose.” Then, click the “OK” button at the bottom of the screen.

• 4. After inputting the number you chose on your computer, fill out the number you chose in the column “The number you chose” in the record sheet. Moreover, please fill out why you chose that number in the column “Reason for your decision” in the record sheet.

• 5. After all subjects click the “OK” button, period 1 is complete. Then, “The number you chose,” “The number the other person chose,” “Your payoff,” and “The other person’s payoff” are displayed on the screen. We ask that you transcribe this information in the record sheet.

• 6. After the transcription, click the “NEXT” button at the bottom of the screen.

Once all subjects click the “NEXT” button, period 2 will start. At the beginning of period 2, you will be paired with the person an experimenter will choose from the other subjects at random. The operation after period 1 is the same as the operation in period 1. The experiment is complete when period 20 is complete.

1.4 I.4 Reward

Your rewards are the sum of your payoffs over all 20 periods. For example, if the sum of your payoffs are 2,580, then your rewards are 2,580 yen. If you earn more payoffs in each period, then you receive more rewards.

This concludes our explanation. Next, you can practice making decisions using the “Payoff table for practice.” After practicing, you will have some time to look at the payoff table for the actual experiment before we begin the experiment. If you have any questions, please raise your hand.

Fig. 12

Payoff table of Type 1 distributed in Treatment VT

Fig. 13

Payoff table of Type 2 distributed in Treatment VT

Fig. 14

Payoff table of Type 1 distributed in Treatment PT

Fig. 15

Payoff table of Type 2 distributed in Treatment PT