Conformity Preferences and Information Gathering Effort in Collective Decision Making

Proof

Supposing that e1=e2=e, and that voter 2 follows his or her signal. When voter 1 has received s1=−h, v1=Y yields an expected payoff 12(1+e)×12(1−e)×G_+12(1−e)×12(1+e)×G‾−be2 that is equal to 1−e24(G‾+G_)−be2 and v1=N yields an expected payoff that is equals to −be2, because of G‾+G_<0, therefore v1=N dominates v1=Y.

When voter 1 has received s1=h, v1=Y yields an expected payoff 12(1+e)×12(1+e)×G‾+12(1−e)×12(1−e)×G_−be2 that is equal to 1+e24(G‾+G_)+e2(G‾−G_)−be2 and v1=N yields an expected payoff that is equal to −be2, therefore v1=Y dominates v1=N if 1+e24(G‾+G_)+e2(G‾−G_)>0 . The analogous argument applies to voter 2.

All in all, supposing a level of effort of e=e1=e2 so that 1+e24(G‾+G_)+e2(G‾−G_)>0, it is optimal for voter i to vote in line with his or her signal: if s1=h, then v1=Y, and if s1=−h, v1=N, given that the other voter votes in line with his or her signal.

Proof

We suppose that e1=e2=e and conformist voter 2 follows his or her signal. When voter 1 has received s1=−h, v1=Y yields an expected payoff of 12(1+e)×12(1−e)×(G_+k)+12(1−e)×12(1+e)×(G‾+k)−be2 that is equal to 1−e24(G‾+G_)+1−e22k−be2, and v1=N yields an expected payoff of 12(1+e)×12(1+e)×k+12(1−e)×12(1−e)×k−be2 that is equal to 1+e22k−be2, because of 1−e24(G‾+G_)<0 and 1−e22k<1+e22k, therefore v1=N dominates v1=Y.

When voter 1 has received s1=h, v1=Y yields an expected payoff of 12(1+e)×12(1+e)×(G‾+k)+12(1−e)×12(1−e)×(G_+k) that is equal to 1+e24(G‾+G_)+e2(G‾−G_)+1−e22k−be2, and v1=N yields an expected payoff of 12(1+e)×12(1−e)×k+12(1−e)×12(1+e)×k−be2 that is equal to 1−e22k−be2, therefore v1=Y dominates v1=N if 1+e24(G‾+G_)+e2(G‾−G_)+e2k>0. The analogous argument applies to voter 2.

All in all, let us assume that a level of effort of e1=e2=e so that 1+e24(G‾+G_)+e2(G‾−G_)+e2k>0, it is optimal for voter i to vote in line with his or her signal: if s1=h, then v1=Y, and if s1=−h, v1=N, given that the other voter votes in line with his or her signal.

Because 1+e24(G‾+G_)+e2(G‾−G_)>0, 1+e24(G‾+G_)+e2(G‾−G_)+e2k>0. Consequently, from Lemma 1, we note that when sincere voting with nonconformist voters exists, sincere voting with conformist voters also exists.

Proof

We suppose that k>0 and the level of effort eC∗ from the function (13), where eC∗=G‾−G_16b−4k−(G‾+G_) and eC∗∈[0,1], satisfies the condition for sincere voting in Lemma 2 (i.e. 1+(eC∗)24(G‾+G_)+eC∗2(G‾−G_)+(eC∗)2k>0). Then, using the functions S1C(eC∗)=S2C(eC∗), it is easy to calculate the total expected social surplus, (SC(eC∗)):

We obtain SC′(ei):

Therefore

Consequently, if G‾+G_2+2k−4b>0,SC′′(ei)>0, and if G‾+G_2+2k−4b<0,SC′′(ei)<0. Therefore, SC′(ei) increases with ei if G‾+G_2+2k−4b>0 and SC′(ei) decreases with ei if G‾+G_2+2k−4b<0.

We assume that eC∗∗ leads to SC′(eC∗∗)=0:

We obtain

From the function S1C′(eC∗)=0, i.e. (G‾−G_2+G‾+G_2+2k)eC∗−8beC∗=0. Moreover, we note that because G‾−G_>0, eC∗/=0. Consequently,

Consequently eC∗∗, which is derived from the function SC′(eC∗∗)=0, cannot be equal to eC∗, because the inequality, SC′(eC∗)>0, is always right. This shows that even when voters have a normative conformity preference, the optimum entire social benefit of collecting information could never be achieved by maximizing each conformist voter’s private benefits. In practice, eC∗ is greater than eC∗∗ if G‾+G_2+2k−4b>0 and eC∗ is smaller than eC∗∗ if G‾+G_2+2k−4b>0. All in all,

In this way, from a social perspective, voters exert more effort when 4k<8b−(G‾+G_), and voters exert less effort when eN∗∗.

We now look for the optimal conformity preference level that resolves the free-rider problem in the benchmark. From the eq. (7) for eC∗ and the eq. (13) for k⪌2b,,

if

it is obvious that

where eC∗.

Furthermore, we assume that the effort level 1+(eC∗)24(G‾+G_)+eC∗2(G‾−G_)>0 satisfies the condition for sincere voting in Lemma 1, i.e. 1+(eC*)24(G¯+G_)+eC*2(G¯−G_)>0. We have therefore proved Theorem 1.