Voting and Communication When Hiring by Committee

Admittedly, if \(h=n+1\), that is, if there exists partisan voters who are never willing to reappoint the agent irrespective of the number of completed tasks, then the informative equilibrium does not exist. We have, however, excluded this case by assuming \(h \le n\).

To see that these are the only possible non-responsive equilibria of the game, assume the contrary. That is, assume the game possesses a symmetric, non-responsive equilibrium in which all the principals do not unanimously vote to fire nor all the principals unanimously vote to reappoint. Then the equilibrium features types \(\theta ^s\) and \(\tilde{\theta }^{s}\), where \(\theta ^s, \,\tilde{\theta }^s \in \varTheta \), and \(\theta ^s \ne \tilde{\theta }^s\) such that \(\upsilon (\theta ^s) \ne \upsilon (\tilde{\theta }^s)\). Since the voting strategy profile is non-responsive, \(\sigma (c)=0\) for all \(c > 0\), and \(\sigma (c)=n\), otherwise. Therefore, each principal who observes an omitted task knows that the agent has not completed any of the tasks. Similarly, each principal who has observed a completed task knows that the agent has completed all the tasks. Since the principals do not know each other’s types, the probability that the vote of an individual principal is decisive for the final outcome, or pivotal, is always positive. As a result, each principal who has observed an omitted (completed) task has strict incentives to vote to fire (reappoint) with probability one. But then the voting strategy profile is responsive. A contradiction.

To see this, note first that if \(\upsilon (h^1) = \upsilon (h^0)\), then the equilibrium is not, by definition, responsive. On the other hand, if it were that \(\upsilon (h^1) < \upsilon (h^0)\), then no agent with positive cost parameter would complete any of the tasks. Therefore, all the principals who receive a bad signal know for sure that none of the tasks has been completed and as a result, \(\upsilon (h^0)=0\). A contradiction.

Note that the same logic does not apply to our original game without the perturbations. Specifically, if all the other principals vote to fire (reappoint), then no individual principal can change the voting outcome given \(q > 1\) (\(q < n\)), and might as well vote along with the others independent of the signal she has received.

In particular, when \(q \ge h\), the game potentially possesses both the informative, and a symmetric, responsive, non-informative equilibrium.

To see why this is the only possible l -pooling equilibrium in pure strategies, note that by definition, in any l -pooling equilibrium, \(\upsilon (l^0)=\upsilon (l^1)=1\). Therefore, in order for the equilibrium to be responsive and in pure strategies, we must have \(0=\upsilon (h^0)<\upsilon (h^1)=1\). In particular, note that a voting strategy profile characterized by \(\upsilon (l^s)=1\) and \(\upsilon (h^s)=0\), for \(s \in \{0, 1\}\) is not responsive.

If we allowed also partially revealing message strategy profiles, or considered more complicated forms of communication – for instance, sequential – then the assumption of binary message space would be restrictive.

To see this, note that if \(\Pr (r \mid {\varvec{s}})\) is constant in \({\varvec{s}}\), then the equilibrium is not, by definition, responsive. On the other hand, if it were that \(\Pr (r \mid {\varvec{s}})\) is decreasing in \({\varvec{s}}\), then no agent with positive cost parameter would complete any of the tasks. Therefore, all the principals who receive a bad signal know for sure that none of the tasks has been completed and vote to fire. But then \(\Pr (r \mid {\varvec{s}})\) cannot be decreasing in s.

Notice that a voting strategy profile in which types \(h^0\) and \(l^1\) randomize cannot constitute an equilibrium. To see this, notice that if it were the case that \(\upsilon (\theta ^s) \in (0, 1)\) for \(\theta ^s \in \{h^0, l^1\}\), then the monotonicity of the voting strategies in thresholds imply that \(\upsilon (h^1)=0\) and \(\upsilon (l^0)=1\). But such a voting strategy profile is not be monotone in signals and therefore cannot be an equilibrium profile.