Consider the housing tax example mentioned at the beginning, and suppose there are two possible world states \( \{L,H\}\) where \( L\) indicates the imposition of housing tax lowers the rent and \( H\) means that the housing tax makes the rent higher. Each resident votes between acceptance or rejection of the housing tax proposal. They do not see the world states, and each of them receives a signal from \( \{l,h\}\) which is correlated to the world state in the following way:

• if the actual world state is \( H\) , a resident receives signal \( h\) with probability \( 0.9\) and receives signal \( l\) with probability \( 0.1\) ;
• if the actual world state is \( L\) , a resident receives signal \( h\) with probability \( 0.4\) and receives signal \( l\) with probability \( 0.6\) .

For simplicity, suppose all the residents are the landlords, who would like to accept the housing tax proposal if the actual world state is \( H\) and would like to reject the proposal otherwise, and suppose the number of the residents/landlords is sufficiently large.

Consider the voting strategy profile where each resident votes for acceptance if receiving signal \( h\) and votes for rejection if receiving signal \( l\) . If the actual world state is \( H\) , about \( 90\%\) (which is more than \( 50\%\) ) of the residents will vote for acceptance, which leads to a good outcome (the house tax proposal will be accepted) for all the residents. If the actual world state is \( L\) , about \( 40\%\) (which is less than \( 50\%\) ) of the residents will vote for acceptance, and the outcome (the house tax proposal will be rejected) is again good for all the residents. Therefore, under this strategy profile, the alternative favored by the residents (i.e., the informed majority decision) wins with high probability.

However, the above strategy profile is not a Bayes Nash equilibrium: if the remaining residents vote following this strategy, a resident’s best response is to always vote for rejection due to the “pivotal voter reasoning”. To see this, a voter’s vote only matters if the remaining voters’ votes over the two alternatives are half-half distributed. Given that the remaining voters are voting according to their signals and the votes are half-half distributed, it is much more likely the actual world is \( L\) (in which case the expected distribution of the votes is \( 0.4\) versus \( 0.6\) , which is much closer to half-half compared with \( 0.9\) versus \( 0.1\) in the case where the actual world is \( H\) ). Therefore, if the remaining voters vote according to their signals, the best response is to always vote for rejection instead of voting according to the signal, and the above-mentioned strategy profile is not a Bayes Nash equilibrium.

When we are considering coalitional agents, we normally focus on groups of voters with non-negligible sizes and disregard the deviation of a single voter (whose behavior is very unlikely to affect the outcome of the election if the number of voters is large). The above-mentioned strategy profile is an (approximate) strong Bayes Nash equilibrium. This is because the “correct” alternative is output with probability approaching \( 1\) (when the number of voters approaches infinity) under this strategy profile, and no group of voters has an incentive to deviate (given that each voter’s “happiness” is almost maximized).

On the other hand, the strategy profile where all residents deterministically vote for acceptance is clearly a Bayes Nash equilibrium, as no resident’s deviation can possibly change the outcome (in this case, a voter can never be “pivotal”, and the pivotal voter analysis is trivial). However, this strategy profile is not a strong Bayes Nash equilibrium when talking about coalitional agents. Under this strategy profile, all residents do not favor the election outcome when the actual world is \( L\) . Therefore, the group of all residents can deviate and choose the vote-according-to-signal strategy profile instead, which is more beneficial to all of them.

A strategy profile \( \Sigma = (\sigma_1, …, \sigma_n)\) is an \( \epsilon\) -strong Bayes Nash equilibrium if no subset of agents \( D\) and alternative profile \( \Sigma' = (\sigma'_1, …, \sigma'_n)\) exist such that

1. \( \sigma_i = \sigma_i'\) for each \( i \not \in D\) ,
2. \( u_i(\Sigma') \ge u_i(\Sigma)\) for each \( i \in D\) ,
3. there exist \( i \in D\) such that \( u_i(\Sigma') > u_i(\Sigma) + \epsilon\) .

In the majority vote mechanism, a critical threshold exists for the majority proportion, denoted as \( \theta_\tt{{ maj}} = \frac1{2M}\) where

For this threshold, there exist functions \( \epsilon(n)\) , \( p(n)\) , and \( \gamma(n)\) all dependent on the number of agents \( n\) , such that as \( n \to \infty\) , \( \epsilon(n) \to 0\) , \( p(n) \to 1\) and \( \gamma(n)\) does not approach to \( 0\) . For these functions, the following two statements hold:

• If the majority proportion \( \alpha\) exceeds this threshold \( \theta_\tt{{ maj}}\) , i.e., \( \alpha > \theta_\tt{{ maj}}\) , an \( \epsilon(n)\) -strong Bayes Nash equilibrium exists. Furthermore, any such equilibrium leads to the informed majority decision with a probability of at least \( p(n)\) .
• Conversely, if the majority proportion is below this threshold, i.e., \( \alpha \le \theta_\tt{{ maj}}\) , there is no \( \gamma(n)\) -strong Bayes Nash equilibrium.

The strategy \( (\delta^*_{{l}},\delta^*_{{h}})\) that maximizes the function \( P(\cdot,\cdot)\) is

where \( P(\delta_{{l}},\delta_{{h}})=\min\{p_\mathbf{A}^{{H}}(\delta_{{l}},\delta_{{h}}),p_\mathbf{R}^{{L}}(\delta_{{l}},\delta_{{h}})\}\) for \( p_\mathbf{A}^{{H}}(\delta_{{l}},\delta_{{h}})\) and \( p_\mathbf{R}^{{L}}(\delta_{{l}},\delta_{{h}})\) defined in (2).

Consider the signal distributions in Table 1, where signal \( {{h}}\) is generally more common.

According to inequalities in (1), any strategy \( (\delta_{{l}},\delta_{{h}})\) satisfying \( \delta_{{h}} < \delta_{{l}} < 3\delta_{{h}}\) will ensure the wish of type-\( {{\mathcal{H}}}\) agents when no type-\( {{\mathcal{L}}}\) agents exist. For example, \( (\delta_{{l}},\delta_{{h}})\) can be \( (\frac{1}{6}, \frac{1}{8})\) . Then agents vote for \( \mathbf{A}\) with a probability of \( \frac12 + \frac18 = \frac{5}{8}\) upon signal \( {{h}}\) and with a probability of \( \frac12-\frac16=\frac{1}{3}\) upon signal \( {{l}}\) . Such a strategy makes the expected vote share of \( \mathbf{A}\) in state \( {{H}}\) , and that of \( \mathbf{R}\) in state \( {{L}}\) be \( 0.75\times \frac{5}{8} + 0.25\times\frac{1}{3} = \frac{53}{96}\) and \( 0.5\times\frac{3}{8} +0.5 \times\frac{2}{3} = \frac{25}{48}\) , respectively.

On the other hand, the optimal strategy for type-\( {{\mathcal{H}}}\) agents is \( (\delta_{{l}}^*,\delta_{{h}}^*) = (\frac{1}{2}, \frac{3}{10})\) . Type-\( {{\mathcal{H}}}\) agents deterministically vote for \( \mathbf{R}\) upon the rarer signal \( {{l}}\) . The expected vote share of \( \mathbf{A}\) in state \( {{H}}\) and that of \( \mathbf{R}\) in state \( {{L}}\) are both \( \frac{3}{5}\) .

The optimal strategy \( (\delta_{{l}}^*, \delta_{{h}}^*)\) satisfies \( p_\mathbf{A}^{{H}}(\delta_{{l}}^*, \delta_{{h}}^*) = p_\mathbf{R}^{{L}}(\delta_{{l}}^*, \delta_{{h}}^*)\) .

For configuration \( \{(\alpha_{{\mathcal{L}}}, \alpha_{{\mathcal{H}}}), (\mu, P_{{h}}^{{H}}, P_{{h}}^{{L}})\}\) where the majority proportion \( \alpha > \frac1{2M}\) and the number of agents \( n \to \infty\) , any strategy profile \( (\sigma_1, \sigma_2, …, \sigma_n)\) such that the majority type-\( {{\mathcal{H}}}\) agents adopt the optimal strategy forms a strong Bayes Nash equilibrium in the majority vote mechanism. Such equilibria lead to the informed majority decision with a probability of \( 1\) .

For the number of agents \( n \to \infty\) , any strong Bayes Nash equilibrium in the majority vote mechanism leads to the informed majority decision with a probability of \( 1\) .

For configuration \( \{(\alpha_{{\mathcal{L}}}, \alpha_{{\mathcal{H}}}), (\mu, P_{{h}}^{{H}}, P_{{h}}^{{L}})\}\) where the majority proportion \( \alpha \le \frac1{2M}\) and the number of agents \( n \to \infty\) , no strong Bayes Nash equilibrium exists in the majority vote mechanism.

Within signal distributions given in Table 1, the maximum vote share of the informed majority decision cast by the majority type agents is \( M = \frac{0.75}{0.5 + 0.75} = \frac35\) .

When the majority proportion, \( \alpha\) , is greater than \( \frac{1}{2M} = \frac56\) (and \( n \to \infty\) ), it forms a strong Bayes Nash equilibrium and aggregates the informed majority decision as long as the majority type-\( {{\mathcal{H}}}\) agents adopt the optimal strategy.

When the majority proportion \( \alpha\) is less than or equal to \( \frac56\) , there is no strong Bayes Nash equilibrium.

For configuration \( \{(\alpha_{{\mathcal{L}}}, \alpha_{{\mathcal{H}}}), (\mu, P_{{h}}^{{H}}, P_{{h}}^{{L}})\}\) where the majority proportion \( \alpha > \frac1{2M}\) , any strategy profile \( \Sigma = (\sigma_1, …, \sigma_n)\) such that the majority type-\( {{\mathcal{H}}}\) agents adopt the optimal strategy forms an \( \epsilon\) -strong Bayes Nash equilibrium in the majority vote mechanism, where \( \epsilon=2B^2 \exp(-2c^2\lfloor\alpha n\rfloor)\) and \( c\) is a constant defined by \( c = \frac{1}{3}(\alpha M - \frac{1}{2})\) .

When the majority proportion \( \alpha > \frac1{2M}\) , if the majority type-\( {{\mathcal{H}}}\) agents adopt the optimal strategy in (3), then the informed majority decision wins the majority vote with probability at least \( 1 - 2\exp(-2c^2\lfloor\alpha n\rfloor)\) .

When the majority proportion \( \alpha > \frac1{2M}\) , consider any strategy profile \( \Sigma\) such that the majority type-\( {{\mathcal{H}}}\) agents adopt the optimal strategy in (3). Let \( i_1\) be an arbitrary \( {{\mathcal{H}}}\) -type agent (i.e., \( t_{i_1} = {{\mathcal{H}}}\) ) and \( i_2\) be an arbitrary \( {{\mathcal{L}}}\) -type agent (i.e., \( t_{i_2} = {{\mathcal{L}}}\) ). Then for any \( \epsilon \ge 2B^2 \exp(-2c^2\lfloor\alpha n\rfloor)\) and any strategy profile \( \Sigma'\) , neither of the following conditions holds in the majority vote mechanism:

1. (1) \( u_{i_1}(\Sigma') - u_{i_1}(\Sigma) > \epsilon\text{ and }u_{i_2}(\Sigma') - u_{i_2}(\Sigma) \ge 0\) .
2. (2) \( u_{i_2}(\Sigma') - u_{i_2}(\Sigma) > \epsilon\text{ and }u_{i_1}(\Sigma') - u_{i_1}(\Sigma) \ge 0\) .

Consider a scenario where both states are equally probable (i.e., \( \mu = \frac{1}{2}\) ). The signal distribution is shown in Table 1. We have three agents, two of type-\( {{\mathcal{H}}}\) and one of type-\( {{\mathcal{L}}}\) . Their utilities are detailed in Table 2 and Table 3.

Consider the following two strategy profiles:

• Strategy Profile \( \Sigma\) : All agents vote for \( \mathbf{A}\) on signal \( {{h}}\) , and \( \mathbf{R}\) on signal \( {{l}}\) .
• Strategy Profile \( \Sigma'\) : All agents adopt an uninformative strategy. They vote for \( \mathbf{A}\) with a probability of \( \frac13\) and vote for \( \mathbf{R}\) otherwise.

Under strategy profile \( \Sigma\) , the probability of the majority vote mechanism outputting \( \mathbf{A}\) in state \( {{L}}\) is \( \lambda_\mathbf{A}^{{L}}(\Sigma) = \frac{1}{2}\) and the probability of outputting \( \mathbf{A}\) in state \( {{H}}\) is \( \lambda_\mathbf{A}^{{H}}(\Sigma) = \frac{27}{32}\) .

Under strategy profile \( \Sigma'\) , the probability of the majority vote mechanism outputting \( \mathbf{A}\) in both states is \( \lambda_\mathbf{A}^{{H}}(\Sigma') = \lambda_\mathbf{A}^{{H}}(\Sigma') = \frac{7}{27}\) .

In the shifting from \( \Sigma\) to \( \Sigma'\) , the change in the probability of outputting \( \mathbf{A}\) in state \( {{L}}\) is \( (1-\mu)(\lambda^{{L}}_\mathbf{A}(\Sigma') - \lambda^{{L}}_\mathbf{A}(\Sigma)) = \frac{1}{2}\times(\frac{7}{27} - \frac{1}{2})\approx -0.12\) . That change in state \( {{H}}\) is \( \mu(\lambda^{{H}}_\mathbf{A}(\Sigma') - \lambda^{{H}}_\mathbf{A}(\Sigma)) = \frac{1}{2}\times(\frac{7}{27} - \frac{27}{32})\approx -0.29\) .

For an agent \( i_1\) of type-\( {{\mathcal{H}}}\) , by Equation (5), this shift yields an ex-ante utility difference of \( \Delta u_{i_1}(\Sigma, \Sigma') \approx -0.29 \times 1 - (-0.12)\times 5 = 0.31 > 0\) . For an agent \( i_2\) of type-\( {{\mathcal{L}}}\) , by Equation (6), the ex-ante utility difference is \( \Delta u_{i_2}(\Sigma, \Sigma') \approx -0.12\times1 - (-0.29) \times 5 = 1.33 > 0\) . Both types of agents find it beneficial to accept the change in strategy, despite their antagonistic preferences.

Any \( \epsilon\) -strong Bayes Nash equilibrium in the majority vote mechanism leads to the informed majority decision with a probability of at least \( 1 - 2\epsilon\) , where \( \epsilon = B\exp(-2c^2n)\) and constant \( c\) is defined as \( \frac{1}{2}(2\alpha - 1)(M - \frac{1}{2})\) .

For configuration \( \{(\alpha_{{\mathcal{L}}}, \alpha_{{\mathcal{H}}}), (\mu, P_{{h}}^{{H}}, P_{{h}}^{{L}})\}\) where the majority proportion \( \alpha \le \frac1{2M}\) , no \( \epsilon\) -strong Bayes Nash equilibrium exists in the majority vote mechanism. Here, \( \epsilon\) is defined as \( \epsilon = \frac{1}{4}\min\{\mu, 1-\mu\} - 2B^2\exp(-2c^2n)\) and constant \( c\) is \( \frac{1}{2}(2\alpha - 1)\min\{M - \frac{1}{2}, \frac{1}{2} - \alpha M\}\) .

There exists an anonymous mechanism \( \mathcal{M}\) with a critical threshold \( \theta^\ast = \frac1{\Delta + 1}\) for the majority proportion such that

• If majority proportion \( \alpha\) exceeds this threshold \( \theta^\ast\) , i.e., \( \alpha > \theta^\ast\) , then truthful reporting becomes an \( \epsilon(n)\) -strong Bayes Nash equilibrium and this equilibrium ensures the informed majority decision with probability at least \( p(n)\) , where \( \epsilon(n) \to 0\) and \( p(n) \to 1\) as \( n \to \infty\) .

Our proposed mechanism provides a solution that aggregates the informed majority decision when agents truthfully report. However, there is no guarantee of agents’ truthful reporting when \( \alpha \le \theta^\ast\) . Our impossibility results in Section 4.3 show that, in such cases, under some mild assumptions, any anonymous mechanism cannot lead to an \( \epsilon\) -strong Bayes Nash equilibrium that aggregates the informed majority decision. This complements the positive result in Theorem 2.

For any configuration \( \{(\alpha_{{\mathcal{L}}}, \alpha_{{\mathcal{H}}}), (\mu, P_{{h}}^{{H}}, P_{{h}}^{{L}})\}\) , it holds that \( \theta_\tt{{ maj}} \ge \theta^\ast\) , i.e., \( \frac1{2M} \ge \frac1{\Delta + 1}\) . Particularly, the equality holds when \( P_{{h}}^{{L}} + P_{{h}}^{{H}} = 1\) .

Given the signal distribution in Table 1 and a prior probability of state \( \mu = \frac{1}{2}\) , the thresholds for our mechanism and Schoenebeck and Tao’s mechanism are calculated as follows.

If all the majority agents report truthfully, the threshold \( \delta\) of ours is \( \frac{P_{{{l}}}^{{L}}}{P_{{{l}}}^{{L}} + P_{{{h}}}^{{H}}} = \frac{0.5}{0.5 + 0.75} = \frac25\) . In contrast, in Schoenebeck and Tao’s mechanism, the threshold values reported by agents are their posterior estimations about the frequency of signal \( {{l}}\) . These estimates come from agents’ beliefs about the true world state, which is either \( \Pr[\omega = {{H}} \mid S_i = {{l}}] = \frac13\) or \( \Pr[\omega = {{H}} \mid S_i = {{h}}] = \frac35\) . So the threshold values are either \( \Pr[s_j = {{l}} \mid S_i = {{l}}] = \frac{1}{3}\times0.25+ \frac{2}{3}\times 0.5 = \frac{5}{12}\) or \( \Pr[s_j = {{l}} \mid S_i = {{h}}] = \frac{3}{5}\times0.25 + \frac{2}{5}\times 0.5 = \frac{7}{20}\) . The collective threshold \( \delta'\) is the median of all the reported thresholds, so it must lie in the range between \( \frac{7}{20}\) and \( \frac{5}{12}\) .

When the true world state is \( L\) , the frequency of signal \( {{l}}\) , which is \( 0.5\) , lies above both the threshold \( \delta\) and \( \delta'\) . On the other hand, the frequency of signal \( {{l}}\) , which is \( 0.25\) , is below the thresholds in state \( {{H}}\) .

The reporting frequency of \( {{l}}\) has the same properties as long as the proportion of the majority type, \( \alpha\) , is large enough. However, when the majority proportion is smaller, for example, when \( \alpha = 0.84\) and \( n\to \infty\) , the reporting frequency of \( {{l}}\) in state \( {{H}}\) may exceed the threshold in Schoenebeck and Tao’s mechanism. That is, when the minority type agents uninformatively claim they receive signal \( {{l}}\) , the reporting frequency of \( {{l}}\) , i.e., \( \alpha P^{{H}}_{{l}} + 1-\alpha = 0.84\times 0.25 + 0.16 = 0.37\) , may exceed the threshold \( \delta'\) , which is in the range between \( \frac{7}{20}=0.35\) and \( \frac{5}{12}\approx0.42\) .

On the other hand, our mechanism correctly infers the true world state if \( \alpha > \frac45\) . In the above case, the reporting rate of signal \( {{l}}\) in state \( {{L}}\) and state \( {{H}}\) are \( 0.84\times0.5 + 0.16=0.58\) and \( 0.37\) respectively. Our threshold \( \delta = \frac{2}{5}\) succeeds in distinguishing two worlds.

Furthermore, for \( \alpha > \frac1{\Delta + 1}\) , we have the following inequalities:

If all agents report truthfully, our mechanism will output the informed majority decision with probability at least \( 1 - 2\exp(-2c^2n)\) . The constant \( c\) is defined as

For configuration \( \{(\alpha_{{\mathcal{L}}}, \alpha_{{\mathcal{H}}}), (\mu, P_{{h}}^{{H}}, P_{{h}}^{{L}})\}\) where the majority proportion \(\alpha > \frac1{\Delta + 1}\) and large enough \( n\) , truthful reporting forms an \( \epsilon\) -strong Bayes Nash equilibrium in our mechanism. Here, we define \( \epsilon\) as \(\epsilon = \max \{B\exp(-2c^2\lfloor\alpha n\rfloor), 2B^2\exp(-2c^2n)\}\), and define the constant \( c\) as

Let \( \Sigma^*\) be the truthful reporting strategy profile. Let \( i_1\) be an arbitrary type-\( {{\mathcal{H}}}\) agent and \( i_2\) be an arbitrary type-\( {{\mathcal{L}}}\) agent. For any \( \epsilon \ge 2B^2 \exp(-2c^2n)\) and any strategy profile \( \Sigma'\) , neither of the following conditions holds in our mechanism:

• (1) \( u_{i_1}(\Sigma') - u_{i_1}(\Sigma^*) > \epsilon\text{ and }u_{i_2}(\Sigma') - u_{i_2}(\Sigma^*) \ge 0\) .
• (2) \( u_{i_2}(\Sigma') - u_{i_2}(\Sigma^*) > \epsilon\text{ and }u_{i_1}(\Sigma') - u_{i_1}(\Sigma^*) \ge 0\) .

The above results show that when \( \alpha > \frac1{\Delta + 1}\) , our mechanism can lead to a “good” strong Bayes Nash equilibrium that aggregates the informed majority decision. Not only that, with \( \alpha > \frac1{\Delta + 1}\) , we can show all the equilibria of our mechanism are “good”, using the same technique for proving Lemma 6.

For configuration \( \{(\alpha_{{\mathcal{L}}}, \alpha_{{\mathcal{H}}}), (\mu, P_{{h}}^{{H}}, P_{{h}}^{{L}})\}\) where the majority proportion \( \alpha \le \frac1{\Delta + 1}\) , \( 0 < \mu < 1, P_{{h}}^{{H}} = 1/2 + \Delta/2\) , and \( P_{{h}}^{{L}} = 1/2 - \Delta/2\) , even the number of agents \( n \to \infty\) , in any anonymous mechanism \( \mathcal{M}\) , truthful reporting cannot form an \( \epsilon\) -strong Bayes Nash equilibrium that leads to the informed majority decision with probability \( 1\) . Here, \( \epsilon\) is defined as \( \epsilon= \frac14\min\{\mu, 1-\mu\}\) .

For \( 0 < \alpha \le 1/(\Delta + 1)\) , \( q(n) =(1 - \Delta\cdot\lfloor\alpha n\rfloor/\lceil(1 - \alpha)n\rceil)/2\) , and \( C_1, C_2\) corresponding to the counting variables in two environments, the total variation distance between \( C_1\) and \( C_2\) is at most \( o(1)\) when \( n \to \infty\) .

Given the conditions

• \( 0 < \alpha \le 1/(\Delta + 1)\) ,
• and \( q(n) =(1 - \Delta\cdot\lfloor\alpha n\rfloor/\lceil(1 - \alpha)n\rceil)/2\) ,

let \( C_1\) be the sum of random variables \( X_1\) and \( Y_1\) , let \( C_2\) be the sum of random variables\( X_2\) and \( Y_2\) where

• \( X_1\) follows binomial distribution \( \mathbf{Bin}(\lfloor \alpha \cdot n\rfloor, 1/2 + \Delta/2)\) ,
• \( Y_1\) follows binomial distribution \( \mathbf{Bin}(\lceil (1-\alpha)\cdot n\rceil, q)\) ,
• \( X_2\) follows \( \mathbf{Bin}(\lfloor \alpha \cdot n\rfloor, 1/2 - \Delta/2)\) ,
• \( Y_2\) follows \( \mathbf{Bin}(\lceil (1-\alpha)\cdot n\rceil, 1-q)\) ,
• \( X_1\) , \( X_2\) , \( Y_1\) and \( Y_2\) are mutually independent.

As \( n\) goes to infinity, the total variation distance between \( C_1\) and \( C_2\) is \( O(\frac{1}{\sqrt{n}})\) .

Let \( Z_{\mu, \sigma^2}\) be the discretization of the Gaussian distribution \( N(\mu,\sigma^2)\) . The probability distribution function of \( Z_{\mu, \sigma^2}\) is given as follows:

where \( Z\) follows the standard normal distribution \( N(0,1)\) .

Let \( X\) be a random variable that follows a binomial distribution \( \mathbf{Bin}(n, p)\) . Define \( \mu\) and \( \sigma^2\) as the mean and variance of \( X\) , where \( \mu = np\) , \( \sigma^2 = np(1-p)\) . The total variation distance between the binomial distribution \( X\) and its corresponding discretized Gaussian distribution \( Z_{\mu, \sigma^2}\) is at most \( \frac{7.6}{\sigma}Ō(\frac{1}{\sqrt{n}})\) .

Consider four random variables \( X_1\) , \( X_2\) , \( Y_1\) , and \( Y_2\) where \( X_1\) is independent of \( Y_1\) , and \( X_2\) is independent of \( Y_2\) . The total variation distance between the sums \( X_1 + Y_1\) and \( X_2 + Y_2\) is bounded by the combined total variation distance of \( X_1\) and \( X_2\) , and \( Y_1\) and \( Y_2\) . Formally,

For configuration \( \{(\alpha_{{\mathcal{L}}}, \alpha_{{\mathcal{H}}}), (\mu, P_{{h}}^{{H}}, P_{{h}}^{{L}})\}\) where the majority proportion \( \alpha \le \frac1{\Delta + 1}\) , \( 0 < \mu < 1, P_{{h}}^{{H}} = 1/2 + \Delta/2\) , and \( P_{{h}}^{{L}} = 1/2 - \Delta/2\) , even the number of agents \( n \to \infty\) , in any anonymous mechanism \( \mathcal{M}\) , no symmetric \( \epsilon\) -strong Bayes Nash equilibrium can guarantee the informed majority decision with probability \( 1\) . Here, we define \( \epsilon\) as \( \epsilon= \frac14\min\{\mu, 1-\mu\}\) .