## 1. Introduction

## 2. The Setting

#### 2.1. The Model

**Definition**

**1.**

**Informationally Efficient Equilibrium.**A limit equilibrium ${\sigma}^{*}$ of an economy $\mathcal{E}$ is informationally efficient if it is true that ${P}_{{\sigma}^{*}}({A}_{i}|{\omega}_{i})=1\phantom{\rule{5.0pt}{0ex}}\forall i$.

#### 2.2. Voter Behaviour

## 3. Results

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Definition**

**2.**

**Informative Types.**For all ${\omega}_{i}$ for which there exists one ${t}^{\prime}$ such that ${q}_{i}(t)\ge {q}_{j}(t)\phantom{\rule{5.0pt}{0ex}}\forall j,j\ne i,$ and all $t\in \mathbf{T}\setminus {t}^{\prime}$, there exists at least one ${\omega}_{k}$ such that ${q}_{i}({t}^{\prime})<{q}_{k}({t}^{\prime})$.

**Theorem**

**3.**

**Proof.**

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Magnitude Theorem and Offset Theorem

#### Appendix A.1. Magnitude Theorem

#### Appendix A.2. Offset Theorem

## References

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1 | Poisson games were first introduced by Myerson [1]. |

2 | The original CJT, introduced by Condorcet [3], is as follows: In a two-alternative election of the above kind, in which voters simply vote for their private information (informative voting), the probability of the correct decision by a committee is always larger than that by a single voter alone, and this probability converges to one if the number of voters converges to infinity. Considering strategic voting, similar results have been shown for small and large two-alternative elections by Wit [4], Feddersen and Pesendorfer [5,6,7,8], Bouton and Castanheira [9], among others.Bhattacharya [10] shows that information aggregation is no longer efficient in two-alternative elections if voter preferences are sufficiently different. |

3 | The familiar reader may wonder how this relates to McLennan [12], who shows for Bayesian games with common preferences and uncertainty about the state of nature, but no population uncertainty, that a strategy that maximises the common utility function is necessarily a Nash equilibrium of the game. McLennan is concerned with the relation between the solution to the maximisation problem and the concept of a Nash equilibrium. In our game, however, we are also interested in whether the strategy satisfies a certain benchmark: efficiency. Besides certain differences in the model (e.g., Bayesian versus Poisson game), McLennan makes no statement about the efficiency of the strategy. It is not necessarily true that the solution to the maximisation problem satisfies our efficiency benchmark. |

4 | Population uncertainty implies that a voter does not know precisely how many other voters there are in the game. This is different from uncertainty about the number of voters that abstain; some of the voters that are in the game may decide to abstain. |

5 | This is an important difference between a Poisson voting game and voting games without population uncertainty: Equilibria in which all voters vote for the same alternative cannot be ruled out by eliminating weakly dominated strategies. So, they remain a viable possibility. |

6 | |

7 | It is possible to assume another tie-breaking rule, such as a fair coin flip, but this makes computations of expected utilities much more cumbersome and computations unnecessarily long. |

8 | Extended Poisson voting games are first introduced in Myerson [2]. |

9 | According to the Law of Large Numbers, the whole mass of probability concentrates in arbitrarily close neighbourhoods around the expected outcomes as $n\to \infty $. |

10 | The second step of the proof is reminiscent of McLennan [12]’s statement (see footnote 3). The important step of our proof is Step 1 which establishes the existence of an efficient strategy. Once we establish that, we use an argument similar to McLennan [12] to argue that this implies the existence of an efficient equilibrium. |

11 | For the sake of this proof, this can easily be seen from the expressions for $pi{v}^{0}$ and $pi{v}^{1}$. In general, the offset-theorem (see Appendix A) can be used to evaluate these types of limits, especially when the different pivotal events have the same magnitude. |

12 | In a model with partisans and simple scoring rules, Goertz and Maniquet [13] point in a similar direction. Of course, their model is different because it includes partisans. In their model, the partisans are the reason why no simple scoring rule aggregates information efficiently. |

13 | This can easily be derived from Equation (3). |

14 | This immediately follows from an inspection of Equation (3). |

15 | The ratio of the probabilities of two pivotal events that occur in the same state of nature but have different magnitudes converges to zero or to infinity because one event is infinitely more likely than the other. |

16 | There are cases in which the most likely subevent does not take the simple form stated above, but these cases do not arise in our analysis. |

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