Study Notes on Jury Theorems

Course Title: POLS 3220: How to Predict the Future
Date: September 30, 2025
Focus: Understanding methods for making predictions using principles from group decision-making theories.

The exercise began with an introductory survey, gathering results from students to assess their predictive accuracy.
Participants were queried about their performance, with a scoring format where each could answer out of 40 questions.
Scores revealed varied accuracy levels among students indicating a range of knowledge and confidence in predictions.

The session prompted students to consider how the answers might differ if the entire class voted on questions.
Questions included:
- "How many correct answers do you think the majority would get?"
- "If only students who were extremely confident in their answers (>95%) were included in voting, would that produce a better result?"

The term Wisdom of Crowds describes the phenomenon where groups often outperform the individuals that compose them. This principle asserts that collective decision-making can yield superior results over individual assessments.
Upcoming discussions will focus on conditions that enable the Wisdom of Crowds to enhance prediction accuracy and when crowd judgments may lead to worse outcomes (termed Madness of Crowds).

An influential philosopher and mathematician recognized for developing the Condorcet Jury Theorem, which describes the effectiveness of majority rule in decision-making scenarios.

Assumption 1: A group is tasked with making a binary (Yes/No) decision.
Assumption 2: Each individual voter possesses a probability, denoted by $p$ , greater than $rac{1}{2}$ , indicating their likelihood of making the correct choice.
Assumption 3: Individual votes are conducted independently.

The theorem states that as the number of voters in the group, represented as $n$ , becomes large:
- The probability that the majority makes the correct decision approaches 100%.

Students are encouraged to ponder if the Condorcet Jury Theorem reminds them of similar models or theories within their broader studies.

The Median Voter Theorem, as articulated by Black (1948), posits that in any dichotomous voting situation, the median value is the position that cannot be marginalized by any majority vote.
- This concept arises from the definition of the median, where:
- 50% of the group is positioned to the left.
- 50% is positioned to the right.
- Thus, any voting bloc securing a majority (>50%) must include the median voter.
The majority judgment can thus be conceptualized as the median voter's position, amplifying the importance of understanding voter distribution.

In scenarios where decisions are not purely binary (Yes/No), different methodologies for crowd voting can be applied.
An example: Ox Weight Judging Competition studied in Tetlock Chapter 3:
- Contestants paid a fee to guess the weight of an ox, illustrating how crowd assessments can vary.

The majority judgment can be approached by iteratively adjusting guesses based on collective inputs:
- If a majority votes that the ox weighs more than a proposed value, the subsequent question is adjusted upwards.
- Conversely, if a majority votes that it weighs less, the proposed value is adjusted downwards.
This iterative guessing process continues until it converges on the median voter’s assessment.

In Galton's notable experiment, the median estimation fell only 9 pounds from the actual weight, indicating the effectiveness of the crowd as an estimator.

The Condorcet Jury Theorem illustrates scenarios in which majority decision making leads to accurate judgments:
- The requirement for individuals to be competent (i.e., p > rac{1}{2}), the necessity for independence in decision-making, and the presence of a sufficiently large group.
The Median Voter Theorem emphasizes the relationship between majority opinion and the median in judgments concerning continuous variables, indicating that the crowd’s perspective often aligns with the median forecast in collective predictions.

Upcoming lectures will address what strategies can be embarked upon when independence in voting assumptions appear questionable, paving the way for deeper understanding of group decision-making.