Does One Vote Matter?

• suppose there are \(2\) candidates \(D\) and \(R\); \(N\) (odd) voters; simple majority
• assume that everyone else votes \(D\) with prob. \(1/2\) and \(R\) with prob. \(1/2\)
• how likely is it that your vote matters?
  • if \(N = 3\), the possible vote outcomes for 2 other voters are
  • your vote is decisive if there's a tie ( others voted DR or RD), which happens with probability \(\frac{1}{2}\)

votes	DD	DR	RD	RR
probability	\(\frac{1}{4}\)	\(\frac{1}{4}\)	\(\frac{1}{4}\)	\(\frac{1}{4}\)

Does One Vote Matter?

• suppose there are \(2\) candidates, \(N\) (odd) voters, simple majority
• how likely is it that your vote matters?
  • if \(N = 10\), then \(prob(tie) = 0.246\)
  • if \(N = 1,000\), then \(prob(tie) = 0.0252\)
  • if \(N = 10,000,000\), then \(prob(tie) = 0.025%\), or \(1\) in \(4000\)

Information Aggregation in Elections

• how are elections at aggregating information?
  • democracy vs dictatorship, when there is uncertainty
• common interest setting: a JURY
  • \(N\) (odd) jurors charged with finding if defendant is \(Innocent\) or \(Guilty\)
  • every juror wants to acquit the \(Innocent\) and convict the \(Guilty\)
  • issue: it is entirely unclear whether the defendant is \(Innocent\) or \(Guilty\)

Information and Signals

• unknown state of the world: (defendant is) \(Innocent\) or \(Guilty\)
• common prior belief: \(prob(Innocent) = \frac{1}{2}\), \(prob(Guilty) = \frac{1}{2}\)
• each juror receives a private signal:
  • value of signal is either \(g\) or \(i\)
  • signal is informative:
    • if \(Innocent\), signal is \(i\) with prob. \(p>\frac{1}{2}\) and \(g\) with prob. \(1-p < \frac{1}{2}\)
    • if \(Guilty\), signal is \(g\) with prob. \(p>\frac{1}{2}\) and \(i\) with prob. \(1-p < \frac{1}{2}\)

Learning from Signals

• if you get signal \(g\), what is probability that (defendant is) \(Guilty\)? \(prob(Guilty | g) = \frac{prob(Guilty) \cdot prob(g | Guilty) }{prob(g)} = \frac{\frac{1}{2}p}{\frac{1}{2}p + \frac{1}{2}(1-p)} = p\)

• if you get signal \(i\), what is probability that (defendant is) \(Guilty\)? \(prob(Guilty | i) = \frac{prob(Guilty) \cdot prob(i | Guilty) }{prob(i)} = \frac{\frac{1}{2}(1-p)}{\frac{1}{2}(1-p) + \frac{1}{2}p} = 1-p\)

Jury

• each juror receives correct signal with prob. \(p>\frac{1}{2}\)
• each juror wants to convict the \(Guilty\) and acquit the \(Innocent\)
• if you (a juror) receive signal \(g\), you vote to CONVICT, because \(p>\frac{1}{2}\)
• if you (a juror) receive signal \(i\), you vote ACQUIT \(p>\frac{1}{2}\)
• note that here we assume that juror vote sincerely (according to their signal only)

Condorcet Jury Theorem

• \(N\) jurors, simple majority, signal precision \(p\); \(Q(N,p)\) is probability of correct decision
• Condorcet Jury Theorem: for any odd \(N\) and any \(p > \frac{1}{2}\)
  • \(Q(N,p) > p\)
  • \(Q(N+2,p) > Q(N,p)\)
  • \(Q(N,p) \to 1\) as \(N \to +\infty\)

Unanimity Rule

• we looked at simple majority rule, how about unanimity rule?

Best Social Choice Function

• Condorcet: simple majority is most efficient because it minimizes (across all SCF) total probability of an error
• he's not wrong, simple majority does minimize total prob. of error:
  • simple majority: \(3p^2 - 2p^3\)
  • unanimity: \(\frac{1}{2}\Big[ 1-(1-p)^3 \Big] + \frac{1}{2} p^3\)

Simple majority vs Unanimity: Ultimate Battle

Connection to Hypothesis Testing

• we can say that the jury is testing hypothesis that defendant is \(innocent\)
• type I error: innocent person goes to jail
  • false positive, null hypothesis is TRUE but it is rejected
• type II error: guilty person walks free
  • false negative, null hypothesis is FALSE but it is not rejected

Simple Majority vs Unanimity: Summary

• unanimity rule has lower type I error
  • fewer innocent people are found guilty (6.4% vs 35.2% when \(p = 0.6\))
• majority rule has lower type II error
  • fewer guilty people are found innocent (35.2% vs 78.4% when \(p = 0.6\))
• majority rule has lower total probability of error
  • \(\frac{1}{2} \text{Type I error} + \frac{1}{2} \text{Type II error}\) (35.2% vs 42.6% when \(p = 0.6\))
• which rule is best?

Assumption we Made

• unknown state of the world (\(innocent\) or \(guilty\))
• each juror receives private signal
• each juror votes according to signal

Strategic Voting: Simple Majority

• same setup as before: \(3\) jurors, \(p>\frac{1}{2}\), simple majority
• you are the "smartest person in the room"
  • you know other 2 jurors are not sophisticated enough to strategize and just vote according to their signal
  • you receive signal \(i\)
  • what is your thought process?

Strategic Voting: Unanimity

• same setup as before: \(3\) jurors, \(p>\frac{1}{2}\), unanimity
• you are the "smartest person in the room"
  • you know other 2 jurors are not sophisticated enough to strategize and just vote according to their signal
  • you receive signal \(i\)
  • what is your thought process?

Strategic Voting in Juries

• under majority rule, there is an equilibrium with sincere voting
  • if everyone else votes sincerely, you do too
  • our assumption that voters vote according to signal is WITHOUT loss of generality \vspace{1cm}
• under unanimity rule, there is NO equilibrium with sincere voting
  • if everyone votes sincerely, you do NOT want to also vote sincerely
  • our assumption that voters vote according to signal is WITH loss of generality
    • technically, we did not properly solve the game and juror's behavior is a lot more complicated (they will strategize)

Connection to Auction Theory

• imagine you are in auction competing with two more people for an object
  • object's value is unknown but similar to every bidder
  • you win, how do you feel about it?

Swing Voter's Curse

• similarly to winner's curse in auctions, we have a swing voter's curse in juries with unanimity rule:
  • if you naively follow you signal and then learn that everyone else voted to convict, then you would "curse" after realizing that your signal was most likely wrong and you spoiled a perfectly good conviction