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. \(1p < \frac{1}{2}\)
 if \(Guilty\), signal is \(g\) with prob. \(p>\frac{1}{2}\) and \(i\) with prob. \(1p < \frac{1}{2}\)
Learning from Signals

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

if you get signal \(i\), what is probability that (defendant is) \(Guilty\)? \(prob(Guilty  i) = \frac{prob(Guilty) \cdot prob(i  Guilty) }{prob(i)} = \frac{\frac{1}{2}(1p)}{\frac{1}{2}(1p) + \frac{1}{2}p} = 1p\)
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: \(\frac{1}{2}\Big[ 1(1p)^3 \Big] + \frac{1}{2} p^3\)
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