# 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 $$\frac{1}{2}$$
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: $$\frac{1}{2}\Big[ 1-(1-p)^3 \Big] + \frac{1}{2} p^3$$

# 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?

• 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