1. Alex Tabarrok

2. Individual Rankings (Inputs) BDA CCC ABD DAB Voting System (Aggregation Mechanism) Election Outcome (Global Ranking) B D A C

3. A Simple Election Example Number of Voters → 392437 1st ACB 2nd CBC 3rd BAA Plurality Rule: A>B>C Borda Count: C>B>A “7,1,0” Count: B>A>C

4.  Assume n=3 candidates then we can write the plurality rule system as (1,0,0) and the Borda Count as (2,1,0).  Clearly, (10,0,0) is equivalent to plurality rule. It's also true although a little bit more difficult to see that (5,3,1) is equivalent to the Borda Count.  Any point score voting system can be converted into a standardized point score system denoted (1-s,s,0), where s ∈ [0, 1/2].  The standardized plurality rule system is s=  The standardized Borda Count is s= 1 0 0 2 1 0 Plurality Rule Borda Count 1-s s 0 Standardized Point Score System 0? ?

5.  A voter may rank three candidates in any one of six possible ways.  The vote matrix can be read in two ways.  Reading down a particular column we see the number of points given to each candidate from a voter with the ranking indicated by that column.  Reading across the rows we see where a candidate's votes come from.  We will write the number of voters with ranking (1) ABC as p₁ the number of voters with ranking (2) ACB as p₂ and so forth up until p₆. We can place all this information in matrix form by multiplying the vote matrix with the voter type matrix. ABCACBCABCBABCABAC A1-s s00s Bs00s C0s s0 p₁ (ABC) p₂ (ACB) p₃ (CAB) p₄ (CBA) p₅ (BCA) p₆ (BAC) 1-s s00s s00s 0s S0 = A’s tally B’s tally C’s tally

6. 0 39 0 24 37 0 110000 000011 001100 = 39 37 24 0 39 0 24 37 0 2/3 1/300 00 2/3 01/32/3 1/30 = 26 32.66 41.33 Plurality Rule Borda Count A Simple Election Example 392437 1st ACB 2nd CBC 3rd BAA p₁ =0 (ABC) p₂=39 (ACB) p₃=0 (CAB) p₄=24 (CBA) p₅=37 (BCA) p₆=0 (BAC)

7. p₁+p₂+(-p₁-p₂+p₃+p₆) ∗ s p₆+p₅+(p₄-p₅+p₁-p₆) ∗ s p₃+p₄+(p₂-p₃-p₄+p₅) ∗ s 1-s s00s s00s 0s s0 = 0.51 A 1 B 1) ABC 2)ACB 3)CAB 5)BCA 6)BAC C 4)CBA p₁ p₂ p₃ p₄ p₅ p₆ Interpret p₁…p 6 as shares of each type of voter then the tallies are vote shares. Vote shares must sum to 100% so one of the equations is redundant. Thus we can graph in 2-dimensions. A Simple Election Example 392437 1st ACB 2nd CBC 3rd BAA Plurality Rule: A>B>C Borda Count: C>B>A “7,1,0” Count: B>A>C 0 39 0 24 37 0 39-39 ∗ s 37-13 ∗ s 24+52 ∗ s

8. Maximum Outcomes from One Ranking CandidatesOutcomes Max. Outcomes from 1 Ranking _____________ Total Outcomes 2210.5 3640.66 424180.75 5120960.8 67206000.833 7504043200.857 840,32035,2800.875 9362,880322,5600.888 103,628,8003,265,9200.9 nn!n!-(n-1)!1-(1/n) Source: Saari (1992) 0.51 A 1 B 1) ABC 2)ACB 3)CAB 5)BCA 6)BAC C 4)CBA Intuition for these results comes from the geometry of the procedure line extended to higher dimensions.

9.  For even small electorates (say 50 or more) and 3 candidates a single profile generates:  7 different rankings (including ties) about 6.7 percent of the time  5 different rankings 18.6 percent of the time,  3 different rankings 41.3 percent of the time  a single ranking 33.3 percent of the time.  A single profile, therefore, generates more than one ranking 66 percent of the time.  As the number of candidates increases the probability that all positional voting systems agree on the winner (K=1) quickly goes to zero.

10. 0.51 Clinton 0.5 1 Bush 1 23 4 56 Perot Pluralityy Rule Borda Count Anti -Plurality Rule Multiple outcomes from the same profile are not always the case. In 1992, conservative commentators emphasized President Clinton's failure to receive more than 50% of the vote and thus his failure, in their minds, to achieve a "mandate.“ An analysis of voter preferences, however, reveals the surprising fact that Clinton would have won under any point-score voting system!

11. 0.51 Clinton 0.5 1 Bush 1 23 4 56 Perot Pluralityy Rule Borda Count Anti -Plurality Rule Approval voting is an increasing popular system where each voter can approve of as many candidates as he or she likes. e.g. if there are 5 candidates the voter could approve 1,2,3, or 4 of them. Approval voting vastly increases what can happen. Note, for example, that with candidates under approval voting each voter has the option of using plurality rule or anti-plurality rule! What could have happened in 1992? Anything!

12.  Group choice is not at all like individual choice.  Groups will always choose in ways that would appear irrational if chosen by an individual.  The voting system determines the outcome of an election at least as much as do preferences.  Voting does not represent the “will of the voters.”  The idea of a group will is incoherent.

13.  Nobel prize winner Amartya Sen has argued that:  “No famine has ever taken place in the history of the world in a functioning democracy.”  “Democracies have to win elections and face public criticism, and have strong incentive to undertake measures to avert famines and other catastrophes.”

15.  Democratic Peace – democracies rarely go to war against one another.  Capitalist Peace – trading countries, countries with private property and capitalist economies rarely go to war against one another.  Democratic and capitalist peace are strongly supported in the data and a consensus has developed in the International Relations literature but less consensus on why.

19.  Democracies don’t kill their own citizens or let them starve.  Democracy is compatible with economic freedom -> democratic/capitalist peace.  Democracies avoid some very bad possibilities.  The threat of throwing politicians out of office is a constraint on what can happen in a democracy.  Dictatorships and oligarchies need only not abuse a minority – in a democracy the standard is higher.  Democracy, however, is not good at representing the will of the voters and in general we should not expect democracy to be a good way of making decisions.  Democracy should be seen as a way of limiting or constraining government.