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Computational Democracy: Algorithms, Game Theory, and Elections Steven Wolfman 2011/10/27

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A voting system is software.

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2001 BC General Election

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Some other oddities: - 1926 MB Federal - 1992 and 2000 US Presidential - 2008 Vancouver Municipal

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MANY Other Algorithms Approval (hand-raising in class, often), Range (IMDB), Party List Proportional (Germany), Cumulative (??), Block (Vancouver municipal), Condorcet (UBC AMS, “ranked pairs” flavour),... But.. what do we really want out of a voting system?

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“Independence of Irrelevant Alternatives” (Intuition. Thanks to Sidney Morgenbesser.) We have apple and blueberry. I’ll take the apple. Oh, wait! We have cherry, too! In that case, I’ll take the blueberry

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Formally: Independence of Irrelevant Alternatives If under one set of votes, A beats B, then... A still beats B under another set of votes with the same relative rankings of A and B.

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General Definition: “Pareto (In)Efficient” If a change in the solution can make everyone better off, then the solution is “Pareto inefficient”. GFEDCBA Candidate (“option”)VoterKey: Question: Which of these is Pareto Inefficient?

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Formal: Pareto Efficient For any two candidates A and B, if all voters prefer A to B, A must beat B. BA Candidate (“option”)VoterKey:

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Formal: Dictator d is a dictator iff for any set of votes, the outcome precisely matches d’s vote. All hail 4!

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Arrow’s Impossibility Theorem (Kenneth Arrow, 1951 Nobel Prize) PE IIA + Dictatorship Let’s prove it. We assume: no ties in votes or outcome. This assumption is unnecessary. We assume: finite #voters, at least three candidates, election is a function. All three are necessary.

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Let’s play with votes. 1.Run the election. 2.Go back in time and change the votes. 3.Rerun the election. (AKA: explore the result of the election function on various inputs.) Scenario: move all B votes to the top PE and all B at top B wins top

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Scenario: all B at bottom 1.Run election 2.Change votes 3.Rerun election PE and all B at bottom B “wins” bottom

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Scenario: B at top or bottom Here, A > B and B > C. Generally, if B is in the middle, something beats B and some-thing loses to B. (Key results will always apply to the general case.) Can B end up in the middle like this? Let’s play!

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Scenario: C just above A PE C beats A IIA A (still) beats B IIA B (still) beats C CONTRADICTION Remember: A beats B B beats C B at top or bottom in all votes B wins top or bottom

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Scenario: find who moves B up Reminder: if every voter puts B at top or bottom, B wins top or bottom. PE left side has B at bottom, right side has B at top

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Scenario: find who moves B up Reminder: if every voter puts B at top or bottom, B wins top or bottom. At some point as we flip B up, it will move to the top.

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Scenario: find who moves B up Reminder: if every voter puts B at top or bottom, B wins top or bottom. At some point as we flip B up, it will move to the top.

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Scenario: find who moves B up Reminder: if every voter puts B at top or bottom, B wins top or bottom. At some point as we flip B up, it will move to the top.

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Scenario: find who moves B up Reminder: if every voter puts B at top or bottom, B wins top or bottom. At some point as we flip B up, it will move to the top. Let’s focus on the voter who “controls” B. (3 in this case, but someone in general.) Starting with the right side.

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Scenario: 3 moves A above B Why is A at the top? Compare to when we discovered 3...

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Scenario: find who moves B up Reminder: if every voter puts B at top or bottom, B wins top or bottom. At some point as we flip B up, it will move to the top. Let’s focus on the voter who “controls” B. (3 in this case, but someone in general.) This time we want the left side.

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Look at the A/B rankings here versus... our “3 puts A above B” scenario. IIA, left says A > B A > B In fact, all but 3 can rearrange their votes, if they don’t move B...

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Scenario: free up all but B and 3’s A As long as we keep voters’ A/B ordering the same, A > B, by IIA. B beats C. Must it? Compare to when we discovered 3...

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Scenario: find who moves B up Reminder: if every voter puts B at top or bottom, B wins top or bottom. At some point as we flip B up, it will move to the top. Let’s focus on the voter who “controls” B. (3 in this case, but someone in general.) Back to the right side.

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Scenario: free up all but B and 3’s A But, must B > C?Yes, by IIA, since B/C rankings match.

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Scenario: free up the B’s Reminder: As long as we keep voters’ A/B ordering the same, A > B and B > C by IIA. Now, everyone moves their Bs around, and 3 moves its A around but keeps it above C. IIA A > C, but the only fixed ranking is 3’s A > C!

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Brief pause for formality: In general, A and C are arbitrary options other than B. So, 3 is a dictator with respect to all relative orderings except those involving B. All (but B) hail 3! Rest of proof is a repeat with different candidate for B; skipping...

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Arrow’s Impossibility Theorem (Kenneth Arrow, 1951 Nobel Prize) PE IIA + Dictatorship

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More algorithms/game theory rangevoting.org Ka-Ping Yee zesty.ca

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Some Other Contexts of Use Usability and interface design Security and voter verifiability District creation and member allocation (and gerrymandering) Design specifications and standards Election auditing and fraud detection Voter files/databases Voter registration systems

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Extra Slides (just in case)

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Scenario: what about B Repeat proof, except with C at top/bottom rather than B. We find a dictator over B vs A (and B vs all but C), but remember...

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Scenario: 3 moves A above B Because 3 moves A to the top, A > B. So, 3 is that dictator as well. And any candidate but B can “play” A. So, 3 is the dictator. All hail 3!

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Canada’s Algorithm, Roughly 1.Start counts for each candidate and for “rejected” at 0. 2.For each ballot: a)If the ballot does not show a single candidate choice or has stray marks that might identify the voter, add one to the “rejected” total. b)Else, add one to the marked candidate’s total. 3.Report the results of all counts. After recounts, ties are do-overs: “deemed by-election” for the “deemed vacant” seat. Based on the Canada Elections Act. (Ties specified in the Parliament of Canada Act.)

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A Harder Case: STV In Single-Transferable Voting, voters rank the candidates, a single election results in multiple candidates elected, and votes are transferred among candidates during the count based on preference order...

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