Batch 2 - Class 296 - Voting Systems

• (Geoffrey - 149) 15 Turks and 15 Christians were aboard a ship caught in a severe storm. The captain decided to throw half of the passengers overboard. To be fair, he asked everyone to stand in the circle, and he would throw every 13th person in the circle. A clever Christian pointed out to his fellow man a way to stand so that only all the turks would get thrown out. Can you find that arrangement?
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• Ask students about elections that they have participated in, or seen - either as candidates or as voters
• What is the mechanism of those elections? How were the winners decided?
• See if anyone describes methods other than plurality (everyone votes for one choice, and the choice with maximum votes wins)
• Is plurality fair? Have kids seen cases where they were not happy with the results of plurality?
• Where have they seen plurality been used? What is used in Indian system of election for Members of Parliament, Prime Minister, President?
• Let's vote for a class trip, and what activity should we choose out of skiing, rafting and caving
• Get each students choices in rank order and tabulate
• Which option should win?
• See if kids still get to plurality?
• To make the problem more interesting, lets get more votes into the mix, take a sample class of 40 students and the following voting pattern.

• Plurality: Satisfies Majority criteria: If a choice is first choice of majority of voters, then it should get chosen
• Which activity would win as per plurality? Why?
• Which activity is liked the least by largest number of people? 
• Why might the plurality method not produce results satisfactory to all voters?
• For example: The winning party BJP, clocked only 31% of the votes in last general elections, but still got majority of the votes
• Violates Condorcet criterion: If a choice wins head-to-head with each other choice, then it must win
• Why is plurality used in most elections?
• Are there ways of voting that allow "better" outcomes? What is better? 
• Lets students come up with the list of methods, and see variations in results. Get the class to describe at least two other voting techniques.
• Borda Count: Let each person specify their first, second, third option. Each first choice is awarded 2 points, each second choice 1 point, and third choice 0 points. Maximum total points win.
• Try Borda count on the initial votes of class. Does it lead to the same result?
• Try Borda count on our extended class of 40. Does it lead to the same result?
• Is this fair? (Note that Borda count can violate the majority criterion)
• Hare System: Take each persons first option initially. Eliminate the choice with least votes, and ask those voters to now exercise their favorite options amongst existing choices, and so on.
• Try Hare system on the initial votes of class. Does it lead to the same result?
• Try Hare system on our extended class of 40. Does it lead to the same result?
• Describe the process stepwise as options are eliminated
• Does Hare system look more fair? Where is it likely to be used? Why is it likely not to be used in large general elections?
• Note that Hare system can violate the Monotonicity criiteria: If a choice X wins, and in a reelection all voters who change their preference do so in a manner favorable to X, then X must still win https://www2.math.upenn.edu/~deturck/m170/wk10/lecture/vote3.html
• Sequential Pairwise Voting: involves a sequence of head-to-head contests. First, the group votes on any one of two options, and then preferred option is matched against the next option which the loser is eliminated. Continue till one option remains.
• Suppose in our extended class, we first vote between skiing and caving, and then the winner against rafting, which option will win?
• Can the result change basis which two options are chosen to go head-to-head first?
• Which of the above methods is fairest? Which is the least fair?
• Suppose your preference is rafting. Devise a voting system that would enable rafting to be chosen. The voting system must be found to be fair to other club members.
• Strategic Voting:
• If a block of voters knew they will not win the elections, could they vote in a fashion that allows them to change the results of the election more in their favor? 
• For example, if amongst three choices A, B, C - the true preferences are ABC (2 people), BAC (3 people), CAB (4 people), who would win?
• If voters of block ABC knew the distribution of voters, what could they do to get a slightly better result?
• Suppose you are one of the ten sports editors whose votes determine rankings of state high school football teams. Borda counting is used (10 points for each first place vote, 9 for second, and so on 1 for tenth place vote)
• You suspect that nine other editors will rank a competing school first, and your team second. Show a possible ranking that will enable your team to earn most points. Explain.
• Consider A, B and C running for office, and following being the voter blocks

• Who will win by plurality?
• Using same method, can the 24% block alter their voting to get more of their liking?
• Who will win using Hare system? Which block of voters would be most disappointed?
• Using same method, devise an alternate ranking they could submit to get more of their liking.

• (Expii) After a deep hundred year slumber, Ant the ant wakes up in an unfamiliar, large, dark square forest. Ant is unaware of its exact location or direction, but it does know that its starting point is exactly 10 meters away from the nearest straight edge. What is the shortest distance Ant must travel to guarantee escape from the forest? Round your answer, in meters, to the nearest hundredth.
• Answer: Not really sure - let kids describe the best solutions they have been able to come up with. One potential is for the ant to go in any one direction for 10m, and then go around the 10m radius circle around the original wake-up point (so, 10 + 2*pi*10). Is there a better answer?