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MAT199: Math AliveVoting, Power, and Sharing
Ian Griffiths
Mathematical Institute, University of Oxford,
Department of Mathematics, Princeton University
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A dinner party• To celebrate the start of the Math Alive lectures we decide to go out to dinner. • Because there are a lot of us, Winberie’s can only offer a set menu. • They give us a choice of chicken or salmon.
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A dinner party• To celebrate the start of the Math Alive lectures we decide to go out to dinner. • Because there are a lot of us, Winberie’s can only offer a set menu. • They give us a choice of chicken or salmon.
• A quick poll shows that more of us prefer salmon than chicken.• So we go with salmon.
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A dinner party• To celebrate the start of the Math Alive lectures we decide to go out to dinner. • Because there are a lot of us, Winberie’s can only offer a set menu. • They give us a choice of chicken or salmon.
• A quick poll shows that more of us prefer salmon than chicken.• So we go with salmon.
• When we call the restaurant they inform us that the salmon is off and it has been replaced with beef.
• We do another quick poll and find that more of us prefer the chicken than the beef.
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A dinner party• To celebrate the start of the Math Alive lectures we decide to go out to dinner. • Because there are a lot of us, Winberie’s can only offer a set menu. • They give us a choice of chicken or salmon.
• A quick poll shows that more of us prefer salmon than chicken.• So we go with salmon.
• When we call the restaurant they inform us that the salmon is off and it has been replaced with beef.
• We do another quick poll and find that more of us prefer the chicken than the beef.
• We call back to confirm we would like chicken and Winberie’s inform us that the chicken is now off but the salmon is back on.
• The choice is now salmon or beef.
• Do we need to do another poll to make sure the majority get the one they prefer?
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Nicolas de Caritat – Marquis de Condorcet
1743 – 1794
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Full English breakfast
1743 – 1794
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Summary of lecture 1
• Paradoxes can arise when ranking preferences (our Winberie’s dinner experience).
• A clear winner is called a Condorcet winner. If there isn’t a clear winner there are various ways of selecting the winner:
• Plurality – the choice with the most first places wins.
• Plurality with run-off – the two people with the most and second-most first places are pitted against one another in another election.
• Sequential run-off (the Hare system) – eliminate the bottom candidate at each round.
• Borda count – assign a score for top choice, second choice, and so on, and add up the total scores. The winner is the one with the highest score.
Voting and social choice
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Holiday destinations
Cornwall
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Holiday destinations
Scotland
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Holiday destinations
Jersey (the real one)
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Kenneth Arrow
Born in 1921
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Table showing fraction of non-Condorcet winners
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Summary of lecture 2• If there isn’t a clear (Condorcet) winner then the way in which you determine the
winner often leads to a different result.
• You cannot produce a universal fair voting scheme (proof by Kenneth Arrow).
• …but provided there aren’t too many choices, it is likely that you get a Condorcet winner anyway so there is no issue.
• You can share a cake between two people in an envy free way by letting one person cut and the other choose.
• Sharing a cake between three people in a fair way is straightforward. Sharing a cake three ways in an envy free way is harder…
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Summary of lecture 3
• A cake may be shared three ways in an envy-free way.
• Generalization for four ways is harder (Brams and Taylor, 1992).
• There is no clear way of generalizing this for five or more people.
• We may divide up several items two ways in an envy-free way (e.g., a divorce settlement).
• We can divide up several items more than two ways in a fair but not envy-free way (e.g., dividing up an estate).
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The Math Alive Barbecue
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The Math Alive Barbecue
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• To be given a burger at the Math Alive Barbecue you must be able to call out the colour of the hat you are wearing. You must call out one at a time (but in no required order).
• You all stand in a queue facing forward. You are not allowed to look behind you.
• You may discuss a strategy before the barbecue. What is the strategy that guarantees at least all but one members calling the correct colour?
The Math Alive Barbecue
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Strategy
• Lab member 1 counts the number of red hats he sees and calls out red if he sees an even number and blue if he sees an odd number.
1 2 3 4 5 6 7 8
The Math Alive Barbecue
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Strategy
• Lab member 1 counts the number of red hats he sees and calls out red if he sees an even number and blue if he sees an odd number.
1 2 3 4 5 6 7 8
Red
The Math Alive Barbecue
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Strategy
• Lab member 1 counts the number of red hats he sees and calls out red if he sees an even number and blue if he sees an odd number.
• Member 2 counts the number of red hats he sees. He sees an odd number (3) so knows that his hat must be red and so calls out ‘red’ and guesses correctly.
1 2 3 4 5 6 7 8
Red
The Math Alive Barbecue
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Strategy
• Lab member 1 counts the number of red hats he sees and calls out red if he sees an even number and blue if he sees an odd number.
• Member 2 counts the number of red hats he sees. He sees an odd number (3) so knows that his hat must be red and so calls out ‘red’ and guesses correctly.
1 2 3 4 5 6 7 8
Red Red
The Math Alive Barbecue
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Strategy
• Lab member 1 counts the number of red hats he sees and calls out red if he sees an even number and blue if he sees an odd number.
• Member 2 counts the number of red hats he sees. He sees an odd number (3) so knows that his hat must be red and so calls out ‘red’ and guesses correctly.
• Member 3 counts and sees an odd number too (3) so knows his hat is blue.
• This proceeds to the end of the line.
1 2 3 4 5 6 7 8
Red Red Blue
The Math Alive Barbecue
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Bending light
• Light will always take the fastest route between two places.