Social Choice Topics to be covered: Use election theory techniques to analyze election data such as majority, plurality, runoff, approval, the Borda method in which points are assigned to preferences, and the Condorcet method in which each pair of candidates is run off head to head. Use fair division techniques to divide continuous objects. Use fair division techniques to solve apportionment problems.

Election Techniques Plurality: Count how many first place votes each candidate got. The one with the most wins. Approval: Each voter can approve as many candidates as they want, and the winner is the one that gets the most votes. Run-Off: Count how many first place votes each candidate gets. Declare all but the two highest as losers. Now, run an election with the two highest. Sequential Run-Off: Eliminate the candidate with the fewest first place votes. Repeat this process until there is a winner. Borda Count: Given a preferential ballot, assign each candidate a number of points equal to the number of candidates below them. Condorcet: Consider all possible two-way races between candidates. The winner, if there is one, is the candidate that can beat each other candidate in a two-way race.

Example Election Consider the following preferential election ballots: Winners: Plurality: A = 18 Runoff: B = 37 Borda Count: A = 18 B = 81 C = 107 D = 136 E = 134 Condorcet: E Sequential Runoff: C Each of the five methods selected a different winner – which one is correct? Election theory is the subject of much debate today.

Analysis of Condorcet Method At first glance, the Condorcet method is very appealing to most people. But, consider this set of preferential ballots and see who the Condorcet winner would be: A beats B 25-14 B beats C 27-12 C beats A 26-13 So there is no Condorcet winner! This is the main flaw with the Condorcet method – a cycle can be created so that there is no clear winner.

Kenneth Arrow Kenneth Arrow, an American economist and Nobel Prize winner, designed a set of criteria to determine a fair election system. In short, no “fair” voting system can be designed to satisfy the following: If every voter prefers X over Y, then the group prefers X over Y. If every voter’s preferences between X and Y remain unchanged when Z is added to the state, then the group’s preference between X and Y will also remain unchanged. There is no “dictator”: no single voter possesses the power to determine the group’s preference.

Fair Division A fair division problem has a set of N players and a set of goods S. We wish to divide S into N shares so that each player gets a fair share of S. A fair share is a share that, in the opinion of the player receiving it, is worth 1/N of the total value of S. Fair division problems can be classified in three ways: Continuous: The set S can be divided infinitely many ways. Examples are large sums of money or cakes. Discrete: The set comprises of indivisible objects (or objects that are not easily divisible). Examples are houses or boats. Mixed: The set contains both continuous and discrete components. Most inheritances fall into this category.

Fair Division Schemes A fair division scheme is a systematic procedure for solving a fair division problem. It must have the following properties: It is decisive. It is internal. It assumes the players are ignorant of each other. It assumes the players behave rationally. We will look at four different methods for solving fair division problems: For continuous sets: The divider-chooser method. The lone divider method. For discrete sets: The method of sealed bids. The method of markers.

The Divider-Chooser Method Alice and Bob win a half-pepperoni half-anchovie pizza. Bob likes pepperonis and anchovies equally well, so he values the pizza at $8 (keeps this value secret). Alice cannot stand anchovies, so she assigns the pizza a value of $5. Note that Bob cannot know that Alice hates anchovies. Suppose that by coin toss Bob become the divider and Alice the chooser. Bob then cuts the pizza in half so that one half is exactly ¾ anchovie and ¼ pepperoni. Now Alice has an easy choice – she choose the half that is ¼ anchovie and ¾ pepperoni. In her value system this is worth $3.75, so in her opinion she has more than a fair share. Bob gets the other half, which is worth $4 to him – a fair share. Note that this may not be the best division, but it is a fair one.

The Lone Divider Method This is an extension of the divider-chooser method for more that two players Step 1. Division: The divider slices the pizza into three pieces. This division is only rational if each piece has an equal value to the divider. Step 2. Declarations: Each chooser declares which pieces he finds acceptable. Step 3. Distribution: What happens here depends on the declarations: Case 1: One chooser declares more than one piece acceptable. The other chooser gets his chosen piece, the chooser gets his other choice, and the divider gets what’s left. Case 2: Both choosers declare one piece, and they are different – obvious division in this case. Case 3: Both choosers declare one piece, and they are the same piece. Here the divider randomly selects one of the two undeclared pieces. Then we put the remaining two pieces back together and apply the divider- chooser method. Everyone gets a fair share.

The Method of Sealed Bids This method can be described as a three step process: Step 1: Bidding: Each player produces a sealed bid in which he attaches a dollar value to each item in S. A player’s fair share is 1/N of his total assessment. Step 2: Allocation: Each item in S goes to the highest bidder for that item. If a player’s assessed value of items received exceeds his fair share, then he must pay the difference into the pool. If the assessed value of items falls short of a fair share, then he is paid out of the pool. Step 3: Dividing the Surplus: The surplus of cash is divided among the players. Unfortunately, due to human flaws there are few common problems that arise with this method.

Problems with Method of Sealed Bids While this method has no mathematical flaws, human imperfections can cause problems to arise: Cash flow: Each player must have enough money to make good on all their bids that are too high. “Priceless” items: No player can insist on getting a favorite item. Everyone must be willing to take a cash substitute for ANY item. Players know each other: Suppose Alice honestly thinks that an item is worth $1000, but she knows that Bob thinks the same item is worth $2000. What is to keep Alice from inflating her bid to $1999, knowing that she will still not get the item?

The Method of Markers This method gets around the problem of cash flow, but there must be a relatively large number of items in S for this to work. We start the process by stringing out all items of S into a line. There are 3 steps in this method: Step 1. Bidding: Each player secretly divides the line into N segments, each of which he considers a fair share. Step 2. Allocation: Find the leftmost marker and give that player everything to the left of it, and remove their other markers. Then find the leftmost marker of the second group of markers. That player gets all of the items in between their first and second marker, and their markers are removed. Finally, the last player gets everything to the right of their second marker. Step 3. Divide the Surplus: There will usually be leftovers, and they can be distributed randomly. If there are enough left, they can use the method of markers again.