Arrow’s Conditions 1.Non-Dictatorship -- The social welfare function should account for the wishes of multiple voters. It cannot simply mimic the preferences of a single voter. 2.Unrestricted Domain -- (or universality) For any set of individual voter preferences, the social welfare function should yield a unique and complete ranking of societal choices. Thus: It must do so in a manner that results in a complete ranking of preferences for society. This is sometimes referred to as Uniqueness. It must deterministically provide the same ranking each time voters' preferences are presented the same way.deterministically

Arrow’s Conditions (continued) 3.Independence of Irrelevant Alternatives -- The social preference between x and y should depend only on the individual preferences between x and y (Pairwise Independence). More generally, changes in individuals' rankings of irrelevant alternatives (ones outside a certain subset) should have no impact on the societal ranking of the subset. For example, the introduction of a third candidate to a two- candidate election should not affect the outcome of the election unless the third candidate wins. Pairwise

Arrow’s Conditions (continued) 4.Unanimity -- If every individual prefers a certain option to another, then so must the resulting societal preference order. This, again, is a demand that the social welfare function will be minimally sensitive to the preference profile. 5.Citizen Sovereignty (or non-imposition) -- Every possible societal preference order should be achievable by some set of individual preference orders. This means that the social welfare function is surjective: It has an unrestricted target space.surjective

Application of the Conditions Arrow inspected many of the common methods of determining a group ranking for their adherence to his five criteria. He also looked for new methods that would meet all five. After doing so, he arrived at a surprising conclusion.

Arrow’s Conclusions Arrow realized that any group-ranking method will violate at least one of Arrow’s conditions in some situations. His proof demonstrated how mathematical reasoning could be applied to areas outside of mathematics. This achievement resulted in Arrow receiving the Nobel Prize for Economics.

Let’s try this! Take a piece of paper and write down and of these soft drinks you feel is acceptable. The choices are: Coke Pepsi Orange Sunkist Dr. Pepper Mt. Dew Yoo-Hoo Sun Drop

Determining Group Ranking Get together with the other people in your row and determine a group ranking. To determine your group ranking all you have to do is count the number of votes for each soft drinks and determine the winner. The winner will be the drink with the most votes. The second place choice will be the one with the second number of votes and so forth. Now let’s put them all together and get a group ranking for the class. Do you think Arrow’s Conditions exist in this “voting activity?”

Approving Voting Although Arrow realized that we could not find the perfect group ranking situation, that does not mean that we can not continue to improve. This led to studies to find the best group-ranking method. It is believed that Approval voting is that method.

Approval Voting (cont’d) In approval voting, you may vote for as many choices as you like, but you do not rank them. You mark all those of which you approve. For example, if there are five choices you can vote for as few as none or as many as five.

Discussion Can you think of a situation where you have been involved in Approval Voting?

Weighted Voting Weighted Voting occurs whenever some members of a voting body have more votes than others have…..can you think of an example? Notation for weighted voting: If voter P1 has W1 votes, voter P2 has W2 votes, and P3 has W3 votes,…..and voter Pn has Wn votes, and the quota is Q, then we will write { q : w 1, w 2, w 3,…, w N }

Weighted Voting A weighted voting system is one in which the preferences of some voters carry more weight than the preferences of other voters. The quota is the number of “votes” needed for a resolution to pass. Example: The three stockholders in a small company form a Board of Directors to oversee the company. John (P 1 ) has 5 votes as the largest stockholder, Ginny (P 2 ) has 3 votes, and Ann (P 3 ) has 2 votes. The quota is 7; that is, it takes 7 or more votes to pass a motion. This weighted voting system is represented mathematically as { 7: 5, 3, 2 }. How many different combinations of voters can you find to reach the quota?