Unit 3 Test Flashcards
Winning coalition/combination
A coalition is winning if they have enough weight to meet the quota
Minimal winning coalition
Winning coalitions where every voter is critical
Banzhaf power index steps
List out all winning coalitions (all combinations), identify each coalition’s critical players, count how many times each player is critical, and the power of a player is the # of times they’re critical
Extra votes principle
The critical voters in a winning voting combination are the “yes” voters whose weights are greater than the number of extra votes in the combination
Equivalent voting systems
When two voting systems have the same power indices
Combination formula
The number of ways to choose k outcomes, without regard to order, from n possibilities is nCk (on calculator) = “n choose k”= n!/k!(n-k)!
Permutation
A specific ordering from first to last of all elements of a set. AB =/BA
Voting permutation/sequential coalition
A coalition where we record the order that players joined
Pivotal player/voter
A player is called pivotal if the coalition goes from losing to winning when they joined
Shapley-Shubik power index steps
List out all sequential coalitions (n! for n players), determine the pivotal player in each, count how many times each player is pivotal, and divide each of these #s by n!
Fair division method
Step-by-step procedure that results in a division of items in a way so that each party feels that they have received what they deem to be fair
Continuous division
Occurs with objects which can be divided into parts as small as we’d like, such as money or cake
Discrete division
Occurs with objects which cannot be divided into smaller parts, like a car or house
Fair share
In Knaster inheritance, the amount for each party when the values of all items are totaled and divided by the number of parties
Adjusted winner procedure steps
Each person is assigned 100 points and distributes these points to the items on the list based on their valuations of the items and the highest bidder of an items gets that item. Ties are broken by giving the items to the person with the most leftover points. After all the points are distributed, total the value of the items awarded to each person by their own point estimations, calculate the point ratio for the initial winner, and transfer the item with the smallest point ratio to the person with the fewest points. Continue until the values of awarded items are equal between the two persons, usually with an algebraic equation
Initial winner
The person after the initial distribution of items with more points according to their own estimation in adjusted winner procedure
Initial loser
The person after the initial distribution of items with fewer points according to their own estimation in adjusted winner procedure
Point ratio
Calculated for each item the initial winner receives initially in adjusted winner procedure. Found as (higher valuation of the item) / (lower valuation of the item)
Equitable
If each player believes they received the same fractional part of the total value
Proportional
If each party believes they received at least 1/n of the total value. Most basic notion of fairness
Envy-free
If no party would prefer another party’s share over their own
Pareto optimal
If no other division would make a party better off without making another party worse off
Sealed bids method/Knaster inheritance procedure steps
Each party assigns each item a bid, the values of all items are totaled and divided by the number of parties, and each item is awarded to the highest bidder. For each party, the value of all items received is totaled, and they pay/get the difference from the fair share to/from a holding pile, then any money left over in the holding pile is divided evenly
Bids
The secret dollar amount each party assigns to each item in a Knaster inheritance procedure
Critical voter/player
If a player is in a winning coalition and them leaving the coalition would cause the coalition to change to a losing one
Dummy voter
A player whose vote is never essential to reach the quota, so have no say in the outcome.
Dictator
A player whose weight is equal to or greater than the quota. Have enough say to always pass, so the election id entirely decided by their vote.
Veto power
A player whose support is necessary to reach the quota. Have enough say to always fail. Possible to have multiple voters with veto power.
Minimal quota
Must be larger than half the number of voters
Maximal quota
Can’t be larger than the number of voters
Taking Turns
Strategy for dividing items by alternately selecting objects with no knowledge of each other’s preferences, so sincere choices
Bottom-Up strategy
A systematic way to choose when both players know the other’s preferences, so they will not willingly choose their least-preferred object available, and won’t waste a choice on an object they know will remain available
Weight
How many votes a player gets. Denoted W1, W2, …
Quota
Minimum weight needed to pass a proposal. Denoted q
Player
Individual/entity casting a vote. Denoted P1, P2,…
Coalition
A group of players voting the same way
Notation for weighted voting systems
[q: W1, …., Wn]
How many possible sequential coalitions for n players?
n!
