The Voting Problem: A Lesson in Multiagent System Based on Jose Vidal’s book Fundamentals of Multiagent Systems Henry Hexmoor SIUC

Voting Problem Plurality Vote: e.g. General election – Vote for favorite option – Tally votes for most popular Runoff election: e.g. The primaries – Vote for favorite set of options – Vote within dominant sets until a winner is determined Pair wise election: – Vote for favorite set of two – Vote within winning sets

Symmetry Refectional Symmetry: – If one agent prefers A over B and another prefers B over A, their votes should cancel each other out Rotational Symmetry: – If one agent prefers A,B,C – Another prefers B,C,A – Third prefers C,A,B Then Cancel votes. All previous votes violate symmetry

The Borda Count With x choices, each agent awards x points to her first choice, x-1 points to her second choice, and so on. The candidate with the most points wins. The Borda Count satisfies both symmetry properties.

Borda Count The Borda count is a single-winner election method in which voters rank candidates in order of preference. The Borda count determines the winner of an election by giving each candidate a certain number of points corresponding to the position in which he or she is ranked by each voter. Once all voter have been counted the candidate with most points is the winner. Because it sometimes elect broadly acceptable candidates, rather than those preferred by the majority, the Borda count is often described as a consensus-based electoral system, rather than a majoritarian one.

The Voting problem A -Set of agents O -Set of outcomes -Preference function of agent i over outcome -Global set of social preferences (i.e., social contract) The problem: What should the final vote reflect and how can an election system preserve social preferences?

Desirable Voting conditions 1. exists for all inputs 2. exist for every pair of outcomes 3. is symmetric and transitive over the set of outcomes 4. should be pareto efficient. If all agents have desired outcomes, should coincide with that. 5. should be independent of irrelevant alternatives 6. No dictators Kenneth Joeseph Arrow (Noble Prize in Economics)

Arrow’s Impossibility Theorem (1951) There is no social choice rule that satisfies all six conditions.

Condorcet method A Condorcet method is any single-winner election method that meets the Condorcet criterion. The criterion always selects the candidate who would beat each of the other candidates in a run-off election – Rank the candidates in order of preference. Tie rankings are allowed, which express no preference between the tied candidates. – Comparing each candidate on the ballot to every other, one at a time (pair wise), tally a “win” for the victor in each match. – Sum these wins for all ballots cast. The candidate who has won every one of their pair wise contests is the most preferred, and hence the winner of the election. – In the event of a tie, use a resolution method.

Mechanism design Designing rules of a game or system to achieve a specific outcome, even though each agent may be self-interested. It tries to achieve four outcomes: 1.truthfulness, 2.individual rationality, 3.budget balance, and 4.social welfare.

Mechanism design Let N be the number of players/participants. Each player i can have a type/signal/valuation e.g. in an auction the type of player would be his valuation/reservation price for the good(s) offered. Depending on her type, the player will pick an action, where is the set of possible actions for player i offered by the mechanism, e.g. an auction would be a bid of a certain amount. Each player has utility,where O is the outcome generated by the mechanism. In auction, the outcome would be the final allocation of goods and the payments each player has to make.

Mechanism design A mechanism M is defined to be a pair (A,g), where is the set of action offered to the players/participants and is the function that maps the player’s actions to an outcome o. A mechanism is direct, if the set of actions equals the set of types for each player, i.e. This is true for auctions, where each player’s action is to announce their valuation of the product. However, there is no need to announce the true valuation if a different strategy yields better utility.

Mechanism design A mechanism is direct truthful, incentive compatible, if it is the dominant strategy to take. All of the participants fare best when they truthfully reveal any private information asked for by the mechanism. A function a social choice function f in dominant strategies, if the set of strategies that lets M generate the same output as f is a dominant Nash Equilibrium

The House Painting example SV(S) AliceYes BobNo CarolineYes DonaldYes EmilyYes Alice lives in a house with four other housemates. The set of people who vote for painting will share equally in the cost of the painters, as long as two or more people vote Yes. The people who voted against painting will pay nothing.

The House Painting example  = {WantPaint, DontNeedPaint} O = {Paint,NoPaint} The cost of painting the house is 20 The agents that want the house painted would get a value of 10 from seeing it painted and 0 if it does not get Painted. Those who think the house is fine as it is get a value of 0 either way. Assume that we want to maximize social welfare. Vickrey solution solves this problem