One-Sided Matching#
Model:
there are \(n\) students, each with some preference over residences
there are \(m\) dorms, each with some capacity
We want to assign students to dorms. One possible solution is to maximize the sum of utilities:
Algorithm 1 (Utility Maximization)
Create a bipartite graph with one side being students and the other side being residencies with with edges from students to dorm with edge weight equal to how much student \(i\) likes dorm \(j\)
Find the max-weight matching (i.e., Hungarian Algorithm)
The issue: each student has an incentive to exaggerate how much they like their favorite dorm and undercut how much they like other dorms.
As such, finding the matching that maximizes happiness was the right solution concept, but failed to take incentives into account. Another possible algorithm is serial dictatorship:
Algorithm 2 (Serial Dictatorship)
Students are sorted by some fixed order (random, seniority, alphabetically, etc.)
Go through the list in order and each student selects their most preferred available dorm
Is this any better? There can still be students who are unhappy with their result under serial dictatorship.
Definition 1 (Mechanism)
A mechanism consists of three things: a method of collecting inputs from agents, an algorithm that acts on the inputs, and an action that is taken based on the output of the algorithm.
Remark 1
All three components are important: for instance, students’ beliefs of how the algorithm works or how the action is taken would affect how they respond to the way inputs are collected.
Properties of Serial Dictatorship#
Definition 2 (Strategyproof)
A mechanism is strategyproof if it is in every agent’s best interest to act truthfully.
Definition 3 (Dominant Strategy Incentive Compatible)
A mechanism is dominant strategy incentive compatible if it is a dominant strategy for each participant to act truthfully. In particular, this means that being truthful is a best response to any possible strategy profile of other players.
Theorem 1
The serial dictatorship mechanism is dominant strategy incentive compatible: it is in every student’s best interest to choose their favorite available dorm in their term.
Proof. Your room choice will not affect what rooms are available by the time your turn comes. When it gets to your turn, what you choose is what you get, so it is best to choose your favorite room among available options.
Definition 4 (Pareto Improvement)
An outcome \(O'\) is a Pareto improvement over outcome \(O\) if all agents either strictly prefer \(O'\) to \(O\) or are indifferent between the two, with at least one strict preference.
Definition 5 (Pareto Optimal)
An outcome \(O\) is Pareto optimal if there are no Pareto improvements from \(O\).[1]
In other words, an outcome is Pareto optimal if you can’t make anyone happier without making someone sadder.
Theorem 2
The outcome under serial dictatorship (with any ordering of students) is Pareto optimal.[2]
Proof. Let the serial dictatorship outcome be \(O\). Towards a contradiction, suppose \(O\) is not Pareto optimal, and there exists some outcome \(O'\) that is a Pareto improvement over \(O\). Consider the first student that gets a different outcome under \(O\) and \(O'\). Since all students before this student are assigned the same room, the set of available rooms for this student is the same under \(O\) and \(O'\). However, the student chooses their favorite room under \(O\), so the room they receive under \(O'\) must be worse.
One issue with this: dictatorship can have a large affect on others. Suppose there are ten students and the first eight have chosen rooms \(1\) to \(8\). Also, suppose student nine has room \(9\) as their \(9\)th favorite room and room \(10\) as their \(10\)th favorite room while student ten has room \(9\) as their favorite room and room \(10\) as their least favorite room. If student nine chooses room \(9\) over room \(10\) they only gain a small improvement, but student ten is now forced to pick room \(10\) over room \(9\) which is a large jump in their preferences.