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A new agent symmetry criterion for social choice

It is time to write something longer than a tweet on this research topic, which in my opinion, trying to be as objective as possible, is very promising and conceptually important. I have worked on this with collaborators Jacky Lo, Geoff Pritchard and especially Rupert Freeman.

Let’s start with the model where we first noticed this new concept, namely one-to-one matching under ordinal preferences (“housing allocation”). Each of N agents has strict preferences over a set of N objects and we aim to find an assignment, namely a bijection between agents and objects. The famous mechanism called Serial Dictatorship fixes an order on the agents, and lets them choose in turn their most preferred object from those remaining. On the other hand, Top Trading Cycles starts with a fixed allocation, and gets each agent to point to the most preferred object. There must be at least one cycle, and we redistribute objects along the cycle so everyone in the cycle is satisfied. After removing these agents and objects, we continue with 2nd preferences, etc, until we reach a final assignment. A famous result of Abdulkadiroglu and Sonmez says that if the choosing order for SD is taken uniformly at random (this is called RSD), as is the initial allocation for TTC (which we call TTC-RE), then the probability distribution over assignments for these different-looking mechanisms is the same.

However, imagine yourself as an agent when RSD is used. No matter how you feel before the choosing order is determined, if you turn out to be at the beginning, you know you will always get your first choice object. If you turn out to be at the end, you know that you will have to take whatever is left. There is clear bias toward the front of the choosing order. On the other hand, it seems clear that no agent in TTC-RE will be as systematically advantaged just by virtue of their position in the mechanism.

A standard social choice concept is anonymity, meaning that a mechanism treats agents symmetrically, and the outcomes depend only on the profile of agent preferences, not the identities of the agents. However full anonymity is not possible in this model. RSD is anonymous, but once the choosing order is determined, the resulting SD is not anonymous. Similarly TTC-RE is anonymous but TTC is not. Yet TTC seems more symmetric than SD.  It seems there is an elusive criterion involving agent symmetry. The randomization discussed above turns out to be a red herring – we can define it in the deterministic case directly.

Our idea is to define such a criterion using average-case analysis. For a fixed probability distribution P on profiles (e.g. the uniform distribution), we consider the expected rank distribution for each agent, and if this is the same for all agents we call the mechanism order symmetric with respect to P. Some restriction on P is needed, because a point mass concentrated on a unanimous profile will prevent all mechanisms from being order symmetric. We show that provided P has sufficient symmetry between agents and between objects, TTC is order symmetric but SD is very far from being so. In other words, no agent is disadvantaged once the randomness in the mechanism is realized, and before preferences are elicited.

Not only does this definition clarify the difference between SD and TTC, it suggests tweaks to existing mechanisms. Full order symmetry may not be achieved in some situations, but we can try to equalize the bias between agents. The Boston mechanism clearly treats those early  in the tiebreaking order much better than those later in the order. A simple idea is to reverse the tiebreaking order after the first round. Simulations show this reduces bias substantially.

I feel that this idea has a bright future, with possible extensions to other social choice models and to practical mechanism design. Now we need a journal or conference to see that – we have had surprising difficulty in getting published. The two types of randomness can be confusing, and many researchers seem to think about fairness in terms of outcomes rather  than a procedural criterion of this sort. Some may just not think it is important, but we should first ensure there is no misunderstanding.

Blog hiatus

It seems it is almost 6 months since the last post. The main reason is that the time spent on Twitter takes up all the “social media” time I can spare. If I do have thoughts that can’t be expressed in 280 characters, I will come back here!

Research news

A few papers are working their way through the journal system:

Once all these are finally done, I can get back to some work on generating functions, after several years, which I am very much looking forward to.

 

Cybermath column NZ Math Society Newsletter August 2017

 

This column is focused on a single specialized topic. For almost the last two years I have been  involved with several international collaborators (from Australia, Netherlands, UK, France and Germany) in a project to accelerate the conversion of mathematics journals to a model involving open access with no direct payments by authors (sometimes called “diamond open access”). Some of these activities have been reported on in this Newsletter in the last several columns.

We created a legally constituted non-profit foundation (Stichting) called MathOA in the Netherlands in order to oversee the “flipping” of subscription journals to open access​.  The advisory board for MathOA includes Timothy Gowers, David Mumford, and several strong mathematicians who haven’t won the Fields Medal. MathOA is modelled on LingOA, a foundation in linguistics that organized the defection of the board of Lingua from Elsevier and the re-founding of the journal under the name Glossa, published by Ubiquity Press.

LingOA and MathOA have since been joined by PsyOA (in psychology) and we intend to create a loose organization called Fair Open Access Alliance. We have formulated what we call the Fair Open Access Principles

We have also invited existing mathematics journals that (essentially) conform to these principles to join an as yet unnamed network which will become part of FOAA. So far the following have agreed to do so: Australasian Journal of Combinatorics, Discrete Analysis, Discrete Mathematics and Theoretical Computer Science, Electronic Journal of Combinatorics, Epijournal de Geometrie Algebraique, INTEGERS: The Electronic Journal of Combinatorial Number Theory, Internet Mathematics, Journal de th\’{e}orie des nombres de Bordeaux, Journal of Computational Geometry, Logical Methods in Computer Science, SIGMA. FOAA is in its infancy and we are investigating ways in which we can create synergy between these independent journals, and make them even better (they are already very good or excellent in many respects).​

After the administrative details above, the \textbf{big news} to report is that the editorial board of Journal of Algebraic Combinatorics, currently published by Springer, has resigned to create a new journal (which is clearly the re-formation of the old one) under the name Algebraic Combinatorics, published in association with Centre Mersenne. This has been assisted from the start by MathOA.

Conversion of journals to open access is accepted by the large publishers only if it doesn’t negatively affect their profits. Thus if they own the journal title, what usually happens is  a refusal to negotiate seriously and an attempt to find a new editorial board to continue the old title. In my opinion, it is an attack on the mathematical community (and the wider public, and science itself) for a researcher to accept an offer to run such a zombie journal. Almost always, journals losing their entire editorial board in this way do cease publication within a few years (see my blog post)  and the new ones thrive.

Although this action by the editors is in some sense obviously ethically correct, and is made  easier by legal and practical help from MathOA, it still requires considerable courage from the editors. Leaving aside them having to forego approximately $2^11 annual stipend, they need to deal with pestering by publisher representatives (who suddenly discover how important the journal is after years of taking it for granted), media attention, learning new editorial software, and general uncertainty. So I salute Akihiro Munemasa, Hendrik van Maldeghem, Christos Athanasiadis and Hugh Thomas, who to my knowledge are the first editors-in-chief to flip their mathematics journal from a subscription model to one run according to Fair OA principles. May they be followed by many, and soon!

Renee Wilson RIP

My mother died 2016-02-02 after a fairly short battle with cancer. Although aged nearly 82 she was very energetic and this was a big shock to all who knew her. It is a strange feeling when someone you thought would be around forever leaves so early. I have a memorial website to which anyone reading this who knew her should ask me for the link.

Being orphaned in middle age is much better than earlier, but still tough to handle.

The logic of journalism?

From BBC:

Intelligence tests are as much a measure of motivation as they are of mental ability, says research from the US.

Researchers from Pennsylvania found that a high IQ score required both high intelligence and high motivation but a low IQ score could be the result of a lack of either factor.

It looks like “P implies A and B” BUT “not A or not B implies not P”. This seems a strange way to write a news story, since the second statement is logically equivalent to the first.

Another interpretation is that the event “low IQ score” is not the complement of  the event “high IQ score”. This gives a little more content: “non-high” is implied a priori by lack of either factor, but in fact “low” is implied.

Of course “could” might mean that “not A and not B is not inconsistent with not P”, but that would be very much less newsworthy. Perhaps we need a standard language for journalists to deal with such issues.

Knowing your own place in the world (of science)

I just found some very interesting graphics that show how relatively (in)active various areas of science are, in terms of citations and volume of publication. I always knew there were a lot of biologists and medical researchers, but just how many was surprising.Now I have a better understanding of how universities work.

http://well-formed.eigenfactor.org/treemap.html

http://well-formed.eigenfactor.org/radial.html

DIMACS 20th birthday

Today I attended the 20th birthday celebrations for DIMACS at Rutgers University. There were some nice talks: Ron Graham on addressing in graphs; Peter Winkler on combinatorics in statistical physics, Joan Feigenbaum on approximate privacy, Michael Trick on the DIMACS Challenges, Richard Karp on implicit set algorithms, Eva Tardos on games in networks, etc. There was also an interesting panel discussion on education (de facto national standards coming in the US are strongly opposed by some here, because of the de-emphasizing of discrete math topics) and industry (the environment is much meaner now, and managers are loath to fund projects without clearly stated benefits to the company). This was my first visit and I can see now why DIMACS has been so successful. Thanks to all the organizers and speakers for a very enjoyable day.

Note: another report by Muthu

Francis Eric Wilson RIP

My father died on Monday. I arrived 3 hours before, and was holding his hand as he died. He lived long enough to see photos of his second grandson, and recognized me when I arrived.

In case there is anyone reading who knew him, I enclose the death notice that my mother and I put in the newspaper. I intend to make a website as a memorial.

WILSON, FRANCIS ERIC passed away peacefully at home [address] on 1 September 2008 aged 77. His generous heart gave out at last. Loved husband of Renee, father of Mark, father-in-law of Golbon and grandad of Yusef and Yahya. Privately cremated. Donations in lieu of flowers to Nelson Regional Hospice Trust or NZ Heart Foundation. He drank deeply from the well of life, and took the final curtain with aplomb.

It is a strange feeling to be the “patriarch” as someone said to me today – childhood now feels officially over. It certainly has given me resolve to look after my health and try to be around until my children are 70.

Straight and crooked thinking

My Otago colleague Mike Atkinson recommended this book which he remembered from his youth. Written in 1930 by Robert Thouless, a psychologist at Cambridge, it is a superb exposition of errors of thought, dishonest rhetorical tricks, and how to combat them (the reviewers on Amazon.com seem to agree with me). Thouless believed in the existence of parapsychological phenomena (ESP) but was severe on purported experimental proofs of this existence. The examples in the book show his frustration with politics. It isn’t often that I find a kindred spirit from the past. Our lifespans overlapped considerably – I wish I had met him.

The book is out of print and hard to find (I found it in our university library). If anyone knows how to get such books reissued, please let me know.