Student perspectives: The Elo Rating System – From Chess to Education

A post by Andrea Becsek, PhD student on the Compass programme.

If you have recently also binge-watched the Queen’s Gambit chances are you have heard of the Elo Rating System. There are actually many games out there that require some way to rank players or even teams. However, the applications of the Elo Rating System reach further than you think.

History and Applications

The Elo Rating System 1 was first suggested as a way to rank chess players, however, it can be used in any competitive two-player game that requires a ranking of its players. The system was first adopted by the World Chess Federation in 1970, and there have been various adjustments to it since, resulting in different implementations by each organisation.

For any soccer-lovers out there, the FIFA world rankings are also based on the Elo System, but if you happen to be into a similar sport, worry not, Elo has you covered. And the list of applications goes on and on: Backgammon, Scrabble, Go, Pokemon, and apparently even Tinder used it at some point.

Fun fact: The formulas used by the Elo Rating make a famous appearance in the Social Network, a movie about the creation of Facebook. Whether this was the actual algorithm used for FaceMash, the first version of Facebook, is however unclear.

All this sounds pretty cool, but how does it actually work?

How it works

We want a way to rank players and update their ranking after each game. Let’s start by assuming that we have the ranking for the two players about to play: \theta_i for player i and \theta_j for player j. Then we can compute the probability of player i winning against player j using the logistic function:

P(Y_{ij}=1)=\frac{1}{1+\exp\{-(\theta_i-\theta_j)\}}.

Given what we know about the logistic function, it’s easy to notice that the smaller the difference between the players, the less certain the outcome as the probability of winning will be close to 0.5.

Once the outcome of the game is known, we can update both players’ abilities

\theta_{i}:=\theta_{i}+K(Y_{ij}-P(Y_{ij}=1))

\theta_{j}:=\theta_{j}+K(P(Y_{ij}=1)-Y_{ij}).

The K factor controls the influence of a player’s performance on their previous ranking. For players with high rankings, a smaller K is used because we expect their abilities to be somewhat stable and hence their ranking shouldn’t be too heavily influenced by every game. On the other hand, players with low ability can learn and improve quite quickly, and therefore their rating should be able to fluctuate more so they have a larger K number.

The term in the brackets represents how different the actual outcome is from the expected outcome of the game. If a player is expected to win but doesn’t, their ranking will decrease, and vice versa. The larger the difference, the more their rating will change. For example, if a weaker player is highly unlikely to win, but they do, their ranking will be boosted quite a bit because it was a hard battle for them. On the other hand, if a strong player is really likely to win because they are playing against a weak player, their increase in score will be small as it was an easy win for them.

Elo Rating and Education

As previously mentioned, the Elo Rating System has been used in a wide range of fields and, as it turns out, that includes education, more specifically, adaptive educational systems 2. Adaptive educational systems are concerned with automatically selecting adequate material for a student depending on their previous performance.

Note that a system can be adaptive at different levels of granularity. Some systems might adapt the homework from week to week by generating it based on the student’s current ability and update their ability once the homework has been completed. Whereas other systems are able to update the student ability after every single question. As you can imagine, using the second system requires a fairly fast, online algorithm. And this is where the Elo Rating comes in.

For an adaptive system to work, we need two key components: student abilities and question difficulties. To apply the Elo Rating to this context, we treat a student’s interaction with a question as a game where the student’s ranking represents their ability and the question’s ranking represents its difficulty. We can then predict whether a student of ability \theta_i will answer a question of difficulty d_j correctly using

P(\text{correct}_{ij}=1)=\frac{1}{1+\exp\{-(\theta_i-d_j)\}}.

and the ability and difficulty can be updated using

\theta_{i}:=\theta_{i}+K(\text{correct}_{ij}-P(\text{correct}_{ij}=1))

d_{j}:=d_{j}+K(P(\text{correct}_{ij}=1)-\text{correct}_{ij}).

So even if you only have 10 minutes to create an adaptive educational system you can easily implement this algorithm. Set all abilities and question difficulties to 0, let students answer your questions, and wait for the magic to happen. If you do have some prior knowledge about the difficulty of the items you could of course incorporate that into the initial values.

One important thing to note is that one should be careful with ranking students based on their abilities as this could result in various ethical issues. The main purpose of obtaining their abilities is to track their progress and match them with questions that are at the right level for them, easy enough to stay motivated, but hard enough to feel challenged.

Conclusion

So is Elo the best option for an adaptive system? It depends. It is fast, enables on the fly updates, it’s easy to implement, and in some contexts, it even has a similar performance to more complex models. However, there are usually many other factors that can be relevant to predicting student performance, such as the time spent on a question or the number of hints they use. This additional data can be incorporated into more complex models, probably resulting in better predictions and offering much more insight. At the end of the day, there is always a trade-off, so depending on the context it’s up to you to decide whether the Elo Rating System is the way to go.

Find out more about Andrea Becsek and her work on her profile page.

  1. Elo, A.E., 1978. The rating of chessplayers, past and present. Arco Pub.
  2. Pelánek, R., 2016. Applications of the Elo rating system in adaptive educational systems. Computers & Education, 98, pp.169-179.

Women and non-binary people in mathematics event

On 5 and 6 November the school hosted its annual Women and Non-Binary People in Mathematics event.

 

 

 

 

This two-day event held in the Fry Building, aimed at encouraging women and non-binary people to consider continuing their studies to PhD level, welcomed participants from around the country, as well as outside of the UK.

The event featured talks from mathematicians working both in universities and industry, giving insight into their current roles and their careers to date. As well as this it also offered ample time for discussion with other participants who are facing the same decisions, and with current PhD students who have recently faced the same questions.

The Tuesday afternoon kicked off with a talk from keynote speaker Delaram Kahrobaei from University of York, followed by a dinner, where participants were able to enjoy discussion and networking in a relaxed environment. Wednesday morning offered an insight into the life of a PhD student, with talks and Q&A sessions with current post-grad students. The final portion of the event was open to to all students and more broadly explored the nature of PhD research, the work environment, the application process, and career options. Talks were given by industry speaker, Katie Russell from OVO Energy and the University of Bristol’s Dr Lynne Walling.

Current PhD student and one of the event organisers, Emma Bailey said ‘My favourite part of the Women and Non-Binary conference is changing assumptions: showing people who maths PhD students are, what we do on a weekly basis, and all of the opportunities out there post-PhD’.

The event successfully illuminated some of the doors a PhD in mathematics can open and hopefully inspired potential PhD students to continue their future in mathematics.

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