At SoCal Swordfight 2023, I was warming up for one of my tournaments with a very good fencer, one that is known for pretty much crushing his opposition. During the few minutes of back-and-forth exchanges, I managed to land a touch on him once. This is something that, given his reputation and record, I thought would be next to impossible. It got me wondering about how likely it is for something like that to happen, and I started working on the math behind Can I Hit That? very shortly afterwards.
Can I Hit That? is a new probability tool that shows four different metrics:
Metric 1: The probability of winning one exchange against an opponent
Metric 2: The probability of winning a 5-exchange match when all touches are worth only 1 point
Metric 3: The probability of winning a 5-exchange match when different targets are different points
Metric 4: The probability of scoring at least one time against your opponent during a 5-exchange match
Follow the link below to play with the tool:
www.tinyurl.com/can-i-hit-that
The Data
Data in this tool comes from all events in HEMA Scorecard except ones set up as test events. Competitors will only have data if they have competed in a tournament of the weapon type chosen. For example, if Competitor A has only competed in rapier tournaments, you will not be able to compare Competitor A for longsword. Similarly, if Competitor B has competed in rapier and longsword, only rapier data will be used to determine rapier probabilities while only longsword will be used to determine longsword probabilities. Additionally, only clean exchanges are used for these metrics (so doubles, exchanges with afterblow, etc are removed).
This tool will not show event information in detail. If you would like to see individual data, you may visit HEMA Scorecard and look under their Stats/Analytics tab at the “Attendance by Fighter” section for information on individual fighters.
Note:
The data contained in this tool is not a direct connection to HEMA Scorecard, so it is not updated in real time. I plan on making updates to the data every couple months. The date of the latest data extract is noted at the bottom of the tool. There are no plans to incorporate data from other sources.
Metric 1: The probability of winning one exchange against an opponent
The first probability metric shown in the tool is the probability of winning an exchange against a given opponent. This probability is calculated by using the idea of Odds. The most common place where people see odds is in betting, which is usually represented as a ratio. For example, someone may say that Person A has a three-to-one chance of winning over person B, or may write it as 3:1. This means that Person A has a three times higher chance of winning than Person B. Another way to think about it is that for every three times Person A wins, Person B will win one time.
In the context of these metrics, a “win” is determined by how often a fighter wins an exchange in a match, or the percentage of wins. This would be calculated by taking the number of exchanges a fighter wins and dividing it by the total number of exchanges:
The equation to figure out the probability of winning from the odds is done by taking the Win Percent of Fighter 1 and dividing it by the sum of Fighter 1’s and Fighter 2’s Win Percents. This will give you the probability of winning one exchange against your opponent:
Metric 2: The probability of winning a 5-exchange match when all touches are worth only 1 point
Metric 2 builds off of Metric 1 by taking exchange winning one step further and applying it to a particular type of match: one where the match consists of five exchanges. In order to win a best out of five match, you can either win 3-2, 4-1, or 5-0. In order to do this, we have to calculate something called the joint probability of each of these scenarios. To calculate the joint probability of winning a match 3-2, you need to multiply the probability of winning one exchange three times for Fighter 1 and winning one exchange two times for Fighter 2. This looks like this:
One more thing that we need to consider is the order of getting the points. For example, in a 3-2 match, the points can be scored in 10 different ways:
You can easily figure out how many different combinations there are by using a Combination calculation. A Combination calculation for winning 3 to 2 would be written as 5C3, which is verbalized as “Five Choose Three”. Basically what this is saying is that of the five exchanges, how many different ways can we choose three different exchanges to get points on. The formula for this is:
The above calculations are just for the scenario where you win 3-2. You also have to consider the scenarios when you win 4-1 and 5-0. You can add all of these scenarios together to get your probability of winning a 5-exchange match:
Metric 3: The probability of winning a 5-exchange match when different targets are different points
Metric 3 is very similar to Metric 2 in that it calculates the probability of winning a 5-exchange, except this time we take into consideration that winning an exchange can be worth more than one point. For this, we use whatever the event organizer has picked as the point values for their tournament, capping the amount of points that can be earned in a single exchange at 5 points (some event organizers award 20 points on a single exchange).
What we do for this is use a statistical concept called Expected Value (EV). In this case, the EV of an exchange is the average amount of points a fighter has scored across all exchanges. However, we are also modifying this a little bit; the points Fighter 1 scores is dependent both on what targets Fighter 1 tends to hit as well as the targets that Fighter 2 tends to give up. We will call these Fighter 1 Targets (F1T) and Fighter 2 Vulnerabilities (F2V). We calculate this as:
Once we have this value, we use it in the equation for winning 1 exchange as follows:
You then substitute this value into the last equation of the Metric 2 section to get your probability of winning a 5-exchange match when you account for weighted targets.
Metric 4: The probability of scoring at least one time against your opponent during a 5-exchange match
The final metric that is included is one that shows you the probability of winning at least one exchange in a 5-exchange match. This means that you don’t leave the match with a 0 on the board!
When we think about this mathematically, this means that Fighter 1 can win 1 exchange, 2 exchanges, 3 exchanges, 4 exchanges, or 5 exchanges. We could create a very long equation where we add up all these scenarios like we did for Metric 2, but we can also think of it a different way. The only situation that cannot occur for this to be fulfilled is for Fighter 2 to score on all 5 exchanges. Since the probability of all scenarios happening is 100%, we can write the equation as follows:
Caveats
There are always things to take into account when thinking about how much you can trust numbers that are thrown out at you. Here are some of the things to consider when thinking about the numbers presented to you.
- These probabilities rely on the idea of independence among exchanges. What this means is that it assumes that no exchange has an effect on any future or prior exchanges. This just simply isn’t how HEMA matches work. Maybe you performed an awesome mutieren, but the judge missed it so you feel frustrated during your next exchange. Maybe you got hit hard and are a little rattled next time. Perhaps you are taking into consideration the way your opponent fenced you during the last exchange, so you adapt how you fight to suit the exchange better. There are many different reasons why exchanges are not independent from each other.
- The caliber of each fighter’s opponents will have an impact on their numbers. If a fighter is constantly fighting easier opponents, their numbers may be inflated. Conversely, if a fighter is regularly fighting harder opponents, their numbers may be lower than what they should be. In HEMA Scorecard, 35,422 out of 176,833 total exchanges come from tournaments that are divided by skill (various difficulty tiers, beginners’ tournaments, advanced divisions, etc). You can view which tournaments a person has participated in on HEMA Scorecard by looking under the Stats/Analytics tab at the “Attendance by Fighter” section.
- The more exchanges a person has, the higher confidence we can have that their probabilities are accurate. Can I Hit That? shows the number of recorded exchanges for the displayed fighters. A good rule of thumb is that 30 exchanges is a decent sample size, with over 100 exchanges being an excellent sample size.
- This tool uses all HEMA Scorecard data available, but does not include other sources. This means that not all tournament data is included in the tool, so perhaps your best (or worst) tournament performances are not included. It also means there are likely regional biases. Additionally, because HEMA Scorecard dates back to 2015, it accounts for a fencer’s entire history since that point in time; realistically, fencers should improve over time but will have all their historical data added to their metrics.
- These numbers do not account for anything other than how a fencer has previously performed in tournaments. There are many different outside factors which could affect their performance in a future fight. Someone may have been training extra hard leading up to their next tournament and outperform their metrics. A fighter could be recovering from illness or injury and not be fighting the way they normally do. Someone could just simply be on fire during their tournament. There are many different reasons not to boil yourself down to a statistic.
Anyway, there’s a lot of caveats about the data. In my mind, there will never be a perfect set of statistics to describe or model the behaviors of fencers, but if we look at all of what’s available, maybe we can get at least a blurry picture. Just remember, it’s all in good fun.
