We played frisbee yesterday. And I am still sore from the extra running and jumping and physical activity. It was a simple, casual game of frisbee, 3 people, just throwing a flying-disc around in circles. A throws to B, B throws to C and C throws back to A.

While playing frisbee, I noticed a few things:

• Plastic frisbees can cut. I’ve got at least 5 cuts on my arms and legs.
• If you throw a frisbee hard, and with deliberate strength, it wobbles (I believe this wobble was what caused Richard Feynman to come up with his quantum electrodynamics work)
• Casual frisbee has no scoring system.

I’ll spare you the whinging on the cuts on my arms and legs (they’re not that bad anyways, I’m just over-dramatizing). I’m not too good with quantum electrodynamics, so I’ll also spare you any errorneous thoughts that might occur. But what was interesting was that frisbee has no scoring system. So I started wondering, if casual frisbee would be any more fun with a scoring system.

Now, there are a number of frisbee games with scoring systems (one was introduced by Kels, called frisbee soccer…), most notably, Ultimate Frisbee. But all those games have horrendously complicated rules. A casual game shouldn’t be complicated.

Why bother Scoring Frisbee, If Its Meant to be Simple?

Well, for one, it makes the game competitive.. Just like Guitar Hero, or Rock Band. They’re both casual music games (which I suck at anyways¹), but, they’re also avenues for competitive play. You can stick 2 people in front of a TV with fake guitars and they’ll compete to see who can get the most accurate frets.

Same theory goes with scoring a casual game of frisbee.

The Actual Scoring

To score frisbee, you’d have to ask, how does one define a perfect game? Well, to me, a perfect game is one where when a player throws to another player, the other player catches without having to move from his/her location².

Casual frisbee is a game of accuracy, of estimation, and of hand-eye coordination. A perfect game of frisbee would have any 2 players have perfect accuracy, perfect estimation, and perfect hand-eye coordination. Because both players are actively involved in both throwing and catching, the scoring should go to both players at the same time.

So, for example, in a perfect game, Player A throws a frisbee to Player B, who catches it without having to run. We assign a score of 10 points³ to each player in a perfect game played. This brings us a maximum of 20 points scorable within each game.

The Imperfect Game

Let’s face it, the odds of a perfect game every time is low. That’s the whole point about scoring. Sometimes, we have bad throwers, sometimes we have bad catchers. Sometimes you have both⁴. So the whole point about scoring is to penalize those bad throwers and bad catchers.

This is how the game tree roughly looks like:

Here is an example outcomes list of the above game tree:

So.. how do we score? We previously agreed on having the maximum score per game set to 20. This leaves us to decide whether a slider score should be used.

Slider Score
A slider score is where you penalize one party, and award the other simultaneously. So, for example, the catcher gets 10 points for catching the frisbee, but because the thrower was a poor thrower, the catcher had to run to the left to catch the frisbee. You should award the catcher for his/her extra effort and penalize the thrower for throwing poorly. So, you take n points from the thrower, and award it to the catcher (catcher:10+n - thrower:10-n). That’s what I mean by a slider scale, so it always adds up to 20. The n points is calculated with a compound multiplier which imply the player’s skill.

Binomial Score
Another method of scoring is binomial scoring. In this case, you’d give points to a good catcher, or good thrower, or both, and deny points to poor players. So, for example, a thrower throws the frisbee well, but the catcher missed it, and it hits the catcher right in the gut. So the thrower gets 10 points, and the catcher gets 0.

Pros and Cons
Slider Score:

• Pro - It gives more incentives to play well (because its possible to lose points).
• Pro - It’s an accurate measure of player skill
• Con - It may get very complicated for scoring, especially if more variables (like run distance, or flight pattern, etc) are involved.
• Con - Problems with scoring the last case: bad throws and bad catches.

Binomial Score:

• Pro - It’s simplistic, easy to use
• Con - It’s a gross measure of player skill.
• Con - Less incentives to play well (nothing to lose)

I originally thought of a complicated slider scoring system, measuring distance travelled (runs and jumps), frisbee flight pattern, etc to form a compound multiplier for the slider score (it goes something like (normalized distance travelled + speed)/catcher accuracy⁵). But as I wrote this article, I realized, hey, this is supposed to be a casual game, not something complicated, so.. I impromptuly switch to a combined slider/binomial scoring system.

So instead, you have these 4 situations and their score:

1. Perfect Game - Both players get 10 points each
2. Good Throw, Bad Catch - Thrower gets 15 points, Catcher gets 5
3. Bad Throw, Good Catch - Thrower gets 5, Catcher gets 15
4. Bad Throw, Bad Catch - Both get no points

More Players, More Games

What happens when you have more players, or more games? Like my game yesterday, we had 3 players. How would one go about? Not to worry, actually, this scoring system is very scalable. For many games, I suggest just summing up the scores and averaging them later.

I would suggest a condorcet method of comparing scores between many players, instead of just using the highest average score. That way, its more fair, and more accurate representation of player skill.

Ultimately, though, it’s just a game - it should be fun. But for a little bit more of competitive fun, it wouldn’t hurt to add scores.

Addendum:

Most duh-ify-ing Nash equilibrium (with the assumption that throws and catches are decisions, rather than skills):

Footnotes

1. the game, not music [↩]
2. A game is defined as one throw and one catch/attempt to catch/picking up [↩]
3. i.e. perfect ten [↩]
4. aka Chewxy [↩]
5. yes, I realize they all have different units, and that made no sense, but the actual formula makes sense, at least to me [↩]

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