Three is a Magic Number

Let’s say you are developing a very simple area majority game, with just two players and two areas - Call them A and B.

Each player has a number of resources and has to allocate them between the two areas.

Whomever has the most in each area gets a point. Ties score a half a point.

Let’s say we both have ten resources. How should I distribute them?

In this case it doesn’t matter. There’s no way to win both. No matter how you distribute, you will always get 1 point.

For example, I distribute 8/2. If you also distribute 8/2 we tie in both and each get 1 point. If you do anything else (7/3 or 0/10 for example) we each win 1 and lose 1.

I’m working on a game now with a mechanic that was essentially analogous to this, and I just couldn’t get it to generate interesting decisions for the player. Putting all your resources into a single box is generally the best strategy.

Some of the rules I tried:

• Hidden semi-random resources

• Having double the resources of your opponent counted as a ‘double-win’ and 2 points.

• Allow an ability that would let sacrifice a resource or two to peek at your opponent’s allocation in one box before you made your choice.

None of them helped. The second one made an even distribution between boxes definitely the best way to go, but that’s not much of an improvement. I tried a lot more, but it just laid there like a dead fish.

Suddenly it occurred to me - what if I had three boxes instead of two?

Seems like a simple change, right?

Well, having three boxes makes a huge difference in this situation.

Now there is counterplay.

For every distribution of resources between the three boxes, there is a distribution that beats it in two out of three.

For example, with my 10 resources:

• 10, 0, 0 loses to 0, 5, 5 (and many others)

• 3, 3, 4, also loses to 0, 5, 5

• 0, 5, 5 loses to 2, 0, 8

and so on.

If you know your opponent’s distribution there’s always a winning play

I have to admit, this blew me away. I’m a little embarrassed by this - I’m guessing that many of you knew this already.

To you, I ask - why didn’t you tell me?

Adding that third box turns this mechanism into Rock Paper Scissors, one of the fundamental tools of the game designer.

Interestingly, if you know your opponent’s distribution, you don’t even have to have as many resources as they do. For example, let’s say you have 10 resources but I only have 8. I can still always win:

• 10, 0, 0 loses to 0, 4, 4

• 3, 3, 4, loses to 4, 4, 0

and so on.

Neat! And it opens up a lot more game play opportunities than I had before.

I still have a ways to go to make the player choices meaningful, but this is a much, much better platform to start with.

Wild West Showdown

There’s another problem that also demonstrates the huge gulf between ‘two’ and ‘three’ - the three-way duel.

Let’s say I have two cowboys in a duel. One of them - The Nameless One - hits 100% of the time. The other, Syd the Kid, hits 75% of the time.

They randomly choose who goes first, and then alternate taking shots until only one cowboy is left.

Obviously, if The Nameless One goes first, they will win, as they hit 100% of the time. Syd the Kid will only win if they go first and hit. If they miss, the turn passes back to the Nameless One, and things won’t go so well for Syd at that point.

Here’s a matrix summarizing the situation:

Now let’s say that we add a third player - One-Legged McGee. McGee only hits the target 50% of the time.

Again, we randomly select someone to go first, and then go around the circle until only one person is left standing.

If every player selects their target to maximize their chances of winning, what happens? What are everyone’s chances to win?

Well, if the Nameless One goes first, they should target Syd the Kid. If they target McGee first, they will have a 75% chance of getting shot instead of 50%. Then McGee will go and have a 50/50 shot (literally) to take out the Nameless One before Nameless gets another turn.

You can perform this type of analysis for each of the scenarios, and what you’ll discover are two very surprising things:

• McGee has the best chances to win

• McGee’s best strategy if they go first (or have a turn when all three are still in the shoot out) is to simply fire into the air and automatically miss.

Here’s a three-player version of the matrix:

This is a shocking result. The worst shot has the best chance to survive. The cause, in this case, is that the two top cowboys will shoot at each other, leaving McGee to pick up the pieces.

Going from two of something to three very often makes a huge difference, be it players or areas to control. Three is a magic number. Never underestimate the power of going to three of something to spice it up.

I did a radio drama about the three-sided duel, featuring my kids, which can be heard here.

On Ludology we also did an entire episode about the difference between two and three player games that is well worth a listen.

I did not invent this three-sided shootout. I believe it may go back to Martin Gardner’s Mathematical Games column in Scientific American, but it may be older than that.

If you want to see all the details about this puzzle, this video has a thorough analysis.

Have you had any experiences in playing or designing games where you experienced the power of three? Please share in the comments!