Using probability to improve your players’ experience

Dave Berlin Mechanics 1 Comment

Graphing for game design

Graphing for game design

In a cooperative game, it’s extremely important to get the balance right. Competitive games are typically easier to balance because the humans balance each other out. As your skills improve, so do mine, and the game continues to feel balanced as we both ramp up in ability. But for co-op games, the humans are playing against the game, and the game needs to respond in kind. Many cooperative games have a difficulty setting, but oftentimes game conditions result in a feast or famine situation where you can lose within a few turns or cruise to victory. These are issues I’d like to avoid in Prism.

A specific problem in Prism version 5 is that the end game is triggered when players find one of the portals on the map. Those portals can be anywhere in the level 4 hexes, and finding one is simply a matter of luck. You can find a portal on the first level 4 hex you flip over, or it could go all the way to the last hex. To be more precise, there’s a 1 in n chance of finding a portal, where n is the number of level 4 hexes on the board (you can use several portals, but let’s keep this simple for this example). If you build a large board, for example with 20 L4 hexes, and you include one portal, you have a 1 in 20 chance of finding a portal on each hex. There’s a Uniform Distribution describing the probability of finding the portal (note: this is the distribution over a large number of games, not the chances of finding a portal as you play one specific game).

Chance of finding a portal

Why is this such a bad thing? Let’s think about what this is saying: if we played 1,000 games, 50 of those would end with players finding the portal on their very first level 4 hex. They may or may not choose to activate the portal at that point, but the rest of the game becomes an exercise in risk mitigation where players are discouraged from flipping over too many level 4 hexes.

Now imagine the other end of the scale: 50 of those games will end with the portal being under the last hex the players flip over. Now it becomes a total slog through every monster on the board. The adventurers will almost certainly burn through the Interrogation deck–the game’s timer–and be in serious jeopardy of losing before they ever find the portal. As they get more desperate to find a way out, they will take more risks and be in more peril. Risk and peril are all well and good, in moderation, but in these cases players will almost certainly lose and feel frustrated because luck beat them.

There’s no way for me as the designer to set the difficulty level appropriately for these two ends of the scale. If I can’t predict whether it’s going to be a first hex portal game or a last hex portal game, there’s very little chance I’ll set the difficulty correctly.

How to fix this? At the last San Diego Board Game Designer meeting, Eric suggested that instead of finding portals directly, I find some number of keys that would open the portal. His key insight was that this would make the game more predictable. Let’s say there are three keys hidden in the 20 hexes, and you need to find these keys to open the portal. Your chances of finding that third key would be:


This definitely fixes some of the problems. Now there’s no way you can find all three keys by flipping over your first portal, so the game will rarely be too easy. By hex 16 you have a 49.16% chance of having found three keys. By hex 17 that number goes up to 59.65% and by 18 it’s up to 71.58%. At first blush this seems great: I don’t have to plan for wizards finding the portal on their first hex, so I can assume each game will be harder than the completely random chance I have now.

But while this is certainly more predictable, it feels like it’s coming down on the “too hard” side of things. After all, 150 of our 1,000 playtests will end with players finding the third key on hex 20. Over half the time–in about 509 of those games–the players will find the third key on hexes 17, 18, 19, or 20. Maybe this is what you want as a designer. But I don’t, and here are a few reasons why:

  • The game will take longer to play if you have to go through every hex
  • Replayability suffers, because you will be exposed to every level 4 hex each game and therefore they won’t feel as fresh
  • The game will actually feel less predictable for players because they know it will often go right down to the end
  • Players’ strategy in responding to the Interrogation deck will be based on this assumption that the game always goes to the last hex
  • This may also result in more games where players lose and feel like the game is just too hard and arbitrary: more monsters equals higher chance of dying

So how can we fix this? Can we make it so that the game ends more in the middle of discovering hexes, so our hexes feel fresher, the game goes quicker, and players could still have a chance of having one of those epic, near-20-hex-flipping adventures? Sure, we can do that, by adding more keys to the hexes while keeping the required number of keys the same. Here’s the graph of your chances of finding your third and final required key when there are five present in the hexes:


This looks like a Normal Distribution, which is great because most games will find the wizards discovering their third key somewhere in the 7-14 hex range. That means I can set the difficulty based on that likelihood and I don’t have to worry about the first hex or last hex scenarios. Over 90% of games will find their third key by the 15th hex. So we have shorter games and fresher hexes. There’s less predictability because a few games will fall outside the normal range, and while the difficulty may not be quite perfect on those games, it’ll keep the players on their toes.

The bottom line, then, is that for this game it’s better for me to include a number of things that players have to collect, and only after collecting all of them will their exit appear before them. Having done the math on this, I’ve started to think of some games that exhibit this behavior. What about you? How many games can you think of that make you collect a bunch of things scattered across the map, rather than finding a single item/exit? And, I wonder if these decisions were made for mathematical reasons, or for theme, fun, or some other reason?

If you’re interested in calculating these probabilities for your game, you’ll want to read up on the Hypergeometric Distribution formula.

Comments 1

  1. Pingback: The Death of Prism | Galvanized Studios

Leave a Reply

Your email address will not be published. Required fields are marked *