Opening the Exploration Potential

(This post probably won't make a lot of sense without reading the previous one)

Knowing that our 2 player game will last on average 32 turns, the 3 player game 24 turns, 4 player game 21 turns and 5 player game 18 turns, we need to know how quickly to open up the game for exploration potential.

This applies to almost any RPG scenario, but it's often something that a lot of designers just don't think about. At the start of play I like to have a limited range of potential for exploration and character development, at this point the game is more about establishing the ouevre. Then as the game progresses we see more opportunity to explore the world in a way that makes sense with what we already know. As play comes to it's conclusion, more choices end up leading to a climax.

You could look at this by saying that we have a few choices at the beginning, more in the middle, and less at the end, but I think of it differently. The range of choices is always decreasing, right from the beginning, the significance of the choices is always increasing. So when you multiply the range by the significance you get a nice trajectory of play.

8 choices x 0 significance = 0
7 choices x 1 significance = 7  
6 choices x 2 significance = 12  
5 choices x 3 significance = 15  
4 choices x 4 significance = 16  
3 choices x 5 significance = 15  
2 choices x 6 significance = 12  
1 choices x 7 significance = 7  
0 choices x 8 significance = 0

Those middle bits of the story are where the meat of the narrative can be found. The earlier bits are just setting it up, and the later bits are using what we know to confront the climax and tie up the loose ends one by one. Note that I would rarely use the zero significance or zero choice options unless there were specific types of stories I were trying to convey. It's usually in that highlighted section where I like to keep my stories.

For the purposes of the scroll game, that means I want the game to open up fairly rapidly, before the horde have had a chance to build momentum, but at the end of the game, the scope of wandering closes up (which naturally occurs as the horde approaches).

Again, I've generated up some average calculations. I've done these for the 2, 3, 4, and 5 player situations of play, but since all those numbers can be a bit overwhelming, I'll focus on the three player game for the moment.

Here's how the horde approaches. After 24 turns the entire scroll has been consumed by their advance.

I'd like to see the scroll most opened up at around the middle of the game. The earlier parts of the game should see more unrolling as exploration occurs quickly and less unrolling as the horde advances slowly, and the later parts of the game should see less unrolling as the ultimate goal is approached (and finally) revealed while the horde continues it's relentless march forward.

I should also note here that I've now added a 22nd stage to the game...so we've ow got 21 steps on the path from death to potential immortality in the "Field of Reeds"...and now we've got the field of Reeds as the final level of transcendence. The horde will never cross the threshold into the Field of Reeds, this is not their domain and powerful gods protect it. But it allows me to also add a new column to the table, "Reeds".

The previous version of the table used calculations to determine the chance of this step being the furthest point to which the horde has advanced. But this calculations falls flat at the end of the scroll, where each turn their should be a higher chance of the horde being here (equal to the chance they had previously advanced to this point, plus the chance they had advanced here on this particular turn). So I've added this calculation into the "Reeds" column which now gives a more accurate projected end game timeframe. In this particular version of the table ("3 players"), the horde is still expected to reach the end of the scroll after 24 turns, but in the 2 player game it has a 50% chance of ending after the 29th turn (instead of the 32nd). The 4 player game still has a 50% chance of ending after the 21st turn, and the 5 player game has a 50% chance of ending after the 19th turn (rather than the 18th). It doesn't make a whole lot of difference especially since user experience will vary based on the players involved.

Focusing back on the 3 player game, now looking at the chance of opening up new areas of the scroll for exploring. For the purposes of absolute simplicity, we'll give every player an equal chance of opening up the scroll by a single step every time they do something.

1 in 4 chance

Let's say a player has two six sided dice, and each of those dice has 3 faces with swords on them. We'll call a sword a success, and two successes are required to open up the scroll. That's a 50% chance of success on each die, two successes have a 25% (1 in 4) chance of occurring.

With 3 players performing actions every turn, there is a 1-in-64 chance that every player will successfully open the scroll during their action, a 9-in-64 chance that two of them will accomplish this feat, a 27-in-64 chance that one of them will do it, and a 27-in-64 chance that none will. It's binomials from high school mathematics if you don't believe me.

I'll make a nice colour coded chart to make things easier to understand.
   

If the cell is red, it's unlikely that we'll be at this point in the game with this much of the scroll unrolled. As the cells move to yellow it becomes more likely that the scroll will be unrolled to this point, as they move to green it becomes even more likely that this is the point here the game will be under the current calculations.

There is a faint chance, incredibly unlikely but within the realms of possibility if every character rolls multiple successes on every action, that the scroll will be fully unwound on the eighth turn. The point where 50% of scrolls should have fully unwound occurs at the end of the 28th turn. Which is a bit tragic because 50% of games will have seen the horde completely consume the scroll by the 24th turn. If we were going for a game of tragic, against the odds survival, this might be a useful set of calculations to apply, but I'm aiming for something a bit more heroic where the players have a reasonable chance of winning, or at least have a variety of meaningful choices available rather than continually being hounded by the horde.  

1 in 3 chance

Maybe a player has a single die, and since there are two shields on the die then one of these opens up the scroll by a step.

Again, if every player successfully opens up the scroll during their action, the whole scroll could theoretically be opened by the eighth turn. It's roughly a billion-to-one longshot, but it's possible and more likely than the previous scenario.
  

Now we see that the 50% threshold is reached after 21 turns, meaning that there will be a significant number of games where the Field of Reeds has been revealed before the horde gets there. Turn by turn, we can see the most likely degree to which the scroll has been opened and the most likely location that the horde has reached. Hypothetically...

Turn 1: Scroll likely opened to step 1, horde likely at step 0.

Turn 2: Scroll likely opened to step 2, horde likely at step 0 or 1.

Turn 3: Scroll likely opened to step 3, horde likely at step 1 or 2.

Turn 4: Scroll likely opened to step 4, horde likely at step 2 or 3.

Turn 5: Scroll likely opened to step 5, horde likely at step 3 or 4.

Turn 6: Scroll likely opened to step 6, horde likely at step 4 or 5.

Turn 7: Scroll likely opened to step 7, horde likely at step 5 or 6.

Turn 8: Scroll likely opened to step 8, horde likely at step 6 or 7.

Turn 9: Scroll likely opened to step 9, horde likely at step 6 or 7.

Turn 10: Scroll likely opened to step 10, horde likely at step 7 or 8.

Turn 11: Scroll likely opened to step 11, horde likely at step 8 or 9.

Turn 12: Scroll likely opened to step 12, horde likely at step 9 or 10.

Turn 13: Scroll likely opened to step 13, horde likely at step 10 or 11.

Turn 14: Scroll likely opened to step 14, horde likely at step 11 or 12.

Turn 15: Scroll likely opened to step 15, horde likely at step 12 or 13.

Turn 16: Scroll likely opened to step 16, horde likely at step 13 or 14.

Turn 17: Scroll likely opened to step 17, horde likely at step 14 or 15.

Turn 18: Scroll likely opened to step 18, horde likely at step 15 or 16.

Turn 19: Scroll likely opened to step 19, horde likely at step 16 or 17.

Turn 20: Scroll likely opened to step 20, horde likely at step 17 or 18.

Turn 21: Scroll likely opened to step 21, horde likely at step 17 or 18.

Turn 22: Scroll likely opened to Reeds, horde likely at step 18 or 19.

Turn 23: Scroll likely opened to Reeds, horde likely at step 19 or 20.

Turn 24: Scroll likely opened to Reeds, horde likely at step 20 or 21.

So in each case, the horde is hot on the heels of the characters. There really isn't a lot of time to explore, or be reflective on the nature of life and death. This might be more appropriate to an scroll game about avoiding pursuit from advancing zombie hordes, criminals on the run, or something which generally aims for more adrenaline.

1 in 2 chance

Accelerating faster than the 1-in-3 version, this twist on the numbers has a slightly higher (but still remote) chance of revealing the Field of Reeds on turn 8 (roughly 1 in 2 million now). The 50% threshold reveals the Reeds by turn 15.
  

 Running through the step by step, we get...

Turn 1: Scroll likely opened to step 1, horde likely at step 0.

Turn 2: Scroll likely opened to step 2 or 3, horde likely at step 0 or 1.

Turn 3: Scroll likely opened to step 4, horde likely at step 1 or 2.

Turn 4: Scroll likely opened to step 5 or 6, horde likely at step 2 or 3.

Turn 5: Scroll likely opened to step 7, horde likely at step 3 or 4.

Turn 6: Scroll likely opened to step 8 or 9, horde likely at step 4 or 5.

Turn 7: Scroll likely opened to step 10, horde likely at step 5 or 6.

Turn 8: Scroll likely opened to step 11 or 12, horde likely at step 6 or 7.

Turn 9: Scroll likely opened to step 13, horde likely at step 6 or 7.

Turn 10: Scroll likely opened to step 14 or 15, horde likely at step 7 or 8.

Turn 11: Scroll likely opened to step 16, horde likely at step 8 or 9.

Turn 12: Scroll likely opened to step 17 or 18, horde likely at step 9 or 10.

Turn 13: Scroll likely opened to step 19, horde likely at step 10 or 11.

Turn 14: Scroll likely opened to step 20 or 21, horde likely at step 11 or 12.

Turn 15: Scroll likely opened to Reeds, horde likely at step 12 or 13.

Turn 16: Scroll likely opened to Reeds, horde likely at step 13 or 14.

Turn 17: Scroll likely opened to Reeds, horde likely at step 14 or 15.

Turn 18: Scroll likely opened to Reeds, horde likely at step 15 or 16.

Turn 19: Scroll likely opened to Reeds, horde likely at step 16 or 17.

Turn 20: Scroll likely opened to Reeds, horde likely at step 17 or 18.

Turn 21: Scroll likely opened to Reeds, horde likely at step 17 or 18.

Turn 22: Scroll likely opened to Reeds, horde likely at step 18 or 19.

Turn 23: Scroll likely opened to Reeds, horde likely at step 19 or 20.

Turn 24: Scroll likely opened to Reeds, horde likely at step 20 or 21.

In this version of the numbers, since the scroll will have opened all the way by turn 15, this leaves nine more turns of exploration before the horde engulfs the deadlands. Much more leisurely. The amount of exploration possible increases through the first half of the game, then contracts as the climax approaches toward the end. I think there is still a bit more tweaking to do here, but it feels like things are on the right track. I think the next thing to look at is an incremental increase of the numbers, perhaps making the first steps of the journey easy to accomplish (possibly a 2 in 3 chance of opening a new step or greater). then things get progressively harder (a 1 in 2 chance, then a 1 in 3, then maybe back to the 1 in 4). This would give the characters more chance to expand the exploration potential early in the game, but would make the horde more of a threat as the game draws to it's conclusion.

Generally the numbers seem to work in a similar manner regardless of the number of players (except that the more players there are, the longer to game tends to play out). There's a lot of background theory going on here, and I have to seriously wonder whether half the game designers out there go to this much trouble...

...for the moment though, back to playing with numbers.