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Dice statistics help - chase odds?

Flatus Maximus

First Post
You should, in this scenario, be able to calculate a non-zero probability of it not having resolved for any arbitrarily large number of steps, which (IIRC) should mean it sums to one only at infinity, not in finite time.

Mind you, that's academic - the chances of it going on for times a mathematician would call "infinite" are not a practical concern. Whether you're rolling dice for 15 minutes to resolve the chase is more the question.

Indeed, the probability that the game will require at least t rounds goes to zero as t goes to infinity. It's relatively easy (with matrix methods) to calculate the probability that the game with require at least t rounds, so one could use that to get a better understanding of "how long?" questions.
 

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prosfilaes

Adventurer
Usually a chase is "ship A tries to catch ship B before it reaches safe haven Q." So infinite dice rolls are unlikely.

Not always; you can have chases across the trackless desert that go on until the target is captured or disappeared. In the first Curse of the Crimson Throne, there's a chase (with more complex and interesting chase mechanics) that could basically go on forever; she could run around the inner city indefinitely if she doesn't get away or get caught by the PCs.
 

Flatus Maximus

First Post
Mind you, that's academic - the chances of it going on for times a mathematician would call "infinite" are not a practical concern.

Actually, it could be a practical concern, and this concern is the reason I mentioned it. (And didn't realize the possible confusion until after re-reading the last few posts.) The fact that the game will end in finite time with probability one means that there won't be any "loops," such as one might find in other games like chess, in which the game can literally go on forever. The fact that we, as humans, don't have infinite time to play such games is a separate issue; the results given above about expected values and variance are more relevant for these concerns.
 

Umbran

Mod Squad
Staff member
Supporter
The fact that the game will end in finite time with probability one means that ....

But, as I pointed out, I don't think that's a fact.

If the game is guaranteed to complete in finite time, there is some maximum finite time T by which it will complete.

But, I can calculate the probability that the game has not completed at T+1, and that probability will be non-zero.

Ergo, this one only is guaranteed to complete in infinite time.
 

prosfilaes

Adventurer
But, as I pointed out, I don't think that's a fact.

If the game is guaranteed to complete in finite time, there is some maximum finite time T by which it will complete.

But, I can calculate the probability that the game has not completed at T+1, and that probability will be non-zero.

That's not what he said. He said "the game will end in finite time with probability one". Given that every seven turns, the number of games remaining uncompleted decreases by about half, the probability of any game not ending in finite time is the limit of t as t goes to infinity of (1/2)^(t/7), which is 0.

I quibble whether that means that this can not go on forever; for another case, if you pick a random number between 0 and 1, the odds you'll pick a rational number is 0, but that doesn't meant you can't pick a rational number. But that's slippery semantics, and certainly irrelevant to real-life questions.
 

Umbran

Mod Squad
Staff member
Supporter
That's not what he said. He said "the game will end in finite time with probability one".

Yes. That means the game cannot go on for infinite time. But, we can see scenarios where it clearly does not complete in finite time.

I agree that this is a mathematical quibble, unrelated to practical matters of running a game. The real question is how long you have to roll dice at the table in order to complete the game. Scenarios that wouldn't happen even if every person on the planet played 24/7 for a century aren't a real concern.

Given that every seven turns, the number of games remaining uncompleted decreases by about half

I am not sure where you're getting that. It seems to me that each game is independent. The number of games remaining to be completed does not depend upon the number already played.
 

Flatus Maximus

First Post
If the game is guaranteed to complete in finite time, there is some maximum finite time T by which it will complete.

The supremum of a set of finite numbers need not be finite. While there is no "longest game," that doesn't mean that an infinitely-long game is possible!

Yes. That means the game cannot go on for infinite time. But, we can see scenarios where it clearly does not complete in finite time.

Can you give an example of a scenario that does not complete in finite time?
 

Radiating Gnome

Adventurer
Not to derail the probability/will-it-end discussion, but is there a way to take a simple mechanic like you've set up, and add some player choice/options to the mechanic?

My concern is that what you've set up doesn't give the players a lot of agency -- they just roll dice and roll dice and have no way to change their odds (without trying something outside the mechanic to change the one modifier in the system, their speed).

There are plenty of more elaborate chase systems out there -- and those mostly include a lot of options. You may not want to go to that much of an extreme -- but to my tastes you've corrected a bit too far over in the other direction.

-rg
 

Each ‘round’ the pursuer and quarry make Command checks and add their ships’ speed ratings. Whoever wins gets a success. If the other ship already has a success, the winner instead uses his success to negate one of his enemy’s successes. The pursuit ends when one side gets three successes.

If the pursuer gets three successes, he manages to catch up. Start a naval combat with the two ships in adjacent stages. The pursuer gets a +5 bonus to his Command checks during the first turn’s Location phase (page xx).

If the quarry gets three successes, it manages to sail beyond sight of the pursuer and gets away.

Based on how far apart the ships are at the start of the chase, one side might begin with successes. The table also gives you a sense of how far apart the ships are over the course of the chase. As ships get farther apart, it’s less likely either side will make much distance over the other, so each round takes longer.

Table x:xx – Stern Chase Distances
Successes Distance Round Length*

Pursuer 2 Quarter-mile 1 minute
Pursuer 1 1.0 mile 4 minutes
*Tied* 3 miles 15 minutes
Quarry 1 10 miles 1 hour
Quarry 2 30 miles 4 hours
*Rough approximation.

Complexity and Complications
Of course a good chase needs to have decision points, or else it just comes down to who’s faster or luckier.

Where To?
The quarry usually has the benefit of being able to choose where to go. They might head for reinforcements, in which case you’ll want to track how far the chase goes. Generally you take a ship’s speed rating to get the miles it sails per hour, but this can vary based on factors of wind and currents, giving the GM a fair bit of wiggle room.

The quarry might choose to head into dangerous terrain, hoping their pursuer gets damaged or stranded (see Terrain and Hazards, page xx). It might head for terrain that offers a lot of hiding places, and the navigator could make a Stealth check opposing the pursuing ship’s look-out’s Perception check, trying to get out of sight, and perhaps even set up an ambush. Or it might simply try to keep its distance until it can effect repairs and come about for battle.

Crew vs Crew
Various crew roles can help during a stern chase, much as they would during naval combat. Bosuns can add their Charisma to the captain’s Command check. Engineers can roll each round to squeeze a little more speed out of the sails. Look-outs and navigators can help avoid hazardous terrain.

If a chase will cover a long distance, navigators also need to follow the rules for Navigation (page xx). Failure by one side might grant the other a bonus of +2 to +5. Perhaps a pursuing navigator could take a -2 penalty to one round’s check to gain a +5 bonus to the next round’s by trying a shortcut.

And of course various magical options can change things drastically.

Complications
If one side is having too easy a time and you want to increase the tension, foul wind might slow each ship to half speed, giving the pursuer more time to catch up before the quarry can reach a safe haven. Another ship might get involved, perhaps prompting a naval combat that will slow down one side or the other. Storms might wash someone overboard, creating a tension of whether to rescue the man or continue the chase.
 

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