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Subject: Battle probabilities rss

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Max Pistilli
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Science is a way of trying not to fool yourself. The first principle is that you must not fool yourself, and you are the easiest person to fool. -Richard Feynman
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I've played Root twice since getting it a week ago and love it so far. While thinking about it I calculated the expected outcomes of battle and figure others may be interested. In case you're not aware (in which case you're probably not reading this anyway), each die has an equal chance of being any digit from 0 to 3, but the attacker (or Woodland Alliance) gets the higher of the two, so the expected distribution of rolls is a little funky.

Defender
Attacker | 0 1 2 3 | Total
0 | 6.25% 0% 0% 0% | 6.25%
1 | 12.50% 6.25% 0% 0% | 18.75%
2 | 12.50% 12.50% 6.25% 0% | 31.25%
3 | 12.50% 12.50% 12.50% 6.25% | 43.75%
Total | 43.75% 31.25% 18.75% 6.25% | 100.00%


The marginal probabilities (the totals at the ends, or "margins") mirror each other such that the attacker's most likely outcome is a 3 (43.75%), and the defender's most likely outcome is a 0 (43.75%), and furthermore the attackers have a 75% chance of a 3 or 2 while the defenders have a 75% chance of a 0 or 1. When you average the results the attacker has an average of 2.125, and the defender has an average of 0.875, for an average difference of 1.250. This is important I think when you consider recruiting vs attacking, because each attack action is worth about the equivalent of 1.25 warriors relative to your opponent; for the cats you still want to recruit because each recruit action will get you as many warriors as you have recruiters, but for the birds an extra attack in your decree can be more efficient than a recruit.

The vagabond can use these to consider how many items he's likely to lose, so for instance if you can afford to lose 2 but really not 3 you know that there's a 93.75% chance that you'll be fine if you attack. Similarly, attacking a vagabond you can know that there's a 75% chance you'll damage at least 2 of his items, which from experience feels far better than wasting a turn just to damage 1 item that he wasn't even using much.

Then of course everything is opposite when you're fighting the Woodland Alliance. If you think a clearing with a base, sympathy token, and only 1 warrior looks tempting, you need to keep in mind that you only have a 25% chance of destroying either the base or the token (and only 6.25% of destroying both), and a 93.75% chance of losing a warrior for your effort. After thinking about these probabilities I have much less sympathy for the small size of the alliance's army!
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Andrés Santiago Pérez-Bergquist
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Nice presentation! See also the joint probability tables that I posted a while back, which show the chance of dealing at least X hits while taking at most Y hits.

Root Probability Tables
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Jay M
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I would like to see a probability table that is styled in terms of number of attackers versus number of defenders. The "caps" affect the probably outcome, and would tell me, for example, my chances or destroying/protecting a particular building in different numerical matchups.
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Max Pistilli
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Science is a way of trying not to fool yourself. The first principle is that you must not fool yourself, and you are the easiest person to fool. -Richard Feynman
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Tiger got to hunt, bird got to fly; Man got to sit and wonder, 'Why, why, why?' Tiger got to sleep, bird got to land; Man got to tell himself he understand. -Kurt Vonnegut
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Race Bannon wrote:
I would like to see a probability table that is styled in terms of number of attackers versus number of defenders. The "caps" affect the probably outcome, and would tell me, for example, my chances or destroying/protecting a particular building in different numerical matchups.


You could make 3 sets of this table for each possible number of attackers where you successively combine the probabilities of each bottom row into the one above it, then you could make 3 more sets of each of those where you combine the probabilities of each rightmost row with the one to the left of it, then you could make two more versions of the 1 and 2 defender tables where all the defenders are non-warriors. That would give you 15 tables to sort through that I don't think would provide any information you couldn't get from the single table above, so I'm not sure it's worth it, but if they would help you then knock yourself out
 
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Jay M
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pistilli wrote:
Race Bannon wrote:
I would like to see a probability table that is styled in terms of number of attackers versus number of defenders. The "caps" affect the probably outcome, and would tell me, for example, my chances or destroying/protecting a particular building in different numerical matchups.


You could make 3 sets of this table for each possible number of attackers where you successively combine the probabilities of each bottom row into the one above it, then you could make 3 more sets of each of those where you combine the probabilities of each rightmost row with the one to the left of it, then you could make two more versions of the 1 and 2 defender tables where all the defenders are non-warriors. That would give you 15 tables to sort through that I don't think would provide any information you couldn't get from the single table above, so I'm not sure it's worth it, but if they would help you then knock yourself out


You're getting tangled on your own feet. You don't need to know the intermediate probabilities of the dice rolls. For a given matchup, you only need to know each outcome, and it's corresponding probability.

If I attack one warrior with three (3-1), there are the following outcomes after piece removal: 3-1, 3-0, 2-1, 2-0. That's it. It could be visually condensed down and would not require sorting through 15 tables. (Because many of the outcomes are flattened out by the cap, and we also don't care about the difference between, say, a 2-2 roll and a 3-3 roll).
 
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Max Pistilli
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Science is a way of trying not to fool yourself. The first principle is that you must not fool yourself, and you are the easiest person to fool. -Richard Feynman
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Tiger got to hunt, bird got to fly; Man got to sit and wonder, 'Why, why, why?' Tiger got to sleep, bird got to land; Man got to tell himself he understand. -Kurt Vonnegut
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If you attack 1 warrior with 3 then you have a 93.75% chance of killing the 1 enemy and therefore a 6.25% chance of not. And he had a 43.75% chance of doing zero hits and therefore a 56.25% chance of doing 1, since all rolls >1 are the same as 1. As I said, If you want to calculate all the joint probabilities of you killing 1 and him killing 1, you killing 1 and him killing 0, and you killing 0 and him killing 0, then again you can add a few numbers in the table and get those far more easily than creating a whole bunch more tables. I agree that the number of combinations of possible attacker-defender numbers would get you tangled over your own feet
 
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Jay M
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I don't want to do anything. I just thought the point of the thread was to discuss probabilities in this new game.

Forget it, carry on.
 
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