Figure 1: Dwarven character creation flow chart

Following yesterday’s post, here I present flow charts for the best survival options for Dwarves (Figure 1) and Halflings (Figure 2). Both charts are based on CART analysis of the simulation data generated for yesterday’s post.

Figure 2: Flow chart for halfling fighter creation

For dwarves, weapon and armour choice is crucial, and weapon finesse is a decision so bad that it actually negatively affects survival: with two feats to choose, wasting one on weapon finesse is a very bad idea. For hobbits, like humans, toughness is only important if the PC doesn’t have good constitution, and weapon choice is only important for clumsy fighters. Note that if a halfling has no strength bonus, constitution is irrelevant to survival but dexterity is important. This is also true for humans, though weapon choice is not important in their case – presumably because they don’t suffer the size penalty on damage dice. For weak dwarves constitution bonus is also not important, but both weapon and armour type make a difference to survival.

Elven decision rules will be posted later…

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Introduction

In previous posts in this series, I showed the differences between fighter builds, and especially that “fast fighters” are a weak decision that is particularly bad for halflings and elves even though they are the more agile races. In this post I will approach the question of fighter builds from a different angle, that of the most effective choice of feats, armour and weapons for given attribute scores. Ultimately, the aim of this work is to develop decision models (expressed as flowcharts) for PC development. We will do this through a generalized version of the simulations run to date, in combination with classification and regression tree (CART) methods.

Methods

For this study a completely random character generation method was developed. This simulation program generated random races, ability scores, weapon and armour types and feats subject to the rules in the online Pathfinder System Reference Document (SRD). Weapons were restricted to three choices: rapier, longsword and two-handed sword. Armour types were studded leather, scale, chain shirt and chain mail. There were eight possible feats: improved initiative, dodge, shield focus, weapon focus, power attack, desperate battler, weapon finesse and toughness. Ability scores were generated uniformly within the range 9 to 18, and racial modifiers then applied: the human +2 bonus was applied randomly to the three physical attributes. Feats were assigned randomly, with humans having three feats and non-humans two. All fighters with a one-handed weapon were given a light wooden shield. Halflings were given size benefits and disadvantages as described in the SRD. Initial investigation revealed that ability score values were only important in broad categories: ability scores that gave bonuses greater than 0 were good, and bonuses of 0 or less were bad. For further analysis, therefore, all ability scores were categorized accordingly into values of that gave a bonus of +1 or greater vs. those that did not.

All fighters were pitted in one-to-one melee combat against an Orc, which had randomly determined hit points and the fully operative ferocity special ability. This happened in a cage deep beneath Waterdeep, so no one could run away. Winners were promised a stash of gold and the chance to buy a farm on the Sword Coast, but were actually subsequently press-ganged into military service in the far south, where most of them died of dysentery. A million fights were simulated.

Once data had been collected it was analyzed using classification and regression tree (CART) models implemented in R. CART models enable data to be divided into groups based on patterns within the predictor variables, which enables complex classification and decision rules to be made. Although it is more complex and less reliable than standard regression, CART enables the data to be divided into classification groups without the formulaic restrictions of classical linear models. Results of CART models can be expressed as a kind of flowchart describing the relationship between variables, with ultimate classification giving an estimate of the probability of observing the outcome. In this case the outcome was a horrible death at the hands of an enraged orc, and the probability of this outcome is expressed as a number between 0 and 1. CART results were presented separately by race, in case different races benefited from different choices of feats.

Some univariate analysis was also conducted to show the basic outline of some of the (complex) relationships between variables in this dataset. Univariate analysis was conducted in Stata, and CART was conducted in R.

Results

Of the million brave souls who “agreed” to participate in this experiment, 498000 (49.8%) survived. Survival varied by race, with 55% of humans surviving and only 45% of halflings making it out alive. Some initial analysis of proportions suggested quite contradictory results for the different feats, with some feats appearing to increase mortality. For example, 47% of those with improved initiative survived, compared to 51% of those without; and 46% of those with shield focus, compared to 52% of those without. This probably represents the opportunity cost of choosing these feats, or some unexpected confounding effect from some other variable.

The three combinations of ability scores and feats with the highest number of observations and the best survival rate were:

  • Dwarf with +3 strength, +3 dexterity, +3 constitution, chain mail armour, rapier, weapon focus and desperate battler (15 observations, 100% survival)
  • Dwarf with +3 strength, +0 dex, +4 con, scale armour, two-handed sword, toughness and weapon focus (13 observations, 100% survival)
  • Dwarf with +3 strength, +2 dex, +3 con, studded leather armour, longsword, desperate battler and power strike (13 observations, 100% survival)

Despite the apparent success of Dwarves, a total of 55% of all unique combinations of ability scores, feats, weapon and armour types with 100% survival were in humans. The majority of the most frequent survival categories appeared to be in non-humans, however – this bears further investigation.

CART results varied by race. For humans, ability scores were most important; for dwarves, weapon type and armour type were important, while constitution was largely irrelevant. For elves and halflings, the only important feat was toughness; weapon finesse was only important for humans, and sometimes only as a negative choice. The key results from the CART analysis were that strength is the single most important variable, followed by dexterity for elves and halflings, or constitution for dwarves; and then by decisions about armour and weapons. Feats are largely relevant only for those with weak ability scores.

As an example, the CART results for humans are presented as a flowchart in Figure 1 (click to enlarge). It is clear that after strength and dexterity, heavy armour and constitution are important determinants of survival. Weapon finesse is only important as a feat to avoid for those with low dexterity – for those with high dexterity it is largely irrelevant. Toughness primarily acts as a counter-balance to poor constitution in those with high dexterity and strength.

Figure 1: Character creation decision model for humans

Decision models for other races will be uploaded in future posts.

Conclusion

This study once again shows that strength is the single most important ability for determining survival in first level fighters, and that feats are largely used to improve survival chances amongst those who already have good ability scores. In previous posts dexterity appeared to be irrelevant, but analysis with CART shows that the absence of a dexterity bonus makes a large difference to survival – those with no dexterity score bonus do not benefit from feat choices, while those who have a dexterity bonus can benefit further by careful choice of armour and feats. Although previous posts found that “tough” fighters have a very high survival rate, this post finds that constitution is not in itself a priority ability score. By following the decision model identified in this study, players can expect to generate a fighter with the highest average survival chance given their ability scores.

As part of my continuing exploration of the statistics underlying Pathfinder, I’ve been comparing mortality for different types of fighters under different types of character generation systems. The basic Pathfinder rules recommend a point-buy system, but also allow for 4d6 choose the best three. I’ve generated PCs under all four point buy systems, 4d6 choose the best three rolled in order, 3d6 rolled in order, and the purposive semi-random system I developed for previous posts on this topic. Between them these cover the gamut of possible ability score generation methods.

For the point buy systems, I spent the points under the following rules:

  • Spend as many points as possible on the most important ability score for the fighter type (strength for strong fighters, constitution for tough fighters, etc)
  • Spend as many of the remaining points on a secondary score: strength for non-strong fighters, and constitution for strong fighters
  • If any points remain, spend them on the last remaining physical ability score
  • If balancing between point allocation is necessary, wherever possible choose ability scores so that they are the minimum value required to achieve a given bonus (so 16 for a +3, not 17)

This guarantees maximization of bonuses and roughly orders scores in the strength/constitution/dexterity priority list.

I ran 1,000,000 simulations pitting a fighter against an orc, with all the orc ability scores randomly determined. Half of all orcs were ferocious (randomly determined). The fighter’s race, class type, and ability score generation method were randomly determined, to ensure a wide spread of ability scores across all types of fighters. Results were calculated as mortality rates – this is really just an addendum to previous research so more detailed analysis was not conducted.

Results

The results are shown in Table 1, as mortality rates for the different ability generation systems for both Meek and Ferocious orcs. Mortality rates are given as percentages of all fighters who participated in the battle.

Stat Generation Method

Orc Type

Meek Ferocious
Rolled in order
  3d6 51 76
  4d6 best 3 36 62
Purposive Random 35 61
Point buy
  Low Fantasy 34 61
  Standard Fantasy 19 44
  High Fantasy 21 46
  Epic Fantasy 12 32

In a somewhat surprising result, 4d6 choose-the-best-3 has a similar mortality rate to the point-buy system labelled as “low fantasy” by Paizo. Mortality rates for fighters pitted against ferocious orcs only reach the 30% mark that one might expect of a CR1/3 monster in the Epic Fantasy scenario.

Conclusion

Rolling 3d6 in order significantly reduces survival rates compared even to a low fantasy point-buy system. Survival of these fighters against ferocious orcs does not differ between standard and epic fantasy builds, suggesting that these categories are essentially irrelevant. New boys fresh out of the village on their first adventure should only consider taking on an orc if they are confident that their genre setting is Epic. Otherwise, they should expect a bloody and gruesome end.

This weekend I continued my work on the epidemiology of Pathfinder, including an expansion of my programs to allow for different types of point buy. In the process I took the advice of some commenters at a related thread on the Pathfinder message boards:

I think for the non human fast fighters dropping weapon finesse makes no sense. Because they can hardly hit if they drop that. I would recommend changing it to dropping improved initiative for the fast non-humans.

In my original simulations I had built non-human fast fighters with improved initiative and weapon focus, but in this revision I changed this around so that non-human fast fighters drop improved initiative and keep weapon finesse. The results, though still not presenting a stirring defense of the decision to play a fast rather than a strong fighter, do bear out the suspicions of those commenting on that board, that for fast fighters weapon finesse is the most important feat to choose. Table 1 compares the results with weapon finesse that I generated today with the previous set of results that dropped weapon finesse in favour of improved initiative. The results in Table 1 are shown for combat with meek orcs (lacking ferocity) to be consistent with the previous post. Similar effects are observed against ferocious orcs, however.
Table 1: Non-human mortality with and without weapon finesse (revised)

Race No Weapon Finesse Weapon Finesse Odds Ratio
Dwarf 43.6 37.0 1.32
Elven Ponce 52.2 44.2 1.38
Halfling Loser 61.6 49.7 1.62

The odds ratios in Table 1 are provided to show which race suffers the most from lack of weapon finesse, and it is no surprise that it is the halflings. This is because they do the least damage, so the loss of hit chances affects them the most.

These results don’t change the fundamental conclusion that fast fighters are a very bad choice, but they do indicate that if one is going to pick this fighter build, weapon finesse is a very important feat to choose.

Continuing my series of posts exploring the epidemiology of Pathfinder, today I will report on the impact of adding ferocity to the orc stat block. Is the orc still a CR 1/3 monster when one accounts for ferocity, and just how tough does a fighter have to be to walk away from a fight with a single ferocious orc?

For this simulation (and all sims from now on) I am going to be using my updated and revised modeling program, which has been subject to some fairly severe stress tests and which I’m now fairly certain perfectly mimics a basic combat exchange between an orc and a fighter. I posted revisions here, showing the basic survival probability for three types of fighter and four races, for an orc with no ferocity. This is the basic program I’ll be working with from now on.

Introduction

Previous analysis of survival in Pathfinder have studied conflict between fighters of the four main races and inferior breeds of orc, but it is likely that serious dungeoneering will bring adventurers into conflict with hardier orcs fighting near their lair. It is well known that orcs who maintain a close cultural connection with their tribe are braver and more determined fighters, and this is usually reflected in their ability to fight even when suffering serious physical injuries. For this analysis, this powerful additional trait of “wild” orcs, ferocity, is included in the analysis. Essentially this analysis compares the survival chance of a lone fighter against a lone orc isolated from its tribe, probably in a city, with a lone fighter in combat with a lone orc near its lair, where it will fight beyond death.

Methods

A set of 200,000 simulated battles between randomly-generated fighters and randomly-generated orcs was analyzed using poisson regression. Orcs and fighters were generated in the standard way, but orcs had a 50% chance of having the ferocity trait, which enables them to continue fighting until they reach -12 hps. A simple main-effects poisson regression model of survival was built, and the effect of orc ferocity on survival reported from this model; subsequently, a model with interactions between ferocity and all the main variables of interest (fighter type, race and ability bonuses) was also built. Results from both of these models are reported selectively for simplicity.

Results

Mortality for the 100,000 fighters against meek orcs was unchanged, at 37.2%; but for fighters battling ferocious orcs mortality increased significantly, to 63%. Patterns of mortality differences by race and class type were similar to those seen previously, but mortality rates were higher in all class types and races. Table 1 shows mortality rates by race and ferocity type.

Table 1: Mortality rates by race and orc ferocity

Race

Orc Ferocity

Meek Ferocious
Human 30.6 57.1
Dwarf 32.4 60.1
Elven ponce 40.8 65.8
Halfling loser 44.9 68.2

Note that, although survival patterns are maintained in battles against ferocious orcs, the mortality ratios decrease: from a 50% increase in mortality between humans and halflings against meek orcs, for example, to a 20% increase against ferocious orcs. The increase in mortality due to ferocity also varies, from nearly a two-fold increased mortality rate in humans and dwarves to only a 50% increased mortality amongst halflings.

In a simple main-effects poisson regression model ferocity was associated with an average relative risk of mortality of 1.7, which was highly statistically significant (Z=80.12, p value <0.0001). That is, the average increased mortality from adding ferocity to an orc stat block was about 70%. However, in a model including interaction terms between orc ferocity and all main variables (fighter type, race, and all three stat bonuses) the role of orc ferocity varied significantly across ability scores. For example, after adjusting for other ability scores, class type and race, the increased mortality amongst fighters with minimum strength bonus was only 20%, while it was 85% for fighters with a strength bonus of +5. This effect is shown in Figure 1, which plots the relative risk of mortality by strength score for meek compared to ferocious orcs. All relative risks are relative to a fighter with a strength of -2.

Figure 1: Mortality by Strength Ability Score for Meek and Ferocious Orcs

Essentially, strength induces a lower gradient of mortality improvements when fighting tough orcs, and combinations of high scores become more important. In fact, it seems highly unlikely that decent survival will be obtainable for fighters of any race and class type generated using Pathfinder’s standard point-buy systems. These systems will restrict most PCs to ability scores in the 14-16 range, which will not guarantee survival against even a single ferocious orcs.

Conclusion

Adding ferocity to an orc’s stat block significantly increases its lethality, with an average increase in mortality risk for fighters in one-to-one combat of about 70% after adjusting for race, class type and ability scores. Even the strongest and most unusual fighters, with ability scores above 18, have surprisingly poor survival of about 30%. Orc ferocity increases mortality across all races and fighter types, with halflings again copping the pointy end of Gruumsh the Bastard’s falchion and incurring death rates of up 70%. This is further evidence that orcs are not CR 1/3 opponents, and suggests that GMs who want to field orcs as cannon fodder against their PCs should judge numbers carefully, or consider treating ferocity as a leader-type trait. It also suggests that – just on the numbers – Pathfinder is the most lethal of the D&D incarnations, especially when ability scores are restricted by point buy options. This will be tested in subsequent analyses.

In preparing an analysis of the effect of orc ferocity, I found I wasn’t able to reproduce the results of my previous post on different types of fighter and different races. The overall mortality in that post was 20%, but I kept getting values of 36%. Because I’m such a stunningly good programmer, I’d overwritten the program I used to produce those results, and it has taken me several days (interrupted by moving house) to dig up the original programs from Time Machine[1]. Checking through them I found a tiny error (three letters in one line of code out of 375[2]) which causes character hit points not to update after a round of combat – so that the only way the orc could win was to kill the PC on its first round of combat. That’s an interesting insight right there – 20% of the time the orc wins in the first round of combat!

So the true mortality rate in that analysis should have been 36%. I’m not going to redo the whole analysis (it’s late and I’m tired and I have a new analysis of ferocity to come), but I will put up the corrected table of mortality rates by race and fighter type, in Table 1.

Table 1: Mortality by race and fighter type (revised)

Race Fighter type
Strong Fast Tough
Human 20.4 36.7 35.1
Dwarf 18.8 43.6 36.2
Elf 30.7 52.2 38.7
Halfling 26.1 61.6 45.9

The general conclusion – that fast fighters are a disaster – is retained, but the effect is even more noticeable in elves and halflings, and high strength is even more important for these races than humans. Mortality rates in fast fighters are 1.8 times higher amongst humans, compared to over 2.5 times higher in halflings. Also, when the orc is not constrained from delivering a second blow, constitution becomes much less important than strength – being able to kill the orc first remains the most important skill.

Dwarves, who in this simulation have dropped power attack if they are strong fighters, benefit hugely from being strong rather than tough, presumably because they already have a constitution bonus.

So, the order of ability scores is: strength, constitution, dexterity. And I need to improve my programming!

fn1: which is awesome, btw.

fn2: which would probably be about 50, if I was any good at this stuff

After taking account of comments here and on the Paizo messageboards, I have adapted my simulation programs to allow for purposive attribute scores, feats and races, and re-analyzed the survival data for a smaller sample of more carefully designed fighters. In this second round of analyses Gruumsh the Bastard doesn’t acquit himself well, but neither do some of the PCs who went against him. This post reports on the updated analyses.

Update (3rd July 2012): In editing my code to incorporate some minor changes, I noticed that I didn’t actually pit 100,000 fighters against 100,000 randomly-generated orcs – I pitted 100,000 fighters against Gruumsh, who only has 6 hit points. Against a full range of Orcs one gets very different results – I will report on this today (3rd July 2012). This post has been edited to remove references to 100,000 randomly-generated orcs.

Introduction

Previous analyses of survival in Pathfinder have relied on randomly generated ability scores assigned in order, and have not incorporated feats, race, fighting styles or weapon types. In this post the analyses are updated to allow for a range of basic feats, four races, purposive rather than completely random assignment of ability scores, and three types of fighter: strong, fast and tough. Survival is compared against Gruumsh again, and results analyzed for insights into possible character creation decisions.

Methods

A sample of 100,000 randomly generated fighters were pitted in battle against Gruumsh, who is still not ferocious. The fighters were generated so as to fall into three types, defined by ability scores, armour and weapon types, and feat choices:

  • Strong fighters: strength was determined randomly from a uniform distribution between 13 and 18, and the fighters were equipped with scale mail and a two-handed sword. Human fighters had three feats: power attack, weapon focus and desperate battler. Humans placed their +2 ability score bonus in strength. Non-human fighters dropped power attack
  • Fast fighters: dexterity was determined randomly from a uniform distribution between 13 and 18, and the fighters were equipped with studded leather armour, a heavy wooden shield and a rapier. Human fighters had three feats: improved initiative, dodge and weapon finesse. Non-humans dropped weapon finesse, and humans put their +2 bonus into dexterity.
  • Tough fighters: constitution was determined randomly from a uniform distribution between 13 and 18, and the fighters were equipped with chain shirt, wooden shield and longsword. Human fighters had three feats: toughness, shield focus and weapon focus. Non-humans dropped toughness (because two of the races already had +2 constitution), and humans put their +2 bonus into constitution.

All other physical stats were generated with 3d6, but scores below 9 were reset to 9. Mental stats were generated using 3d6 in order, but nobody cares if their meat shield has read Shakespeare, so the details aren’t reported here. The hapless 100,000 were then thrown against Gruumsh, with the promise that anyone who survived would get to meet Salma Hayek. Needless to say, I lied: for unknown reasons, Hayek only dates bards. All fighters with power attack were assumed to be using it for every strike, and you would too if you met Gruumsh.

Results

After incorporating racial bonuses and feats, and assigning ability scores purposively rather than randomly, overall survival increased significantly: only 20% of the newly trained fighters died. However, variation in survival was significant and depended heavily on race and fighting style. Table 1 shows the mortality rates by race and fighter types.

Table 1: Mortality Rates by Race and Fighter Type
Race Strong Fast Tough
Human 17.1 26.5 0
Dwarf 11.1 21.8 1.1
Elven Ponce 27.3 46.9 16.4
Halfling Loser 21.2 45.3 8.3

From Table 1 it is clear that elves and halflings are not good fighters, and Dwarves are excellent in this particular role. The small difference in mortality between humans and dwarves is probably due to the reduced number of feats that dwarves have relative to humans. In fact, once feats and purposive ability score selection are included in character development, constitution becomes an extremely important score: 0% of fighters with constitution bonuses above 3 died. This is probably because CON bonuses of 3 or more guarantee a fighter cannot be killed in a single blow by an Orc (maximum damage 12) and the increased damage and hit stats of these fighters mean the orc will not survive to deliver a second blow. This is indisputably a good thing.

It is clear from table 1 that the least successful form of fighter is the fast fighter, and indeed some perverse results obtain. Figure 1 shows the mortality rate by dexterity score: mortality increases with increasing dexterity in this dataset. This is probably because higher dexterity scores are more likely in the “fast fighter” choice, and amongst halflings, both of which deliver less damage than other races and class types.

Figure 1: Mortality by dexterity score

A similar perverse result is visible with armour class. Figure 2 shows the relationship between mortality and armour class, which is positive.

Figure 2: Mortality by Armour Class

Again, it is likely that the highest armour class values are only achieved by halflings (who have size bonuses), and higher AC is associated with lower damage and attack values. Note that fast fighters have very high initiative values (up to +9!) but these don’t seem to say the battle: for fighters who start with a minimum of 8 hit points, starting the battle first is less important than being able to hit your opponent and do massive amounts of damage.

Conclusion

Dexterity is useless, and a fighting style based on light armour and fast weapons is a waste of time. As a result, weapon finesse is the ultimate wasted feat: it could have been used to get 3 more hit points, which for a first level fighter guarantees that one strike from an Orc will not be fatal. After incorporating feats, the best option for a first level fighter is to choose toughness, shield focus and weapon focus, and pour as many points as possible into constitution. 17 hit points, chain armour and a shield at first level are vastly more useful than a fancy fighting style and a leather skirt!