In this post I will use some basic probability theory to show that, in essence, the Warhammer 2nd edition combat system is not deadly, as I think is often claimed, but is actually really slow and boring, and inherently survivable.

This assumption of deadliness arises, I think, from the fact that PCs at low levels are poor at doing anything, and the assumption is that if you’re bad at stuff then you’ll die quickly doing that stuff if it’s also dangerous stuff. I think this assumption also lies beneath claims that early D&D was deadly, an assumption which I don’t test here (due to lack of familiarity with early D&D rules) but which is probably somewhat better placed than any assumptions about Warhammer’s relative riskiness.

I came to this comparison because on Friday and Sunday last week I role-played respectively in Pathfinder and Warhammer 2nd edition, and I was struck in both instances by the length and inevitable dreariness of the combat, and by the fact that both combats had to be ended by a non-combat act of the GM’s. This post, about the probability of survival in each of three systems, will serve to show how this comes about and also I think reveals some obvious conclusions about tactical combat rules in role-playing. I aim to expand on this post in future with a proper simulation and statistical analysis, complete with survival curves, but that will take a bit of time.

Introduction

The probability of surviving a single round, and the cumulative probability of surviving multiple rounds, are calculated here based on the underlying combat mechanic of three systems – Warhammer 2nd edition, D&D 3.5, and my own Compromise and Conceit modifications of the d20 system. All three are compared with a putative “control” system in which the mechanics are not specified, but are assumed to result in a 50% probability of a hit in any given round, and death after 3 successful hits. The chief conclusion for each system is the number of rounds required to fight before reaching a 50% chance of death, referred to hereafter as the “median survival time,” though strictly speaking this is not a median survival time. In practice of course time to death varies according to the good or bad luck of the player, and how much they lie about their rolls to the GM, so survival time should here be assumed to be roughly representative of a long-run probability. The methods presented here also use various simplifications and approximations, specifically ignoring the role of criticals, fate points, and the death spiral in the Compromise and Conceit system, which makes the order of hits important for survivability.

In all cases, the survival probability is calculated for a fighter-type PC attacking an NPC with exactly the same skills as themselves.

The fundamental mechanics assumed are set out below. The fundamental problem with Warhammer can be seen to derive from the number of defensive manoeuvres available to a fighter in a standard combat round. Once a successful hit has been scored, the defender can then roll a defensive roll using their own combat skill, and then (if a fighter-type character) can roll a damage reduction check against their constitution. For a typical fighter we will see that this reduces a fighter’s successful hit chance to just 15%, and in a series of binomial trials requiring 3 successes, this can significantly extend the run of rolls required.

Method

For each system, a typical build of first level fighter was generated, using average statistics that might be expected for such a system, and pitted against exactly the same fighter character. No special feats were assumed in D&D or Compromise and Conceit (C&C), and the special feat of “Damage Reduction” was assumed for the Warhammer fighter (though as we shall see, it is not an enormously important feat). Other assumptions are outlined in detail below.

The combat method for each of the systems was summarised as a single probability of successfully scoring damage against an opponent. Damage was assumed to be the average for the type of character, and the number of hits required to kill the PC for the given average damage was used as the number of hits required before the PC or their opponent was killed. In each round, the cumulative probability of death was calculated as the probability that the given number of hits occur by that round, which is practically given as 1-P(less than that number of hits occurred). Formally, given a requirement of x hits to achieve death, the probability that a character survived to round k is the probability that they have received at most x-1 hits in k trials. The adjusted probability is the probability that they have survived to round k, or that they killed their opponent in round k-1. This probability in turn is given as the probability that they survived to round k-1 and they delivered 3 or more hits by round k-1.

This problem reduces to a simple binomial distribution for a given probability of a hit. Note that inclusion of critical hits, special moves, fate points, or death spiral effects renders this calculation completely different, and will be handled subsequently in a simulation.

Assumptions for each system are set out below.

Warhammer

A fighter-type character (for example, mercenary or watchman) is assumed to have rolled an average attack and constitution value on 2d10, giving values of 30 in each. The character is further assumed to have added 5 to the attack score, giving a value of 35. The chance of a successful attack is thus 35%, the chance of a successful defence is also 35%, and the chance of a successful damage reduction is 30%. The character is assumed to absorb 3 points of damage (30/10), and does 1d10+3 damage, and so final average damage is the average damage on a d10, or 5.5. The character is assumed to have 13 hit points, and be wearing leather armour (AP 1), so overall average damage is 4.5. Probability of doing any damage in one round is given as the Probability of a successful attack AND a failed defense AND a failed damage reduction. Since the opponent is exactly the same, this gives us the following results vis a vis the PC:

  • Chance of being damaged by the opponent in one round=0.16
  • Number of hits required to die: 3

D&D3.5

The D&D fighter is assumed to have a +2 strength bonus, BAB of 1, and weapon focus, for a total attack bonus of 4. Armour is chain with a shield, +2 dexterity bonus, and +1 dodge bonus, for a total AC of 19. The fighter is assumed to have maximum hit points, the Toughness feat and a +1 constitution bonus, giving 14 HP. Damage is from a longsword with +2 strength bonus, giving average damage of 6.5, so 3 hits are assumed to be required to kill the fighter. No other feats are assumed. This means that the chance of a successful hit is 25%, because the PC needs to roll over 15 on a d20, giving a 25% chance of success. This gives the following results:

  • Chance of being damaged by the opponent in one round=0.25
  • Number of hits required to die: 3

Compromise and Conceit

The Compromise and Conceit (C&C) fighter is assumed to have 4 ranks in attack, with a +3 strength bonus, and 4 ranks in defense, with a +3 agility bonus. The fighter is assumed to be wearing armour with Damage Reduction 3, and to have a maximum damage of 5 wounds. The fighter is also assumed to have 4 ranks in fortitude, with a total of 7 wounds. When fighting against himself, this means the fighter would need to roll a 10 to hit, but a 14 to do damage. Calculating average damage is tricky because the probability distribution is truncated between 1 and 5 with uneven probabilities, so for now we assume it is weighted towards the lower boundary of the damage distribution (due to the nature of the 2d10 roll), so assign an average damage of 2. Recall that this system uses a 2d10 attack roll, so we have a final result of:

  • Probability of successfully doing damage = 0.34
  • 4 hits required to kill the PC

Control system

This system assumes a 50% chance of doing damage, and 3 hits required to kill.

With these results we construct the probability distributions.

Results

The median unadjusted survival time for each system is:

  • Warhammer: 17 rounds
  • D&D: 11 rounds
  • C&C: 11 rounds
  • Control: 5 rounds

Figure 1 shows the unadjusted survival times (D&D has been misnamed AD&D).

Figure 1: Unadjusted survival times

The adjusted times were:

  • Warhammer: 23 rounds
  • D&D: 15 rounds
  • C&C: 14 rounds
  • Control: 7 rounds

and the probability curves are plotted in figure 2.

Figure 2: Adjusted survival curves

Recall that these are not true survival curves, but simply cumulative probability distributions.

Conclusion

It actually takes a long time to die in Warhammer, with a concomitant number of die rolls. At the unadjusted median survival time, if the player wins, he or she will have rolled 17 attack rolls and 3 damage rolls (on average); he or she will also have suffered an average of 6 attacks that required defensive rolls, giving a total number of defensive rolls of between 6 and 12, for a total of 26 – 32 rolls. The D&D player will have rolled 11 attacks and 3 damage rolls, for a total of 14 rolls. The C&C player will have rolled just the 11 attack rolls, and the control player will have rolled 5 attacks and 3 damage rolls for a total of 8 rolls.

It’s worth noting that, fiddling with the underlying parameters of the game assumptions for warhammer shows that damage reduction is a significant factor in the slowness – losing this feat increases the base hit chance to 23%, similar to D&D. However, the relative ability scores of the enemy are not that important. If the enemy has only a defense score of 15, half that of the PC, hit probability increases to 20% and the survival time drops (for the person with the higher skills) to 13 rounds, only shaving off 4 rounds. Also, if both fighters have an attack ability of 55%, the overall chance to hit remains roughly similar, at 17%, so gaining levels doesn’t significantly speed up combat.

Even if we assume that the warhammer system represents reality in its long drawn-out slugfests, we have to ask if this is a system that we want to actually play – fights this long are very boring. Also we note that a player has fate points to spend, and that in the “low power” world of warhammer these are one of the player’s main advantages over NPCs. But the average player will have 3 fate points, which can be used to reroll a single roll. Given they have to roll 26 – 32 times to win, it seems that these fate points aren’t going to make a significant difference to the battle’s progress. Also, unlike in D&D and C&C, the absence of other powers and magic means that the player has little else to do in combat but roll to hit, making these 26 rolls considerably less interesting than in other systems.

We also can note that there is no particular reason for a given number of rolls to be made for one attack. Combat systems abstract combat, so we could in essence reduce combat for the Warhammer case to a single roll against a 15% hit chance, and have the same result as described here, at the cost of 6-12 rolls less. Players want a certain amount of argy-bargy in combat, but I think most people would argue (and I think certainly the people I’ve played Warhammer with have agreed) that a little less argy bargy and a bit more fun could be had from a different system.

In a subsequent post, I will consider a full simulation for a set of sample fights, include criticals and death spirals, and give a statistical analysis.

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