This post describes the probabilistic structure of the feint in 2nd edition Warhammer, and its effects in combat, and concludes that it is a highly risky manoeuvre not worth using in any situation. The feint increases the opponent’s attack benefit more than the user’s in all cases, with the penalty for its use increasing with the increasing weapon skill of the user. This post will also show some more remarkable facts about the structure of probabilities in Warhammer 2, particularly the effect of the rules on attack probabilities for evenly-matched antagonists with very high or very low ability scores.

Introduction

The feint is a risky manoeuvre in Warhammer 2nd edition, which enables the user to disable the target’s parry/dodge ability, rendering them easier to strike. It is risky because it uses a half action, requiring the user to give up their own defensive parry/dodge ability in order to make the manoeuvre. Because the parry ability relies on weapon skill, it seems reasonable to assume that the manoeuvre should provide most benefit when the target’s weapon skill is higher than the user’s, since in this case the user’s parry ability will be of lower utility to them than the target’s parry ability is to the target. It is not clear that a straight trade – dropping one’s own parry for the chance to remove an opponent’s – is practically a very sensible suggestion. Here we test this.

Method

To test the effect of the feint ability we consider two fighters, one with weapon skill p, the other with weapons skill q, and, for simplicity, both having equal toughness t. We then calculate the probability of doing damage on a standard attack, and the probability of doing damage for a feint. We also calculate the odds ratio of a hit for the user vs. the target, i.e. the odds that the user hits with a feint divided by the odds that their target subsequently hits them. For the feint to be a useful manoeuvre, the odds ratio needs to be greater than one (indicating that the feint has increased the user’s hit chance more than dropping the parry has increased the target’s hit chance). All values of p,q and t are here represented as values between 0 and 1, to represent their probabilistic nature – on the character sheet they are of course expressed between 0 and 100.

The probability of succesfully doing damage with a feint is the probability that the feint succeeds and the subsequent attack succeeds OR the feint fails but the attack succeeds anyway. This can be expressed as

p*p*(1-q)*(1-t)+p*(1-p)*(1-q)*(1-t)+p*p*q*(1-q)*(1-t)

which is the probability that a standard attack hits plus p*p*q*(1-q)*(1-t). This in turn can be expressed as (1+pq) times the probability that a standard attack hits. The feint increases the chance of a successful attack hitting by a factor of pq. In practice this is a very small amount, and is maximized when both the attack and defense values are high. However, as we shall see, the normal chance of success is smallest when the attack probabilities are highest, and this is the point in time when dropping one’s own parry is most dangerous.

Tables of probabilities were calculated in Microsoft Excel for attack probability p ranging from 0.3 to 0.9, target attack probability q ranging from 0.1 to 0.9, and three values of toughness of 0.3, 0.5 and 0.75. Odds ratios (OR) were calculated as follows:

Odds(Feint user hits) = P(Feint User hits)/P(Feint User misses)

Odds(Target hits)=P(Target hits)/P(Target misses)

Odds Ratio =Odds(Feint user hits) / Odds(Target hits)

with an OR greater than 1 indicating that the feint has increased the success probability for the feint user to a greater extent than dropping the parry has benefited the target (and vice versa for an odds ratio less than 1).

For the remainder of this post, the word “hit” should be taken to mean a blow that penetrates all defenses (parry and damage reduction) to do actual damage.

Results

First we present the curve of successful hit probabilities without using feint for two cases: the attacker has a weapon skill of 0.3, against all target weapon skills between 0.1 and 0.9; and the case of equal weapon skills for attacker and target for every value between 0.3 and 0.9. In both cases, toughness is assumed to be 0.3. These two curves are shown in Figure 1 (click on the figure to enlarge).

Figure 1: Probability of attack success for two cases, no feint

It is clear from Figure 1 that fights between characters with very low or very high weapon skill will last for a very long time. Previous analysis showed that a total attack success probability of 16% is associated with a 50% survival of 17 rounds; with attack success less than half this, we can expect a battle between two antagonists with WS 0.9 to last considerably longer, perhaps as long as 30 or 40 rounds, in 50% of cases.

Next we compare these curves separately for attacks with and without a feint. Figure 2 shows the probability of success with and without feint for a character with WS 0.3, against a range of weapon skills from 0.1 to 0.9 (click to enlarge).

Figure 2: Probability of attack success with and without feinting, wS=0.3

Clearly the benefits of feint for this character are marginal and not equal across targets, with the largest benefit lying at roughly the middle of the range of target Weapon Skills. In Figure 3 we show the probability of successful attack with and without feint for antagonists with equal weapon skill ranging from 0.3 to 0.9 (again, click to enlarge).

Figure 3: Probability of successful attack with and without feint, equal WS

This chart clearly shows that the benefits of a feint technique are greatest for Weapon Skills around 0.5, when both antagnoists have similar values of WS. However, this is not the complete story, since in all cases the attacker is giving up a parry move, thus lowering their own defenses. To consider the full effect of the feint an estimate is required of the relative benefits to attacker and defender. We represent these benefits as the odds ratio of a successful attack, that is, the odds that the attacker is successful divided by the odds that the defender is successful. Ideally, applying the feint should lead to an increased odds ratio relative to an attack without the feat, in some circumstances. Note that an odds ratio of 1 indicates equal chance of attack success for both attacker and target, and should occur in the case of two antagonists with equal WS and toughness attacking each other without a feint. To illustrate this, we first present the odds ratio curves for two situations – an attacker with WS 0.3, against a range of target WS from 0.1 to 0.9; and an attacker with WS 0.4 against a similar range. This is shown in Figure 4 (click to enlarge).

Figure 4: Odds of successful attack without feint, two common WS

A grid line has been placed in this chart at an OR of 1 to show that the equal success chance lies at the point where the WS are equal. Note that a WS of 40 significantly improves the odds of a successful attack compared to the target when the target has a lower WS, but makes little difference against large weapon skill values.

Finally, figure 5 plots the odds ratio of success without feint against the odds ratio with feint for a weapon skill of 0.3 (click to enlarge).

Figure 5: Odds of success with and without feint, WS=0.3

Using a feint reduces the success rate relative to a non-feinting enemy for all values of the enemy’s Weapon Skill. In the case of a WS of 30, the reduction in relative success is about 30-40% across all values of target WS; however, for an attacker with a WS of 90, the reduction is about 90% across all weapon skills. For example, against a WS of 0.1, an attacker with a WS of 0.9 would have an odds ratio of 185 without feint; this drops to 21 with feint.

Conclusion

Although superficially an appealing mechanism, the feint technique leads to a significant loss of relative success in combat against an opponent not using the feint skill. This loss of success occurs regardless of the target’s Weapon Skill, and increases with the increasing weapon skill of the attacker, so that the penalty for using the feint technique increases as the combat skill of the users increases. This penalty at high attack values is extremely large, but even at standard WS values (of about 30) the penalty is significant, representing an approximately 30% reduction in effectiveness relative to the opponent.

An additional conclusion of this post is that the probability of success in a single combat round for equally matched antagonists is not consistent across all weapon skills, with the antagonists most likely to hit each other being in the middle of the weapon skill range (about 50), while those with much higher or lower weapon skills are doomed to very, very long combats regardless of the attack techniques they use.