AD&D Combat Round Versus D&D Combat Round

I recently purchased the re-issue of the 1st edition AD&D core rule books.  I realized in retrospect that I only thought I was playing AD&D back in the day.  Really, I was playing a sort of mish-mash of D&D and AD&D.  One of the things I'm struck by is the differences in the two combat systems.  In D&D there's a combat round checklist, and everything happens in a specific order.  In this respect, it's very much like playing a wargame like Panzer Blitz.  I ended up writing a short MATLAB script for predicting the probability of various outcomes of differently armed and armored opponents.  It doesn't take into account special abilities.  I was hoping that it'd be easily modified for the AD&D game.  Unfortunately, that hasn't proven to be the case due to some of the following differences in the combat systems.

Time:  D&D combat rounds are 10 seconds long.  AD&D rounds are 1 minute long which are further subdivided into ten 6 second segments.

Surprise:  In D&D the DM rolls 1d6 for both sides.  A 1 or 2 indicates that side is surprised.  It's fairly easy for both sides to be surprised and spend a round doing effectively nothing, which, in practice, most of the time means that nobody is surprised since if both sides are surprised and do nothing, the DM just marks off a round on his time record and keeps the game moving.  With AD&D, while both sides can be surprised, it's possible for one side be less surprised.  The degree to which one side or the other is less surprised depends on the difference in their surprise die rolls.  Additionally, if both sides are surprised equally, nobody is surprised, negating the effects of surprise entirely.  Interestingly, in certain cases, extremely surprised parties can be surprised for multiple segments.

Initiative:  In D&D, initiative is determined each round.  In AD&D, achieving surprise is also equivalent to winning initiative, so it's only determined on rounds where nobody is surprised, or all sides are equally surprised.  This means that surprise is more deadly.  Not only is there a potential to get free attacks, but you might get multiple free attacks (in the event of both sides being surprised but there being a large difference in their surprise rolls), and on top of that, you get to attack first each segment after the surprise segments are resolved.  YIKES!  Additionally, initiative is determined at the beginning of the round, hence the advantage of intitiative is carried through all ten segments. 

Surprise in D&D feels to me almost like an optional rule, inserted and often employed at the DM's discretion.  In AD&D achieving surprise is an integral part of combat.  Additionally the advantage of initiative is much more decisive. 

Weapon Speed Factors:  This one has me really perplexed.  Most combats are not 1 on 1 encounters, hence they are not uniformly armed.  I can't help but wonder if these rules were only meant to be employed in the event of a duel.  It makes me think an early D&D player must have been an SCA wire-weenie. 

I can't help but wonder if this would have the effect of making shrewd players very focussed on achieving surprise, since the advantages are so much greater.  I'm not sure a fast weapon matters very much, though, since I can only see it mattering in one-on-one encounters.

Other Initiative Bonuses and the Next Step

This afternoon, in a spasm of post-computer modeling boredom, I computed the remaining probabilities for +2, +3, and +4 initiative bonuses.  A +5 initiative bonus is uninteresting because it is equivalent to the orc automatically losing initiative unless he has some bonus that negates the elf's bonus.  In that case, though, it becomes statistically equivalent to the 0, +1, +2, +3 or +4 case.  The results of these and my previous investigations are summarized below.  They were all arrived at by methods identical to the previous posts.

Figure 1: The probability of the elf winning initiative as a function of any bonuses.

Figure 2: The probability of the elf tying initiative as a function of any bonus.

To proceed any further in my effort, I need to define some more of the properties of my elf and my orc.  I need to determine their weapons, hit points, and armor.  The orc is easy:

Orc(1): AC: 6, HD: 1, 8 hp, MV: 120'(40'), ATT:  1 sword, DAM: 1d8, Save As: F1, ML: 8, ALIGN: Chaotic.

That's just reading straight out of the rules and assuming maximum hit points.  I figure the orc is a good "standard" bad guy against which to gauge the capabilities of a character.  Similarly, I'll assume a "standard" level 1 PC elf with max hit points for now.  The PC elf will be wearing leather armor with a shield giving him an armor class of 7, which is slightly inferior to the orc.

According to the Labyrinth Lord attack tables, the Orc must roll a 12 or better to hit the elf, and the elf must roll a 13 or better to hit the orc.  This implies P(elf hits orc) = 40%, and P(orc hits elf) = 45%.  This places the orc at a slight advantage in terms of hitting ability. Given the previous observations that what really makes the difference in initiative, besides bonuses, is the ability to survive in a tied initiative round, I can't help but wonder if this slight advantage will translate into a noticeable advantage in the end for the orc.

At this point, the system is completely defined.  Neglecting magic and missiles (which I left out), I have everything necessary to fill out a probability tree for the turn.  The goal will then be to create a process so that I can analyze trade offs.  Then I can ask questions like, rationally, when is a two handed weapon worth losing initiative?  Irrational, "I think battle axes are cool!" is perfectly fine.  Lord knows, PCs should not feel constrained to act rationally, and resolving all of a character's choices to a series of optimized tactical trade offs is probably one of the least interesting and fun ways to play the game.  None the less, the question remains there to be answered what's the best decision?

Whatever results I get from this experiment are going to be driven by the assumptions going into them.  What's the best weapon to carry against an orc, might not be the best weapon to carry against another elf.  I'm also neglecting magic.  Really, whether the results of this experiment gives any indication of the "best" choice anything is a subject of debate.  Any "best" choice would certainly be highly situation dependant.  I imagine one day compiling a list of optimized equipment for various foes.  I doubt I'll do it, though.

For my next step, I'm going to boldly forgo proving that the combat turn is a Markov process and just fill out the tree assuming each state of the system is annotated by the elf's hit points and the orc's hit points.  Hence, on turn one, the beginning state is E(6)O(8) for "Elf: 6 hp, Orc 8hp."  There are 63 possible states.  I guess now I need to fill out my probability tree...

Initiative in a One on One Combat with a +1 Bonus

What is the impact of an initiative bonus on combat?  I'll start to answer this question using the same sorts of observations I made previously. Assuming a +1 bonus, the initiative matrix looks like this:

Figure 1:  Initiative Outcomes Assuming a +1 Bonus to the Elf

The PC elf now has an appreciable advantage in the intiative phase.  P(Elf Wins) = 21/36 = 58.33%, which is a pretty substantial edge if you can maintain the advantage through the later phases of combat round.  It is, by no means decisive.

Figure 2: Elf Wins Initiative 58.33% of the Time with a +1 Bonus

The advantage would be even more substantial if the elf enjoyed further combat advantages enabling good survivability in the event of tied initiative.

Figure 3: Elf Wins Ties Initiative 13.89% of the Time with a +1 Bonus

Does a +1 bonus enable a serious "first strike" advantage?  Probably not.  But over several combat rounds would the cumulative effect become noticable?  You'd notice it. particularly if you were strong enough to withstand a tied round now and then.  If that was the case, then it'd be pretty decisive.   

Beginning with the Combat Round

To begin my analysis, I decided to start with the combat round.  I plan to start off very simply and describe a notional encounter between a PC elf and an orc.  This one-on-one renders questions about group versus individual initiative irrelevant for now.  I expect I'll get back to that some other time.  I'll flesh out their characteristics further as it's necessary.  The first step of a combat round is initiative.    To determine who wins initiative, we roll 1d6 and who ever's roll is higher wins.  If both sides roll the same number, actions are assumed to occur simultaneously.  There are 6^2=36 possible outcomes in the initiative phase.

Figure 1: Initiative Outcomes in a One-on-One Encounter

 
What is the probability of the elf winning initiative and getting one up on the orc?  As the above table shows, the elf wins, 15 out of 36 times.  Therefore P(Elf Wins) = 15/36 = 41.67%.  Since the system is symmetric the orc has an equal probability of winning initiative.

Figure 2: Elf Wins Initiative in 15 Possible Outcomes

 The diagonal elements of the matrix of possible outcomes constitutes ties.  There are 6 possible ties.  Therefore P(Tie) = 6/36 = 16.67%. 




Figure 3: Elf and Orc Tie Initiative in 6 Possible Outcomes


 The first effect of group initiative is that neither the orc nor the elf is likely to win initiative.  This isn't to say that neither side has an advantage due to initiative.  If for some reason (armor, spells, special abilities, etc.) the elf might have an advantage in the event of a tie, the initiative might actually slightly favor the PC because the probability of winning or tie-ing, P(win ^ tie)=58.33%. 

Assuming that's not the case, however, most of the time a PC is not going to get any "first strike" advantage, therefore to be really effective in combat, a PC needs to survivable against a monster's first strike or the outcome of a tied round. A one-on-one combat in D&D, therefore is about survivability.  With no initiative bonuses, combat as an ambiguous scenario, where if any side has any advantage at all it results from other combat related advantages such as armor class, hit points or saving throws. 

Certain special abilities entail the player or DM applies a small bonus to their initiative rolls.  A dexterity score of 13 entails a +1 bonus.  How large are the effect of bonuses in initiative?  I'll have to play with that calculation later, though.