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...