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.