The computer game Dungeon Raid is quite complicated, but for our purposes we can consider the following simplified version.
The player has a current health $\def\hcur{h}\hcur$ and a maximum health $\def\hmax{h_{max}}\hmax$ with $0\le\hcur\le\hmax$, and a current armor $\def\acur{a}\acur$ and a maximum armor $\def\amax{a_{max}}\amax$ with $0\le\acur\le\amax$.
The playing field contains groups of monsters, potions, and shields. On each turn, the player may attack some monsters, gather some potions, or gather some shields. Killing monsters scores points. Gathering potions raises $\hcur$, possibly all the way to $\hmax$, but no higher. Similarly, gathering shields raises $\acur$, but not past $\amax$.
After the player's turn, any monsters still alive attack the player, doing $d$ points of damage, where $d$ is independent of the other quantities and is calculated in a way that need not concern us. The damage is divided into a portion that is absorbed by the armor, $d_a = \min(\acur, d)$, and the remaining portion that is applied to the player's health, $d_h = d-d_a = \max(d-\acur, 0)$. The armor is degraded as follows: $d_a$ fair coins are flipped, and $\acur$ is reduced by one point for each tail. The player's health $\hcur$ is reduced by $d_h$. If $\hcur$ is now zero or below, the game is over; if $\hcur$ is positive the player gets another turn.
At intervals the game will offer the player a choice of equipment upgrades. One such is a health upgrade, which raises both $\hcur$ and $\hmax$ by 5 points; another is an armor upgrade, which raises both $\acur$ and $\amax$ by 1 point.
I would like to analyze whether the health or armor upgrade is likely to be better.
To this end, I want to engage in intelligent, directed collection of statistics. For example, it seems important to know the distribution of $d_a$ under typical play. Similarly, it might be useful to know the expected increase to $\acur$ from gathering shields, which depends on how often $\acur$ is close to $\amax$; when $\acur = \amax - 1$ gathering any number of shields can only raise $\acur$ by 1.
I don't know what information to collect, how to decide what to collect, or what to do with the information once I have collected it.
My question is:
On what parameters might the comparison of the armor and health upgrades depend, and what would I need to know in order to compare them?
Each point of armor on average reduces the damage to your health by $2$ points (assuming the battle lasts a long time), so that is the trade between potions and shields and shows that health upgrades are better than armor upgrades. Can we assume that $a,h$ recover to maximum between battles? A harder question is the trade between offense and defense. If you kill a monster presumably you don't take damage from it next round, but you may have to look ahead several rounds to know whether you are better off.