There are at least some fingerprintable gladiators who keep gladiating, and who need to be Accounted For (the presence of such people makes all archetypery suspect: are Dwarven Knights really that Good, or are there just a handful of super-prolific Dwarven Knights who give everyone an unfairly good impression?). This includes a Level 7 Elven Ninja, almost certainly Cadagal's Champion, who inexplicably insists on always wearing black (even though it doesn't seem to make a difference to how well ninjas ninj).

Level 4 Boots and Level 4 Gauntlets are super rare in the dataset. The Gauntlets are always worn by a pair of hypercompetent Level 7 Dwarven Monks; the Boots are always worn by the Level 7 Elven Ninja.

Despite this, Cadagal's Champion is facing us with Level 2 Boots.

We have some Level 4 Boots.

. . . we robbed this guy, didn't we? And if we wear the boots - our most powerful equipment - he'll flip out and set his House against us whether we win or lose? Dammit . . .

A is a Human Warrior. Warriors lose to Fencers, Humans lose to Fencers, Humans lose to Elves. We have an Elven Fencer on call; send Y.

B is a Human Knight. Rangers are best vs Knights, so send W. (Not super confident in this one)

C is an Elven Ninja. Ninjas are super weak against Knights. Send Z, the Elven Knight. (Slightly concerned by how underrepresented Elves are in the sample of gladiators who managed to beat this guy but I'm assuming that's either noise or an effect which Z will be able to shrug off with the Power of Friendship and/or Urgency)

D is a Dwarven Monk. Monks are weak to Ninjas; send U.

I haven't managed to figure out how equipment works beyond "higher number good"; if there's specific synergies with/against specific classes/races/whatever they elude me. For that reason:

Y and Z are my best shots. I'll have them both wear what their opponents are wearing, to reduce the effects of uncertainty and turn those fights into "who wore it better?" contests. (So +3 Boots and +1 Gauntlets for Y, +2 Boots and +3 Gauntlets for Z.)

U vs D looks pretty solid so I'll give him the remaining +2 Gauntlets and +1 Boots.

W vs B is my most tenuous guess, I hope she won't hold a grudge after I send her out unequipped to boost everyone else's chances.

After initial observations that e.g. higher numbers are correlated with winning, I switched to mainly focus on race and class, ignoring the numerical aspects.

I found major class-race interactions.

It seems that for matchups within the same class, Elves are great, tending to beat dwarves consistently across all classes and humans even harder. While Humans beat dwarves pretty hard too in same-class matchups.

Within same-race matchups there are also fairly consistent patterns: Fencers tend to beat Rangers, Monks and Warriors, Knights beat Ninjas, Monks beat Warriors, Rangers and Knights, Ninjas beat Monks, Fencers and Rangers, Rangers beat Knights and Warriors, and Warriors beat Knights.

If the race and class are both different though... things can be different. For example, a same-class Elf will tend to beat a same-class Dwarf. And a same-race Fencer will tend to beat a same-race Warrior. But if an Elf Fencer faces a Dwarf Warrior, the Dwarf Warrior will most likely win. Another example with Fencers and Warriors: same-class Elves tend to beat Humans - but not only will a Human Warrior tend to beat an Elf Fencer, but also a Human Fencer will tend to beat an Elf Warrior by a larger ratio than for a same-race Fencer/Warrior matchup???

If you look at similarities between different classes in terms of combo win rates, there seems to be a chain of similar classes:

Knight - Warrior - Ranger - Monk - Fencer - Ninja 

(I expected a cycle underpinned by multiple parameters. But Ninja is not similar to Knight. This led me to consider that perhaps there is an only a single underlying parameter, or trade off between two (e.g. strength/agility .... or ... Speed and Power)).

And going back to the patterns seen before, this seems compatible with races also having speed/power tradeoffs:

Dwarf - Human - Elf

Where speed has a threshold effect but power is more gradual (so something with slightly higher speed beats something with slightly higher power, but something with much higher power beats something with much higher speed).

Putting the Class-race combos on the same spectrum based on similarity/trends in results, I get the following ordering:

Elf Ninja > Elf Fencer > Human Ninja > Elf Monk > Human Fencer > Dwarf Ninja >~ Elf Ranger > Human Monk > Elf Warrior > Dwarf Fencer > Human Ranger > Dwarf Monk >~ Elf Knight > Human Warrior > Dwarf Ranger > Human Knight > Dwarf Warrior > Dwarf Knight

So, it seems a step in the race sequence is about equal to 1.5 steps in the class sequence. On the basis of pretty much just that, I guessed that race steps are a 3 speed vs power tradeoff,  class steps are a 2 speed and power tradeoff, levels give 1 speed and power each, and items give what they say on the label.

I have not verified this as much as I would like. (But on the surface it seems to work, e.g. speed threshold seems to be there). One thing that concerns me is that it seems that higher speed differences actually reduce success chances holding power differences constant (could be an artifact, e.g., of it not just depending on the differences between stat values edit: see further edit below). But, for now, assuming that I have it correct, speed/power of the house champions (with the lowest race and class in a stat assumed to have 0 in that stat):

House Adelon:  Level 6 Human Warrior +3 Boots +1 Gauntlets - 14 speed 18 power

House Bauchard: Level 6 Human Knight +3 Boots +2 Gauntlets - 12 speed 21 power

House Cadagal: Level 7 Elf Ninja +2 Boots +3 Gauntlets - 25 speed 10 power

House Deepwrack: Level 6 Dwarf Monk +3 Boots +2 Gauntlets - 15 speed 18 power

Whereas the party's champions, ignoring items, have:

• Uzben Grimblade, a Level 5 Dwarf Ninja - 15 speed 11 power
• Varina Dourstone, a Level 5 Dwarf Warrior - 7 speed 19 power
• Willow Brown, a Level 5 Human Ranger - 12 speed 14 power
• Xerxes III of Calantha, a Level 5 Human Monk - 14 speed 12 power
• Yalathinel Leafstrider, a Level 5 Elf Fencer - 19 speed 7 power
• Zelaya Sunwalker, a Level 6 Elf Knight - 12 speed 16 power

For my proposed strategy (subject to change as I find new info, or find my assumptions off, e.g. such that my attempts to just barely beat the opponents on speed are disastrously slightly wrong):

I will send Willow Brown, with +3 boots and +1 gauntlets no gauntlets, against House Adelon's champion (1 speed advantage, 3 4 power deficit)

I will send Zelaya Sunwalker, with +1 boots and +2  +1  gauntlets, against House Bauchard's champion (1 speed advantage, 3 4 power deficit)

I will send Xerxes III of Calantha, with +2 boots and +3 +2 gauntlets, against House Deepwrack's champion (1 speed advantage, 3 4 power deficit)

And I will send Varina Dourstone, with +3 gauntlets no items, to overwhelm House Cadagal's Elf Ninja with sheer power (18 speed deficit, 9 12 power advantage).

And in fact, I will gift the +4 boots of speed to House Cadagal's Elf Ninja in advance of the fight, making it a 20 speed deficit.

Why? Because I noticed that +4 boots of speed are very rare items that have only been worn by Elf Ninjas in the past. So maybe that's what the bonus objective is talking about. Of course, another interpretation is that sending a character 2 levels lower without any items, and gifting a powerful item in advance, would be itself a grave insult. Someone please decipher the bonus objective to save me from this foolishness! 

Edited to add: It occurs to me that I really have no reason to believe the power calculation is accurate, beyond that symmetry is nice. I'd better look into that.

further edit: it turns out that I was leaving out the class contribution to the power difference when calculating the power difference for determining the effects of power and speed.  It looks like this was causing the effect of higher speed differences seeming to reduce win rates. With this fixed the effects look much cleaner (e.g. there's a hard threshold where if you have a speed deficit you must have at least 3 power advantage to have any chance to win at all), increasing my confidence that effects on power and speed being symmetric is actually correct. This does have the practical effect of making me adjust my item distribution: it looks like a 4 deficit in power is still enough for >90% win rate with a speed advantage, while getting similar win rates with a speed disadvantage will require more than just the 9 power difference, so I shifted the items to boost Varina's power advantage. Indeed, with the cleaner effects, it appears that I can reasonably model the effect of a speed advantage/disadvantage as equivalent to a power difference of 8, so with the item shift all characters will have an effective +4 power advantage taking this into account.

I haven't yet gotten into any stats or modeling, just some data exploration, but there's some things I haven't seen mentioned elsewhere yet:

Zeroth: the rows are definitely in order! First: the arena holds regular single-elimination tournaments with 64 participants (63 total rounds) and these form contiguous blocks in the dataset with a handful of (unrelated?) bonus rounds in between. Second: Maybe the level 7 Dwarf Monk stole (won?) those +4 boots by winning a tournament (the Elf Ninja's last use was during a final round vs that monk!) and then we acquired the boots from that monk? They appear to have upgraded their boots once before from +1 to +3 when defeating a Dwarf Ninja, though that was during a bonus round, not a tournament.

Does the fact that we see the winners of tournaments 6x more often than those eliminated in round one matter for modeling? It might; if e.g. gladiators have a hidden "skill" stat but for some reason the house champions don't have very high skill, we'll be implicitly significantly overestimating their hidden skill stat.

 Looking at how various combinataions of race and class do against one another when their levels are the same there are clearly some combinations that do a lot better than others. Increasing the level helps to an extent, but the race/class combination looks like it is easily the most important factor. Special items do help, but less than the level. In most cases boots seem a bit more useful than gauntlets.

Manually scanning through the data suggests that the following combinations hopefully won't be too bad:

DWARF NINJA  v HUMAN WARRIOR + 1 boots of speed + 2 gauntlets
ELF KNIGHT     v HUMAN KNIGHT + 3 boots of speed +1 gauntlets
DWARF WARRIOR  v ELF NINJA + 3 gauntlets + 2 boots of speed
HUMAN MONK     v DWARF MONK + 4 boots of speed

As for the bonus obective. It looks like someone is threatening us if we do well in one or more of the fights, however I can't establish the details with any reliability. And as we have no idea who sent, and what their real intentions are, we would probably be wise to ignore it ad do what we would have done anyway, at least for now.
 

• Adelon looks to have rough matchups against Elf Monks. Which we don't have. They are however soft to even level 3-4 challengers sometimes. Maybe Monks and/or Fencers have an edge on Warriors?
• Bauchard seems to have particularly strong matchups against other Knights, so we don't send Velaya there. They seem a little soft to Monks and to Dwarf Ninjas and especially to Knights, so maybe Zelaya? Boots should help here.
• Cadagal has precious few defeats, but one of them might be to a level 2(!) Human Warrior with fancy +3 Gauntlets. Though it seems like there's a lot of combats where some Cadagal-like fighter has +4 Boots instead? Not sure if that's the same guy.
  • And on that note, the max level is 7, and the max bonus for Boots and Gauntlets both is +4.
  • Max Boots (+4) is always on a level 7 Elf Ninja with +3 Gauntlets (but disappears altogether most of the way through the dataset).
  • Max Gauntlets (+4) is on either a level 7 Dwarf Monk who upgraded from +1 Boots to +3 Boots halfway through, or else there's two of them. Thankfully we're not facing them.
• Deepwrack poses problems. They have just as few defeats, and one of them even contradicts the ordering I derived below! Ninjas are meant to lose to Monks. Maybe the speed matters a lot in that case?
• It looks like a strict advantage in level or gear - holding all else constant - means you win every time. If everything is totally identical, you win about half the time. (Which seems obvious but worth checking.)
• Looking through upsets - bouts where the classes are different, the losing fighter had at least 2 levels on the winner, and the loser's gear was no better than the winner's - we generally see that:
  • Fencers beat Monks and Rangers and lose to Knights, Ninjas, and Warriors
  • Knights beat Fencers and Ninjas, tie(???) with Monks and Warriors, and lose (weakly) to Rangers
  • Monks beat Ninjas, Rangers, and maybe Warriors, tie (?) with Knights, and lose to Fencers
  • Ninjas beat Fencers and (weakly) Rangers, and lose to Knights, Monks, and Warriors
  • Rangers beat Knights (weakly), Ninjas, and Warriors, tie with Fencers, and lose to Monks
  • Warriors beat Fencers, Ninjas, tie(?) with Knights, and lose to Rangers and maybe Monks

Willow v Adelon, Varina v Bauchard, Xerxes v Cadagal, Yalathinel v Deepwrack.

If I had to guess what gear to give to who: Warrior v Knight is a rough matchup, so Varina's going to need the help; the rest of my assignments are based thus far on ~vibes~ for whether speed or power will matter more for the class. Thus:

Willow gets +2 Boots and +1 Gauntlets, Varina gets +4 Boots and +3 Gauntlets, Xerxes gets +1 Boots and +2 Gauntlets, and Yalathinel gets +3 Boots.

• Race affects how good you are at a class. Elves might be best at rangering, say.
• Race and/or class affect how much benefit you get out of boots and/or gauntlets. Being a warrior might mean you get full benefit from gauntlets but none from boots.
• Color might affect how well classes do. Ninjas wearing red might win way less often.
  • The color does not actually seem to affect ninjas all that much if at all - 6963 vs 6762 wins. Could still be a tiebreaker?
  • Color doesn't affect things much overall either: 40136 vs 39961 wins.
• There's some rank-ordering of class+race+level matchups, maybe an additive one.
  • Alternatively there could be some nontransitive thing going on with tiebreaks sometimes from levels, races, and gear?
  • On further reflection that totally seems to be what's going on here.
  • Maybe there's something about the matchup ordering being sorted over (race, class)? D's loss (as a L6 Dwarf Monk) to a L4 Dwarf Ninja is... unexpected to say the least!

• If you [use the +4 Boots in combat and beat Cadagal then they'll know you were] responsible [for] ????? ?????? [Boots from his/her/the] House.  [You will gain its] lasting enmity, [and] [people? will?] ???????? ???? ??? ???? ?? ??? ???? ??????? ?? ?? ????? ?? ????????? ?? ??? [upon] your honor [if] ????????? ???? ?? ???? ??? ???? ??? ??? friendship ???? ?? ??? ???? ?? ??? ?????? ??????? ?? ?? ?????.
  • So maybe we're OK to use the +4 Boots as long as it's not against Cadagal?
  • No idea how to even guess at what's going on in that second sentence apart from "bad things will happen and everyone will hate you, you dirty thief".

 There might probably be some effects in the dataset that are only relevant at lower levels, and so exist mainly as a red herring, since all our fights are between people of at least level 5. I therefore doublechecked everything looking at only the subset of data with all fighters in high level.

The classes seem to work in a such of Rock/Scissor/Paper way, some being much stronger against others.

I plan to try to beat the:   Human Warrior with my Ranger / Human Knight with my Monk / Elf Ninja with my Knight / Dwarf Monk with my Fencer.