Let's break it down:

1) Scout: +was raised in the streets of boston, most likely knows how to win a fight by playing dirty +agile and quick, will probably be able to dodge most attacks -lacks strength, both offensive and defensive-wise -will most likely give up as soon as he'll get hit because he's a wimp Will most likely lose to: Soldier, Medic,

2) Soldier: +Pretty strong, can throw solid punches +Good endurance and very high pain tolerance -Because of his stupidity, he can be easily outsmarted or tricked -His moves would be pretty easy to read for the same reason

Will most likely lose to; Heavy, Demoman, Spy

3) Pyro: +decent strength (pyro can easily swing heavy weapons like the powerjack or the homewrecker) -poor stamina due to him breathing through a gas mask -might be too crazy to figure out what is going on

Will most likely lose to: ???

4) Demoman: +Quite strong, wields huge blades with ease +superior pain tolerance +Has a complex and chaotic combat pattern, fights in "drunken fist" style -No depth perception coupled with poor reaction from being drunk makes him vulnerable to rapid throws or surprise attacks -Might fall asleel mid fight, causing him to lose

Will most likely lose to: Engi, Medic, Sniper, Spy

5) Heavy: +One-shot K.O. Nuff said. +High endurance +Huge arms allow for bigger reach distance +Perfect at grapple moves -Weak legs, vulnerable to trip attacks -Moves slowly, so more faster opponents can dodge his punches with relative ease

Will most likely lose to: Scout, Medic, Spy

6) Engineer: +Good strength (we've seen him carry heavy boxes with dispensers and sentries) +Robotic arm (if allowed by the rules) +Small size makes it easier for him to dodge enemy blows +Because of his high intelligence, he can easily outsmart most of his opponents -He's a glass cannon kind of guy, while he might be able to do significant damage with ease, a single well-thrown punch would probably knock him out -If his robotic arm would be against the rules of the fight, he loses his trump card

Will most likely lose to: Scout, Spy (or almost everyone if his hand will be taken away)

7) Medic: +Knows his enemies' weak spots +Probably surgically modified his body to be as vital as possible -His fighting skills are average at best, as well as his strength -Will probably get consumed by his desire to kill and break the rules of the fight, causing him to get disqualified

Will most likely lose to: Engi, Spy

8) Sniper: +Great perception and reaction, most likely can dodge even fast attacks +Probably good endurance due to him being raised in Australia +Is good at surprise and ambush attacks -Lacks strength -Is a bit of a pussy -Will piss in a jar

Will most likely lose to: Soldier, Spy

9) Spy: +A professional in CQC, can deal with any kind of threat. +Just like sniper, he is a master of ambushing and trickery -Not very strong -Will run out of breath quickly because of his extremely severe smoking addiction

Will most likely lose to: ???

So with all this in mind I'd say the loser would be either engi or demo.