So, as I think most people know, there's a very noticable jump in difficulty between the Normal and Hard tournaments. With multiplayer not working for me, I've been exploring the single-player tournament a bit more, and after paying a bit more attention to the game I think I've figured out a very simple way to tilt the playing field in your favor:
Buy every gold upgrade obsessively, even the ones that are never worth it in multiplayer. Also, buy +XP upgrades to at least one or two levels.
The mistake I made the first few times I tried to play the tournament was to play as a solo player. Sure, I had teammates, but they're so dumb that it didn't seem worth worrying about them; they'd die over and over, and I just had to kill the other team faster to win. I actually managed to win all my tournaments this way, but it was really tough to do it; every tournament had a few rounds where I just got crushed and didn't know how to avoid it.
Here's the thing about the AI. Your AIs are more or less as dumb as the other side's AIs; it's just that the other side gets a few buffs (to starting gold and maybe a few stats, I've forgotten exactly what). This means that even a very minor advantage to the other team will cause your AIs to lose most battles. Since the AI isn't clever enough to run away from more powerful adversaries, this means that they get killed a lot, and the other team has a constant source of gold unless you do something to stop this pattern, ideally ASAP.
You can't order the AI around and it's hard to effectively protect them (unless, perhaps, you're Sedna or QoT), but there is one thing you can do to buff your AIs up: buy gold and XP upgrades at the Citadel. As I noted above, all the AIs seem to do more or less the same thing:
1. Choose target.
2. Walk towards target.
3. Attempt to use random skills on target and/or autoattack it.
Then, one of:
4a. You win!
4b. Run away.
4c. Fail to run away and die.
Your allies and the enemies are doing exactly the same thing; the only way you can get an advantage is to change how effective they are in step 3. That way, the enemy AIs will execute step 4c more than your AIs do. Even a relatively small increase in the gold income that your guys get means they can buy a few more potions, some of the cheap armor, etc, and this can turn around individual AI-vs-AI battles. Once your AIs start winning battles instead of losing, they'll buy better items and kill the other side more, and the slippery slope that usually kills you in Hard will work in your favor instead.
Similarly, your AIs are just about as effective at acquiring experience as the enemy AI. Buying +XP upgrades means that your team will level faster than the other team (all else being equal); this is particularly effective since the AI isn't coded to ever purchase the upgrade, so you'll be at a permanent advantage relative to the other team.
Once you've bought up these upgrades (I assume of course that you're killing the enemy AI at every opportunity to get the money), you can start equipping your Demigod as usual and crush them, without having to worry about one of your enemies becoming ultimately powerful through constantly killing your allies.