I had a very interesting experience with +25 luck today. However, the luck didn't work in my favor.
I was playing tiny/rare/intelligent. Each race had 2-3 planets. The Drengin, who were in first, had 2 planets in 2 different systems, with 4 useless planets in each of those systems. The Drengin got the "all unihabited planets within 2 sectors get their PQ raised by 14" event. That's right, the Drengin now had *EIGHT* PQ 20 planets that they had exclusive colonization rights to. I managed to get one upgraded planet in another sector. I'm wondering if the PQ event wound up helping me; the Drengin stopped their military buildup to develop their new worlds.
Earlier in the same game, the Drengin had allied up with everyone else and the Drengin declared war on me. (All aliens were set to pure good.) Even my trading partner (the Altarians), for whom I was 50% of their income, declared war. I managed to buy peace with everyone after a few turns. I was careful to buy peace with everyone on the same turn to end the war. (I couldn't buy peace right away; the Drengin had to be struggling with their war before willing to give peace.) (Note: if you send a freighter to a planet you're at war with, you get an interesting message.)
Still in the same game, the Carinoids were about to get eliminated. They were in the same sector as the Arceans. I moved in my invasion fleet so I could get the planet before anyone else did. But the Arceans had too much influence for me. I had to build a culture starbase to keep my new planet from flipping to the Arceans. But the Arceans demanded the starbase. I couldn't give them the starbase without losing the planet (and the game), so I found myself at war with everyone again. Again I managed to buy peace after a few turns of war. Still later, the Arceans started demanding my culture starbase again, threating war. I turned down their demand, and without leaving the diplomacy screen, I extracted 500 BC in exchange for a peace treaty! So I managed to avoid a war. (This is a bug that has been noted elsewhere. The AI will make a demand and declare war, and then immediately give concessions for peace. I think the fact that I had the Diplomatic Translators is part of the reason.)
I did finally manage to pull out a victory. Now that I has a better military than the Arceans, and enough military to keep the Drengin away, I built up 2 party palaces to flip the Arcean planet. After that, it was easier to flip a few more planets. By then the survivors were close to me, so I got an alliance win. That was the first time I ever saw Avatars on tiny, without intentionally prolonging the game.