A bounty on orphaned blocks doesn't make sense. As already mentioned noone has to and noone probably would include a transaction to the miner who mined the orphaned block.
And another point is that it wouldn't increase the "fairness" because normally everyone should have the same rate of orphans in average. The miners who are much above the average rate are doing something wrong they are wasting hashing power for which we shouldn't give a bounty since it doesn't increase the network security.

I like the lottery analogy. Let's assume every block starts a lottery round then normally the only one who wins is the one who is the first that brings his lottery ticket to the lottery company. With your proposal the one who is a bit late would also get a small amount of money. In the long run it's nothing else than giving the first one more money. For Bitcoin giving a bounty for orphaned blocks is the same as increasing the block reward to 51BTC because everyone has the same chance to find a block that gets orphaned. If we would still want to limit the total BTC amount we would have to change the block reward to 49BTC and the bounty for an orphaned block to 1BTC.
But then we didn't change anything we only decreased the variance for miners and we didn't increase the fairness for miners. And to decrease variance which is the only thing your proposal could do there are much better methods like pools or even decreasing the average time to find a block.

In the end your proposal doesn't help anyone and I have no idea how to implement it and that's enough to simply drop it.