Although the material distribution algorithm is fair in the long run, it can be brutal in the short term. I have proposed methods for assigning materials after the run before, but I may have hit upon a better idea that works better with the "assign-as-you-go" model. The method is as follows:

1) Each player starts with a percentage value for each * level material. When a material is picked up, the RNG rolls to determine who gets that material. The person receiving the material has their percentage value decreased accordingly, and everyone else gets theirs increased- but only for that * level.
2) When a person joins a run, they are assigned a percentage value of P = 1/N, where N is the number of people. Everyone else has their percentage value subtracted by P/(N-1). This is done for each * level.
3) When a person leaves a run, their current percentage value is divided by the number of remaining players, and each player gets that number added to their percentage value. This is also done for each * level.

To illustrate, here is an example. I will use the format (playername)[0*, 5*] to denote each player's percentage values. (I will omit the values for 1*-4* from now on to make reading easier, since my example will only involve two material pickups, a 0* and a 5*)
1) Players A and B start a run. A[.5, .5]; B[.5, .5].
2) A blue shard is picked up (0*). The RNG rolls, and determines that player A gets it. Player A now has their 0* value decremented, say be 10%, and the amount is distributed to all other players (in this case, just B). Thus, A[.45, .5]; B[.55, .5]
3) Player C joins the run. They are assigned a percentage value of 1/3: C[.33, .33] Since percentages must add up to 1, Players A and B have (.33/2) subtracted: A[.28, .33]; B[.39, .33]. Notice how it maintains fairness- the next time a shard (0*) is picked up, player A, who has already picked one up, is less likely to receive it, and player B is more likely to receive it.
4) Now, a flame soul is picked up. The RNG determines it goes to player C. The percentage values for 5* are modified accordingly- perhaps a greater penalty of 50% is assessed because 5* mats are rarer. Thus, C's 5* percentage value becomes (.33*.5), and the 0.165 percentage points are distributed to A and B: A[.28, .41]; B[.39, .41]; C[.33, .17]
5) Lastly, Player B leaves. His percentage values (.39, .41) are divided by the remaining number of players, and distributed evenly. The percentage values now look like this: A[.48, .62]; C[.52, .38].

Notice that a 'memory' that A picked up a blue shard, and C picked up a flame soul, are maintained by the percentage values. Because the system still relies on the RNG to make the distribution decision, and maintains a separate percentage value for each material, one cannot game the system as in a round-robin distribution scheme. I believe this weighted distribution method could reduce the amount of perceived unfairness in the material distribution system.