Rav and Shmuel argue in a case where 2 people are holding contracts that they were gifted a particular piece of land, and both contracts contain the same date. Rav holds that they divide the land, whereas Shmuel holds "shuda d'dayni"- the dayan decides who gets it. Tosafos explains that if we were speaking about the type of contract that would go into effect at the moment the day ends, such as a shtar of shi'bud, so that both of them have equal rights to the land, even Shmuel would agree that it is divided. Similarly, Tosafos explains that according to R' Meir who says that witnesses who sign the contract make it go into effect, so that the contract would only go into effect at the moment the day ends, even Shmuel would agree that it is divided equally. Basically, Shmuel would only hold his position of "shuda d'dayni" when: 1. we are speaking about a shtar matana that goes into effect when it is received, and 2. according to R' Eliezer that eidei mesirah make the kinyan so that Reuven deserves the property if he received his contract earlier on that day, and Shimon deserves it if he received his contract earlier on that day.
The Rashbam disagrees with Tosafos and holds that even though the contract will only go into effect at the moment the day ends, so that they both have an equal claim on the property, Shmuel will still say "shuda d'dayni", rather than say to divide.
It seems that the Rashbam and Tosafos go li'shitasam:
Tosafos and the Rashbam argue about the definition of "shuda d'dayni". Rashbam says that the dayan has to give it his best guess to try to figure out who really deserves it in din (such as who is the closer friend that the giver would have more likely gifted it to). Tosafos holds that shuda is literally the dayan doing whatever he wants. Tosafos holds that the dayan has such liberty in his decission that Tosafos entertains the possibility that the dayan can receive payment for deciding this case, since anyway he can do what he wants (just that if he accepts payment he forfeits his status as a dayan). The Rosh explains that Tosafos position is supported by the Rabbeinu Chananel that Tosafos quotes that only an "expert dayan" has the power of shuda. Based on the Rashbam's approach, Rav sheishes, although not an "expert dayan" should have been able to decide who rightfully deserves it, but since it is completely in the hands of the dayan to do whatever he wants, he requires the authority of an "expert dayan".
In short, Tosafos holds that shuda is not even an attempt at justice. Therefore, whenever their is a possibility of justice, we would choose that over shuda. In any case where the contract goes into effect at the moment the day ends (either a shtar shi'bud, or any contract according to R' Meir), they both have an equal claim to the property, so it is better to divide. That is why Tosafos holds that shuda is only a possibility when one of the two deserve it because one contract went into effect before the other so that dividing is not justice, so we allow the dayan to choose who he wants to give it to. But, according to the Rashbam that shuda is an attempt at justice, it applies even to a type of contract that goes into effect at the moment the day ends so that they both have an equal claim on the property since their contracts went into effect simultaneously. Certainly dividing would be a just option, but being that shuda is an attempt at justice also, the rashbam holds that according to shmuel we assume that shuda will have a better chance at achieving justice than dividing.