[qiemem]

First, accumulation would be the wrong way to think about it. Once the two agents are engaged in transaction, it doesn't matter which is which.

Let D be the difference in wealth post-transaction divided by the total. The expected value of D is 1/2. That means that we should expect that one agent will walk away with 75% of the pot and the other with 25%. Thus, although the expected outcome for an agent is 50% of the pot, we're still pretty much guaranteed to get very uneven splits.

Next, note that a person's wealth affects the payout of the next transaction. In other words, the person who takes 25% is almost guaranteed to be able to make less on the next transaction than the person who walked away with 75%. Suppose all our agents start out with 100, and that the 75%/25% splits are guaranteed. If each agent engages in exactly one transaction, then half our agents will have 50 and half will have 150. A 50's next transaction will either be (50, 50) or (50, 150). A 150's next interaction will either be (150, 50) or (150, 150). The expected outcome for (50, 50) is (25, 75). The expected outcome for (50, 150) is (50, 150). The expected outcome for (150, 150) is (225, 75). None of the expected outcomes even things up. It just gets more and more spread out.