Suppose the Banker always offered the expected value of the remaining cases. Could a player somehow make smart choices to increase his or her expected winnings? No. Let X be the random variable representing the value of the briefcase held by the player. Let Y be the random variable describing the briefcases open so far. A well known equality states E(E(X|Y))=E(X), i.e., the expectation of the expected value of the briefcase given the current game situation is just the original expectation of the briefcase. Any strategy by the player will yield exactly the same expected winnings, about $131,477.54.
Usually the Banker gives an offer below the current expected value of the briefcase. Why? As I mentioned in the previous post, the players are risk adverse and may accept a smaller guaranteed amount now. But more importantly a lower amount will increase the chances that a player will not accept the deal and play longer. The Banker pays an expected $131K per player not per episode and thus pays out less per episode the longer each player plays.