1. Second-sourcing.  You are a monopolist selling a durable good that requires periodic upgrades.  Think enterprise software.  The monopoly price extracts the lifetime user-value of the product.  To maximize the lifetime user-value of the product you should set the price for future upgrades at cost.  The  problem is that your users don’t trust you.  They foresee that at the time of the upgrade, when the original purchase price of the software is a sunk cost you are not going to set upgrade prices at cost.  Indeed you will again try to extract the (remaining) lifetime user-value of the product with the upgrade price.   You need a device to commit yourself not to try to exploit your customers in the future so that you they will submit to your exploitation today.  By spinning off a division of your company you can create a competitor for upgrades.  This competition guarantees that you cannot act like a monopolist for upgrades and the upgrade price will be competitively priced at cost.
  2. Pre-emption.  You are currently a monopolist in an industry that can accommodate at most two firms.  It is inevitable that there will be an entrant eventually and your monopoly profits will turn into duopolist profits.  Since you are going to have a competitor anyway why not create your own competitor?  You could sell half of your company to the highest bidder.  He will be willing to pay up to the duopolist profits and then he will compete with you driving the profits of your remaining half down to the duopolist profits.  In total your profits are equal to the sum of the two duopolist’s profits rather than just a single duopolist profit.  That’s necessarily less than the monopoly profit but that wasn’t going to last anyway.