tl;dr Basically ICANN on the blockchain but owned and controlled by UIA holders and also where new names can either be put on auction immediately or be purchased by the name claimer for a fixed fee that depends on the quality of the name (usually meaning the length). The fixed fee structure (and other properties of the system) can be controlled by a committee elected by the UIA holders (assuming a necessary quorum agree and with a 2 week delay just like with delegates), and it would be designed to force name claimers of shorter names (which are likely to be higher valued) to always prefer the auction route instead but name claimers of longer names (which are more likely to be used for account names) to be purchased for a reasonable fixed fee. The committee also can collectively act as a judge (or more realistically delegate responsibility to multiple trusted judges) who run their own private (and profitable) court system to listen to claims of trademark infringement (if that is the policy UIA holders choose to adopt) and can enforce their ruling by forcefully taking away a name from a current owner and giving it to someone else (assuming the necessary quorum is reached and again with the 2 week delay which gives the UIA holders the opportunity to veto the action by voting out members in the committee).
I would love to see the following experiment in one of the namespaces that could be potentially used for domains and even account names (see my post
discussing various economic models that could be used for different namespaces). This namespace could be used to point to web sites and/or point to accounts (possibly in addition to whatever other account name system also exists).
The idea behind this naming system is to not have ownership only be determined by a set of rules on the blockchain but also by decisions by a group of elected people. These people can act as arbitrators in matters that the blockchain either cannot have knowledge of and/or cannot have the intellect to decide what is fair and just according to the vague human policy it is supposed to follow. That means this naming system could in theory respect existing trademarks making it more attractive to corporations, but it does not need to be limited to following the policies that ICANN currently follows or even necessarily follow the IP laws of a particular country (in that case, you better hope the necessary quorum to make decisions on the blockchain cannot be reached by a subset of the known elected arbitrators residing in the jurisdiction of that country).
There would be a special UIA that represents ownership of this naming system itself. The UIA holders can vote for the number of elected arbitrators (N) to have and the arbitrator candidates themselves. This would be the same system used for selecting delegates and the number of delegates to have. The top N arbitrator candidates would become the elected arbitrators and their weight would be determined by their relative approval votes (just like in the delegate system). If the necessary quorum (weight greater than some defined threshold) of elected arbitrators agree, they can forcefully move ownership of one of the names to another account. This could be useful in giving the legitimate trademark owner their name. Another possible action they can take with the proper quorum is to take away an existing name from an owner and putting it up for auction. Just like with the delegates, any action taken by these electors has a 2 week delay allowing the stakeholders the option to veto the action by voting out enough of the arbitrators.
All new names would be classified by a computer algorithm into the class they belong to. The algorithm would typically make the decision based on the length of name. A user trying to register a new name would either have the option of paying the fixed fee specific to the class the name belongs in to own the name, or putting it up on an open auction where they could then try to win the name by being the highest bidder. The user would have to pay a very small fixed fee (potentially growing with the length of the name) just for the privilege of bringing a new name into the system and getting the exclusive choice to either pay the fixed fee for the name or put it up on auction (this fee is to always at least compensate the network for the technical expense of managing the names, even if the market price ends up being 0). If the user believes the market will value the name at a lower price than the fixed fee for its class, it is rational to choose to put it up on the auction rather than paying the fixed fee. The fees for each class can be set and adjusted by the arbitrators (again with the 2 week delay).
I would expect really long names to belong to a class with a low fixed fee, so regular people could afford to get their full names (maybe with a number at the end) for low cost and use them as their account names. People with short first and last names (or a short pseudonymous handle) that are nevertheless uncommon would likely put the name up on auction and end up paying around the same fee as the low fixed fee (they just would have to wait 30 days before they were able to get their name). Short names would belong to a class with an intentional outrageously high fixed fee, forcing everyone who wants to get such names to choose the auction route instead. The auction system would try to fairly price the name. If the name is a high-valued name already owned by a big company, it is likely that the auction will not fairly capture its market value because the people with the big money to bid it up would not yet exist on the network by the time the naming system opens up. Because these would be well established names of big companies, the arbitrators would likely decide to take the name away later when the trademark owner complained, and this fact would likely cause squatters to think twice about paying a moderate fee to "own" this name.
I would like to see how such a system evolves. The arbitrators would likely act as judges (or rather delegate that task to selected judges they trust, which also allows them to scale their operation) running their own small courts where they listen to both parties and examine the evidence to see whether the prosecutor has a legitimate claim on the name (legitimate by the policy set by the UIA holders of course). The UIA holders would want to set a policy that was reasonable and mostly respected existing trademark laws since it would make their name system more attractive to people and therefore give it more value. The judges would get paid by the prosecutors who would have to put up funds to compensate the judges for taking the time to listen to their case as well as to set aside funds that would pay the defendant (the existing owner of the name that they claim belongs to them) for their lost time in the case the judges rule against the prosecutor. A few other major differences I would expect with this system compared to regular courts is that the entire thing could be done virtually using the internet (making it easier for all parties involved to "meet") and of course the defendant would not go to jail for "contempt of the court" due to not showing up or being rude (the worst possible thing that can happen to them is that they lose the name with absolutely no compensation). So I think this would be an exciting use case to try out the concept of private justice and see how well it actually works in practice (at least in this narrow field of trademark disputes).
The fees on the prosecutor may be set so that after compensating everyone needed there would be extra funds that could be placed into this naming system's special reserve pool (different than the BitShares reserve pool). I would expect the fees to be set pretty high to avoid spreading the judges too thin and to just focus on the high value cases dealing with the trademarks of medium to large companies (smaller trademark claims could probably be more cost effectively settled out of the courts between the existing name owner and the name claimer, but would also be less likely to be necessary since lower value names are less likely to be taken by squatters and also due to other methods we can devise ). The reserve pool would also have an income stream coming from the paid fixed fees and the funds collected from the auctions. It would also have an expense stream that paid the arbitrators some daily salary (which could be adjusted by the necessary quorum of the arbitrators but with the 2 week delay) to compensate them (and also allow them to compensate judges they may have delegates tasks to) for the ongoing costs they have regardless of whether any cases come up (and to motivate them to be available to do their job properly when a new case does come up). Finally, the arbitrators could (again with the 2 week delay) pay out a dividend from the reserve pool to the UIA holders. Ideally, I would like to see dividend support built in to the blockchain as an operation for any UIA. But short of that, the 2-week delayed operation might just move the funds from the reserve pool into another account collectively managed by the arbitrators without delays that would then be responsible for manually transferring those funds in proportion to a snapshot of the UIA holders taken on the block in which those dividend funds were first moved out from the reserve pool.
 One other thing the arbitrators might do (assuming the UIA holders allow) to help reduce squatting is to run an automated system on the side that does the following. It allows someone to submit a form requesting unauthorized registration of an already established domain name by providing the ICANN domain name of the same name (with some set of accepted TLDs). If the claimed name meets certain conditions (perhaps less than a certain length) and also the domain name was registered in the ICANN system (with one of the appropriate TLDs) before
the name was registered on the blockchain, it then moves on to the next stage. The next stage involves providing a random large number to the claimer and requesting that they put that number in its own file accessible from a specified URL using the ICANN domain name. After the claimer does this and provides the URL back to the automated system, the automated system checks that the numbers match and if so moves on the next stage. This next stage involves waiting for some period of time for any counter claims. People who own the same name with another higher-ranked TLD (the appropriate set of TLDs this system recognizes will be ranked) can go through a similar process for their counter claim to either maintain existing ownership on the blockchain or to take ownership of the name themselves if they don't have it already. If the period ends without any legitimate counter claims, the system automatically coordinates with all other arbitrators (each one could be running this automated system and the claimer would be using a client that submits this info to all of their servers) to sign a delayed transaction that automatically takes that name from the current owner and puts it into an auction. This allows the claimer to own the domain by paying the fair market price for the name (which could be cheaper than the fees they need to pay to try to take it to arbitration courts directly). It means that the existing owner of the domain cannot extort the claimer by demanding them to pay higher than the fair market price. For this reason, the existing owner will be willing to sell the name to the claimer (assuming they prove they actually own the ICANN domain) for much lower than the market price because at least then they get some money out of it. This reality will make squatting on existing domain names far less likely even for small companies that could not afford the court fees. And the squatters speculating on names that are currently not owned by anyone in the ICANN system would be paying for the fair market price (at the time they claim it) because of the auction. So if they make any profit at all from that name, it is because they had the foresight to claim a name that no one else was using at the time and that no one else thought would be as valuable as it ended up being. And that isn't evil squatting but rather legitimate speculation. Also, the existence of this mechanism doesn't cannibalize the revenue source for courts. Those are focused on high value names where the market price of them would likely be higher than the court fees (therefore it is rational for those name claimers to settle the dispute in court and get the name rather than using this automated system to then have to bid on the name). However, it is also important to consider that from a squatters perspective they know that a name can be taken from them by the legitimate owner for only the cost of the high court fees. Therefore, it is in their best interest to just sell the name to the legitimate owner for a lower cost than the court fees. This also means that it wouldn't be rational to pay more than the court fees in an auction for a name legitimately owned by someone else. Thus these mechanisms effectively put an upper bound (worth approximately the cost of the court fees) on the cost of any given name.