The Republican Party's convention rules, drafted in 2012, can't be changed until the 2016 convention begins. In theory, they can be changed in any way the delegates like, as Gerard points out.
In practice, there are good reasons to think that the convention won't change them in ways that would benefit opponents to either Trump or Cruz, the two leading candidates.
That means that Trump and Cruz will likely be the only two names the delegates will ever get to vote on.
Why is this? After all, for weeks the pundits have been spinning out scenarios in which a host of other candidates emerge and fight things out on the convention floor. (John Boehner just endorsed Paul Ryan.)
The problem is that Trump and Cruz will use their muscle to prevent this from happening.
Rule 40(b) of the current rules says that no one can be placed on the ballot for nomination unless they have won a majority (not a plurality) of delegates in eight different states (or state equivalents, like the Northern Mariana Islands, recently won by Trump).
This rule was originally adopted by Romney supporters to head off the nomination of Ron Paul at the 2016 GOP convention in the 2016 reelection campaign of Mitt Romney. As you can see from that last sentence, things didn't turn out precisely as expected.
As of now, only Trump has gotten over Rule 40(b)'s hurdle. Cruz has won a majority of delegates in four state delegations, Kasich, only one.
Cruz still hopes that he can win a majority in four more states; Kasich is hoping for a rules change. So too are Mitt Romney, Paul Ryan, and almost everyone else in the party who wants to avoid nominating Donald Trump.
But if Trump enters the convention with a little less than 1237 delegates (a majority), and Cruz has something like 600 delegates or more, they will control three quarters or more of the delegates.
At that point they might support changing any number of convention rules. But they don't have an interest in changing Rule 40(b) in a way that would allow anyone else to be eligible to be nominated for president.
That's because they want to avoid the nomination of anyone whom the delegates might believe is a more attractive general election candidate.
This is obvious for Trump. It is also true of Cruz, for different reasons.
Assume once again that neither candidate has a majority of the delegates, but that Trump has close to a half and Cruz has a quarter or more.
Suppose that Cruz gets majorities in 8 jurisdictions. Then he wants only two nominations from the floor--himself and Trump. He hopes that, faced with the choice of The Donald or him, delegates on the first ballot or thereafter will swallow hard and choose him. (Note that Rule 40(b) is not limited to the first ballot).
Suppose that Cruz falls just short of majority support in 8 jurisdictions, but that he is far closer to satisfying Rule 40(b) than Kasich. Then we have a crucial divergence of interests among Trump's opponents. All of them-- controlling a little more than a majority of the delegates--want to amend Rule 40(b) and open up the nominations process. But they want this to differing degrees.
Cruz still wants as little competition as possible, and if he controls over a quarter of the delegates (that is, over half of the delegates needed to outvote Trump on a rules change on the convention floor), his views are likely to dominate. The last thing he would agree to is a rule that allows someone like Paul Ryan to be nominated from the convention floor. So he will likely only agree to changes in the rules that allow him to pass the new requirements but (almost) nobody else.
We will see if this logic holds at the GOP Convention.
UPDATE: Several readers have pointed out that the fact that Trump or Cruz have been assigned delegates who will vote for them on the first ballot does not mean that these delegates will support Trump or Cruz in rules changes when they come to the floor of the convention. These changes will be voted on before the first ballot. So it is theoretically possible that even if Trump and Cruz collectively control, say, 80 percent of the delegates who will vote on the first ballot, many of these delegates will desert them on rules votes. In theory, then, the two candidates could still lose a rules battle on the floor, opening up nominations to a flood of new candidates.
This is certainly true as a formal matter. But it overlooks the fact that Trump and Cruz also know this, and that they have been busily working at installing as many of their allies as possible as convention delegates, both in state conventions and in members of state delegations chosen separately from presidential preference primaries. (My understanding is that Cruz, in particular, has been particularly attentive to these features of the system.) Trump and Cruz are also making and will continue to make side deals with delegates in order to win their support in potential rules battles.
Moreover, convention delegates understand that flouting both Trump and Cruz and handing the nomination to a third party risks breaking the party apart. Trump has not been particularly subtle about this, noting the possibility of "riots" if he isn't nominated. Like a mafia don, he has been saying in effect: "Nice little political party you have there. It would be a shame if something were to happen to it." (In fact, I'm surprised that people haven't started calling him Mafia Don.)
Perhaps even more important, both Trump and Cruz (and Kasich) are using their influence to gain as many seats as possible on the all-important rules committee, which submits proposed rules changes to the convention as a whole. Supporters of open nominations have to win both in the rules committee and on the convention floor. Trump and Cruz only have to win in one location.
In order for Trump and Cruz to have a practical veto on rules changes that would open up the nomination process, it is not necessary that all of the delegates pledged to vote for them on the first ballot must be loyal to them in votes on convention rules. It is only necessary that, taken together, at least half of the total delegates are loyal to one or to the other of them. By June, I predict that that burden will be pretty easy for them to meet. Moreover, if Trump and Cruz together are able to gain significant support on the rules committee--and there is no reason to think that they aren't paying extra special attention to this--nothing damaging their interests will ever make it to the convention floor in the first place.
The collective action problem of the anti-Trump forces is still in play. Cruz wants to be the only person nominated on the floor other than Trump. So he has no reason to support anti-Trump forces who want more than this.
Again, we will see what happens when the convention opens.