There are many ways in which the world championship can be changed, and in fact has already been changed over the past 132 years since the first official championship in 1886. Most of the time it has been contested in a match, but the rules have varied greatly. For starters, some have been first to n wins, and some have been best-of-n games, and n has varied in each case. I think, but might be mistaken, that the two subtypes have been combined before: first to n wins, but with a cap on the total number of games. (Which was the case in the Candidates final between Anatoly Karpov and Viktor Korchnoi in 1974 that wound up as the de facto world championship match, but I think it happened in at least one official match as well.)
There's also the question of tied matches. Many world championship matches have finished their official course in a tie, with some ending there leaving the champion as champion, while others have gone on to a playoff. And there too there have been differences. The 1892 match between Wilhelm Steinitz and Mikhail Chigorin continued with classical games played at the same time control (though its rules were different enough that the parallel is imperfect), but recent playoffs have gone to rapid games, and had they been tied would have continued with blitz. This may be an improvement on giving the champion draw odds (indeed, this is a big plus in my opinion), but it does raise a question: does the winner of such a match deserve the title of classical world champion? If not, what exactly is the nature of the winner's championship title?
Another format is a round-robin event; this was used in 1948 after Alexander Alekhine's death in 1946 left the champion's title vacant, and then again in 2007 as part of the reunification process bringing the Garry Kasparov-Vladimir Kramnik legacy portion of the title back under the auspices of FIDE. Still another format is the knockout tournament, used five times by FIDE during the divided era to determine a champion. Those too would mix time controls, and indeed the first final in that format required a rapid playoff between Anatoly Karpov and Viswanathan Anand to determine a champion. (Karpov won that playoff 2-0.)
Many questions and proposals can come to mind, but I want to focus on two. First, what does it mean to say that someone is the world champion? Second, how can one arrange for this particular championship title to be fairly and appropriately contested? Let's consider each in turn.
What does it mean, or should it mean, to be the world champion? Does it mean that the person is the strongest player in all forms of chess? If so, why is there a world correspondence championship, a world rapid championship, and a world blitz championship? No doubt a world bullet championship would find lots of interested parties as well. There's little doubt that Magnus Carlsen would be great at all of these other disciplines (and that he is is already well-known in the case of rapid and blitz, where he has won world championships, and in bullet too he has proven to be a beast in online competitions), but those titles are kept distinct anyway. "The" world championship really seems to be the classical world championship. This is fine--but then why is it being settled with rapid games?
So to turn to the second question, I'll propose a series of ways this situation can be resolved, noting difficulties with each. (That there are difficulties doesn't mean that any option is fatally flawed, only that it has its own distinctive problems.)
1. Co-Champions!
There are competitions that allow for this; why not the world championship? If the match comes to its natural terminus without a winner, then instead of arbitrarily allowing the champion to keep the title or switching to a non-classical time control, declare both players champion.
Problems: It's unlikely to be a fan favorite - we tend to want winners and losers. For many chess players, draws are a bane even when they are rare, hard-fought, exciting, well-played and full of content. Calling the world championship match a draw will be even worse in their eyes. Nor will the outside media think much of it, and so it's bad for publicizing the game. Another difficulty is that it's ahistorical: we haven't done things this way in the past. We like having the totemic figure of a single world chess champion ruling over the chess world. Still another problem, even more serious than the foregoing: what happens in the next cycle? Will there be a three player match-tournament, and if they finish in a tie a four-way event, a five-way event, and so on ad infinitum? Or will the co-champions get thrown back into the pool?
About this latter idea: while the suggestion of co-champs is unlikely to find many if any takers (well, maybe Mikhail Chigorin, Karl Schlechter, David Bronstein, Vasily Smyslov, Anatoly Karpov, Peter Leko, Boris Gelfand, Sergey Karjakin, and Fabiano Caruana would like it, if retroactively applied), the reductio ad absurdum of bigger and bigger world championship match-tournaments could be mitigated if the title is no longer determined by head-to-head matches but round-robins, knockout events, or even a sort of grand prix system. More on that later.
2. The Champion Keeps The Title
This is entirely arbitrary and has nothing to do with chess skill. It may reflect nothing more than the greater age of the incumbent champion. Suppose players X and Y follow the exact same trajectory in their careers, but player X is older and thus peaked first. Upon reaching the age of X's peak, Y hits that same peak, and both players remain alone on that plateau for years to come. How would it make any sense for X to be the world champion that whole time despite never being stronger than Y, and never defeating Y? There is no chess-based justice to incumbency for its own sake. While the co-championship idea is the least practical solution, this one seems the most morally problematic. (Incidentally, I put my money where my mouth is on this one. I won a king-of-the-hill competition, toppling an incumbent who had draw odds in our match, and immediately renounced that privilege for any future title defenses as well as one other built-in advantage held by the reigning champion. The prestige of the champion's title, and the fact that there's no need to qualify for the next championship event, are enough. The aim shouldn't be to turn a championship title into a tenure track position.)
3. Unlimited Matches!
This may have been a good idea a long time ago, when top players were weaker than they are today and, more to the point, knew an awful lot less. Nowadays we might all die of old age before such a match finished, if the players don't die of exhaustion first. There are also logistical problems with venues - do organizers want to commit to a venue for six months? That's how long the first, unfinished match between Karpov and Kasparov went before it was abandoned, replaced by a 24-game rematch.
4. Best-of-X Games, Followed by Two-Game Mini-Matches
Carlsen and Caruana are tied after 12 games? No problem: they play two more classical games. Still tied? Two more games, and so on. This is a version of the previous solution, but a much more practicable one. The earth may be swallowed by the sun before one of them managed to win six games, but the first to win one? That's doable. There could still be a logistics issue, but it's a lot easier to see this approach resulting in a winner in a manageable time frame. It's not guaranteed, but it's reasonable. By the way, I think this was the policy in the world championship match between Steinitz and Chigorin back in 1892, so it wouldn't even be a novelty.
5. Other Classical Formats.
I'll mention three options here. First, a tournament (like Mexico City in 2007); second, a knockout event (like the ones FIDE used during the split era, and which live on as the World Cup); finally, a Grand Prix system (a series of classical tournaments followed by a final with fewer players, whether just two or a greater number).
5a. Regarding the first, the strength is that more players are involved, and the world champion's advantages as the incumbent are greatly reduced. It's nice for the fans, too, as there are likely to be more exciting games than in a one-on-one match, and more openings will be tested. A drawback is that this becomes just one more tournament. We've already got the Grand Chess Tour and other round-robins for the super-elite; this would only be one more event of a very familiar kind. It also undermines the glamour of the world championship title, for what that's worth to us as chess fans and to the outside world when it comes to attracting interest from the mass media.
5b. Regarding the second, this really makes for an exciting and dramatic event. There will be tons of games, so until the very end chess fans will have loads of games to look at and loads of interesting openings to examine. But it also devalues the title by turning the event into a semi-random crap shoot. The event is almost certain to be won by an elite player, but it's also almost certain to see many elite players flame out very early on. (Remember Carlsen's early exit from the last World Cup, in round 3, at the hands of Bu Xiangzhi.)
5c. The third system would be new - or at least new to chess - as a way to determine the champion. This method has the virtues of 5a and 5b while avoiding their vices. A fair number of players will be involved up until the final event, which makes for more games and more spectator interest, and because the Grand Prix system involves multiple preliminary tournaments before the final it avoids the randomness of the World Cup-style event. And because it ends in a small final - four players, or maybe just two - the final maintains at least something of its traditional gravitas. (If it is a final four, is it all-play-all or a knockout to reach the final two?) There are still some questions: will the final be long enough? And what if it finishes in a tie? There's something to be said for this method, but it may not be a full answer in itself when it comes to the problem that motivated this post.
6. A Broader Format: The All-Around World Championship.
That label is not quite correct, as correspondence and (perhaps?) bullet won't be included, but as the people behind the Grand Chess Tour have introduced a universal rating system that incorporates one's classical, rapid, and blitz results in some weighted fashion, perhaps it's time to conduct a world championship that explicitly and intentionally turns the event into a kind of triathlon. All sorts of options are available for choosing the number of games and how to weight them, but I'll offer the following to get ball rolling.
Classical games: 12 (as now), with a 4x weight. (In other words, they are scored on a 4-2-0 system.)
Rapid games: 12, with a 2x weight (i.e. a 2-1-0 system).
Blitz games: 24, weighted normally (i.e. on the traditional 1-½-0 system).
To sum up: the blitz games are each worth half of what rapid games are worth, but because there are twice as many that segment counts for as much as the rapid segment, and the blitz and rapid segments together are worth as much as the classical portion: 48 points are up for grabs in each.
The strengths of this system are that it takes all three time controls into account, and since rapid and blitz events are common these days, no longer limited to training games and mere recreation, skill at each will be rewarded and considered part of what it is to be a total player. It will be more exciting for the fans and the general media, at least once the faster games are underway. It also reduces the pressure for any one particular game while creating a greater sense of urgency for the player who is comparatively weak at one of the disciplines.
There are drawbacks, but most are easily managed. For instance, one might complain that my version will take too long: the 2½-3 weeks for the classical games, and then another week or so for the rapid, and then a couple more days for the blitz - not counting rest days. Fair enough; the event could be shortened a little, or we might just say that it's a world championship and you're playing for a million dollar prize fund. Tough it out!
Another objection: All these proposals, including this radical one, have been offered because the specter of a tied match is such a terror. Well, this format could also result in a tie, though it's admittedly rather unlikely. But since it is possible, how is this a good solution? In reply I would first emphasize the unlikeliness of a tied finish, especially compared to what we have at present. Still, the tie is possible, so what then? Here I think an Armageddon game would make sense--but I'm not sure what time control would be best for it. I'm inclined to think that something like the "bid Armageddon" game used in some U.S. Championships a few years back could work nicely. Those were rapid games, and since that's the intermediate time control it seems fairer than a blitz Armageddon battle. The way the bid version works is that both players say how much - or rather, how little - time they're willing to take with Black, together with draw odds, against White's full hour (or whatever amount is decided upon). The low bidder then gets Black, draw odds, and exactly the amount of time on the bid.
Still another objection: This is too radical. The current system also gives some, limited weight to one's rapid and blitz abilities, but the proposal here weights it far more heavily. Indeed, even if, say, Caruana had won the classical portion of the match by a game or two, it's very possible, given the disparity in their rapid ratings and especially their blitz ratings that Carlsen would have come back in the second and third parts of the match and still managed to win going away. I'm ambivalent about this criticism. On the one hand, I'm sympathetic to it; on the other, I'm inclined to think that if Carlsen (or someone else someday) is so much better than his peers in shorter time control games then that's just how it is. His challengers will have to step up their game. Still, one can argue about how much weight should be given to each time control; my suggestion above was meant only to get the ball rolling.
Let's bring this long post to an end. I trust you'll have some feedback on all of this, and will have thought of (or know of) some alternative systems of your own that might be even as good or even better. There may well be other games, sports, and other game theory scenarios that face similar problems, and better solutions are already known but haven't been implemented or even considered for chess. So have at it, especially you mathematician types out there!
Tl;dr version: If the world championship match is really the classical world championship match, it shouldn't be settled by rapid and blitz games. But it should be settled in some way. I proposed a wide variety of options, and the three I found most interesting and palatable were these: (1) A Conservative Approach: Two-game classical mini-matches after the tie until there's a winner; (2) A Grand Prix Format with preliminary tournaments followed by a final four and/or a final two (possibly needing to incorporate (1) in case of a tie); (3) change the world championship to an all-around world championship with classical, rapid, and blitz segments, with a rapid bid-Armageddon game in the very unlikely event that a playoff is needed.