People occasionally wonder why there are still so many Republicans running for president. We can make a simple model of the situation to understand this.
Any candidate who decides to drop out of the race will probably drop out in favour of some other candidate, throwing their support to that candidate. They can expect some kind of reward from that candidate in the event of success. A candidate X will decide to drop out if the expected value of staying in the race falls below the best offer they could get from another candidate.
The value of staying in the race is interesting; it consists of a fairly predictable, risk- or debt-like term which represents their chance of winning based on the current polls, plus an uncertainty or equity-like term which represents the residual value of just being in the race.
While X stays in the race, it’s possible that some other candidate will drop out in X’s favour. It’s also possible that some event will knock out a major candidate and transform the others’ fortunes. This being America, the classic example is a wild-eyed loser sneaking into a Trump rally with a pink AR-15. Public opinion drives the first term, while the other one is basically exogenous, with the important proviso that its value declines as we approach polling day – there is less and less time available for our wild-eyed loser to intervene. This implies that the chance of dropout is asymptotically increasing with elapsed time. It also, interestingly, implies that greater generalised uncertainty – more wild-eyed losers – predicts more candidates, and I think you can actually see this effect in world politics (compare, say, Italian and British political parties).
The potential offer is also interesting. Candidates with less support will be willing to bid higher, because they need additional votes more intensely. We would therefore expect to get the highest offer from the next candidate ahead of us in the race. This would suggest that we’d see a cascade of exits, as the weakest candidates exited first, until only Donald Trump and one non-Donald are left, providing only that non-Donalds all prefer non-Donald candidates to the Donald, and that they can make credible promises of reward.
This last point is where it gets complicated. The offer consists of two components – the reward candidate Y offers candidate X, and the probability that Y will deliver. This itself consists of two components, the probability that Y will be President and therefore in a position to make X Ambassador to the Netherlands or whatever, and the probability that Y will betray X.
It’s possible that social trust is different between the parties, even probable, but this risks letting prejudice cloud our thinking, so let’s assume that political cynicism is evenly distributed for the moment. Therefore, the second component is a constant representing the average level of political deceit.
Now, the risk-adjusted value of the reward will be higher the more likely Y is to be elected President. The face value of the reward will be higher the less likely Y is to become the candidate. If the gap between the leading non-Donald and the rest is large, the credibility effect will dominate; if it is small, the reward effect will dominate. But in the aggregate, the value of any reward offered will be greater the more likely the Party of X & Y is to win the general election, because the winning candidate is more likely to be in a position to deliver.
(Y could also drop out in their turn, and transfer the promise to X along with their vote to some candidate Z, but we can deal with this by pointing out that such promises are only likely to be weakly transitive, and further by thinking about the party’s chances rather than Y or Z’s.)
Compare the Democrats. Most people expect Hillary Clinton to be the candidate and to win. Not surprisingly, all the other potential candidates skipped, accepting either an implicit or explicit offer of office or support in the knowledge Clinton is likely to be able to deliver on it. (Bernie Sanders chose to run in order to make a point, rather like John McDonnell in 2007 or Jeremy Corbyn this year.)
Upshot: the less likely your party is to win, the more candidates will stay in the race to be the candidate, and more generalised chaos tends to cause more candidates. Paradoxical! Further, if there is differential social trust between parties, the more trustful party will tend to have fewer candidates in their race.