Candidates S and R are competing in the opposing party’s primary, and your candidate awaits the winner in the general election. Your candidate beats S in the general election with probability s and beats R in the general election with probability r<s.  You would like S to win the primary since s>r. But S is currently the underdog, he beats R in the primary with only probability p. Should you spend money to help S?

Every percentage point you can add to S’s chance of winning the primary increases your candidate’s odds in the general election by s-r < 1.

(you win the general election with total probability ps + (1-p) r. an increase in p by one unit increases this probability by s-r.)

If you save your money for the general election, every percentage point you add to your own chance of winning raises your own chance of winning by, well, 1 percentage point.

1. The same analysis goes through with any number of candidates in the primary.  So you can add G and P and it won’t change anything.
2. This is about the marginal value of influencing a 1 percent change in the election probabilities.  That value is larger in the general election.  But there may be differences in the marginal cost of influencing a primary versus the general.
3. In particular, if the primary is a three-candidate race there may be a lumpy return on your investment.  For example, if you increase s by a little bit that could cause G to drop out of the race and then hope that a big chunk of his probability of winning goes into increasing p.
4. However, with G currently at about 3% probability at Intrade, at most you can get 3 times s-r.  For this to outweigh the 1 you get from the general it must be that s-r > 33%