According to the evolutionary theory of sexual selection, what are three prerequisites for a trait to evolve by sperm competition?

The quick answer is that you need conditions where (1) females mate with multiple males, (2) different variants of the trait must result in different degrees of reproductive success, and (3) these variants must be heritable.

Let's unpack what all that means by going into more detail. First I'll define sperm competition, and then give examples of the sorts of traits that might evolve by sperm competition. 

Sperm competition is competition "between males to prevent each other from fertilizing eggs" (Krebs and Davies, below). It is a type of male-male competition, which, in turn, is a type of intra-sexual selection, i.e., sexual selection that consists of members of one sex competing for access to the other sex.

When we hear the words "sperm competition," we might think first of sperm battling each other in the reproductive tract of a female who has mated with multiple males in succession. We envision traits specific to the sperm themselves.

While sperm competition could, in principle, lead to the evolution of such traits -- like faster swimming speed (to beat the sperm of rival males to the egg) -- sperm competition can lead to the evolution of many other traits, including

• mate guarding, where a male hangs around the female to thwart her from mating with his rivals


• frequent copulation, to help ensure that the male enters more sperm in the "lottery" of fertilization


• increased sperm production, to help ensure more entries in the lottery, and


• anatomical characteristics that help a male remove a rival's sperm from a female's reproductive tract (for example, a penis with a flange that, during intercourse, helps remove previously-deposited sperm

To demonstrate that a trait is sexually selected, three conditions must be met.

1. There must be competition for mates. In the particular case of sperm competition, this means that females of the species must at least occasionally engage in polyandrous matings -- i.e., fertile females must mate with more than one male. 

2. Different variants of the trait must result in differences in reproductive success. In other words, you need to show that having one version tends to result in more offspring than another. An example here would be the trait of high sperm count per ejaculate. You'd need to show that males with higher sperm counts are more likely to father offspring.

3. These variants must be heritable -- i.e., variation in the trait must be related to variation in genes. Returning to our example about sperm count, you'd need to show that differences in sperm count between individuals reflects differences in their genes.

How important is sperm competition? In the old days, people assumed that polyandrous mating was relatively rare. Nowadays, we know differently, and we've learned that extra-pair matings occur even in species that pair-bond.

For instance, field workers have proven that female birds engage in multiple matings, even in species characterized by a social system of pair-bonding and bi-parental care. Paternity studies have found that the offspring weren't always sired by the male who feeds and fathers them: The mothers engaged in "sneaky copulations" when their pair-bonded mates were away, and, as a result, some of their eggs were fertilized by other males.

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References

Andersson, Malte. Sexual Selection. Princeton University Press.

Krebs, J.R. and Davies N.B. An introduction to behavioural ecology. Blackwell.