I have wondered if competition for mate among males and the race among sperm cells would result in healthy offspring,
why no such mechanisms exist among females and egg cells?
(Even females are genetically responsible for the offspring.)
As @Dexter said, there are examples of species where female compete for the access to the males but most often it is the other way around; males compete for the access to the females.
The reason for this discrepancy is mainly explained by Bateman's principle. Citing wikipedia:
Bateman's principle suggests that in most species, variability in reproductive success, or "reproductive variance," is greater in males than in females. This is ultimately a consequent of anisogamy. Females, especially mammalian females, almost always invest more energy into producing offspring than males invest. Bateman's principle anticipated and is consistent with Robert Trivers's theory of Parental investment—in most species females are a limiting factor over which males will compete. This competition results in some males being more successful than others, leading to greater reproductive variance among males than females
Actually there are lot of examples in which females are competing for mating which also have evolutionary consequences. It is also widely spread in nature. This competition may arise at cellular, molecular, genetic, behavioral level. Quoting from (Stockley and Bro-Jørgensen 2011)
female competition is associated with many diverse adaptations, from overtly aggressive behaviour, weaponry, and conspicuous sexual signals to subtle and often complex social behaviour involving olfactory signalling, alliance formation, altruism and spite, and even cases where individuals appear to inhibit their own reproduction.
Following table taken from (Gwynne 1991) which shows lot of examples in which females are competing
There are many more you can find by just searching on pubmed!
Males in monogamous species don't compete much for females. According to "The selfish gene", monogamy is an evolutionary stable strategy. They tend to only produce the number of kids they can get enough food to rear because those that produce more kids almost certainly can't rear all of them and maybe even 2 of them will starve because of the extra energy of producing the extra kid and of the kids fighting with each other for food. If any of them fight for more than their fair share of food, those who fight back those who fight for more than their fair share will be selected for because there's nothing to lose in fighting back for more than their fair share let alone their fair share. After that, those that start a fight for more than their fair share will be selected against because of the possibility of losing the fight and getting even less food, which is why they don't start a fight for more than their fair share of food in the first. At an evolutionary stable population size, each individual can on average get enough food to rear 2 offspring. It's an evolutionary advantage for a female to pick a male that doesn't already have another mate because if she picks a male who already has another mate, there will be 3 parents so they can get enough food to rear 3 offspring but each one will only have a 50% chance of being her own, making it pretty impossible for a male to get a second mate. Since there are about the same number of males as females, almost all males get one mate with almost no left over males or females. Males who fight against any male who tries to take their mate away from them are selected for which in turn causes males of a monogamous species who fight another male for a second mate to be selected against.
According to https://www.dailymotion.com/video/x1e19qg_bonobo-ape-our-closest-relative-nature-documentary_tech, bonobos are also peaceful. All the members of a group have sex with each other all the time and no male knows which baby is his removing the advantage in males fighting. They're probably also peaceful for a similar reason, that those that fight back to not get less than their fair share of food are selected for which in turn causes those that fight to get more than their fair share to be selected against.