Much of my time at present is consumed by setting up an experimental test of what is known as the grandmother hypothesis. The grandmother hypothesis, in short, is the best guess we have as to why the females of humans and a few other species can live well past the age of reproductive cessation. In most species, and indeed in human males, there is no significant post-reproductive lifespan. Individuals are physiologically capable of reproducing for as long as they live. But the females of humans, a few other primates, a couple of cetations and maybe elephants go through menopause and then can live a significant portion of their lifespan after that. Our closest relatives, the chimps and bonobos, apparently go through menopause at the same age as our females, but live at most a few years after that.
Evolutionarily, this makes sense. If one is no longer increasing one's lifetime reproductive success, staying alive offers no obvious selective advantage. No point in investing in physiologies and structures that will last 100 years if one is only going to reproduce for 50 years. Better to put those resources into having more kids now.
But under a certain set of circumstances, reproduction does not end with, or shortly after, childbirth. If your young aren't really able to take care of themselves for a decade or two, you aren't done reproducing until they don't need you any more. In most hunter gatherer societies, the survival rate of five year olds whose mothers die is quite low. So for human women, having a kid in the last several years of life was likely a waste of time.
Worse, the kid who didn't make it took time and resources that could have been put into other kids, and childbirth, particularly late in life, is dangerous. Plus, elder human females are important for helping their daughters raise their own young, and learn how to do so. It has been shown that young mothers in several societies have a higher success rate raising kids if their mothers are around. The women who stopped having kids and focussed on the kids and grandkids they already had, and avoided the risk of late life childbirth, are thought to have ended up getting more of their genes into future generations than women who kept giving birth as long as they lived. If so, and if this variation in life history was heritable, which seems likely, this differential reproductive success would inevitably lead to a population with more and more women stopping early and fewer and fewer giving birth late in life. This is, we think, why we ended up with this "grandmother effect" of women living well past reproductive age.
The benefit of having a grandmother around seems to be restricted to maternal grandmothers. And this observation, that paternal grandmothers don't seem to make as much of a difference (at least in the societies studied) to the survival of their grandkids, points to at least two possible reasons why we don't see a "grandfather effect" to go along with this "grandmother effect."
First, in most societies, at least those studied in this context, males are providing less in the way of vital care. So if a women has a son who has kids, perhaps she is less involved in care, or in teaching how to care, because her son is not as involved as his mate in that care, and the daughter-in-law is not nearly as likely to look for advice and help from her husband's mother than her own mother. And perhaps this same logic applies to grandfathers on both sides. If they are not who the primary caregiver can go to for help and advice, the advantage of having them around to help is smaller.
Second, paternal grandmothers are less certain of which is really their genetic grandchild. If a woman gives birth to a daughter, and watches that daughter give birth to babies, she can be very confident that those are her descendants. If a woman gives birth to a son, and then watches that husband's mate give birth, there is a significant chance (and we have the genetic data to substantiate this) that the baby was fathered by some other man, and those babies aren't her genetic kin. So investing in them heavily may not be doing her any good. This argument is doubly true for grandfathers. The daughter who is giving birth may not even be his. A couple of generations removed, and who can be sure?
A final reason males may not have evolved to have a post reproductive period comes back to that risk in late life childbirth. Men don't give birth, so the risk to late life survival posed by late life reproduction may be greatly reduced, or completely absent. Without that trade-off, why not keep on breeding as long as possible?