Chapter 12 Nices Guys Finish First

In this chapter Dawkins explains an experiment preformed by a mathematician. He set up a program where different plays would play a game of Prisoners challenge. Several people were invited to enter in players to see which, at the end of 100 rounds, would have the most points. Though several complex players were submitted one that was most successful was Tit for Tat which would COMPLY on the first round then copy the opponent. In fact, "Nice" players (ones that cooperated on the first round) were universally the most successful.
After this game the mathematician preformed another round. He warned the programmers that the most successful ones had been nice. He got many more submissions, 88 of them. Many programmers made nice ones because they had preformed the best in the first round, many made mean ones preparing to take advantage of the all the nice ones they were expecting to be submitted. The winner was Tit for Tat.
Then the mathematician changed up the rules. Instead of being awarded points the players were awarded offspring and they played for 100 generations. At first the mean ones nearly drove the nice ones to extinction, taking advantage of them. However once the means were in large numbers they ended up screwing themselves over, so to speak, and were pushed back by the nice. At the end the player with the most offspring was Tit for Tat. This is because even though the nice players can be hurt in numbers they will form a small bastion where they are able to be nice to each other and then able to spread back out and conquer the mean who are destroying themselves from within.

Chapter 11 Memes: The New Genes

I'll ignore this chapter since it doesn't fit in with the rest of the book. Essentially Dawkins claims memes replicate in minds as genes replicate in bodies.

Chapter 10 You Scratch My Back I'll Ride Yours

In this chapter Dawkins explains that in any case of individuals working together they must gain more then they put in. In herding all creatures are trying to minimize their zone of danger by flocking with other animals so that their companions are closer to areas where predators could be lurking then they are. Birds give warning calls, which seems to be altruistic, but its actually to increase their likely hood of survival. They want to run for cover but if they run by themselves they will easily be picked out and eaten. They must run for cover but assure that everyone else runs to so that they can be lost in the crowd.
Social insects are much more intensely organized and altruistic but that is because they have one reproducer that they're all protecting. Each individual worker is sterile and therefore uninterested in their own survival, only the good of the collective.
Survival machines do not group for the good of the group, but for the good of themselves. Even members of different species will cooperate if they have more to gain from the grouping then they give.

Chapter 9 Battle of the Sexes

Richard Dawkins theorizes that species originally differentiated into sexes when one group began to develop larger sex cells because the larger endowment of resources would be beneficial to the development off the offspring. This was ripe for the exploitation. Another group was able to develop smaller sex cells. This would benefit them because they could create more because they were smaller. This caused the large sex cells to get larger because they were getting less help from the other sex cells, eventually becoming the egg. The smaller sex cells evolved to become smaller and more mobile to reach the egg, becoming the sperm.
However this is not the only conflict of interests between sex. Neither party would hang around and invest resources in the child if they could get away with leaving the other partner to do it. Dawkins claims the male is more likely to do this because they began with a smaller investment. This is clearly true as we can tell by the number of single mothers in our society. The female could leave the male with the child but the danger of him abandoning it would be too great because it would be easier for him to impregnate another woman then to raise the child by himself. This would be detrimental to the woman because of the amount of resources already poured into the child.
So how does the woman get the man to hang around and raise the child? One strategy is to court the male for a long time before allowing him to mate in order to ensure that he's committed. They may also make him invest a certain amount of resources before being allowed to mate, therefor it would beneficial for him to raise the child then go and search for another mate because he knows he would be forced to invest again. An example of this is making the male build a nest (or buy dinner.)
Another strategy is to give up hope of the male helping to raise the child and simply picking out a male who will give the child beneficial genes. This will quickly become women simply picking out men who are attractive because they will then have children who are also attractive and therefore likely to have children of their own.

Chapter 8 Battle of the Generations

There is an inherent conflict between the generations of parent and child. The mother is interested in having the optimum number of off spring and the children are interested in receiving a majority of the resources so long as its not at to much of a detriment to their siblings. While a cuckoo will throw all other eggs out of the nest upon hatching the swallow will only throw off a couple. The swallow is related to eggs discarded so it would be damaging to its genes to throw them all out. Instead it only tosses a couple. This will help keep resources concentrated on itself and still have siblings to carry on genetic material.
Because of this conflict children will evolve to better at exploiting their parents. We definitely see this in human society. Many children are experts at manipulating their parents. Older siblings often try and divert resources from siblings to themselves.