No, they don't. That's completely against what Frans de Waal has to say.
“Alpha male chimps also have an obligation to keep the peace in the group. They try to prevent fights from breaking out, and typically support the underdog in any conflict, even if that means not favoring a relative or friend. This provides security for the lowest-ranking members of the group, which helps the alpha’s popularity. Alpha male chimps are also the ones who demonstrate the most empathic behavior, spending a lot of time comforting others. As de Waal characterizes them, alphas are the “consoler in chief.” In general, females demonstrate more of this sort of behavior, but the alpha male has been shown to be the top comforter of others in the troop.
De Waal says that males who become very good at these two things become extremely popular, which helps them to stabilize their position as alpha. What it takes to be an alpha chimp is good, fair, empathetic leadership. Conversely, bullies are not liked or usually tolerated, and if they have managed to gain position are apt to lose it through this sort of behavior — completely unlike what is purported by Pick-Up Artists and other proponents of the alpha male idea. Alpha females also hold very important, central positions within the community, and this is also quite different from human alpha male believers, who generally hold rather low opinions of women.”
Humans do much more food sharing and cooperation than any other primate species, but I think you kind of oversold the “high ranking animals act selfishly” aspect of this.
