Betrayal is a painful topic. In a game theoretic sense, we assume that all players are rational, which means they always out for their own best interest. With that in mind, betrayal is almost inevitable! The real question then is "why does anyone ever cooperate?"
When two players cooperate, they are often losing out on personal gain in favor of team gain. Possibly because the team has come to some kind of agreement to all stand by each other. But unless this agreement is binding, any player can back out at any time. We don't usually keep legal contracts for "we'll take turns taking out the trash," but we do for more important things where we need all players to stick with the team.
Consider a game of Risk, where I take North America, you take South America, and we agree to only keep single armies on our border in Central America. This is a very common occurrence, but in the time it took you to read this paragraph, you couldn't help thinking about how you'd happily drop a bunch of troops onto Central America and storm through my Northern Empire to beat me at Risk. We can't help it. As long as we're looking out for ourselves, betrayal is in our nature.
Betrayal in games is inevitable as soon as any player has more to gain by fighting against a team than standing with it. Sadly, being inevitable doesn't make it hurt any less.
Episodes featuring Betrayal