A classic case in economics is explained very clearly in Mancur Olsen's Logic of Collective Action, which explains why group-based decisions are often at odds with the interests of the group as a whole.
Another is Arrow's Impossibility Theorem, which hods that there is no method for constructing social preferences from arbitrary individual preferences. In other words, there is no rule, majority voting or otherwise, for establishing social preferences from arbitrary individual preferences -- so long as you also require unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. In other words: democracy not only may not, but cannot deliver an optimum outcome.
There are many other such paradoxes and conflicts, many of which I'm becoming increasingly convinced undermine key and manadatory assumptions of a properly functioning free market economy.
What changes might fix these is of course a rather longer discussion....