As was true of his earlier work on social choice, the magnitude of Professor Arrow’s theoretical insight was staggering. But, he made clear, his powerful conclusions about the workings of competitive markets held true only under ideal — that is to say, unrealistic — assumptions.His assumptions, for example, ruled out the existence of third-party effects: The sale of a product by Harry to Joe was assumed not to affect the well-being of Sally — an assumption routinely violated in the real world by, for example, the sale of products that harm the environment.

Take “learning by doing,” a notion that Professor Arrow examined in the early 1960s. The basic idea is straightforward: The more that a company produces, the smarter it gets. Decades later, economists incorporated this idea into sophisticated theories of “endogenous growth,” which have a country’s rate of economic growth depending on internal policies that promote innovation and education — the very forces that Professor Arrow’s writings anticipated.

in the early 1960s, he teased apart the complexities that asymmetric information creates in the market for health insurance. He pointed to incentives for patients and their physicians to agree to medical procedures of questionable value when a third party, the insurer, pays the bills.

Professor Arrow proved that their system of equations mathematically cohere: Prices exist that bring all markets into simultaneous equilibrium (whereby every item produced at the equilibrium price would be voluntarily purchased). And market competition puts society’s resources to good use: Competitive markets are efficient, in the language of economists.Professor Arrow’s theorems set out the precise conditions under which Adam Smith’s famous conjecture in “The Wealth of Nations” holds true: that the “invisible hand” of market competition among self-serving individuals serves society well.

What Professor Arrow proved in his book “Social Choice and Individual Values” (1951) was far more sweeping. Not only would majority-voting rules prove unsatisfactory; so, too, would nonvoting systems of making social choices if, as was fundamental to his way of thinking, those choices were based on the preferences of the individuals making up the society.

majority voting can produce arbitrary outcomes.