After a pleasant dinner at a pricey restaurant, I was presented with an unusually informative bill. It displayed neatly printed calculations of what the tip would be at the 15, 20 or 25 percent level. How considerate!

Curious, I asked the waiter whether these informationally enhanced bills actually made any difference. He looked at me as if I had lost my mind: “All of our tips have gone up. People now know when they’re stiffing us.”

Clever. A tiny dose of cheap information significantly boosts the average tip. The value of this simple computational innovation is extraordinarily high. The waitstaff and the restaurant owners are obviously happier. And the customers don’t appear to be unhappy even though they are, on average, paying more than they used to. (To be sure, the practice would have been swiftly discontinued if tips had declined.) At first glance, this seems to be a win-win info-innovation.

Then again, it also highlights a conflict for innovators struggling to strike a healthy balance between business profitability and customer sensitivity. Do innovations succeed because they make customers smarter or because they better exploit customer ignorance? To put the question more provocatively: do companies make more money from smart customers or stupid ones?

Financial-services firms, professional-services providers and health-care companies are constantly tormented by the trade-offs posed by taking advantage of customer ignorance. Introducing innovations that educate may be the dumbest thing a firm can do. Information becomes the enemy. It’s one reason credit card companies, for example, resist legislation requiring them to disclose how long it would take cardholders to pay off their balances if only monthly minimums were paid. Concealing that information is more lucrative than revealing it.

Likewise, pharmaceutical companies are growing more concerned with compliance-ensuring that people take the right doses of their medication at the right time. For both regulatory and legal reasons, drug firms need to disclose the risks associated with improperly taking their medicines. Is delivering better information the best investment in compliance? Or as consumers become better educated about the drugs they’re taking, do they switch to less expensive generic versions? Billions of dollars in health-care budgets depend on the answers.

Perhaps this recalls George Bernard Shaw’s cynical epigram that “All professions are conspiracies against the laity.” The conundrum here is that some innovations become more valuable the more customers know, while other innovations become worthless as customer IQ-Innovation Quotient-rises.

Most dangerous, innovators often don’t know in advance whether keeping customers ignorant is more profitable than investing in educating them. Are pharmaceutical companies better off providing doctors with everything they need to know about a new therapeutic or providing just enough information to treat people safely? Proprietary innovators have tough choices.

For commodity offerings, however, the answer is clear: smart customers consistently pay lower prices. That’s why all the large accountants and auditors launched consulting businesses. They had to offer value-added services to complement their commodities. That can create all kinds of conflicts.

But smarter customers also create smarter opportunities. Design, manufacturing and process engineers are always on the lookout for computer-aided-design tools that facilitate cost-effective creativity and integration. The more an engineer knows about using a tool, the more valuable that tool becomes.

Online investment companies such as Charles Schwab and E*Trade are the logical, digital extensions of a business bet that smarter customers are more ready, willing and able to pay a premium for an investment innovation. The virtual financial-services firms have economic incentives to provide information about innovative trading strategies so that customers can take better advantage of their improved ability to execute timely trades in the equities, options and derivatives markets.

As a general rule, however, ignorance can be profitable bliss for innovators who know far more than their customers do. Indeed, the role such “information asymmetries” play in distorting markets won the 2001 Nobel Prize in economics for Joseph Stiglitz, George Akerlof and Michael Spence. Their work confirms that when sellers know far more than buyers, unusual opportunities for great profits can arise.

This is especially true in the technology marketplace, where successful marketers may feel the incentive to confuse and complicate rather than to inform and educate. They aspire to asymmetry. Precisely because a Microsoft or an IBM may make more money from its “stupid” customers than from smart ones, it may have decidedly mixed feelings about how clever it wants its customers to become. Creating fear, uncertainty and doubt makes economic sense. That’s why they do it.

Ultimately, the real leaders in innovation don’t just educate their best customers, they learn how to collaborate with them. They collectively play with possibilities. They understand that innovation is about coevolution. But only the stupidest and most arrogant innovators believe that “smartest customer” equals “most profitable customer.” There is a reason why P. T. Barnum made his money, even if he never did say “there’s a sucker born every minute.” Check please.