Will diners turn their nose up at no-tip policies?

In response to minimum wage increases set for many cities, restaurants are experimenting with no-tipping policies to manage rising labor costs, according to a report by the New York Times.

While new rules could bring sweeping changes to employees, New York Times Op-Ed columnist and former restaurant critic Frank Bruni said the "real X-factor" is how diners will respond.

"I think a lot of customers are deeply unsettled when they get into a restaurant and they find out they're not going to tip," Bruni said Tuesday on "CBS This Morning."

What's the right amount to tip?

He said it's very difficult to change decades of diner behavior that involves tipping.

"I think the average diner thinks tipping gives him or her some sort of power at the table and is going to be reluctant to give that up," Burni said.

For restaurant workers, new policies could bring much-needed change.

"There's been a revolt over recent years in what's called the 'back of the house' of a restaurant," Bruni said.

Dishwashers and line cooks are usually paid less because they often don't receive tips, although some restaurants on the West Coast, Bruni said, have begun experimenting with two lines of tips -- one for the servers and one for the kitchen staff.

U.S. states, cities raising minimum wage

But those in the front of the house are not treated equally across the board.

The Fair Labor Standards Act requires an employer to only pay $2.13 per hour if that amount, in addition to tips, equals the federal minimum wage. The minimum hourly wages for tipped employees varies state by state, however, and studies reveal disparities in tip amounts.

"There's been some studies showing there's discrimination in tipping. Blonde waiters get more than brunettes. There's a study that showed African American waiters get less," Bruni said.

Now, as states across the country increase their minimum wages, restaurants are working to level the pay playing field. Bruni said some restaurants are rolling the tip money into the overall price of the meal while others are adding a 20 percent administrative charge line.

"If you could just take those tip prices, put them into the rest of the menu in some way, then you as management have control over how that money is dispersed and you can even the wage gap between the front and back of house," he said.