To understand the anxiety inside American Express — which has announced its second consecutive year of declining revenue — you need to know about a story that made its way through the company’s headquarters last fall.

A few Amex executives heard that a rival company had enticed a group of young, affluent professionals — the kind of millennials whom credit card companies salivate over these days — to a fancy meal. At the evening’s conclusion, everyone was asked which of the many credit cards in their wallets they might consider using to pay.

“I don’t think it would be American Express,” one diner said. “I feel like that would be braggy, like I’m trying to prove I’m a big shot.”

Others nodded in agreement. “I’d probably use this,” said another, pulling out a blue-tinged credit card. “An Amex says you’re rich, but this says you’re interesting.”

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That blue card, executives at American Express knew, was the Chase Sapphire Reserve, a wedge of plastic and metal that was already giving the Amex brass night terrors. In August, Chase released the Sapphire Reserve with an audacious sign-up bonus of 100,000 points and a smorgasbord of cardholder benefits, setting off an avalanche of applications from millennials and a flood of social media postings exalting the awesomeness of Chase.

A credit card, mystifyingly, had suddenly become cool. And what really got under Amex executives’ skin was that Chase was succeeding by, essentially, copying the American Express playbook and chasing the same up-and-coming elites who had traditionally joined Amex’s ranks.

A few months later, one of American Express’s top executives confessed at a conference that “I am deeply paranoid about these types of competitive assaults on our customer base.” Then, the company announced a slew of new cardholder perks that seemed suspiciously timed to beat back Chase’s enticements.

The paranoia flowed even deeper within American Express’s offices, according to seven current and former executives at the company who shared their insights on the condition of anonymity because they were fairly certain they would get fired if they were named.

It was one thing for the company to fight a rewards war by offering ever-sweeter travel points and bonuses to attract new cardholders. Amex had waged such battles plenty of times.

This challenge, however, seemed different. Inside Amex’s plushly carpeted New York offices and between rounds of squash, the company’s leaders began asking one another: Could it be that American Express, the card that had defined ostentatious luxury and capitalist striving since the 1980s, was on the brink of becoming passé? What kinds of hoops would Amex need to jump through to attract these new hoodie-wearing moguls and young tycoons?

Was it possible — and this really gave everyone the sweats — that millennials would never be convinced that income inequality was something they should aspire to?

For more than 30 years American Express has reaped enormous profits by telling its customers that they are successful, elite, the cream of the moneyed crop — and, with mottos like “Membership Has Its Privileges,” that there’s no better way to make certain everyone knows just how special you are than by pulling an Amex out of your wallet.

And for decades, that pitch worked marvelously. American Express has become a financial behemoth with more than 109 million cardholders around the world. Nearly every year — at least until recently — more and more people paid American Express up to $7,500 for the privilege of carrying cards that are very similar to the ones Visa and MasterCard give away free.

But in the last few years, Amex’s hold on our affluent fantasies has started to wobble. Last year, for instance, the number of American Express cards in use declined by almost 18 percent, according to industry analysts. Amex’s 2016 revenue was down more than $2 billion from two years before. And the company’s relationships with Costco and JetBlue — long big sources of new cardholders — summarily ended when those firms found alternative credit card partners.

One of the big reasons for those stumbles is that other financial companies, most notably Chase and Citibank, have started beating Amex at its own game, often by hiring the same executives who built Amex. The head of Citibank’s credit card division as well as the heads of its branded cards, global rewards, customer acquisition, proprietary products and analytics all came from American Express. The woman credited with creating Chase’s Sapphire Reserve is an Amex alum, as are her boss and two top colleagues.

But there is also the problem of simple demographics. Amex, like all credit card companies, desperately needs new customers each year, preferably ones with decades of spending ahead of them. Hence the focus on millennials.

“Traditionally, American Express succeeds because of its customer service,” said David Robertson, publisher of the industry newsletter The Nilson Report. “They can book travel for you, they have concierges to recommend the best restaurants. If you leave your reading glasses inside a hotel room in Budapest, Amex will get them mailed back to you. No one else does that.”

Millennials, however, don’t really need travel agents or concierges: They have Priceline and Yelp. Nor are they traditionally fans of opera, ballet, Dom Pérignon tastings, or the other high-culture events Amex can get cardholders into. If you leave your reading glasses in Budapest, it’s probably faster to order new ones from Warby Parker, which is pretty cheap to begin with.

This is the real reason Amex’s executives became so worried when they heard about the dinner, one of multiple held by Chase, when a cardholder said his Sapphire card made him more interesting. American Express, for decades, has essentially sold snob appeal. But for millennials, snobbery isn’t quite as appealing as it once was. Or, more precisely, snobbery has to be hidden and camouflaged as something else, like an Instagram post from your Iceland spelunking adventure or a lament about how hard it is to find a charging station near Burning Man for your electric sports car.

The Chase Sapphire Reserve — which in the seven months since it was released has signed up more than a million cardholders, half of whom are under 35 — is all about emphasizing what cardholders can do, rather than what they can buy.

“The message we send is, this isn’t your father’s credit card,” said Pam Codispoti, who created the Chase Sapphire Reserve after 18 years at Amex. “For millennials, travel might mean taking an Uber to a hole-in-the-wall restaurant in Chinatown, and then riding the subway to karaoke, and then catching a taxi home. So we’re going to give you accelerated travel points on all that. This is a card for accumulating experiences.”

Amex, it should be noted, is fighting back valiantly. The company has increased sign-up bonuses and spending rewards, and now gives cardholders an annual $200 credit with Uber. The firm is quick to note that more than a third of its new cardholders last year were millennials. Soon, the Amex Platinum will look sleeker, less flashy. Amex’s commercials, which once featured businessmen (but no businesswomen) rushing from the boardroom to a first-class flight, now present Tina Fey teaching viewers how to meet their boyfriend’s parents.

But it’s an uphill battle for Amex, in part because the company is fighting an image of conspicuous consumption it has cultivated for decades. Amex sold a dream about what success looks like, a universe where handsome men and glittering women murmur about second homes over decanters filled with expensive booze. The company declined to put executives on the phone for this column, but it did send an email explaining that cardholder benefits included “premium access to the Grand Prix de Monaco, a wine enthusiast’s dream trip through French wine country and a global fashion tour with Harper’s Bazaar” — not exactly the hipster’s bucket list.

The other challenge is that many people who work at American Express aren’t all that millennially minded themselves. If you visit Amex’s headquarters in Lower Manhattan, you’ll find squared-jawed men in bespoke suits and fashion model-glamorous women, but not a lot of young people in the uppermost ranks. The company’s chief executive, Kenneth I. Chenault, has led Amex for almost 16 years, an eternity in corporate America. In one Amex brainstorming session, according to an executive I spoke with, participants spent 10 minutes trying to figure out what FOMO meant before turning to Google.

They discovered it stands for “fear of missing out.” It is unclear if the group recognized the irony.

That said, there’s every reason to believe that American Express will eventually solve this puzzle. Amex, it should be remembered, created a world where leaving your glasses in a Budapest hotel seems like a realistic concern — even to its many customers (myself among them) who have trouble finding Budapest on a map. A few years from now, who is to say that your Amex won’t get you into a foam party at the Electric Daisy Carnival as easily as a cocktail party at the Met?

And until then, the real winner of all these battles is the cardholders. The long dark winter of the financial crisis hangover is, finally, being made glorious summer by the tussles between Amex and its competitors. You might think millennials are spiritually poorer for substituting the dream of the Grand Prix de Monaco with the reality of a subway ride to the karaoke bar. But, don’t feel too bad for them.

In the meantime, they — and, by extension, all of us — are almost getting paid to spend.