The thrango and the tiger

Imagine standing in the toiletries aisle, staring at a bottle of your favorite face wash. This shop has it for €11; you’re almost certain that another had it – same brand, same size – for €6 the week before. You feel some paralysis: do you drive across town for the cost savings, or just eat the difference?

It feels like chaos, but there is a story behind that €5 gap. It is a choreographed competitive dance, and understanding its steps turns out to say something important about innovation, market dominance, and the awkward limits of consumer protection.

For our research “Asymmetric Models of Sales” published in the American Economic Journal: Microeconomics, my co-author David P. Myatt and I analyze that choreography. Why do identical products carry wildly different price tags? Who decides when to run a sale, and who sits back and charges a high “regular” price? The answers emerge from a deceptively simple idea: firms are balancing two very different types of customer, and the interaction of costs, captive bases, and investment incentives produces a surprisingly structured – and sometimes brutal – market outcome.

Captives, shoppers, and the fundamental dilemma

Picture two kinds of buyers. There are the “captives”: people who, for reasons of habit, convenience, or geography, almost always buy from the same seller. They will pay a bit more rather than shop around – think of airport water or your usual neighborhood grocer. Then there are the “shoppers”: price-comparison hunters who check apps and comparison websites and buy from whoever posts the lowest price.

For a firm, this is a painful trade-off. Set a high price and you extract margin from captives, but you lose the shoppers. Set a low price and you win the shoppers, but you leave money on the table from captives who would have paid more. No single fixed price does both well, so firms “randomize”: sometimes posting high prices, sometimes low, producing the dispersion we observe every day.

The classic theoretical treatment – Hal Varian’s 1980 model of sales – captured this elegantly but assumed near-total symmetry across firms. A later development by Baye, Kovenock, and De Vries (1992) showed that, when captive-audience sizes differ, only the two firms with the fewest captives compete for shoppers: they “tango,” while everyone else stays off the dance floor and charges high prices. That is a neat story, but one that assumes the marginal costs of firms are all the same.

From “tango” to “thrango”

Our paper generalizes this picture by letting sellers’ costs differ too – a gap the literature had never fully closed for more than two firms. The central concept we introduce is “aggression.” A firm is aggressive if it is willing to set very low prices to capture shoppers. Aggression rises for two reasons. On the supply side, a lower marginal cost lets you survive lower prices without losing so much money. On the demand side, having few captives means you have less regular revenue to sacrifice when you discount.

A striking result emerges: in any equilibrium, there is typically a uniquely most-aggressive firm – think of it as the tiger. That tiger stays on the dance floor at all price levels, always willing to fight for shoppers. Its opponent, however, may change depending on the price range. At the bottom rung the tiger duels with one rival; higher up the price ladder, a different opponent may tag in. This partner-swapping – the tiger holding its position while competitors cycle in and out – is what we call a “thrango.” Other firms may dip into particular price bands from time to time, otherwise hanging back as “wallflowers” – quietly selling to captives and skipping the sale of the day.

That structure explains the seemingly wild spread of prices you observe on any supermarket shelf or online marketplace. The bottom rung is fiercely competitive; further up, different firms enter or retreat; some never bother. What looks like random dispersion can be understood as a sequence of rational, structured choices about who is willing to fight at which rung of the ladder. A wallflower is not passive by accident – it has simply calculated that the expected gain from joining the fight is less than the captive revenue it would sacrifice.

How the incumbent becomes the tiger

Here is where the story gets counterintuitive. Suppose firms can invest in process innovation: paying a large fixed cost today – buying robots, rewriting logistics, building superior software – to lower their marginal cost of production tomorrow. Intuition might suggest the scrappy newcomer, hungry for market share, would invest hardest. The theory reveals a different force entirely.

The firm that expects to serve the most customers – the one that will often win shoppers as well as keep its loyal base – gains far more from any reduction in marginal cost than a firm expecting only captive sales. A cost reduction is worth more when it applies to a larger volume of customers. This creates a “kink” in the innovation payoff: crossing into the most-aggressive position does not just improve outcomes marginally; it changes the entire scale of returns on every euro spent reducing costs.

So the sleepy incumbent, given the right technological conditions, can become the tiger. Its captive revenues finance the upfront investment that makes it the low-cost leader. That cost advantage draws shoppers, which justifies further investment. Even if firms begin completely symmetric, the theory predicts that exactly one innovates significantly more than the others, takes the tiger role, and entrenches its position. Asymmetry is not a starting condition; it is the equilibrium destination.

There is an efficiency twist that makes this paradox uncomfortable for policymakers. If the tiger genuinely has the lowest cost, shoppers benefit and resources are allocated well at the bottom of the market. But the overall picture can be distorted: the tiger tends to over-invest relative to the social optimum – it innovates as though it will serve all shoppers with certainty, even though in equilibrium it only sometimes does. Rivals that also step onto the dance floor, meanwhile, under-invest. These findings show how uneven innovation outlays can be understood through the lens of strategic interactions.

The law of monopoly pricing

What happens at the other extreme, when captive audiences nearly disappear and almost everyone compares prices? You might expect that if everyone compares prices, firms would simply undercut each other down to cost. In fact, the opposite tends to happen.

In the limit as captive audiences vanish, we show that one highly efficient (lowest-marginal-cost) firm anchors the low end, pricing at the second-lowest marginal cost and winning nearly all shopper sales. But the other firms do not follow it down. Unable to match that price profitably, they instead post high prices – often the monopoly price – hoping for the occasional inattentive buyer. The market bifurcates: a fiercely competitive low end and a quietly predatory high end. We call this the “law of monopoly pricing,” a result that upends the textbook intuition that near-perfect consumer information drives all prices to cost (Myatt and Ronayne 2025).

What this means in practice

For shoppers, the takeaway may be that there is method to (what may seem like) madness. A mid-range sale price can represent a firm calculating whether the expected gain from winning your business justifies sacrificing margin on the customers who would have paid the full price. When you see a sale, you might ask yourself whether you are looking at the tiger taking a breather, or a high-cost firm tentatively stepping onto the dance floor with an unusual discount.

For managers, the framework reframes the competitive significance of a large loyal customer base. Captive revenue is not just comfort; it is the financing mechanism for the investment that can make a firm the tiger. The strategic question is not only whether to invest in cost reduction, but whether your captive base is large enough – and your technological opportunity strong enough – to make that investment pay off at the scale required.

For policymakers, one takeaway is that asymmetric market outcomes need not signal anti-competitive behavior. The theory shows that even a perfectly symmetric starting field can produce a single dominant innovator as the natural equilibrium outcome. That does not make the outcome benign – the welfare analysis is clear that the wrong investment levels are made relative to the social optimum.