''Diamond Paradox'' by Diamond (1971)
This is a "less-known paradox," usually put as a counter to famous Bertrand paradox. It is a starting point in the literature on informational frictions in consumer markets, and the scientists in the field agree on its significance.
Its idea is diametrically opposite to that of Bertrand. Consider the following simple example. There are $2$ firms which produce homogeneous goods at zero marginal cost and compete in prices, $p$. This simultaneously set prices. Also there is a single consumer who supplies a demand given by $1-p$. Importantly, the consumer does not observe prices set by firms and, therefore, needs to search for them sequentially, where search is costly. Suppose that cost of visiting a firm is given by $0 < c \leq \frac{1}{2}$. Then, the unique equilibrium of the market is that both firms charge monopoly price
$$p^M= \frac{1}{2}.$$
This is a diametrically opposite result to that of Bertrand.
The reasoning behind the result is as follows. Suppose both firms charge $p=0$. Then, the consumer randomly visits one of the firms, say firm $i$, and buys. However, firm $i$ could have charged $c$ and made positive profits as the consumer would have bought goods anyway because she would have suffered cost $c$ had she left firm $i$ in order to buy from the rival firm. By the same argument, one can see that $p=c$ cannot be an equilibrium as now firm $i$ can charge $c+c$ and improve its profit. Continuing this way, it is easy to arrive to an equilibrium where both firms charge $p^M$. A firm does not want to charge $p^M+c$ simply because its profit is maximized at $p^M$.
Formal Analysis of the Example
Timing: First, the firms simultaneously set prices. Second, the consumer without knowing prices engage into sequential search. The first search is free and the consumer visit each firm with equal probability. The consumer can come back to the previously searched firm for free. The consumer has to observe a price of a firm to buy goods from that firm.
Beliefs: In equilibrium, the consumer has correct belief about strategies of firms. If, upon visiting a firm, she observes a price different from an equilibrium one, the consumers assumes that the rival firm has deviated to the same price too. Thus, the consumer has symmetric (out-of-equilibrium beliefs). Note: the results of the game does not change if the consumers has passive beliefs.
Strategies: Strategies of the firms are prices. As mixing is allowed, let $F(p)$ represent the probability that a firm charges a price no greater than $p$. Strategy of the consumer is whether to search for the second price, upon observing the first one. This strategy is given by a reservation price $r$, such that upon observing a price lower than $r$ she buys outright, upon observing a price greater than $r$ she searches further, and upon observing a price equal to $r$ she is indifferent between buying immediately and searching further.
Equilibrium Notion: Concept of Perfect Bayesian Equilibrium (PBE) is employed. A PBE is characterized by price distribution $F(p)$ for each firm and the consumer's reservation price strategy given by $r$ such that $(i)$ each firms chooses $F(p)$ that maximizes its profit, given the equilibrium strategy of the other firm and the consumer's optimal search strategy, and (ii) the consumer searches according to the reservation price rule $r$, given correct beliefs concerning equilibrium strategies of firms.
Theorem: For any $c>0$, there exists a PBE characterized by triple $(p^M, p^M, r)$, where $p^M$'s are charged with probability $1$ and
$$r=1.$$
Proof: First, I prove that $r=1$, or that the consumer buys outright when she observes any price lower than $1$. Clearly, if she observes a price greater than $1$ she does not buy from that firm as this yields a negative payoff to the consumer. Now, suppose she observes price $p'<r$. Then, she expects the rival firm to charge $p'$ too. Thus, if she buys outright her payoff is $\int_{p'}^{1}(1-p)dp$, and if she searches she expects a payoff equal to $\int_{p'}^{1}(1-p)dp - c$. As the former is greater than the latter, she better-off when she buys immediately. This proves that $r=1$.
Next, I prove that both firms charge $p^M$. Clearly, firms never charge above $1$ as they will never sell. Then, the expected profit of a firm is $\frac{1}{2}(1-p)p$ because the consumer visits a firm half of the time. It is easy to see that the profit is maximized at $p^M$. QED.