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An oligopoly is a market structure where a handful of large firms dominate the market. The small number of firms mean each firm has market power and this generates mutual interdependence. Mutual interdependence is a situation where an action by one firm will cause a reaction in the other firms.
Firms in an oligopoly can produce either identical or differentiated products. As a result, firms in an oligopoly will compete through non-price competition. The other firms are likely to reduce its price when you reduce yours, making it pointless to reduce price. It is better to improve your products which your rivals may not be able to do so. Advertising and research and development is important in oligopoly in order to engage in non-price competition.
Barriers exist in an oligopoly. It is not easy to enter the market

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Price and output decisions for an oligopolist

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Price and output decisions for an oligopolist

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It is difficult to analyse the demand curve for an oligopolist because of mutual interdependence; when you change your price, your rivals may follow. Several theories explain different demand curves for the oligopolist.
If no collusion exist, the demand curve will be a kinked one. Collusion is a situation where firms act together and collude to fix prices or restrict competition.

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If no collusion exist within the other firms, the firms in an oligopoly will have a kinked demand curve. It states that if you increase price, your rivals will not follow; but if you decrease price, your rivals will follow.
This result in price rigidity where it is hard to change the price. When drawing the demand curve, the current price and quantity must be known
The point of the current price and quantity is where the bent is. When the firm increase its price, the demand becomes elastic because the other firms will not follow. Many customers will switch to buy from another firm. On the other hand, when the firm decrease its price, the demand becomes inelastic because the other firms will follow. The firm will still gain customer but it would be a small amount only because some will buy from the other firms which also decreased its price.
Due to this kink, oligopolist will usually not change the price as it means decreased revenue.
The bend will cause the marginal revenue curve to have a discontinuity because each demand curve has its own marginal revenue curve. A change in marginal cost will not change the profit maximising output as long as the MC curve cut the MR curve at the discontinued portion of the MR curve.

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Cartel is a formal agreement made by the group of firms to control the price and output of a product.
Mutual interdependence encourage firms to collude to maximise their profits especially if the products are identical. When firms establish a cartel, they act like a monopoly and they will cooperate to control the output so as to increase the price.
However, firms find it difficult to agree on the amount of output that each firm will decrease in order to increase the price. It is also difficult to enforce the agreement and no penalties are imposed if a firm betrays the rest; one firm can betray the rest by lowering the prices and producing more than the agreed amount with the intention to increase profit. This will destroy the monopoly power.

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Price leadership is an informal collusion where every other firm follows the price set by the dominant firm. The price is usually matched (not always) against the largest firm.