This article will explore an overview of pricing strategy. We begin by explaining the simple Cournot and Bertrand games, which are game theoretic analyses of a firm’s pricing strategy. With this information we proceed by explaining what strategic complementarities and substitutes are before looking at a paper by Fudenberg and Tirole entitled “The Fat-Cat Effect, the Puppy-Dog Ploy, and the Lean and Hungry Look”.

**Cournot Game**

In this scenario we have (at least) two firms who compete on the amount of output they produce, choosing the quantity simultaneously and independently whilst taking the output decision of the other firms as given. Therefore in making this decision firm 1 will need to contemplate what firm 2 will do: if both firms produce a large quantity then we would expect price to fall and so profits will fall. Therefore each firm has a choice; they could independently produce low, hoping that the other party will do this (and so keep prices and therefore profit high) or they may realise that one firm has an incentive to deviate and so both firms deviate.

In other words, both firms know that a lower overall quantity is desirable to keep prices and profits high. However it isn’t a Nash equilibrium for both firms to produce low quantities. If Firm 1 expects Firm 2 to produce a low output, then it will have an incentive to increase its output, knowing that it can sell more goods without seeing a large change in the price. Therefore it will capture more of the market, and make greater profits (since both q and p are higher).

But Firm 2 realises that this is an outcome and hence decides to produce a large quantity (deviate). The result is that both firms produce a low quantity and the price is low. Note, however, that the price doesn’t necessarily equal the marginal cost.

The outcome of this is that prices will be above the perfectly competitive outcome, but lower than the monopoly outcome and as the number of firms tends to infinity the price tends to the marginal cost (perfectly competitive outcome). Output will be higher than under monopoly but not as high as with perfect competition.

We point out that this model makes the following assumptions:

- All firms are producing a homogenous (same) good
- Firms have market power such that each firm’s output decision affects the good’s price
- Firms do not cooperate
- The firms are rational profit maximisers

A disadvantage of this model is that it assumes that firms can easily alter quantity, but this may not be the case if a firm is a producer as it can be difficult (and/or costly) to change quantity.

**Bertrand Game**

The Bertrand game has similar assumptions to Cournot, in that all firms are producing a homogenous good, and we assume they are rational profit maximisers, but now they set **prices** simultaneously, not quantities, and consumers wish to purchase from the firm with a lower price. We assume that if firms set the same price then demand is split equally.

The outcome of such a game is that prices must equal the marginal cost, and hence prices and output are the same as in the perfectly competitive model. We can examine why this must be the case by considering a firm’s potential pricing strategies:

A: Charge the same price as competitors where p>MC

B: Charge the same price as competitors where p=MC

C: Charge the same price as competitors where p<MC

D: Charge a price below competitors where p<MC

E: Charge a price above competitors where p>MC

It is easy to see that C and D cannot be equilibria because charging a price below the marginal cost will bankrupt a firm*. Furthermore, E isn’t an equilibrium because all consumers will flock to the other firm, and the market share of our firm will be zero. Hence under E a profit-maximising firm would seek to match its competitor’s price in order to make a profit.

Of course, if the competitor price is already equal to the MC then profits will still be zero and as such may mean that the firm decides not to alter prices, because either way it can’t make profit; but the overall outcome in this case will be that p=MC.

Finally we can see that A cannot be an equilibrium because each firm has an incentive to slightly reduce its price, below that of its competitor, in order to fully take the market and make profit; this process continues until p=MC.

The above analysis has assumed continuous prices. If we take the more realistic assumption of integer pricing (i.e. each price has to be to the nearest penny) then the equilibrium is a price one penny above the marginal cost. This is because at this price neither firm can undercut the other whilst increasing profit (by undercutting to the MC the firm will now make zero profit), and no firm can increase the price without losing market share. Hence both firms would prefer to stick at p=MC+0.01 and make strictly positive profits.

A disadvantage of this model is that it ignores capacity constraints, i.e. it assumes that a firm is able to produce as much quantity as it wants, and therefore can set whatever price it likes and be able to provide to the market. However in reality, a firm may be limited in the amount it can produce. The Edgeworth paradox tells us that with capacity constraints there may not exist any pure strategy Nash equilibrium.

*This simple model precludes loss-leading or predatory pricing strategies. In the former, a firm may price below marginal cost on a certain, high-salience, product with the knowledge that customers will flock to it but also purchase other goods, where it can recoup the loss and make a profit. Research suggests that supermarkets employ such a strategy, pricing milk and bread (staple goods, which consumers recognise the price of) very low, knowing that if more customers come to buy these products they will also purchase other goods in the shop, which have a higher profit margin (cross-subsidisation). With predatory pricing, an incumbent firm prices less than an entrant to force them out of the market, and then increases the price highly (recouping the loss) once they have left the market.

**Stackleberg Game**

As an interesting aside, the Stackleberg game is when one firm moves first and then the follower firms move sequentially. Such an outcome may occur when there is a first-mover advantage: if there is an advantage to moving first, such as the hope of capturing a market, then we can study such a scenario as a Stacklberg game. The firms choose an output, so we can compare with the Cournot model.

We solve such a model by using backward induction to find the subgame perfect Nash equilibrium (SPNE): we start by looking at what the leader considers the best response function of the follower: how the follower will respond once it has observed the quantity of the leader. The leader will then pick a quantity to maximise its payoff, anticipating the predicted response of the follower. This output is what the follower actually observes. The outcome is that there is an advantage to moving first in that the leader firm makes greater than Cournot level profits.

We are assuming that after the leader has selected its equilibrium quantity that the follower does not deviate from the equilibrium and choose some non-optimal quantity. By doing this it would hurt itself (and also the leader), and therefore – at least in a single game – would violate our assumption of rationality and profit maximisation.

In a single game there would be no advantage in the follower firm adopting such a strategy: it might announce to the leader that unless the leader chooses a Cournot-level quantity that it will produce a much larger quantity than its best response in order to lower the profit of the leader (meanwhile hurting itself). However this policy is not credible and so the leader would disregard it. It isn’t credible because if the leader chooses a quantity above Cournot-level (as we would expect) then once this decision has been made it makes no sense for the follower firm to try and punish the leader, because it has absolutely no incentive to do this in a single game.

On the other hand, in a repeated game, there may be an advantage to pursuing a punishment strategy and such a threat could be credible.

Without the assumption of perfect competition – that the follower observes the quantity of the leader – the game reduces to the Cournot version.

So far we have learnt that when firms compete on quantity, we expect competitive prices to be the outcome when the number of firms is large (i.e. price equals marginal cost). When firms compete on price we expect the price to be pushed down to the marginal cost (perfect competition equilibrium), and for output to be the same as under the perfect competition model. Neither the Bertrand model, nor the Cournot model can be considered “better”, we choose which model to use based on the situation we are faced with. Also remember that when we have a large number of firms the Bertrand and Cournot results are the same.

Next, we ask how a firm will react to a competitor’s pricing strategy? Will it increase its own price and therefore increase its profit, or will it decide to maintain its price and try and capture more of the market. Obviously the outcome depends on which strategy maximises profits and for this we would need individual profit functions, however we can predict what will happen depending on whether the market is characterised as being complementary or substitutory.

**The Fat-Cat Effect, the Puppy-Dog Ploy, and the Lean and Hungry Look**

We want to consider how our competitors will react to us changing our prices, and this depends whether the market is characterised as complementary or substitutory, terms coined by Bulow, Geanakoplos, and Klemperer. If the goods in the market are complements then the competitor will respond in kind. On the other hand, strategic substitutes mean that the competitor firm will acquiesce.

A complements market is typically associated with the Bertrand game: price cuts by one firm results in matching behaviour by competitors, whilst a substitutes market is associated with the Cournot game: if I capture a larger share of the market then my competitor’s share will go down.

The following taxonomy comes from Fudenberg and Tirole’s paper and is based on the notion of a market being strategic complements, or strategic substitutes:

Puppy Dog Ploy: your competitor will fight back if you fight and so you don’t commit to playing tough

Top Dog: you play tough, expecting your competitor not to react Fat Cat Effect: you don’t play tough because you know your competitor will also play easy Lean and Hungry Look: you look tough because you knew that if you played soft that your competitor would play tough

If the market is characterised as strategic substitutes then a firm’s strategy can either be top dog or the lean and hungry look. Conversely, if the market is characterised as strategic complements then a firm’s strategy can be the puppy dog ploy or the fat cat effect.

The Fat Cat Effect occurs under a strategic complements market and can lead to the counter-intuitive result that if the incumbent firm in an industry increases their price that there is a benefit to them AND their competitors as both see profits rise. This is because firms in the industry follow each others actions and so when the incumbent firm increases its prices, it sees it profits rise without losing much market share. The second firm acknowledges this (and realises that it has nothing to gain by keeping prices low) so increases its prices also. In the advertising market this can have the effect that greater investment in advertising doesn’t deter entry into the market and that incumbent firms should under-invest in advertising if they want to deter new entry (Schmalensee, and Fudenberg and Tirole). This makes sense when we consider that advertising increases overheads and demonstrates that there is profit to be made in such markets.

The puppy dog ploy occurs when the incumbent decides to play tough and reduce prices. This encourages an aggressive response by the competitor, who reduces its own prices to try and undercut the incumbent. This price competition means that in the long-term there are less profits to invest in technology and so technological innovation in that industry falls.

Under a Top-Dog strategy, the firm will increase production of their product to try and crowd out the competitor. This increased production will lead to greater quantity supplied and hence lower prices (the law of demand), but for the firm increasing production the higher sales volume will result in increasing profits even whilst prices fall. This assumes that the competitor can’t respond quickly and also increase their sales volume which would likely result in a price war where both firms retain their existing market share but see lower prices and therefore lower profits. Such a strategy may therefore be wiser to conduct when the firm knows that its competitor can’t increase production, perhaps due to limitations in their factory size (which would limit production in the short run until factory size could be expanded).

Bulow, Geanakoplos, and Klemperer furthermore point out that a firm which operates in one market expanding into another market can have strategic effects on the first market, as marginal costs could increase (if there are diseconomies of scale or scope).