The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Oligopoly and Game Theory

  • Hugo Sonnenschein
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_1890

Abstract

Oligopoly theory is concerned with market structures in which the actions of individual firms affect and are affected by the actions of other firms. Unlike the polar cases of perfect competition and monopoly, strategic issues are fundamental to the study of such markets. In this entry we will explain some of the central themes of oligopoly theory, both modern and classical, and emphasize the connection between these themes and developments in the noncooperative theory of games.

Oligopoly theory is concerned with market structures in which the actions of individual firms affect and are affected by the actions of other firms. Unlike the polar cases of perfect competition and monopoly, strategic issues are fundamental to the study of such markets. In this entry we will explain some of the central themes of oligopoly theory, both modern and classical, and emphasize the connection between these themes and developments in the noncooperative theory of games.

Section 1 presents a simple static oligopoly model and uses it to discuss the classic solutions of Cournot (1838), Bertrand (1883) and Stackelberg (1934). Section 2 contains an introductory account of a modern line of research into a class of dynamic oligopoly models. In these models, firms and consumers meet repeatedly under identical circumstances. An example is presented to illustrate the important result that a firm’s behaviour in such situations can drastically differ from that in the static model. Section 3 is concerned with a ‘folk theorem’ which states that with free entry, and when firms are small relative to the market, the market outcome approximates the result of perfect competition. Novshek’s Theorem (1980) gives a precise statement of this result, and in doing so provides an important bridge between oligopoly theory and the theory of perfect competition.
  1. I.

    We consider a market in which n firms (n > 1) produce a single homogeneous product. The quantity of output produced by the ith firm is denoted by qi and the cost associated with production of qi by Ci(qi). Demand is specified by an inverse demand function F(⋅): F(Q) is the price when Q ( = Σiqi) is the aggregate output of firms. Let q and qi denote the vectors (q1,…, qn) and (q1,…, qi−1, qi+1,…, qn) respectively. The profit of the ith firm is given by

     
$$ {\varPi}_i(q)=F(Q){q}_i-{C}_i\left({q}_i\right). $$
The interdependence of firms’ actions is reflected in the fact that the profits of the ith firm depend not only on its own quantity decision but also on the quantity decisions of all other firms.
A Cournot equilibrium is an output vector \( \overline{q} \) = (\( \overline{q} \)1,… \( \overline{q} \)n) such that
$$ \forall i, \forall {q}_i, {\varPi}_i \left(\overline{q}\right)\ge {\varPi}_i \left({q}_i,{\overline{q}}_{-i}\right) $$
where (qi, \( \overline{q} \)i) denotes the vector (\( \overline{q} \)1,…, \( \overline{q} \)i−1, qi, \( \overline{q} \)i+1,…, qn). The equilibrium \( \overline{q} \) is symmetric if \( \overline{q} \)1 = ⋯ = \( \overline{q} \)n.

In the Cournot model, firms make quantity decisions. A single homogeneous good is produced, which all firms sell at the same price. At equilibrium, no firm can increase its profit by a unilateral decision to alter its action. A Cournot equilibrium is illustrated in the following example.

Example 1: Let n firms have identical linear cost functions: Ci(qi) = cqi for all i. Assume that the inverse demand function is linear: F(Q) = abQ, where a, b > 0, and a > c. Thus,
$$ {\varPi}_i (q)=\left(a- bQ\right){q}_i-c{q}_i. $$
At Cournot equilibrium \( \overline{q} \),
$$ \frac{\partial {\varPi}_i\left(\overline{q}\right)}{\partial {q}_i}=0 \mathrm{for}\;\mathrm{all} i. $$
Therefore,
$$ a-b\sum_i{\overline{q}}_i-b{\overline{q}}_i-c=0 \mathrm{for} \mathrm{all} i. $$
It follows that equilibrium is unique and symmetric and
$$ {\overline{q}}_i=\frac{a-c}{b\left(n+1\right)} \mathrm{for} \mathrm{all} i. $$
Equilibrium aggregate output is (ac)/b(1 + 1/n); thus with two or more firms it is greater than monopoly output (ac)/2b but less than competitive output (ac)/b. (The competitive output is defined by the condition that inverse demand price is equal to the constant per unit cost.)

It can be argued that Cournot incorrectly deduced from the fact that, in equilibrium, a homogeneous commodity can have only one price, the conclusion that an oligopolist cannot choose a different price from one charged by its competitors (see Simon 1984). Bertrand observed that if firms choose prices rather than quantities, then the Cournot outcome is not an equilibrium. For the case in which prices rather than quantities are the strategic variable, the analysis proceeds as follows. Assume that all n firms (n > 1) have linear cost functions as described in Example 1 and that demand is continuous. Since the good being produced is homogeneous, a firm charging a price lower than that of other firms can capture the entire market. (To be specific we assume that all sales are shared equally among the firms that charge the lowest price.) Let \( \overline{p} \) and \( \overline{\varPi} \) denote price and individual profits respectively at the symmetric Cournot equilibrium. A firm can earn profits arbitrarily close to \( n\overline{\varPi} \) (and hence, greater than Π) by lowering its price by a little from \( \overline{p} \). The same argument can be used to show that in Bertrand equilibrium there is only one price at which sales are made. This price equals marginal cost and aggregate output is the competitive output. In Bertrand equilibrium, no firm can make a higher profit by altering its price decision.

An alternative equilibrium concept, due to Stackelberg, will be applied to the case of duopoly. There are two firms, labelled 1 and 2. The function H2 (⋅), called the reaction function of firm 2 (see Friedman 1977), is defined by
$$ {q}_2={H}_2\left({q}_1\right) \mathrm{if} \forall {\tilde{q}}_2, {\varPi}_2\left({q}_1,{q}_2\right)\geqslant {\varPi}_2\left({q}_1,{\tilde{q}}_2\right). $$
The output vector q = (\( \widehat{q} \)1, \( \widehat{q} \)2) is a Stackelberg equilibrium with firm 1 as the leader and firm 2 as the follower if firm 1 maximizes profit subject to the constraint that firm 2 chooses according to his reaction function; that is,
$$ \forall {q}_1, {\varPi}_i\left[{\widehat{q}}_1,{H}_2\left({\widehat{q}}_1\right)\right]\geqslant {\varPi}_1\left[{q}_1,{H}_2\left({q}_1\right)\right] \mathrm{and} {\widehat{q}}_2={H}_2\left({\widehat{q}}_1\right) $$
In the model of Example 1, the Stackelberg equlibrium is
$$ \left({\widehat{q}}_i,{\widehat{q}}_2\right)=\left(\frac{a-c}{2b},\frac{a-c}{4b}\right) \mathrm{and} \widehat{p}=\frac{a+3c}{4}. $$
The Stackelberg equilibrium is interpreted as follows. The leader decides on a quantity to place on the market: this quantity is fixed. The follower decides how much to place on the market as a function of the quantity placed on the market by the leader. Again, equilibrium requires that neither firm can increase its profit by altering its decision.
Despite the fact that for the same model the Cournot, Bertrand and Stackelberg outcomes differ from each other, there is an important respect in which they are similar. In particular, they can all be viewed as the application of the Nash equilibrium solution concept (see the entry on NASH EQUILIBRIUM) to games which differ with respect to the choice of strategic variables and the timing of moves. Thus, Cournot and Bertrand equilibria are Nash equilibria of simultaneous move games where the strategic variables are quantities and prices respectively. The Stackelberg equilibrium is the subgame perfect equilibrium of a game where firms make quantity choices but where the leader moves before the follower. This observation points to a general characteristic of oligopoly theory; the results are very sensitive to the details of the model. Nash equilibrium is the dominant solution concept in the analysis of oligopolistic markets and because its application is so pervasive one might expect substantial unity in the predictions of oligopoly theory. Unfortunately, as the preceding analysis makes clear, this is not so.
  1. II.

    It was observed in Example 1 that aggregate output in Cournot equilibrium exceeds monopoly output. This holds generally and it implies that aggregate profit in a Cournot equilibrium is less than monopoly profit. Thus, there exists a pair of (identical) quantity choices for firms such that with these choices each firm earns a higher profit than in Cournot equilibrium. Since such choices do not form a Cournot equilibrium it would be in some firms’ interest to deviate unilaterally from the choice assigned to it. In other words, without the possibility of binding contracts, the higher profit choices cannot be sustained, at least not in a static model. In this section, an extended example is presented to illustrate that if firms and consumers meet repeatedly, then it is possible for them to act more collusively than would be the case if they met only once. This result is very general and its importance for oligopoly theory was first pointed out by Friedman (see Friedman 1971).

     
There are two firms labelled 1 and 2. Each firm has three pure strategies L, M and H which can be thought of as representing ‘low’, ‘middle’ and ‘high’ quantities of output respectively. The payoffs are indicated in the matrix shown in Fig. 1, where (L, L), (M, M) and (H, H) may be thought of as the monopoly, Cournot and competitive outcomes respectively. In this game, (M, M) is the unique Nash equilibrium: given that one’s opponent plays M, the best that he can do is play M himself.

Oligopoly and Game Theory, Fig. 1

Consider now the game which is an infinite repetition of the game described above. The point that we wish to develop is that with repeated play it is possible to sustain outcomes that are much more collusive than (M, M). Strategies in the repeated game are more complicated than in the single period game. Specifically, the play of firm i in period t is a function of the ‘history’ of the game; i.e., of the plays of both firms in all periods preceding t. This allows a firm to ‘punish’ or ‘reward’ other firms. An outcome of the infinitely repeated game is a pair of infinite streams of returns, one for each firm. These infinite streams can be evaluated according to various criteria: two examples are considered. The stream \( {\left\{{x}_t\right\}}_{t=0}^{\infty } \) is preferred to the stream \( {\left\{{y}_t\right\}}_{t=0}^{\infty } \) according to the limit of means criterion if
$$ \underset{T\to \infty }{\lim}\left(1/T\right)\sum \limits_{t=0}\left({x}_t-{y}_t\right)>0. $$
In the case where there is discounting, \( {\left\{{x}_t\right\}}_{t=0}^{\infty } \) is preferred to \( {\left\{{y}_t\right\}}_{t=0}^{\infty } \) if the former has a higher present value; that is, if
$$ \sum \limits_{t=0}^{\infty}\frac{x_t-{y}_t}{{\left(1+r\right)}^t}>0, $$
where r is the discount rate.
Consider first the case where outcomes are evaluated according to the limit of means criterion. The strategies in which both players choose M, no matter what the history, is easily seen to constitute an equilibrium. However, strategies in which both players choose L in every period (call this (L, L)) provided there has been no deviation also form a subgame perfect Nash equilibrium. If there is a deviation (L, L), then the equilibrium strategies call for players to play the subgame perfect Nash equilibrium (M, M). A firm contemplating a unilateral deviation from (L, L) at time t must weigh an immediate gain of 6 against a loss of at least 3 from t + 1 onwards. The deviation is unprofitable according to the limit of means criterion since a gain of 6 today becomes arbitrarily small when averaged over an increasingly large number of periods. The mean gain from the deviation is thus zero, while the mean loss from the deviation is 3. This argument can be used to demonstrate that any feasible payoff which dominates (M, M) can be realized by some equilibrium. (Strategies which involve reversion to Nash equilibrium forever cannot be used to characterize the entire set of subgame perfect Nash equilibria utility outcomes. In fact, the shaded area in Fig. 2 can be obtained). These ideas are developed further in Aumann–Shapley (1976), Friedman (1971) and Rubinstein (1979). See also Axelrod (1984).

Oligopoly and Game Theory, Fig. 2

It is considerably more difficult to characterize the set of subgame perfect equilibria in the case where outcomes are evaluated according to their present value. However, Abreu (1986) provides results which help to determine the amount of collusion that is possible with various amounts of discounting. Of course this amount depends on the interest rate. It also depends on punishments that are a good deal more subtle than the threat to repeat the single period Nash equilibrium in the event of any deviation. To introduce you to this work we return to Fig. 1 and consider first the case where r = 1/4. The threat of playing (M, M) forever if there is a deviation from (L, L), sustains (L, L) as an equilibrium. To see this, note that a firm by deviating gains 6 immediately and loses 3 forever, thereafter. This loss has a present value of 3/r = [3/(1/4)] = 12, so deviation is not profitable. On the other hand, if r = 3/4 present value of the loss is 3/r = [3/(3/4)] = 4, which is less than the gain from deviating. Therefore, deviation is profitable. But note that (L, L) can be sustained by a pair of subgame perfect Nash equilibrium strategies which are recursively defined as follows:
  1. (a)

    The prescribed initial play is L for both players.

     
  2. (b)

    If both players act according to the prescription in t, then they are both to play L in t + 1.

     
  3. (c)

    If one or both do not play according to the prescription in t, then they are both to play H in t + 1.

     
To verify that this is a subgame perfect equilibrium, it has to be checked that no pattern of unilateral deviations is beneficial to a firm for any history of the game. The required argument is somewhat technical and is not given here (see Abreu 1986); however, we will show that no one-period deviation is profitable for any history of the game. There are two cases to consider:
  1. (a)
    No firm has deviated in period t − 1. In this case, the other firm is considered to be playing L at t so that the gain from deviation at t, is at most 6. At t + 1, a loss of 15 (= 15 − 0) will occur, which has a discounted value of
    $$ \frac{15}{1+r}=\frac{15}{1+\left(3/4\right)}=\frac{60}{7}, $$
    which is greater than 6.
     
  2. (b)

    Some firm has deviated in period t − 1. In this case, the equilibrium strategy requires both firms to play H in t. A firm by deviating (to L) can receive 3 in period t rather than 0; however the loss of 15 in the next period, as before, has present value 60/7, which is greater than 3.

     
  1. III.

    The theory of perfect competition assumes that all agents are price takers. We can improve our understanding of that theory by developing foundations for it that have firms behave strategically, in that they appreciate their market power, but nevertheless find themselves forced into actions that are well explained by the price taking assumption.

     

Consider the simple case where the demand function is linear and all firms have identical cost functions of the type C = cqi. Recall from Example 1 that aggregate output in Cournot equilibrium is (ac)/b(1 + (1/b)) and the equilibrium price is therefore a − (ac)/1 + 1/n. As the number of firms n increases, equilibrium price converges to c, which is the competitive price. This result does not generalize to the case of U-shaped average cost curves; furthermore, it has the defect that the number of firms in the market is fixed exogenously rather than being the result of a competitive process of free entry. These deficiencies are remedied in the work of Novshek.

Novshek’s model

Novshek considers economies of the type described in Fig. 3.

Oligopoly and Game Theory, Fig. 3

In the figure, F denotes an inverse demand function and AC an average cost curve associated with the employment of any one of an unlimited number of available units of an entrepreneurial factor. The price P* and the output Y* are the (perfectly) competitive price and the (perfectly) competitive output respectively. An intuitive argument for the convergence of equilibrium to P* runs as follows. Suppose price \( \overline{P} \) exceeds P*. A firm can now enter the market and make a profit by producing at minimum average cost provided that it does not change prices by ‘too much’. If the minimum efficient scale is small relative to the market, price will not change by ‘too much’ when the firm enters. Since there is an inexhaustible supply of potential entrants, \( \overline{P} \) is not viable. Prices below P* are not viable since firms are free to leave the market.

Novshek’s theorem may be interpreted as a formalization of the intuitive argument presented above. The theorem states that there exists a quantity-setting Cournot equilibrium with entry when efficient scale is small relative to demand and that in this case the equilibrium output and price are approximately competitive. We conclude with a formal statement of the result. Assumptions: All firms have the same cost function C:
$$ C\left({q}_i\right)=0 \mathrm{if} {q}_i=0, $$
and
$$ C\left({q}_i\right)={C}_0+v\left({q}_i\right) \mathrm{if} {q}_i>0, $$
where C0 > 0 and for all qi ≥ 0, v′ > 0 and v″ > 0 . Assume further that average cost is minimized uniquely at qi = 1.

The inverse demand function F(Q) is assumed to be twice continuously differentiable, with F′ < 0 whenever F > 0, and there exists Y* > 0 such that F(Y*) = C(1) (price equals minimum average cost). Definitions: An α(α > 0) size firm corresponding to C is a firm with cost function Cα(qi) = αC(qi/α). Average cost for an α size firm is minimized at qi = α. For each α, C, and F, one considers a pool of available firms, each with cost function Cα, facing inverse market demand F.

Given C, F and α, an (α, C, F) market equilibrium with free entry is an integer n and an output vector \( \overline{q} \) = (\( \overline{q} \),…, \( \overline{q} \)n) such that (a) \( \overline{q} \) is an n firm Cournot equilibrium (without entry), that is,
$$ {\forall}_i=1,\dots, n, \forall {q}_i, {\varPi}_i\left(\overline{q}\right)\geqslant {\varPi}_i\left({q}_i,{\overline{q}}_{-i}\right), $$
where Πt(⋅) is the profit function for firm i described in Section 1 and (b) entry is not profitable, that is,
$$ \forall {q}_i,F\left(\sum \limits_{j=1}^n{q}_j+{q}_i\right){q}_i-{C}_{\alpha}\left({q}_i\right)\leqslant 0. $$

The set of all (α C, F) market equilibria with free entry is denoted by E(α C, F).

Novshek’s theorem states that Cournot equilibrium exists provided that efficient scale is sufficiently small relative to demand, and furthermore, that it converges to the competitive output as efficient scale becomes small.

Novshek’s theorem: Under the above hypotheses, for each C and F there exists α* > 0 such that for all α ∈ (0, α*], E(α, C, F) is non-empty. Furthermore, \( \overline{q} \)E(α, C, F) implies \( \sum \limits_{j=1}^n{\overline{q}}_j\in \left[{Y}^{\ast }-\alpha, {Y}^{\ast}\right] \) and so aggregate output and price approximate the perfectly competitive values P* and Y*.

It is perhaps reasonable to believe that the perfectly competitive result will also hold under conditions that allow for only a relatively small number of firms. No claim is made here that a large number of firms is necessary for firms to act as if they are unable to influence price. Novshek’s Theorem, which relates well to the classical analysis of Cournot, provides a framework in which the perfectly competitive result obtains in the limit because in the limit firms cannot influence price.

See Also

Bibliography

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© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Hugo Sonnenschein
    • 1
  1. 1.