A model of market competition as a prize contest or a model of strife for market domination

Abstract

This article assumes that firm behavior concentrates on market share reflecting a view of the market as a contest prize. Perfectly competitive firms induce thus Cournot competition while monopolistically-competitive firms prompt the appearance of a differentiated Cournot environment. In any case, monopoly or a winner-takes-all result cannot be a Nash equilibrium. In terms of aggregative games, the equilibrium is a draw, and despite the equivalence between contests and best-response potential, Cournot outcomes are found to differ from contest ones. The result that diversity is superior to homogeneity is confirmed here, too. Competition intensity is independent of market concentration and product homogeneity or heterogeneity. The novelty of the paper in terms of methodology is that these results are obtained within the context of a process of market-share rather than quantity competition under perfect or monopolistic competition that produces a quantity–oligopoly equilibrium. Aggregative-game treatments of such an oligopoly take its presence as a given, they do not motivate its emergence, and arrive at equilibrium based on quantity competition.

Introduction

It is a standard political-economy observation that “No company wants perfect competition. It’s tough… Instead, what companies strive for is market dominance” (Kim and Mauborgne 2016). In this spirit, this article investigates whether the Cournot theorem still obtains if seller market interaction starts from perfectly and monopolistically-competitive conditions rather than from monopoly circumstances as is typically assumed. From a sequential-move perspective, homogeneous and differentiated Cournot outcomes are usually derived based on a scenario according to which free entrance in a monopolized market provokes a sequence of responses between the monopolist and the newcomer, converging to Cournot-Nash equilibrium, upon which Cournot theorem rests. This paper asks whether this theorem can be derived as a result of perfectly or monopolistically-competitive firms seeing market competition as a prize contest for relative payoff maximization (Sheremeta 2017). Perfect competition is linked to homogeneous-product Cournot interaction while monopolistic competition is associated with heterogeneous-product Cournot antagonism.

Methodologically, it is the exit rather than the entry of firms which is of analytical concern. There has never been a research agenda trying to approach Cournot outcomes starting from a decentralized state of affairs rather than from the centralization associated with a monopoly or duopoly. For example, aggregative-game analyses of quantity (and price) oligopoly competition do not motivate its emergence (Corchón and Marini 2018). Also, the “market-as-a-prize-contest” is an approach, which implies that the consequences of joint strategic interaction may be a reason by itself for endogenous exit decision. From this point of view, this approach belongs to the list of endogenous market-structure ones as reviewed, for instance, by Etro (2014). Perfect competition includes many small price-taking firms producing under constant returns to scale while firms under monopolistic competition are isolated price setters, possessing each some market power in its own product but disregarding any strategic interaction between itself and its competitors. It will be seen below that viewing the market-as-a-prize takes away this disregard of interaction, introducing an element of oligopolistic behavior; and in so far as perfect competition is concerned, such a perception of the market by the sellers introduces an element of monopolistic competition. Exit (and entry) in monopolistic competition becomes subsequently possible at any time and not only in the long-run as postulated by the models of this market structure; and, the monopolistically-competitive element brought about by market manipulation starting from perfect competition does not take away the free movement of resources to and from such an industry.

The novelty and significance of this article are clear from the viewpoint of (i) political economy, (ii) microeconomics/industrial economics, and (iii) industrial policy. From the viewpoint of political economy, the market-as-a-prize-contest approach to competition reconciles capitalism and competition in that capitalism is premised on the accumulation of capital, but competition does not imply that profits get competed away. From the viewpoint of microeconomics/industrial economics, such a market-dominance approach is novel by itself, especially when the present decentralized approach to Cournot outcomes does not lead to results that are symmetric to those derived from the typical centralized treatment of these outcomes. From the viewpoint of industrial policy, one key conclusion of the analysis is that the higher market concentration that our decentralized approach can produce is a duopoly, which in turn implies that competition in private goods markets cannot by itself justify the emergence of a monopoly. A monopoly cannot be the result of competition however fierce may be; it takes the presence of additional factors like patents, exclusive resource access, or institutional arrangements like those mentioned by Stuart and Harborne (2007). Consequently, there is ample room for government intervention in monopolies, and if monopolistic price-making does not improve welfare as Miao (2006) maintains, there are legitimate concerns for competition policy.

In the next section, firm behavior concentrates on market share reflecting a view of the market as a contest prize. Perfectly competitive firms induce thus Cournot competition while monopolistically-competitive firms prompt the appearance of a differentiated Cournot environment. A market-as-a-prize contest encompasses certainly market manipulation. Putniņš (2012) offers a taxonomy of the ways of how exactly firms might manipulate in practice the market to induce other firms to abandon it, but this is a matter which is of no concern here. What is the heart of the matter in this article is its foundation institutionally on Adam Smith’s (1994 [1776]) perception of competition as a race between competitors to gain market share rather than as a state or situation. Free competition is related to individual firm freedom rather than exclusively on firm numbers. The competition process ‘arises whenever two or more parties strive for something that all cannot obtain” (Stigler 1957) and acts as a selection mechanism prompted by rivalry, compatible with market power and “abnormal” profit rates as the Austrian School maintains. In any case, monopoly or a winner-takes-all cannot be a Nash equilibrium. Section 3 concludes with the observation that since the emergence of a monopoly should be attributed to non-price factors, monopoly rents and welfare costs should be linked with these factors.

Formal considerations

Let there be \(n\) identical perfectly-competitive sellers of a homogeneous good, each selling at disequilibrium, quantity \({q}_{i}, i=\mathrm{1,2},\dots n,\) costlessly in a market in which \(Q\) totally would be demanded under a zero price. Consequently, total supply is: \({q}_{i}+\left(n-1\right){q}_{-i}\), where \({q}_{-i}\) is the quantity sold by all other, \(n-1\) firms. Suppose now that the \(i\)th seller aims at monopolizing the market to be offering \(Q/2\). To do so, control of the entire market should be acquired first. That is, each seller individually sees \(Q\) as a prize, and \({q}_{i}\) as the effort, i.e. as the non-Cournot output, required to gain this prize taking for granted that the effort by all others is \(\left(n-1\right){q}_{-i}\). The “effort” is the action taken by agents before the prize is allocated; it is output marketed for the sake of rivalry and hence, in suboptimal quantities. There is Cournot, naive strategic substitutability, but in market share rather than quantity. Cournot attitudes do not also allow for the emergence of the output sub-optimality that would be needed to claim the entire prize; it is a sub-optimality which is more extensive than the one required to reach a Nash equilibrium, because: “optimality under monopolization aspirations, entails effort-maximizing structure of multi-stage sequential elimination contests, with pooling competition in each stage.” (Fu and Lu 2012, 353). From the viewpoint of multi-stage interaction, players are aware of their backward-induction rather than “winner-take-all” game, because they want for some reason to avoid the extensive output sub-optimality associated with the latter game. It is finally assumed that a positive market shock such as a market entry leads to an increase in the smallest and largest equilibrium aggregates.

Under these circumstances, following Sheremeta (2017), the expected payoff from this effort will be then:

$$E\left( {{\Pi }_{i} } \right) = \frac{{q_{i} }}{{q_{i} + \left( {n - 1} \right)q_{ - i} }}Q - q_{i}$$
(1)

The fraction in (1) denotes the probability of left eventually operating alone in a market where the long-run perfectly-competitive equilibrium quantity is \(Q\). Put differently, this fraction is the Tullock (1980) probability of getting the prize. The multiplication of this probability with \(Q\) minus \({q}_{i}\), is the expected payoff of the \(i\) th seller. The first-order profit maximization condition is:

$$\frac{Q}{{q_{i} + \left( {n - 1} \right)q_{ - i} }} - \frac{{Qq_{i} }}{{\left[ {q_{i} + \left( {n - 1} \right)q_{ - i} } \right]^{2} }} - 1 = 0$$
(2)

At equilibrium, all firms sell the same quantity, exert the same effort to seize the prize, each selling \(q\), which renders (2):

$$\begin{gathered} \frac{Q}{nq} - \frac{Qq}{{\left( {nq} \right)^{2} }} - 1 = 0 \Rightarrow nQ - Q - n^{2} q = 0 \Rightarrow \hfill \\ q^{*} = \frac{n - 1}{{n^{2} }}Q \hfill \\ \end{gathered}$$
(3)

Total quantity sold is hence:

$$nq^{*} = \frac{n - 1}{n}Q \propto \frac{n}{n + 1}Q$$
(4)

which is the same as under Cournot competition once 1 is added to the left-hand fraction. Nevertheless, the quantity sold by each firm is less than that under Cournot competition, namely, smaller than \(Q/n+1\), because comparing this expression with the right-hand side of (3), yields that:

$$\frac{n - 1}{{n^{2} }}Q < \frac{1}{n + 1}Q \Rightarrow - 1 < 0$$

which is true. Figure 1 is based on (3), with \(y\equiv {q}^{*}\) and \(x\equiv n\) for \(Q=100\) (lower black curve) and \(Q=1000\) (upper gray curve); \(y\) can be defined only for \(n\ge 2\), taking on its maximum value at \(n=2\). A duopoly is the most seller behavior under the view of the market-as-a-prize can produce; in terms of contest and game theory, the outcome is a draw rather than winner-takes all. Note that the results are inconsistent with the alleged equivalence between n-player lottery contest and a best-response game (Ewerhart 2017) in that firms do not sell at equilibrium the same quantities they would under a Cournot game.

Fig. 1
figure1

Market share (vertical axis) and number of homogeneous-product firms (horizontal axis), under contest

Suppose now that the competing firms are monopolistically-competitive ones, each with a constant cost, \({c}_{i}\), due to product differentiation. (2) may be rewritten as follows:

$$\begin{gathered} \left( {n - 1} \right)q_{ - i} Q^{\prime} - \left[ {q_{i} + \left( {n - 1} \right)q_{ - i} } \right]^{2} = 0 \Rightarrow \hfill \\ q_{i}^{2} + 2\left( {n - 1} \right)q_{i} q_{ - i} + \left[ {\left( {n - 1} \right)^{2} q_{ - i}^{2} - \left( {n - 1} \right)q_{ - i} Q^{\prime}} \right] = 0 \hfill \\ \end{gathered}$$
(2′)

where \(Q{^{\prime}}\) is the long-run monopolistically-competitive equilibrium output. (2′) is a quadratic equation, which when solved for \({q}_{i}\) in terms of \({q}_{-i}\), yields the two solutions:

$$q_{i,1,2} = - \left( {n - 1} \right)q_{ - i} \pm \sqrt {\left( {n - 1} \right)q_{ - i} Q} ^{\prime}$$

Since \({q}_{i}>0\), only the solution:

$$q_{i,1,2} = - \left( {n - 1} \right)q_{ - i} + \sqrt {\left( {n - 1} \right)q_{ - i} Q^{\prime}}$$
(5)

with

$$\left( {n - 1} \right)q_{ - i} Q^{\prime} > \left( {n - 1} \right)^{2} q_{ - i}^{2} \Rightarrow Q^{\prime} > \left( {n - 1} \right)q_{ - i}$$

becomes admissible. This solution, i.e. (5), comprises the reaction function of the \(i\)th seller with respect to the output-effort of the other sellers. Figure 2 illustrates this function for n = 2, n = 3, and n = 30, given Q’ = 100; \(x\equiv n-1\) and \(y\equiv {q}_{i}\). Again, a duopoly at most can be given rise by the contest, in which case equilibrium occurs at the intersection point \(\Gamma\). The reaction function becomes increasingly steeper, tending to the textbook illustration of the reaction functions, as the number of rivals becomes larger. One concludes then out of these considerations that a contest to seize the market-as-a-prize, originating in monopolistic competition, results in a differentiated Cournot rather than Bertrand solution. From the reaction functions, one gets the solution:

$$\hat{q}_{i} = \left\{ {\frac{{\left( {n - 1} \right)^{2} Q^{\prime} + \sqrt {\left( {n - 1} \right)^{4} Q^{{\prime}{2}} + 4n^{2} \left[ {\left( {n - 1} \right)Q^{\prime}} \right]^{\frac{3}{2}} } }}{{2n^{2} \left( {n - 2} \right)}}} \right\}^{\frac{2}{3}} = \hat{q}$$
(6)
Fig. 2
figure2

Reaction Functions for n = 1, n = 3, and n = 30, and Q = 100

At equilibrium, all sell the same quantity. Figure 3 illustrates (6), with \(y\equiv \widehat{q}\) and \(x\equiv n\) for \({Q}^{^{\prime}}=100\) (lower black curve) and \({Q}^{^{\prime}}=1000\) (upper gray curve). Again, a duopoly at most can be given rise, with each firm selling \(Q{^{\prime}}/4\), which is what point \(\Gamma \left(\mathrm{25,25}\right)\) in Fig. 2 captures given a \(Q=100\). The similarity between Figs. 1 and 3 is striking and according to them, market share under homogeneity is higher than the share under heterogeneity! The reason, of course, is that sellers of differentiated products who see market domination as a prize, become aware of their interaction even though the underlying market structure is monopolistically competitive.

Fig. 3
figure3

Market share (vertical axis) and number of heterogeneous-product firms (horizontal axis), under contest

From the viewpoint of the contest literature, these results are a mixture of proportional-prize contests in which rewards are shared in proportion to performance, and winner take-all contests in that a multi-stage contest eliminates some players based on performance merit at each stage, but the “economics” of the contest produces an optimal prize allocation at equilibrium that is not consistent to a single winner taking over the entire prize. Of course, this is the case because the ‘winner-take-all’ mentality on the part of firms, a complete-sequence ‘Pyramid’ contest has been ruled out ex hypothesi. The output waste needed to claim the entire prize is greater than the waste required to reach a Nash equilibrium, and players do not want the greater waste. Moreover, the superiority of monopolistic over perfect competition outcomes attest to Jensen's (2018) proposition that diversity may be superior to homogeneity.

Concluding remarks

In general, the approach of this article to market concentration has been in the vein of the structure-conduct-performance hypothesis of industrial economics from the viewpoint that higher concentration is assumed to lead to higher profits disregarding the role firm efficiency and managerial incentives/disincentives play in influencing profitability. Armstrong and Green (2007) are among those who question the realism of market-share targets when these targets are seen as competitor-oriented objectives, and Keen and Standish (2006, 84) maintain that “instrumentally rational profit-maximizers do not play Cournot-Nash games”. This article has also been in this vein but only to the extent that perfect competition is contemplated. The firms from such more or less market backgrounds, seeking to increase market penetration, are too many; and, each hence believes that the effects of individual firm behavior on the market are too small to make competitors worry about each other. This is a plausible assumption adopted also when the market background is monopolistically competitive and firms do care about market shares. But, when market dominance is set as a target in mature markets where, as Weber and Dholakia (1998) claim, market-share growth can be of critical importance to further business development, firm interaction can be shown to become explicit.

The strategic substitutability surrounding competition above, is one concerning market shares in a homogeneous-product market, which bears no resemblance to the Walrasian, price-taking assumption that may be confused with the naive, lacking foresight, Cournot-seller expectation according to which competitors do not change their market output and hence, price. As a matter of fact, Cournot limit theorem states that the two decision rules, Cournot's best-response behavior, and the Walrasian behavior of deciding next-period profit-maximizing output by acting as a price taker, eventually coincide in terms of competition outcome when the number of firms is sufficiently large. We do not need to enter into the debate regarding this matter (Radi 2017); the point made is that it is irrelevant in so far as our approach here is concerned.

The model used was highly stylized and may not be related directly to any actual market. It intends to capture in a simple manner some crucial elements of trades in the centralized and decentralized markets. It follows that the model can be extended in several dimensions as suggested by the industrial organization literature. In any case, competition intensity is independent of market concentration and product homogeneity or heterogeneity. Competition intensity is always present in the effort of firms to seize as much as possible control of the market. Therefore, market concentration indexes should be seen by the policymaker as an index of how close an industry is in approaching the target of turning into an oligopoly variant. Finally, it should be reinstated that since monopoly cannot be a result of competition, non-price non-quantity factors should be held responsible for the emergence of a monopoly. Monopoly welfare costs are consequently costs linked to such factors.

Data availability

Data sharing is not applicable to this paper.

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Correspondence to Gerasimos T. Soldatos.

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Soldatos, G.T. A model of market competition as a prize contest or a model of strife for market domination. SN Bus Econ 1, 28 (2021). https://doi.org/10.1007/s43546-020-00035-4

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Keywords

  • Market as a prize-contest
  • Perfect competition
  • Monopolistic competition
  • Cournot oligopoly
  • Monopoly

JEL Classification

  • D41
  • D42
  • D43