The Dynamics of Market Competition

Abstract

This chapter, which is mainly based on (Forrest et al. 2017b), studies the dynamics of a coordinate monopoly with m incumbent risk-neutral firms regarding how these firms compete by adjusting prices and when new competition(s) will enter the market with expectations of making more profits than any of the incumbents. Major findings include (1) how risk neutrality in a developed marketplace can lead to stagnation in expected profits and irrational decision on pricing, (2) a sufficient and necessary condition under which new competitor(s) will enter the market, although the market is coordinately monopolized, etc.

Appendix: Proofs of Theoretical Results

1.1 The Proof of Theorem 3.1

Let F_k(P) be the price distribution of Firm k, k = 1, 2. Then the profits for Firm 1 from its loyal consumers are α₁P, and those from its share of the switchers are [1 − F₂(P)]βP. So, Firm 1’s objective function is

$$ {\mathit{\max}}_{F_1(P)}E\left({\Pi}_1\right)=\underset{-\infty}{\overset{+\infty}{\int}}\left\{{\alpha}_1P+\left[1-{F}_2(P)\right]\beta P\right\}{dF}_1(P) $$

(3.2)

where Π₁ is Firm 1’s profits, E(Π₁) is the expected profits, and Firm 1 likes to maximize its expected profits by selecting its appropriate price distribution.

Because Firm 1 likes to attract as many switchers as possible to potentially increase its profits from the guaranteed level α₁ from its loyal consumers by charging them the reservation value 1, the following holds true

$$ {\alpha}_1P+\beta P\ge {\alpha}_1. $$

So, P ≥ α₁/(α₁ + β). Because P = 1 is the reservation value of the loyal consumers, the objective function of Firm i in Eq. (3.2) becomes

$$ {\mathit{\max}}_{F_1(P)}E\left({\Pi}_1\right)=\underset{\frac{\alpha_1}{\alpha_1+\beta}}{\overset{1}{\int}}\left\{{\alpha}_1\times P+\left[1-{F}_2(P)\right]\beta \times P\right\}{dF}_1(P). $$

(3.3)

The equilibrium indifference condition for Firm 1 is

$$ {\alpha}_1\times P+\left[1-{F}_2(P)\right]\beta \times P={\alpha}_1\times 1, $$

which leads to

$$ {F}_2(P)=1-\frac{\alpha_1\left(1-P\right)}{\beta P}, $$

(3.4)

satisfying

$$ {F}_2\left(\frac{\alpha_1}{\alpha_1+\beta}\right)=0\kern0.5em \mathrm{and}\kern0.5em {F}_2(1)=1. $$

(3.5)

Similarly, one has

$$ {F}_1(P)=1-\frac{\alpha_2\left(1-P\right)}{\beta P}, $$

(3.4a)

satisfying

$$ {F}_1\left(\frac{\alpha_2}{\alpha_2+\beta}\right)=0\kern0.5em \mathrm{and}\kern0.5em {F}_1(1)=1. $$

(3.5a)

So, Firm 1’s expected profits are

$$ {\displaystyle \begin{array}{c}E\left({\Pi}_1\right)=\underset{\frac{\alpha_1}{\alpha_1+\beta}}{\overset{1}{\int}}\left\{{\alpha}_1P+\left[1-{F}_2(P)\right]\beta P\right\}{dF}_1(P)\\ {}=\underset{\frac{\alpha_1}{\alpha_1+\beta}}{\overset{1}{\int}}{\alpha}_1{dF}_1(P)\\ {} = {\left.{\alpha}_1{F}_1(P)\right|}_{\frac{\alpha_1}{\alpha_1+\beta}}^1\\ {}={\alpha}_2.\end{array}} $$

(3.6)

Similarly, the expected profits of Firm 2 are α₁.

If α₁ ≠ α₂, without loss of generality we can assume α₂ > α₁. Then from the assumptions about the loyal consumers, one has α₁ = E(Π₂) ≥ α₂ > α₁, a contradiction. So, this end means α₁ = α₂. QED

1.2 The Proof of Theorem 3.2

When these incumbent firms compete by using prices, there is no pure strategy Nash equilibrium. (Note: for the case that no symmetric pure strategy equilibrium exists, please consult with Narasimhan (1988) and Varian (1980)). In fact, for any pure strategy portfolio (x₁, x₂, …, x_m), if there is a unique index i ∈ {1, 2, …, m} such that

$$ {x}_i<{x}_j $$

(3.7)

where j ∈ {1, 2, …, m} and j ≠ i, then Firm i has successfully attracted all the price switchers and can therefore slightly raise its price x_i to bring in additional profits as long as the new price still satisfies the condition in Eq. (3.7). So, the pure strategy portfolio (x₁, x₂, …, x_m) is not a Nash equilibrium.

If the cardinality |I| of the set

$$ I=\left\{i\in \left\{1,2,\dots, m\right\}:{x}_i={\min}_{j=1}^m\left\{{x}_j\right\}\right\} $$

is greater than 1, then Firms k, k ∈ I, have absorbed all the switchers. Because everything in this scenario is set up symmetrically, each of these firms would have taken in β/|I| portion of the switcher segment. So, Firm k’s profits, k ∈ I, are

$$ \alpha {x}_k+\frac{\beta {x}_k}{\left|I\right|}. $$

So, one of these firms, say Firm k, can lower its price slightly to ${x}_k^{\prime}$, satisfying

$$ \frac{x_k^{\prime}}{x_k}>\frac{\alpha +\beta /\mid I\mid}{\alpha +\beta}, $$

to bring in additional profits by attracting all the switchers. That is, the portfolio of pure strategies (x₁, x₂, …, x_m) is not a Nash equilibrium.

A similar argument can show that even when not all firms k ∈ I share the same portion of the price switchers, the firm with the fewest price switchers can slightly lower its price to increase its profits by attracting all price switchers. That is, we once again show that the portfolio of pure strategies (x₁, x₂, …, x_m) is not a Nash equilibrium.

Lastly, if for any i ∈ {1, 2, …, m}, x_i = 1, then Firm j, for any chosen j ∈{1, 2, …, m}, can slightly lower its price from the reservation value 1 to anywhere in the interval (α/(α + β), 1) to increase its profits by taking in the entire segment of switchers. So, (1, 1, …, 1) is not a Nash equilibrium, either. That is, this particular game does not have any pure strategy Nash equilibrium.

However, this game does have a symmetric mixed strategy Nash equilibrium (Zhou et al. 2015). To this end, let F_i(P) be the price distribution of Firm i, i ∈ {1, 2, …, m}.

First, assume that there are only two competing firms i and j, that is, m = 2. At price P, Firm i’s profits from its loyal consumers are αP, and its profits from switchers are [1 − F_j(P)]βP. So, the objective function of Firm i is

$$ {\displaystyle \begin{array}{c}{\mathit{\max}}_{F_i(P)}E\left({\Pi}_i\right)=\underset{-\infty}{\overset{+\infty}{\int}}\left\{\alpha P+\left[1-{F}_j(P)\right]\beta P\right\}{dF}_i(P)\\ {} = \underset{0}{\overset{1}{\int}}\left\{\alpha P+\left[1-{F}_j(P)\right]\beta P\right\}{dF}_i(P)\end{array}} $$

(3.8)

where Π_i = Π_i(P) represents Firm i’s profits at price P, E(Π_i) Firm i’s expected profits for all possible prices, and for this Firm i’, its objective is to maximize these expected profits by appropriately choosing its particular price distribution F_i(P). The reason why the upper and lower limits of the integral are changed, respectively, from +∞ and −∞ to 1 and 0 is because when P < 0 or when P > 1, the profits are zero.

Assume that there are three incumbent firms, namely i, j, and k, that are involved in the price competition. Then, at price P, Firm i’s profits from its loyal consumers are αP. The portion of switchers Firm j does not get is γ = [1 − F_j(P)]β, which is still available for Firms i and k to take. Now, [1 − F_k(P)]γ= [1 − F_k(P)][1 − F_j(P)]β is the portion of switchers taken up by neither Firm j nor Firm k. So, they are left for Firm i to take in. So, the objective function of Firm i is

$$ {\mathit{\max}}_{F_i(P)}E\left({\Pi}_i\right)=\underset{0}{\overset{1}{\int}}\left\{\alpha P+\left[1-{F}_j(P)\right]\left[1-{F}_k(P)\right]\beta P\right\}{dF}_i(P). $$

(3.9)

So, mathematical induction implies that when there are m incumbent firms that compete through pricing, the portion of switchers Firm i is able to take in is

$$ {\prod}_{j\ne i}^m\left[1-{F}_j(P)\right]\beta. $$

Therefore, the profits Π_i Firm i generates when the firm sells its product at price P are given by

$$ \alpha P+{\prod}_{j\ne i}^m\left[1-{F}_j(P)\right]\beta P, $$

and Firm i’s objective function is

$$ {\mathit{\max}}_{F_i(P)}E\left({\Pi}_i\right)=\underset{0}{\overset{1}{\int}}\left\{\alpha P+{\prod \limits}_{j\ne i}^m\left[1-{F}_k(P)\right]\beta P\right\}{dF}_i(P). $$

(3.10)

Firm i can earn α by charging the reservation value 1, because the firm’s loyal consumers will purchase its product at that maximum price. However, to potentially maximize profits, each of these incumbent firms adjusts its price P in order to take in as many of the switchers as possible. At the same time, no firm has incentive to price its product below α/(α + β), because any selling price below α/(α + β) will yield profits less than α despite of attracting all switchers, where

$$ \alpha P+\beta P\ge \alpha \to P\ge \frac{\alpha}{\alpha +\beta}. $$

The Nash equilibrium indifference condition for Firm i is

$$ \alpha \times P+{\prod}_{j\ne i}^m\left[1-{F}_k(P)\right]\beta \times P=\alpha \times 1,\mathrm{when}\ \frac{\alpha}{\alpha +\beta}\le P\le 1, $$

(3.11)

for i, j = 1, 2, …, m, and i ≠ j. So, the symmetric equilibrium price distribution is the following continuous function

$$ F(P)={F}_i(P)={F}_j(P)=1-{\left[\frac{\left(1-P\right)\alpha}{P\left(1- m\alpha \right)}\right]}^{\frac{1}{m-1}},\mathrm{when}\ \frac{\alpha}{\alpha +\beta}\le P\le 1, $$

(3.12)

satisfying the boundary conditions:

$$ F\left(\frac{\alpha}{\alpha +\beta}\right)=0\ \mathrm{and}\ F(1)=1. $$

(3.13)

In this unique mixed strategy Nash equilibrium, there is no mass point of prices that the firms charge with positive probability. Hence, each firm’s expected profits are

$$ {\displaystyle \begin{array}{c}E\left(\Pi \right)=\underset{-\infty}{\overset{+\infty}{\int}}\left\{\alpha P+{\prod \limits}_{j\ne i}^m\left[1-{F}_k(P)\right]\beta P\right\} dF(P)\\ {}=\underset{\frac{\alpha}{\alpha +\beta}}{\overset{1}{\int}}\alpha dF(P)\\ {} = {\left.\alpha F(P)\right|}_{\frac{\alpha}{\alpha +\beta}}^1\\ {}=\alpha.\end{array}} $$

(3.14)

In short, in the Nash equilibrium of symmetric mixed strategies, each incumbent firm’s expected profits do not change, although the firms try to attract as many switchers as possible, while exploits its loyal consumers by charging them an as high price as possible. QED

1.3 The Proof of Theorem 3.3

From Theorem 3.2, we know that in the symmetric mixed strategy Nash equilibrium, the expected profits for each incumbent firm are α. That is the same as whether or not an incumbent firm simply charges its loyal consumers the reservation value 1 without putting in any effort to entice switchers. To this end, there are two possibilities:

• (i) Each incumbent firm charges expectedly the reservation value P = 1.

• (ii) Each firm charges expectedly a price P less than 1.

If case (i) is true, then the magnitude of consumer surplus is expectedly equal to β = 1 − mα because all the switchers are waiting for discounts. If case (ii) is true, from its loyal consumers, each incumbent firm’s profits are αP (< α). So, the additional expected profits α − αP need to come from switchers so that

$$ \alpha P+\frac{\left(\beta -\gamma \right)P}{m}=\alpha, $$

(3.15)

where γ stands for the proportion of those consumers who are still waiting for deeper discounts from the prevalent price P, and each of the m incumbent firms acquires the same percentage of the switchers’ segment of the market, because all the incumbent firms are assumed to be identical as how they are set up earlier. Now, Eq. (3.15) implies that

$$ \gamma =1-\frac{1-\beta}{P}. $$

(3.16)

So, the proportion of consumer surplus is an increasing function of price P satisfying that when P = 1, γ = β; and when P = 1 − β, γ = 0. QED

1.4 The Proof of Theorem 3.4

(⟹, the necessity) Suppose that a new enterprise enters into the coordinately monopolized market occupied by m incumbent firms. So, each of these m firms establishes its selling price after taking into account the price of the new firm and those of all other existing firms. So, the equilibrium indifference condition of Firm k is

$$ \alpha \times P+\beta \times P{\prod}_{j\ne k}^m\left(1-P\right)\left[1-{F}_j(P)\right]=\alpha \times 1. $$

(3.17)

So, for these m incumbent firms, (3.17) provides the following symmetric equilibrium pricing strategy:

$$ F(P)=1-{\left(\frac{\alpha}{\beta P}\right)}^{\frac{1}{m-1}}. $$

(3.18)

However, for the expression F(P) in Eq. (3.18) to be a well-defined probability distribution, we must have

$$ 1-{\left(\frac{\alpha}{\beta P}\right)}^{\frac{1}{m-1}}\ge 0, $$

(3.19)

which implies that α/β ≤ P ≤ 1. That is, the consumer surplus β ≥ α.

(⟸, the sufficiency) Assume that the magnitude β = 1 − mα of the consumer surplus is greater than or equal to α. It suffices to show that there is one business enterprise that will profit expectedly by competing in this coordinately monopolized market with the incumbent firms through employing a uniformly randomized price strategy over the interval [0,1], where the marginal cost of the entrant is also assumed to be 0.

From the assumption that β = 1 − mα ≥ α, we have α/β ≤ 1. So, for any price P in the closed interval α/β ≤ P ≤ 1, the expression F(P) in Eq. (3.18) will be a well-defined mixed strategy for each of the m incumbent firms. And this strategy satisfies the equilibrium indifference condition in Eq. (3.17). This fact implies that for each of the m incumbent firms, its lowest allowed price is α/β.

To complete this proof, it suffices to show that the entering firm actually expects to make profits in this new market. In fact, because

$$ \underset{P\to {1}^{-}}{\lim}F(P)=1-{\left(\alpha /\beta \right)}^{\frac{1}{m-1}}\ne F(1)=1, $$

the cumulative price distribution function F(P) has a jump discontinuity at the reservation value P = 1. The amount of jump is ${\left(\alpha /\beta \right)}^{\frac{1}{m-1}}$. That is, F(P) has a mass point of size ${\left(\alpha /\beta \right)}^{\frac{1}{m-1}}$ at the reservation value P = 1. So, the expected profits of the entering firm are equal to the following:

$$ {E}_e\left(\Pi \right)=\underset{0}{\overset{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$\kern0.15em \beta $}\right.}{\int}}\beta P dP+\underset{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$\kern0.15em \beta $}\right.}{\overset{+\infty}{\int}}\beta P{\left[1-F(P)\right]}^m dP $$

(3.20a)

$$ =\underset{0}{\overset{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$\kern0.15em \beta $}\right.}{\int}}\beta P dP+\underset{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$\kern0.15em \beta $}\right.}{\overset{1}{\int}}\beta P{\left[1-F(P)\right]}^m dP+\beta {\left(\frac{\alpha}{\beta}\right)}^{\frac{m}{m-1}} $$

(3.20b)

where the first term on the right-hand side of Eq. (3.20a) stands for the expected profits of the entering firm when its selling price is the lowest in the marketplace and when it captures the entire segment of switchers. The second term of Eq. (3.20a) is equal to the entering firm’s expected profits when it is in direct competition with the m incumbent firms.

Evidently, the first term in the right-hand side of Eq. (3.20b) satisfies

$$ \underset{0}{\overset{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$\kern0.15em \beta $}\right.}{\int}}\beta PdP=\frac{\alpha^2}{2\beta}>0, $$

the second term is ≥0, because the integrant is positive, and the third term is positive. So, the expected profits of the entering firm E_e(Π) is greater than 0. In other words, this argument implies that if the magnitude of the consumer surplus satisfies β = 1 − mα ≥ α, there will be then at least one new firm that will enter the market to compete with the incumbent firms. QED

1.5 The Proof of Theorem 3.5

(Continued from the proof of Theorem 3.4.) Let us compute the expected price of the m incumbent firms as follows:

$$ {\displaystyle \begin{array}{c}{E}_m(P)=\underset{-\infty}{\overset{+\infty}{\int}}P\times {F}^{\prime}(P) dP\\ {}=\underset{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$\kern0.15em \beta $}\right.}{\overset{1}{\int}}P\times {F}^{\prime}(P) dP+1\times {\left(\frac{\alpha}{\beta}\right)}^{\frac{1}{m-1}}\\ {} = \left\{\begin{array}{ll}{\left(\frac{\alpha}{\beta}\right)}^{\frac{1}{m-1}}+\frac{{\left(\frac{\alpha}{\beta}\right)}^{\frac{1}{m-1}}-\frac{\alpha}{\beta}}{m-2},& \mathrm{if}\ m\ge 3\\ {}\frac{\alpha}{\beta}\left(1-\mathit{\ln}\frac{\alpha}{\beta}\right),& \mathrm{if}\ m=2\end{array}\right.\end{array}} $$

(3.21)

and the expected profits of any of the m incumbent firms are

$$ {\displaystyle \begin{array}{c}{E}_m\left(\Pi \right)=\underset{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$\kern0.15em \beta $}\right.}{\overset{1}{\int}}\left\{\alpha \times P+\beta \times P\left(1-P\right){\prod \limits}_{j\ne i}^m\left[1-F(P)\right]\right\} dF(P)+\alpha \times {\left(\frac{\alpha}{\kern0.15em \beta}\right)}^{\frac{1}{m-1}}\\ {} = \underset{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.}{\overset{1}{\int}}\alpha dF(P)+\alpha {\left(\frac{\alpha}{\beta}\right)}^{\frac{1}{m-1}}=\alpha.\end{array}} $$

(3.22)

On the other hand, the expected price of the new entering firm is E_e(P) = 1/2, and the expected profits of the firm, based on Eqs. (3.20a) and (3.20b), is

$$ {E}_e\left(\Pi \right)=\left\{\begin{array}{ll}\frac{-m}{2\left(m-2\right)}\frac{\alpha^2}{\beta}+\frac{m-1}{m-2}\frac{\alpha^{\frac{m}{m-1}}}{\beta^{\frac{1}{m-1}}}+\beta {\left(\frac{\alpha}{\beta}\right)}^{\frac{m}{m-1}},& \mathrm{if}\ m\ge 3\\ {}\frac{\alpha^2}{2\beta}-\frac{\alpha^2}{\beta}\mathit{\ln}\frac{\alpha}{\beta}+\beta {\left(\frac{\alpha}{\beta}\right)}^{\frac{m}{m-1}},& \mathrm{if}\ m=2\end{array}\right. $$

(3.23)

From Eq. (3.21), it follows that

$$ \frac{\partial}{\partial \alpha}{E}_m(P)=\left\{\begin{array}{ll}\frac{1}{\left(m-2\right){\beta}^2}\left[{\left(\frac{\alpha}{\beta}\right)}^{\frac{1}{m-1}-1}-1\right],& \mathrm{if}\ m\ge 3\\ {}\frac{1}{\beta^2}\left(-\mathit{\ln}\frac{\alpha}{\beta}\right),& \mathrm{if}\ m=2\end{array}\right. $$

(3.24)

So, we have

$$ \frac{\partial}{\partial \alpha}{E}_m(P)>0,\mathrm{for}\ 0<\alpha \le 1/\left(m+1\right), $$

because α ≤ β = 1 − mα. So, E_m(P) is an increasing function on the interval (0, 1/(m + 1)]. It can be readily checked that

$$ {\left.{E}_m(P)\right|}_{\alpha /\beta =1}=1>1/2={E}_e(P). $$

Hence, there is ${\alpha}_P^{\ast}\in \left(0,1/\Big(m+1\right)\Big)$ such that when $\alpha \ge {\alpha}_P^{\ast}$, the expected price E_m(P) of the incumbent firms is greater than that E_e(P) of the entering firm.

On the other hand, it is ready to check that

$$ \frac{\partial}{\partial \alpha}\left[{E}_e\left(\Pi \right)-{E}_m\left(\Pi \right)\right]>0 $$

(3.25)

and that when α = 1/(m + 1) = β, E_e(Π) − E_m(Π) > 0. Hence, there is ${\alpha}_{\Pi}^{\ast}\in \left(0,1/\Big(m+1\right)\Big)$ such that when $\alpha \ge {\alpha}_{\Pi}^{\ast}$, the expected profits E_e(Π) of the entering firm are greater than that E_m(Π) of any of the incumbent firms.

By letting ${\alpha}^{\ast}=\mathit{\max}\left\{{\alpha}_P^{\ast},{\alpha}_{\Pi}^{\ast}\right\}\in \left(0,1/\Big(m+1\right)\Big)$, the conclusion of Theorem 3.5 follows. QED