Incumbents’ defense strategies: a comparison of deterrence and shakeout strategy based on evolutionary game theory

Abstract

This paper addresses the question of how an established firm can successfully defend its market against current and future competitors. Previous studies on this issue are surprisingly scarce and typically concentrate on only a single generic defense strategy. Thus, little is known about the degree and the manner in which different generic defense strategies, such as a deterrence strategy (pursued before competitor market entry) and a shakeout strategy (pursued after competitor market entry), differ in effectiveness and efficiency and about the corresponding role of product and market conditions. As these strategies tend to be costly, an established firm must decide which of these strategies to focus its scarce resources on. Drawing on evolutionary game theory and an empirical calibration and validation study, this paper seeks to fill these research gaps. While both strategies turn out to be viable options for market defense, the authors find that in general, a shakeout strategy tends to be superior to a deterrence strategy. However, the authors also identify product and market conditions under which an established firm is better off focusing on a deterrence strategy. In methodological respects, the paper contributes to the marketing discipline by introducing evolutionary game theory, which has not been used previously for analyzing marketing issues, as well as an evolutionary approach to research on market defense.

Appendices

Appendix A

Long-term basic structure of the model

As stated in the main text, the dynamics of the model is given by the linear recurrence equation

$$ {n^{{t + 1}}} = A{n^t},{ }t \in \mathbb{N}, $$

(A.1)

with a non-negative matrix A. According to the Perron-Frobenius theorem for non-negative matrices (Horn and Johnson 1990), which ensures the existence of eigenvalues and eigenvectors of a real positive matrix, Eq. A.1 always converges towards the equation

$$ \lambda u = Au, $$

(A.2)

where u is a right eigenvector with respect to eigenvalue λ and describes the firm’s long-term market structure u = (u _M , u _O , u _RM ). The condition A.2 corresponds to the system of equations

$$ \begin{gathered} \lambda {u_M} = P(1 - \eta ){u_M} + \zeta \left( {{F_M}{u_M} + {F_O}{u_O} + {F_{{RM}}}{u_{{RM}}}} \right), \hfill \\ \lambda {u_O} = P\eta {u_M} + P\sigma (1 - \rho ){u_O}, \hfill \\ \lambda {u_{{RM}}} = P\sigma \rho {u_O} + P{u_{{RM}}}. \hfill \\ \end{gathered} $$

(A.3)

The last two equations determine the long-term market structure u up to a constant factor:

$$ {u_M}:{u_O}:{u_{{RM}}} = \frac{{\lambda - P\sigma \left( {1 - \rho } \right)}}{{P\eta }}:1:\frac{{P\sigma \rho }}{{\lambda - P}} $$

(A.4)

The dominant eigenvalue λ reflects that once a firm’s long-term market structure u is reached, the firm’s total number of markets will grow with a factor λ per period. This factor is introduced to resolve the trade-off between positive and negative consequences of defense investments and thus to identify the optimal amount of defense investment x*.

Because the firm’s total number of markets cannot infinitely increase (e.g., owing to merely a finite number of potential markets as well as prohibitively high transaction and coordination costs in reality), it is reasonable to assume that λ does not exceed 1. In contrast, λ can be less than 1, as the firm’s total number of markets may well shrink to zero in the long run. However, there are two different types of possible causes of this decrease—defense-related causes (i.e., whether and how the firm defends its markets against competitors) and non-defense-related causes (e.g., declining product lifecycles). Unfortunately, the mixture of the effects of these two different types of causes does not allow separate analysis of the effect of defense-related causes (i.e., of the firm’s amount of defense investment x) on the firm’s number of markets in the long run and thus on the firm’s long-term survival (which is, however, the goal of the model).

Hence, to be able to identify a firm’s optimal amount of investment x*, it is necessary to assume that (only) in case of the optimal amount of investment x* (which enables the firm to successfully defend its markets against current and future competitors and thereby maximizes the probability of long-term survival), the firm’s number of markets does not shrink to zero in the long run (for a similar logic, see van Boven and Weissing 2004; Caswell 2001). Thus, in this case only, we need to normalize λ to 1 by means of the parameter ζ. In all other cases (i.e., a suboptimal amount of investment), λ is less than 1. This is also in line with our focus on the firm’s long-term survival. Specifically, if the firm defends its markets in a suboptimal way, it will not be able to withstand competitive forces in the long run. Thus, its market position will sooner or later erode, which in turn forces the firm’s market exit (Karakaya 2000; Madsen and Walker 2007). The requirement on ζ corresponding to the condition λ = 1 results from inserting Eq. A.4 into the first equation of A.3:

$$ \zeta = \frac{{1 - P\,\eta }}{{({F_M} + \frac{{P(1 - \eta )}}{{1 - P\sigma (1 - \rho )}}({F_O} + \frac{{P\sigma \rho }}{{1 - P}}{F_{{RM}}}))}} $$

(A.5)

In the following, we normalize the right eigenvector u so that:

$$ {u_M} + {u_O} + {u_{{RM}}} = 1 $$

(A.6)

Such a standardization is useful because then u _O directly corresponds to the proportion of the firm’s oligopoly markets. Using the last two equations of A.3, Eq. A.6, and the assumption λ = 1, a straightforward calculation yields the explicit expressions for u _M , u _O , and u _RM :

$$ \begin{gathered} {u_M} = \frac{{(1 + P( - 1 + \rho )\sigma )}}{{gP}} \hfill \\ {u_O} = \frac{{(1 - P)\{ P(g + \sigma (1 - \rho )) - 1\} }}{{gP(1 - P(1 - \sigma \rho ))}} \hfill \\ {u_{{RM}}} = \frac{{\rho \sigma ( - 1 + P(g + \sigma - \rho \sigma ))}}{{g(1 + P( - 1 + \rho \sigma ))}} \hfill \\ \end{gathered} $$

(A.7)

Appendix B

Calculation of the optimal defense investment

To derive the optimal defense investment, i.e., the evolutionary stable strategy (ESS), we compare two alternative investments x and y competing with each other. It is important to recognize that the success of the investment y also depends on the investment x due to reputational effects. Thus, according to Eq. 4 in the main text, the probability η of a market entry attempt is represented by η(y, x) = g(y) u _O (x). Overall, we obtain the following matrix A(y, x) that describes the dynamics of incumbent’s market structure, when the incumbent pursues the investment strategy y (van Boven and Weissing 2004):

$$ A(y,x) = \left( {\matrix{{*{20}{c}} {P(1 - \eta (y,x)) + \zeta (x){F_M}(y)} & {\zeta (x){F_O}(y)} & {\zeta (x){F_{{RM}}}(y)} \\ {P\eta (y,x)} & {P\sigma (y)(1 - \rho (y))} & 0 \\ 0 & {P\sigma (y)\rho (y)} & P \\ } } \right) $$

(B.1)

The parameter ζ depends on x for technical reasons. λ is normalized to 1 in the case of an optimal incumbent’s investment x*. To ensure the comparability to an alternative investment y, both long-term growth rates have to be normalized by the factor ζ (x*).

As stated in the main text, the optimal investment x* is characterized by the fact that no alternative investment y leads to a higher λ when competing with x*, i.e., the investment x* “is a strict best response to itself” (Samuelson 2002, p. 49). In other words, the defense investment x* is optimal (i.e., evolutionary stable), if the following condition holds:

$$ \lambda \left( {y,{x^{*}}} \right) < \lambda \left( {{x^{*}},{x^{*}}} \right) = {1},{\text{for\, all}}\, y \ne {x^{*}} $$

(B.2)

It is worth mentioning that λ(x*, x*) = 1 owing to the normalization by the parameter ζ (see Eq. A.5). Thus, x* is evolutionary stable, if the function λ(y, x*) attains a maximum in y at y = x*, i.e.,

$$ \frac{{\partial \lambda (y,{x^{*}})}}{{\partial y}}\left| {_{{y = {x^{*}}}} = 0.} \right. $$

(B.3)

By means of the second-order condition

$$ \frac{{{\partial^2}\lambda (y,{x^{*}})}}{{{\partial^2}y}}\left| {_{{y = {x^{*}}}}} \right. < 0 $$

(B.4)

it can be examined whether x* does indeed correspond to a maximum. The second-order condition

$$ \frac{{{\partial^2}\lambda (y,{x^{*}})}}{{{\partial^2}y}}\left| {_{{y = {x^{*}}}}} \right. + \frac{{{\partial^2}\lambda (y,{x^{*}})}}{{\partial x\partial y}}\left| {_{{x = y = {x^{*}}}}} \right. < 0 $$

(B.5)

is also relevant, because it allows examining whether x* is convergence stable. Convergence stability ensures that the evolutionary stable strategy can be attained by a series of strategy substitution events and thereby provides a dynamic perspective of the ESS concept (Eshel 1983; Taylor 1996). To apply Eq. B.3, we need the dominant eigenvalue λ(y, x) of the Matrix A(y, x), which is, in practice, difficult to calculate. However, we simplify Eq. B.3 using left eigenvectors (Caswell 2001). The left eigenvector v with respect to eigenvalue λ = 1 is given by the equation v = vA, and can be calculated up to a constant factor:

$$ {v_M}:{v_O}:{v_{{RM}}} = 1:\frac{{1 - P(1 - \eta ) - \xi {F_M}}}{{P\eta }}:\frac{{\xi {F_{{RM}}}}}{{1 - P}} $$

(B.6)

The eigenvalue λ = λ (y, x*) of Matrix A = A(y, x ^*) as well as its derivative with respect to y can be calculated by means of right and left eigenvectors u = u(y, x ^*) and v = v(y, x ^*):

$$ \lambda = \frac{{vAu}}{{vu}};{ }\frac{{\partial \lambda }}{{\partial y}} = \frac{{v({{{\partial A}} \left/ {{\partial y}} \right.})u}}{{vu}} $$

(B.7)

Consequently, because of the conditions B.1 and B.7, x ^* can be calculated based on the equation

$$ {v^{*}}\frac{{\partial A(y,{x^{*}})}}{{\partial y}}\left| {_{{y = {x^{*}}}}} \right.{u^{*}} = \sum\limits_{{i,j}} {v_i^{*}u_j^{*}} \frac{{\partial {a_{{ij}}}(y,{x^{*}})}}{{\partial y}} = 0, $$

(B.8)

where $v_i^{*} = {v_i}({x^{*}})$ and $u_i^{*} = {u_i}({x^{*}})$ denote the elements of the left and right eigenvector of the Matrix A(x ^*, x ^*), respectively. Inserting the elements of Matrix B.1 in Eq. B.8, we derive the ESS condition (see Eq. 5 in the main text).

The second-order conditions B.4 and B.5 can be controlled by means of left and right eigenvectors (for further details, see Caswell 2001, chap. 9.4). By normalizing the left and right eigenvectors so that vu = 1, we obtain:

$$ \begin{gathered} \frac{{{\partial^2}\lambda (y,{x^{*}})}}{{{\partial^2}y}}\left| {_{{y = {x^{*}}}}} \right. = {v^{*}}\frac{{{\partial^2}A}}{{{\partial^2}y}}\left| {_{{y = {x^{*}}}}} \right.{u^{*}} + \frac{{\partial v}}{{\partial y}}\left| {_{{y = {x^{*}}}}} \right.\frac{{\partial A}}{{\partial y}}\left| {_{{y = {x^{*}}}}} \right.{u^{*}} + {v^{*}}\frac{{\partial A}}{{\partial y}}\left| {_{{y = {x^{*}}}}} \right.\frac{{\partial u}}{{\partial y}}\left| {_{{y = {x^{*}}}}} \right. \hfill \\ \frac{{{\partial^2}\lambda (y,{x^{*}})}}{{\partial x\partial y}}\left| {_{\begin{subarray}{l} y = {x^{*}} \\ x = {x^{*}} \end{subarray} }} \right. = {v^{*}}\frac{{{\partial^2}A}}{{\partial x\partial y}}\left| {_{\begin{subarray}{l} y = {x^{*}} \\ x = {x^{*}} \end{subarray} }} \right.{u^{*}} + \frac{{\partial v}}{{\partial x}}\left| {_{{x = {x^{*}}}}} \right.\frac{{\partial A}}{{\partial y}}\left| {_{{y = {x^{*}}}}} \right.{u^{*}} + {v^{*}}\frac{{\partial A}}{{\partial y}}\left| {_{{y = {x^{*}}}}} \right.\frac{{\partial u}}{{\partial x}}\left| {_{{x = {x^{*}}}}} \right. \hfill \\ \end{gathered} $$

(B.9)

Appendix C

Table 3 Questions for model calibration and validation