Strategic Rivalry for Market Share: A Contest Theory Approach to Dynamic Advertising Competition

Abstract

The paper extends the Lanchester model of advertising competition to a setup in which the rate at which a firm attracts customers from its competitors depends not only on the firm’s own advertising effort, but also on the efforts of its rivals. Doing so enables us to use attraction rate specifications borrowed from the economic theory of contests. Exploiting the fact that the sum of attraction rates equals one, we show that the differential equations that define the evolution of market shares in the Lanchester model can be considerably simplified. This makes the optimization problems of the firms considerably easier to analyze. Finallly, to illustrate how the above extensions work, three alternative specifications of attraction rates are studied: the Tullock ratio formulation, a linear transformation of the Tullock ratio, and a specification that incorporates an exogenous bias.

Appendix: Proofs

Proof of Lemma 1

Market share dynamics are $\dot{X}_{i}(t) =g_{i}(a(t))-X_{i}(t)$ which implies $\sum _{i=1}^{N} \dot{X}_{i}(t) =1-\sum _{i=1}^{N}X_{i}(t)$. Defining $Y(t)\triangleq \sum _{i=1}^{N}X_{i}(t) ,$ the dynamics can be written as $\dot{Y}(t)=1-Y(t)$. This differential equation has the particular solution $Y(t) =1\Leftrightarrow \sum _{i=1}^{N}X_{i}(t) =1$. We conclude that $X_{i}(t) \le 1$. Integrating the market share dynamics yields $X_{i}(t)=x_{i0}e^{-t}+e^{-t}\int _{0}^{t}g_{i}(a(t))e^{t}dt$ and hence $X_{i}(t)\ge 0$ for all t. $\square$

Proof of Proposition 1

The HJB equation for firm i is

$$\begin{aligned} -\dot{\alpha }_{i}(t)-\dot{\gamma }(t)X_{i}=\max _{a_{i}\ge 0}\left\{ p_{i}X_{i}-a_{i}+\gamma _{i}(t)\left[ g_{i}(a_{i},a_{-i})-X_{i}\right] \right\} . \end{aligned}$$

Collecting coefficients of $X_{i}^{1}$ and $X_{i}^{0}$ readily shows that value function parameters $\alpha _{i}(t)$ and $\gamma _{i}(t)$ must satisfy

$$\begin{aligned} \dot{\gamma }_{i}(t)= & {} -p_{i}+\gamma _{i}(t);\,\, \gamma _{i}(T)=0 \\ \dot{\alpha }_{i}(t)= & {} -\gamma _{i}(t)g_{i}(a(t))+a_{i}(t);\,\, \alpha _{i}(T)=0 \end{aligned}$$

which completes the proof. $\square$

Proof of Proposition 2

The equations in [18] have the following solution, valid for $t\in [t,T):$

$$\begin{aligned} a_{i}(t) =\frac{\gamma _{i}^{2}(t) \gamma _{j}(t) }{\left( \gamma _{i}(t) +\gamma _{j}(t) \right) ^{2}}>0;\quad a_{j}(t) =\frac{\gamma _{j}^{2}(t) \gamma _{i}(t) }{\left( \gamma _{i}(t) +\gamma _{j}(t) \right) ^{2}}>0. \end{aligned}$$

Inserting from (11), advertising rates can be expressed as

$$\begin{aligned} a_{i}(t)=\frac{p_{i}^{2}p_{j}}{\left( p_{i}+p_{j}\right) ^{2}}\left( 1-e^{t-T}\right) ,\quad a_{j}(t)=\frac{p_{j}^{2}p_{i}}{\left( p_{i}+p_{j}\right) ^{2}}\left( 1-e^{t-T}\right) \end{aligned}$$

from which the equilibrium attraction rates follow. The market shares in (20) are found by integration. $\square$

Proof of Proposition 3

The optimal profits of the firms can be rewritten as

$$\begin{aligned} V^{i}(x_{i0},0)= & {} \frac{p_{i}^{3}\left( e^{-T}-1+T\right) +p_{i}x_{i}\left( p_{i}+p_{j}\right) ^{2}\left( 1-e^{-T}\right) }{\left( p_{i}+p_{j}\right) ^{2}} \\ V^{j}(x_{j0},0)= & {} \frac{p_{j}^{3}\left( e^{-T}-1+T\right) +p_{j}x_{j}\left( p_{i}+p_{j}\right) ^{2}\left( 1-e^{-T}\right) }{\left( p_{i}+p_{j}\right) ^{2}} \end{aligned}$$

and, given the two assumptions of the proposition, the result of the proposition follows. $\square$

Proof of Proposition 4

Noting that the attraction rates are

$$\begin{aligned} g_{i}(a(t))=A_{i}+\frac{p_{i}(t)}{p_{i}(t)+p_{j}(t)} \end{aligned}$$

in which the second term on the right-hand side is the attraction rate valid for the unbiased case. Inserting the above attraction rates into the market share dynamics and integrating yields the equilibrium market shares. $\square$

Proof of Propositions 5, 6, and 7

The calculations essentially proceed in the same way as in Sects. 6.1 and 6.2 and it seems safe to omit the details. $\square$