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# Global and local advertising strategies: A dynamic multi-market optimal control model

• Differential games have been widely used to model advertising strategies of companies. Nevertheless, most of these studies have concentrated on the dynamics and market structure of the problem, neglecting their multi-market dimension. Since nowadays competition typically operates on multi-product contexts and usually in geographically separated markets, the optimal advertising strategies must take into consideration the different levels of disaggregation, especially, for example, in retail multi-product and multi-store competition contexts. In this paper, we look into the decision-making process of a multi-market company that has to decide where, when and how much money to invest in advertising. For this purpose, we develop a model that keeps the dynamic and oligopolistic nature of the traditional advertising game introducing the multi-market dimension of today's economies, while differentiating global (i.e. national TV) from local advertising strategies (i.e. a price discount promotion in a particular store). It is important to note, however, that even though this problem is real for most multi-market companies, it has not been addressed in the differential games literature. On the more technical side, we steer away from the traditional aggregated dynamics of advertising games in two aspects. Firstly, we can model different markets at once, obtaining a global instead of a local optimum, and secondly, since we are incorporating a variable that is common to markets, the resulting equations systems for every market are now coupled. In other words, one's decision in one market does not only affect one's competition in that particular market; it also affects one's decisions and one's competitors in all markets.

Mathematics Subject Classification: Primary: 91A23, 91A25, 91A35, 91A40, 91B26.

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• Figure 2.  $\mu_{i}(t)$ for the global case in table 2

Figure 3.  $\lambda^{i}(t)$ for the global case in table 2

Figure 4.  Phase space diagram ($\lambda^{i}$ versus $x$) for the global case in table 2

Figure 1.  $x(t)$ for the global case in table 2

Figure 5.  Game description

Figure 6.  Case 1 in table 3

Figure 7.  Case 2 in table 3

Figure 8.  Case 3 in table 3

Figure 9.  Case 4 in table 3

Figure 10.  The pure-global case, case 0

Figure 11.  The quasi-global case, case 1

Figure 12.  The local case, case 2

Table 1.  Notation

 $J_{i}$ Profit function of player $i$ $x_{ik}(t)$ Market share of player $i$ at location $k$ $q_{ik}$ Gross profit rate per unit of market share of player $i$ at location $k$ $Q_{ik}$ Second order gross profit rate per unit of market share of player $i$ at location $k$ $b_{ik}$ Linear local advertising cost of player $i$ at location $k$ $B_{ik}$ Second order local advertising cost of player $i$ at location $k$ $e_{i}$ Linear global advertising cost of player $i$ at location $k$ $E_{i}$ Second order global advertising cost of player $i$ $\sigma_{ik}$ Effectiveness of local advertising of player $i$ at location $k$ $\sigma_{i}$ Effectiveness of global advertising of player $i$ at location $k$ $r_{i}$ Discount rate of player $i$ $\mu_{ik}(t)$ Local advertising effort of player $i$ at location $k$ $\mu_{i}(t)$ Global advertising effort of player $i$

Table 2.  Data for the pure global game.

 Global parameters $q_{11}$ 0.8 $q_{21}$ 0.3 $Q_{11}$ 0.0 $Q_{21}$ 0.0 $e_{1}$ 0.0 $e_{2}$ 0.0 $E_{1}$ 0.5 $E_{2}$ 0.3 $\sigma_{1}$ 0.95 $\sigma_{2}$ 1.6 $r_{1}$ 0.01 $r_{2}$ 0.05

Table 3.  Data of numerical examples

 Case 1 Case 2 Case 3 Case 4 Firm 1 Firm 2 Firm 1 Firm 2 Firm 1 Firm 2 Firm 1 Firm 2 1 2 1 2 1 2 1 2 $q_{i1}$ 3 3 3 3 3 3 3 3 $q_{i2}$ 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 $B_{i1}$ 1 1 1 1 1 1 1 1 $B_{i2}$ 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 $E_{i}$ 1 1 1 2 1 1 1 1 $\sigma_{i1}$ 0.1 0.1 0.1 0.1 0.7 0.1 0.7 0.8 $\sigma_{i2}$ 0.1 0.1 0.1 0.1 0.7 0.1 0.7 0 $\sigma_{i}$ 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

Table 4.  Data for pure-global (case 0), quasi-global (case 1) and local effects (case 2).

 Case 0: pure-global Case 1: quasi-global Case 2: local effects Firm 1 Firm 2 Firm 1 Firm 2 Firm 1 Firm2 \hline $q_{i1}$ 0.8 0.3 0.8 0.3 0.8 0.3 $q_{i2}$ 0.8 0.3 0.8 0.3 0.8 0.3 $Q_{i1}$ 0 0 0 0 0 0 $Q_{i2}$ 0 0 0 0 0 0 $b_{i1}$ 0 0 0 0 0 0 $b_{i2}$ 0 0 0 0 0 0 $B_{i1}$ 0 0 0.001 0.001 5 0.1 $B_{i2}$ 0 0 0.001 0.001 1 2 $e_{i}$ 0 0 0 0 0 0 $E_{i}$ 1 2 1 2 1 2 $\sigma_{i1}$ 0 0 0 0 0.1 0.3 $\sigma_{i2}$ 0 0 0 0 0.6 0.1 $\sigma_{i}$ 1 1.9 1 1.9 1 1.9 $r_{i}$ 0.01 0.05 0.01 0.05 0.01 0.05
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