Advertising Strategies for a Duooly Model with Duo-markets and a budget constraint Ernie G.S. Teo Division of Economics, Nanyang Technological University Tianyin Chen School of Physical and Mathematical Sciences Nanyang Technological University June 6, 21 Abstract In the Colonel Blotto game of resource allocation, the army Colonel needs to otimally allocate troos to several battle fronts. This model has been alied to scenarios such as residential elections (Merolla, Munger and To as, 24). Insired by the resource allocation game, we investigate a di erential game where duooly rms strategically allocates advertising budget between two markets. This is an extension of the Lanchester duooly model of advertising. Each rm is constrained by a budget to be allocated to either market. The amount of advertising (relative to the rival rm) determines the rm s market share at any oint of time. We found that the equilibrium advertising budget allocated in a market decreases when its share in the other market increases. to advertise more when the market share is lower. This is in line with marketing institution The otimal strategy is to lace more resources in the weaker market. Given initial conditions, there are two ossible outcomes. The rst is where market shares converge to 5% for both rms. Another ossible outcome is one where each rm gets the full share of one market. This result is similar to the static General Blotto game as described by Golman and Page (26). Keywords: Advertising budget allocation, Lanchester model, Blotto games JEL Codes: C61, C73, G31, L1, M37 Corresonding Author. Tel.: +6565138179. Address: Division of Economics, Nanyang Technological University, Nanyang Avenue, Singaore 639798 E-mail address: gsteo@ntu.edu.sg 1
1 Introduction When undertaking economic activities, most rms in real life are constrained by some kind of budget. One may argue that the rm can borrow if it is otimal to send more that they have. This is not exactly true in real life, esecially in a ost 28 credit crisis world where banks are extra careful in their granting of loans. The idea of resource/ budget allocation is not entirely new. In game theory, the Colonel Blotto game is used to analyze scenarios with limited resources. The game traditionally deals with the allocation of troos over several battle elds. Deending on the seci cations of the game, there may not be any ure strategy equilibria. The next section discusses this grou of literature. The same story can be alied to cororate situations where rms are limited nancially on how much they can send on given rojects. Firms may need to choose which roducts to allocate resources for R and D, factors to consider include the robability of success, exected ro ts if successful and the number of other rms engaged in similar R and D. Another examle where rms may be limited by a budget is that of advertising in di erent geograhical markets. Advertising strategies are discussed frequently in the industrial organization literature, these usually assume a single market and unconstrained budgets. One classic examle is Grossman and Shairo (1984) s model of Informative Advertising with Di erentiated Products, this models a static game of advertisement choice. The advertising decision is dynamic in nature and the Lanchester model is frequently used analyze di erential games where advertising evolve over time according to market share, some examles of this are Chintagunta and Vilcassim (1992) and Erickson (1995). Using ideas from the Colonel Blotto literature, we modify the Lanchester duooly model to one with two markets and a budget constraint. We nd that the Markov erfect equilibrium where (deending on initial market shares) rms allocate similar budgets to each market and where rms concentrate all resources on a single market. The amount of advertising sent by a rm in a articular market is increasing with the amount of advertising send by the rival rm, decreasing with the share of that market and increasing with share in the other market. We start by discussing the relevant literature and roceed to our model of advertising in a duooly with duo-markets and a budget constraint. 1.1 Colonel Blotto Games The colonel blotto game of resource allocation is described as follows: Two army colonels (x and y) from oosing sides need to otimally distribute their troos over m battle- elds (or fronts). The arty that has allocated the most troos in a front will win the battle. Both arties do not know how many troos the oosing arty will allocate to each front, the side that wins the majority of the m fronts is the overall winner. x i 2
is the fraction of the budget (total number of troos) to be allocated by layer x to front i. Payo to x against y is where function mx sgn(x i y i ) (1) i=1 8 >< sgn(x i y i ) = >: 1 if x i y i > if x i y i = 1 if x i y i < This is a two erson zero sum game. A mixed strategy equilibrium always exists where each arty choose to strategically mismatch allocations. Gross and Wagner (195) show that the Colonel Blotto game has a mixed strategy equilibrium in which the marginal distributions of x i are uniform on [; 2 m ] along all fronts1. Roberson (26) de nes the Blotto Game Equilibrium using n-variate distributions or n-coulas. There is always a mixed strategy equilibrium to the Blotto game but not a ure strategy. Golman and Page (26) suggests a more general setu to include other ossibilities to the blotto game. They assume the following seci c form for the scoring function: (2) f() = sgn(): jj (3) The margin of victory () becomes imortant when >. When <1, it was found that there are no ure strategy Nash equilibrium. When =1, there is a ure strategy Nash equilibrium where 1 of the budget is allocated to every front. When >1, there is m a Nash equilibrium when all resources are allocated to any one front. In our dynamic model of advertising budget allocation where advertising decisions a ect market shares, we nd that both a mixed and ure strategy equilibrium exists, the equilibrium deends on the initial market share at time zero. This aer modi es the Lanchester duooly model (a commonly used dynamic market setu) and extends it to two markets with an advertising budget constraint. The next section discusses the Lanchester model and its results. 1 If n is the number of troos then the distribution will be uniform on [; 2 mn]. For examle, if there 2 are 1 Troos and 5 Fronts, m n = 2 51 = 4. Each layer s strategy is to choose between [,4] for each front. 3
1.2 The Lanchester Duooly Model and its Equilibrium Advertising Strategies The Lanchester model is a dynamic model of advertising cometition in a duooly. The following setu is taken from Jarrar, Martín-Herrán and Zaccour (24). There are 2 rms (1 and 2) and rm i battles for market share by choosing an amount of advertising (a i (t)) at each oint of time. M(t) denotes rm 1 s market share, thus Firm 2 s market share is (1 M(t)). The change in market share at any oint of time deends on the amount of advertising taken by each rm, each rm s advertising e ectiveness (given by k i ) and current market share (M(t)). This is given by the di erential equation: _M(t) = dm(t) d(t) = k 1 a1 (t)(1 M(t)) k 2 a2 (t)m(t); M() = M (4) The initial market share M 2 (; 1). Each layer maximizes a stream of discounted ro ts over an in nite horizon. Their objective functions are as follows (the time argument is omitted where no confusion may arise): max max a 1 Z 1 a 2 Z 1 e rt [g 1 M a 1 ]dt (5) e rt [g 2 (1 M) a 2 ]dt (6) g i is the gross margin er oint of market share realized at time t. To determine the stationary Markov erfect Nash equilibria (MPNE) one need to nd bounded and continuously di erentiable functions V i (M) satisfying the Hamilton-Jacobi-Bellman (HJB) equations: rv 1 (M) = max u 1 [g 1 M u 2 1 + V 2(M)(k 1 u 1 (1 M) k 2 u 2 M)] (7) rv 2 (M) = max u 2 [g 2 (1 M) u 2 2 + V 2(M)(k 1 u 1 (1 M) k 2 u 2 M)] (8) where u 2 i = a i ; V i (M) is the value function for layer i and V i (M) its derivative. To obtain the MPNE, the stes are to assume interior solutions, di erentiate the right-handsides of equations 7 and 8 and equate to zero to obtain: u 1 = k 1 (1 M)V1 (M); u 2 = k 2 2 2 MV 2 (M) (9) For M 2 (; 1); the above can be written as: V 1 (M) = 2u 1 k 1 (1 M) ; V 2 (M) = 2u 2 k 2 M (1) 4
Next equation 9 is substituted into equations 7 and 8. To solve for the MPNE, one need to ostulate a functional form for V i (M). This can be obtained numerically or the HJB equations can be simli ed so that they become analytically tractable. Literature following this second aroach, such as Chintagunta and Vilcassim (1992), assumes a zero discount rate and nd that advertising strategies are increasing functions of their own market shares. This result is counterintuitive as rms should invest more in advertising when they hold a smaller market share. Jarrar, Martín-Herrán and Zaccour (24) takes a numerical aroach to comute the Markov erfect Nash equilibrium advertising strategies, this method does not require layers to discount future earnings at a zero rate. This aroach entails rewriting the HJB equations as a system of two ordinary di erential equations for the value function which can be solved numerically by a standard mathematical ackage. It is shown that each layer s equilibrium advertising strategy is a decreasing function of its own market share. This is an intuitive result from a marketing oint of view. 2 The allocation of advertising budget in a duo-market duooly We extend dynamic model of advertising cometition in a duooly in a single market to one with two markets (x and y) and an advertising budget ( A). Each rm is restricted by an advertising budget ( A) to allocate between two markets, A = a i (t) + a i(y) (t), where a i (t) is the amount rm i sent on advertising in market x at time t and a i(y) (t) is the amount rm i sent on advertising in market y at time t. We assume 2 markets, rm 1 s market share in each market is given by M x and M y. Firm 2 s market share in market j is given by (1 M j ) Where the change in market share at any given oint in time is given by: _M x (t) = dm x(t) d(t) = k 1x a1 (t)(1 M x (t)) k 2x a2 (t)m x (t); M x () = M x (11) _M y (t) = dm y(t) d(t) = k 1y q A a1 (t)(1 M y (t)) k 2y q A a2 (t)m y (t); M y () = M y (12) Similar to the Lanchester model, k i reresents the advertising e ectiveness of rm i in market j. Each rm s objective function are as follows: max a 1 Z 1 e rt [g 1x M x + g 1y M y A]dt (13) 5
max a 2 Z 1 e rt [g 2x (1 M x ) + g 2y (1 M y ) A]dt (14) g ij is the gross margin er oint of market share of rm i in market j. We can determine the Markov erfect Nash equilibrium (MPNE) by using the HJB equation and assuming interior solutions. Proosition 1 a)the equilibrium advertising strategy by rm 1 in market x is given by: a 1 = My(1 Mx) M x(1 Mx) [ M ]2 Aa2 y) : (15) A + [ My(1 M x(1 M ]2 y) a 2 a 2 i) This is increasing with the rival rm s advertising in market x (a 2 ), ii) increasing with market share in market y (M y ) iii) and decreasing with market share in market x (M x ). b)the equilibrium advertising strategy by rm 2 in market x is given by: a 2 = Aa 1 A[ My(1 Mx) M x(1 M ]2 My(1 Mx) y) [ M x(1 M y) ]2 a 1 + a 1 : (16) i) This is increasing with the rival rm s advertising in market x (a 1 ), ii) increasing with market share in market y (1 M y ) iii) and decreasing with market share in market x (1 M x ). Proof. Firms maximize the following functions: max a 1 Z 1 e rt [g 1x M x + g 1y M y A]dt (17) subject to max a 2 Z 1 e rt [g 2x (1 M x ) + g 2y (1 M y ) A]dt (18) _M x (t) = dm x(t) d(t) = k 1x a1 (t)(1 M x (t)) k 2x a2 (t)m x (t); M x () = M x (19) _M y (t) = dm y(t) d(t) = k 1y q A a1 (t)(1 M y (t)) k 2y q A a2 (t)m y (t); M y () = M y (2) To determine the Markov erfect Nash Equilibrium, we set u the Hamilton-Jacobi- 6
Bellman (HJB) equations as follows: rv 1 (M x ; M y ) = max u 1 [g 1x M x + g 1y M y A (21a) + @V 1 (k 1x A sin u1 (1 M x ) k 2x A sin u2 M x ) + @V 1 (k 1y A cos u1 (1 M y ) k 2y A cos u2 M y )] rv 2 (M x ; M y ) = max u 2 [g 2x (1 M x ) + g 2y (1 M y ) A (21b) + @V 2 (k 1x A sin u1 (1 M x ) k 2x A sin u2 M x ) + @V 2 (k 1y A cos u1 (1 M y ) k 2y A cos u2 M y )] V i (M x ; M y ) is the value function for layer i. We assume a i = A sin 2 u i for comutation uroses. Assuming interior solutions, we di erentiate the R.H.S of equations 21a and 21b w.r.t u 1 and u 2 resectively and equate to zero to obtain: and = @V 1 k 1x A cos u1 (1 M x ) @V 1 k 1y A sin u1 (1 M y ) (22) ) tan u 1 = @V 1= : k 1x(1 M x ) @V 1 = k 1y (1 M y ) = : k 1x(1 M x ) k 1y (1 M y ) = @V 2 k 2x A cos u2 M x + @V 2 k 2y A sin u2 M y (23) ) tan u 2 = @V 2= @V 2 = : k 2xM x k 2y M y = : k 2xM x k 2y M y : k Equations 22 and 23 gives: tan u 1y (1 M y) 1 = @My k k 1x (1 M x) = tan u 2y M y 2 k 2x M x, so tan 2 u 1 tan 2 u 2 = ( k 1xk 2y k 1y k 2x ) 2 [ M y(1 M x ) M x (1 M y ) ]2 (24) Since a i = A sin 2 u i = A 2 (1 cos 2u i) = A 2 (1 1 tan 2 u i 1+tan 2 u i ) = A tan 2 u i 1+tan 2 u i ; tan 2 u i = a i A a i : (25) 7
Substituting equation 25 into equation 24, we get: a 1 ( A a 2 ) a 2 ( A a 1 ) = (k 1xk 2y ) 2 [ M y(1 M x ) k 1y k 2x M x (1 M y ) ]2 (26) Assuming that k 1x = k 1y and k 2x = k 2y (both rms are symmetric in both markets), we can solve for the equilibrium advertising strategies, a 1 and a 2: a 1 = My(1 Mx) M x(1 Mx) [ M ]2 Aa2 y) (27) A + [ My(1 M x(1 M ]2 y) a 2 a 2 A a 2 = 1 a 1 M M 2 (M x 1) (28) 2 x 2 y (M y 1) A 2 a1 + 1 @a Comarative statics are as follows: 1 @a 2 > ; @a 1 < ; @a 1 > ; @a 2 @a 1 > ; @a 2 > ; @a 2 < : When M y > M x ( rm 1 s osition is relatively stronger in market y than market x), the a 1 is lesser than a 2 (choose less advertising in market x than rm 2). Each rm ool resources to maintain osition in the stronger market. Conversely, Firm 1 will choose to advertise more than Firm 2 in Market x if Firm 1 s relative osition is stronger in Market x comared to Market y. This result is consistent with Jarrar, Martín-Herrán and Zaccour (24) and marketing intuition. This result is also artly due to budget limitations, as increasing advertising in one market means less advertising in the other. Since advertising budget is xed at A, it is otimal for the rm to divert resources to the market where market share is low. Firm 1 will choose to advertise more than rm 2 in the market where it wants to maintain the stronger osition. We assume that the advertising budget must be fully utilized, funds are diverted to the market where market share is relatively low comared to the other market. a 1 is an increasing function of a 2 and vice versa, rms behave cometitively with markets. The otimal strategy is to increase advertising in the market if the rival rm increase theirs. 2.1 Stability of equilibrium The equilibirum advertising strategy found in Proosition 1 may not yield a stable market share over time. In this section, we check the stability of the advertising strategies over market share. We assume that A=1 and k 1x = k 2x = k 1y = k 2y = 1 ( rms are symmetric in both markets) and examine di erent combinations of initial market share and advertising budget (M x (); M y (); a 1 () and a 2 ()). We nd that these initial states a ect the stability of market shares over time. When M x () = M y () = :5 and a 1 () = a 2 () = :5, market share is stable at.5 er market given this strategy, see Table 1 shows the evolution of market share over time when each rm emloys the equilbirum 8
advertising strategy in Proosition 1. Mx My a1 a2 t.5.5.5.5 t1.5.5.5.5 t2.5.5.5.5 t3.5.5.5.5 t4.5.5.5.5 t5.5.5.5.5 t6.5.5.5.5 t7.5.5.5.5 t8.5.5.5.5 t9.5.5.5.5 t1.5.5.5.5 Market Share.6.5.4.3.2.1. Mx My 1 2 3 4 5 6 7 8 9 1 11 Time Period Table 1, M x () = M y () = :5 and a 1 () = a 2 () = :5 When we check other ossible values of M x (); M y (); a 1 () and a 2 () (see aendix for some examles), we observe the following: i) When M x () = M y () = :5 and a 1 () = a 2 (), market shares will remain stable over time at.5. ii) When a 1 () = a 2 () = :5; and M x () = M y (), market shares will converge to.5 over time. iii) When M x () 6= M y () and/or a 1 () 6= a 2 (), over time, rms will reach full market share in one market and zero share in the other. This is not stable as the advertising strategy (a 1; a 2) requires all the advertising budget be allocated to the market with zero share. This will result in market shares iing between zero and one if rms follow the advertising strategy. This is artly because there is a binding condition to utilize the whole budget and the money is most e ciently sent in the market with zero market share. In this case, we suggest an additional condition for market shares to be stable; when share in market j reaches 1, the rm will change strategy to A amount sent on advertising in market j and zero sent in market j. 3 Discussion The otimal advertising strategy is to lace more resources in the weaker market (this is in line with tradition marketing intuition). A rm will also increase advertising in a market (all else being equal) when the rival rm increases their advertising. If both rms start with equal market share and allocation of budgets, then the advertising budget should be left unchanged, both rms share the two markets equally. With an initial state where either market shares or amount sent on advertising is equal between the two rms, equal market share will result in the long run when both rms follow the equilibrium advertising strategy. This result mirrors the ure strategy Nash equilibrium of the General Blotto game in Golman and Page (26) where each layer will allocate half of their budgets to each front if the two layers have symmetric 9
ayo s. Golman and Page (26) also suggests an equilibrium where all resources are allocated to any one front (when the margin of victory is imortant) this is similar to the case where M x () 6= M y () and/or a 1 () 6= a 2 () discussed in section 2.1. Although both the General Blotto game and our model are games of resource allocation, our model is dynamic in nature and the General Blotto game is not. The similarities in results suggests that there may be linkages between the two which can be further exlored. 4 Aendix 4.1 Market share over time (based on di erent initial states) Mx My a1 a2 t.5.5.2.2 t1.5.5.2.2 t2.5.5.2.2 t3.5.5.2.2 t4.5.5.2.2 t5.5.5.2.2 t6.5.5.2.2 t7.5.5.2.2 t8.5.5.2.2 t9.5.5.2.2 t1.5.5.2.2 Market Share.6.5.4.3.2.1. Mx My 1 2 3 4 5 6 7 8 9 1 11 Time Period Table A, M x () = M y () = :5 and a 1 () = a 2 () = :2 Mx My a1 a2 t.5.5.2.8 t1.28.72.99.1 t2.98.2. 1. t3. 1. 1.. t4 1... 1. t5. 1. #DIV/! #DIV/! t6 #DIV/! #DIV/! #DIV/! #DIV/! t7 #DIV/! #DIV/! #DIV/! #DIV/! t8 #DIV/! #DIV/! #DIV/! #DIV/! t9 #DIV/! #DIV/! #DIV/! #DIV/! t1 #DIV/! #DIV/! #DIV/! #DIV/! Market Share 1.2 1..8.6.4.2. Mx My 1 2 3 4 5 6 Time Period Table B, M x () = M y () = :5 and a 1 () = :2; a 2 () = :8 1
Mx My a1 a2 t.8.8.5.5 t1.38.38.5.5 t2.55.55.5.5 t3.48.48.5.5 t4.51.51.5.5 t5.5.5.5.5 t6.5.5.5.5 t7.5.5.5.5 t8.5.5.5.5 t9.5.5.5.5 t1.5.5.5.5 Market Share 1..8.6.4.2. Mx My 1 2 3 4 5 6 7 8 9 1 11 Time Period Table C, M x () = M y () = :8 and a 1 () = a 2 () = :5 Mx My a1 a2 t.8.2.5.5 t1.38.62.88.12 t2.83.17. 1. t3. 1. 1.. t4 1... 1. t5. 1. #DIV/! #DIV/! t6 #DIV/! #DIV/! #DIV/! #DIV/! t7 #DIV/! #DIV/! #DIV/! #DIV/! t8 #DIV/! #DIV/! #DIV/! #DIV/! t9 #DIV/! #DIV/! #DIV/! #DIV/! t1 #DIV/! #DIV/! #DIV/! #DIV/! Market Share 1.2 1..8.6.4.2. Mx My 1 2 3 4 5 6 Time Period Table D, M x () = :8; M y () = :2 and a 1 () = a 2 () = :5 11
References Jarrar, R., Martín-Herrán, G. and Zaccour, G., 24, Markov Perfect Equilibrium Advertising Strategies of Lanchester Duooly Model: A Technical Note, Management Science, 5,. 995-1 Whittaker, J. C., 1996, The allocation of resources in a multile trial war of attrition con ict, Advance Alied Probability, 28, 933-964 Erickson, G., 1995, Di erential game models of advertising cometition, Euroean Journal of Oerations Research, 83, 431-438 Gross, O., and Wagner, R., 195, A continuous colonel blotto game, U.S. Airforce Project RAND Research Memorandum Grossman, G., and Shairo, C., 1984, Informative Advertising with Di erentiated Products, The Review of Economic Studies, 51,. 63-81 Golman, R., and Page, S. E., 29, General Blotto: game of allocative strategic mismatch, Public Choice, 138, 279-299 Roberson, B., 26, The colonel blotto game, Economic Theory, 29, 1-24 Chintagunta, P. K., and Vilcassim, N. J., 1992, An emirical investigation of advertising strategies in a dynamic duooly, Management Science, 38 123-1244 12