The Sequential Hotelling Game with One Parameterized Location

Abstract

This article studies the location–price Hotelling game. Numerous studies have been conducted on the Hotelling game with simultaneous decisions; however, in real-life scenarios, decisions are frequently sequential. Unfortunately, studies on the sequential Hotelling (SHOT) game are quite scarce. This article contributes to the study of the SHOT game by considering the case in which the location of one of the players, either the leader or the follower, is externally fixed. The game is studied analytically and by numerical simulation to address scenarios where mathematical analysis is cumbersome due to the discontinuous nature of the game. Simulation is found to be particularly useful in evaluating the subgame perfect equilibrium (SPE) solution of these SHOT games, where the follower outperforms the leader as a very general rule, with very few exceptions. This article complements a previous study of the SHOT game where the two locations are parameterized and paves the way to address the analysis of more sophisticated formulations of the SHOT game, such as those with reservation cost and with elastic demand.

1. Introduction

The spatial competition model defined by H. Hotelling in [1] involves two vendors located on a line selling an identical product with customers equally spread along this line. These firms compete on location and price in the proposed homogeneous market, so that a customer decides to buy the product of a firm depending on the price and the transportation cost to the point of sale, assumed to be linear with the distance in the initial model. The sum of the price of the product and the transportation cost associated with a customer represents the costs associated with purchasing the product by this customer. He established what is known as the principle of minimum differentiation, which means that firms make product that tends to be more equal or, as Hotelling said in his article: “an undue tendency for competitors to imitate each other in quality of goods, location, and in other essential ways”. In the concrete case of Hotelling’s initial model, it implies that companies tend to select similar locations for their stores. Subsequently, a large body of literature on spatial competition and product differentiation emerged, inspiring the development of spatial models of political competition and becoming an indispensable part of economics teaching [2].

Although the Hotelling game caused great criticism due to its limitations, it could not be proven that the principle of minimum differentiation is invalid until fifty years later in [3], where it was proven that a Nash equilibrium (NE) does not exist when the players are too close, which was not noticed in the seminal analysis by Hotelling [1]. In order to circumvent this problem of the nonexistence of NE, the quadratic instead of linear transportation cost was introduced in [3], although some authors are reluctant to adopt this variant of the game, facing its relevance and/or validity [4].

In the usual approach to the Hotelling game, both players decide simultaneously. In contrast to this, the aim of this article is the study of the behavior of the game when the players decide in a leader–follower sequential manner and the location of one of the players, either the leader or the follower, is externally fixed.

The sequential approach to the Hotelling game has been very little studied. This article contributes to its study in the conventional formulation of the game from a simulation-based perspective. This approach can be applied to more sophisticated formulations of the Hotelling game and to other continuous games with discontinuities in their payoff functions.

Section 2 introduces the Hotelling game both with simultaneous choices and with sequential choices (SHOT game in Section 2). The numerical simulation technique implemented in this article is introduced in Section 3. The SHOT game with linear transportation cost is simulated in Section 4, and the SHOT game with quadratic transportation cost (SHOT2) is simulated in Section 5.

2. The Hotelling Game

In the Hotelling (HOT) game introduced in the seminal article [1], two players (1 and 2) are located in a line of length ${\displaystyle L}$ at locations ${\displaystyle x_1=a\le L/2}$ and ${\displaystyle x_2=L-b\ge L/2}$. They sell a homogeneous product at prices ${\displaystyle p_1}$ and ${\displaystyle p_2}$ to consumers uniformly distributed throughout the line. If the transportation cost is linear with respect to the distance to the player, the expenses (or full prices) of a generic consumer located at ${\displaystyle s}$ are as follows: ${\displaystyle e_i(s)=p_i+t|s-x_i|,i=1,2}$ . As a result, the indifferent consumer, where ${\displaystyle e_1(s)=e_2(s)}$, is located at ${\displaystyle \overline{s}=\frac{1}{2}\left(s_x+\frac{p_2-p_1}{t}\right),s_x=x_1+x_2=L+k;k=a-b}$, so that the demands to both players are ${\displaystyle d_1=\overline{s}}$, ${\displaystyle d_2=L-\overline{s}}$.

The HOT game is sketched in Figure 1, where the players are presented as “ice cream dealers” [4]. In the figure, two consumers have been identified: one who buys from player 1, and the indifferent one, maybe the young Hotelling (inset).

The payoff functions (${\displaystyle u}$) in the HOT game are given in Equation (1), which also take into account the capture of the entire market by a player who charges a very low price, namely, that undercuts the other player. Note that there are two discontinuities in the ${\displaystyle u}$-formulas of Equation (1)  (e.g., Figure 6b,c). The existence of such discontinuities is at the origin of the difficulties that arise in the study of the HOT game.

$$u_1=\left\{\begin{array}{ccc}L{p}_1& \mathrm{if}& p_1<p_2-t{d}_x\\ d_1{p}_1& \mathrm{if}& |p_1-p_2|<t{d}_x\\ 0& \mathrm{if}& p_1>p_2+t{d}_x\end{array}\right.,u_2=\left\{\begin{array}{ccc}L{p}_2& \mathrm{if}& p_2<p_1-t{d}_x\\ d_2{p}_2& \mathrm{if}& |p_1-p_2|<t{d}_x\\ 0& \mathrm{if}& p_2>p_1+t{d}_x\end{array}\right.d_x=x_2-x_1$$

(1)

The Nash equilibrium (NE) in prices in the HOT game with given locations found in [1] is given in Equation (2)1. That is, ${\displaystyle p_1^{\star }}$ is the best response to ${\displaystyle p_2^{\star }}$ and ${\displaystyle p_2^{\star }}$ is the best response to ${\displaystyle p_1^{\star }}$.

$$(p_1^{\star },p_2^{\star })=t{\displaystyle \frac{1}{3}}(3L+k,3L-k)⟶(d_1^{\star },d_2^{\star })={\displaystyle \frac{1}{6}}(3L+k,3L-k)$$

(2)

Because ${\displaystyle u_1^{*}=d_1^{*}{p}_1^{*}}$ increases with ${\displaystyle a}$ and ${\displaystyle u_2^{*}=d_2^{*}{p}_2^{*}}$ with ${\displaystyle b}$, both players would tend to coincide in their location, a phenomenon that in [1] is referred to as the minimum differentiation principle. However, long after the seminal article by Hotelling, it was proven that an NE only exists under constraints that impede such an unrestricted approach in the player locations in NE [3,5,6,7]. In order to circumvent the serious problem of the nonexistence of an NE, quadratic transportation cost with respect to the distance, i.e., ${\displaystyle e_i(s)=p_i+t{(s-x_i)}^2,i=1,2}$ was proposed in [3]. The HOT game Quadratic transportation cost is also considered in this study, albeit some authors are reluctant to consider this somehow ad hoc variant of the game. For example, T.Puu stated in [8]: “Further, to escape some consequences of the paradox, the unrealistic and contrived idea of quadratic transportation costs was launched and became popular”.

Note that the HOT game is not studied in a global way, that is, considering simultaneously location and price variables, a scenario in which no NE exists [9]. In contrast to the global approach, the game is analyzed in a two-stage process, first analyzing the effect of the choice of prices with fixed locations and then considering the locations as variable parameters. It is also important to note that in the standard approach to the game, both players decide simultaneously.

There exists an immense body of literature dealing with the original Hotelling game and with its variants that is not reviewed here. However, we refer to the recent overview in [2]. References dealing with sequential moves in the HOT game in particular are quite scarce. Among them, the article by S.Anderson, to which we will refer in this study, stands out. In reference [10], only the linear transportation cost is considered. In contrast to this, most of the not numerous articles dealing with the SHOT game stray to quadratic transportation cost, easier to analyze. This is the case of the reference [11]. By the way, in the HOT review reference [2], sequential HOT is not mentioned, in fact the article [10] does not appear in its bibliography.

The Sequential Hotelling Game

At variance with an implicit assumption in the seminal formulation of the HOT game, in this article the players do not decide simultaneously but sequentially. Thus, one of the players, the leader (or first mover) decides first, then the other player, the follower (or second mover), adopts the best response to the known decision of the leader. The sequential Hotelling game will be referred to as the SHOT game, where according to the backwards induction principle, in the SHOT game the leader assumes that the follower player 2 would react optimally to any choice of player 1. Figure 2 sketches the backwards induction principle in the price variables, where ${\displaystyle \beta }$ stands for best response and the ^⋆-marked prices denote the SPE solution.

In a previous study [12], the two-stage approach (first prices, then locations) was scrutinized in the SHOT game. In this work, an advance is made in the direction of a global approach to the study of the SHOT game by analyzing two complementary SHOT game scenarios: (${\displaystyle i}$) the leader decides not only the price but also the location, while the location of the follower is fixed exogenously (scenario referred to as SHOT-b), and (${\displaystyle ii}$) the follower decides not only the price but also the location, whereas the location of the leader is fixed exogenously (SHOT-a).

The mathematical analysis of the SHOT game is complicated in general due to the discontinuities in the response functions. This is why we will resort to simulation in a lattice, as explained in the next section. Additionally, the backwards induction principle will be programmed in a direct way (as explained when commenting Figures 13 and 18), although this approach demands very high computational resources.

3. Numerical Simulation

In the numerical simulations in this article, a large number of players of type 1 (leader) and of type 2 (follower) are arranged in a two-dimensional ${\displaystyle N\times N}$ lattice. Every ${\displaystyle (i,j)}$ cell of the lattice is occupied by a player, alternating in site occupation in a chessboard form, so that if ${\displaystyle 0\equiv (i+j)(mod2)}$ the cell ${\displaystyle (i,j)}$ is occupied by a player 1, and if ${\displaystyle 1\equiv (i+j)(mod2)}$ the cell ${\displaystyle (i,j)}$ is occupied by a player 2.

Consequently, every player is surrounded by four partners (${\displaystyle 1}$-${\displaystyle 2}$, ${\displaystyle 2}$-${\displaystyle 1}$), and four mates (${\displaystyle 1}$-${\displaystyle 1}$, ${\displaystyle 2}$-${\displaystyle 2}$) as Figure 3 illustrates.

The initial prices ${\displaystyle p}$ are assigned to both types of players at random from a uniform distribution in the ${\displaystyle [0,p_{max}]}$ interval, where ${\displaystyle p_{max}}$ denotes the maximum available price. Thus, initially it is ${\displaystyle \overline{p}\simeq p_{max}/2}$ and ${\displaystyle {\sigma }_p\simeq \sqrt{p_{max}^2/12}}$ for both players. In the SHOT-b game, the initial locations ${\displaystyle x_1}$ are assigned to the leader players at random from a uniform distribution in the ${\displaystyle [0,L/2]}$ interval, while all the player 2 followers in the lattice are assigned to a fixed ${\displaystyle x_2}$ location. Therefore, it is initially ${\displaystyle \overline{x_1}\simeq L/4}$ in the SHOT-b game. In the SHOT-a game, the initial locations ${\displaystyle x_2}$ are assigned to the follower players at random from a uniform distribution in the ${\displaystyle [L/2,L]}$ interval, while all the player 1 leaders in the lattice are assigned to a fixed location ${\displaystyle x_1}$. Therefore, it is initially ${\displaystyle \overline{x_2}\simeq 3L/4}$ in the SHOT-a game. In both scenarios, it is initially ${\displaystyle {\sigma }_x\simeq \sqrt{L^2/28}}$,

The game is iterated in a cellular automata (CA) manner [13], i.e., with uniform, local, and synchronous interactions. The arrows in the generic players in Figure 3 aim to make clear that the interactions are local, i.e., they involve only nearest-neighbors. Thus, in the updating steps (Figure 3a,b) both types of player scrutinize their NE–NW–SE–SW mate neighbors (Please note that NE stands here for north–east and not Nash equilibrium), whereas playing concerns (Figure 3c) the N–S–E–W partner neighbors.

At every time-step,

• Every leader player 1 in the lattice will act first and will locate which state variables among that of himself and those of his mate neighbors would provide him the highest payoff applying the backwards induction principle. Such a generic leader will adopt the best local state variables (Figure 3a).
• After updating the state variables of every player 1 in the lattice, each follower player 2 will locate among that of himself and those of his mate neighbors the state variables that provide the best payoff when playing with his partner neighbors: The generic follower will adopt the best values of the state variables (Figure 3b).
• Once the moves are made, every player plays with his four adjacent partners such that the payoff ${\displaystyle u_{i,j}^{(T)}}$ of a given individual at time-step ${\displaystyle T}$ is the average over these four games (Figure 3c).

The simulations performed in this article have been implemented using Fortran code with double precision variables.

Table 1. Leader update of the (i,j) cell in the SHOT-b game.

subroutine SHOTBLEAD(bc,WPP,WXX,p1p,x1x,i,j,n)

double precision WPP(n,n),WXX(n,n);integer bc(0:n+1)

common /hot/rL,r2L,xx2

ux=0.d0;p1p=WPP(i,j);x1x=WXX(i,j)

DO jj=j-1,j+1;DO ii=i-1,i+1;ik=bc(ii);jh=bc(jj)

if(mod(ik+jh,2)==1)cycle

pp1=WPP(ik,jh);xx1=WXX(ik,jh)

p2x=0.d0;u2x=0.d0

diffx=xx2-xx1;sumx=xx2+xx1

p2=pp1-diffx-0.001d0;if(p2x<0.d0) goto 1       !U

u2=p2*rL

if(u2>u2x)then;p2x=p2;u2x=u2;endif

1   p2=(pp1+r2L-sumx)/2.d0                                  !M

call HOT(rL,pp1,p2,xx1,xx2,u1,u2)

if(u2>u2x)then;p2x=p2;u2x=u2;endif

p2=pp1+diffx-0.001d0                                         !N

call HOT(rL,pp1,p2,xx1,xx2,u1,u2)

if(u2>u2x)then;p2x=p2;u2x=u2;endif

call HOT(rL,pp1,p2x,xx1,xx2,u1,u2)

if(u1>ux)then;ux=u1;p1p=pp1;x1x=xx1;endif

ENDDO;ENDDO

end

shows the Fortran code that implements the leader update in the SHOT-b game with ${\displaystyle t=1}$. The HOT subroutine implements the Hotelling game with ${\displaystyle L\equiv rll}$, ${\displaystyle 2L\equiv r2L}$ and ${\displaystyle x_2\equiv xx2}$, whose values are accessible via the common sentence. The subroutine performs the updating of the location and price of the leader in the (i,j) cell site up to x1x and p1p supported in the current values of the locations and prices stored in the WXX and WPP working matrices. The application of the backwards induction principle is supervised in Table 1 because only the three best potential responses to the (${\displaystyle p_1,a}$) choice of the leader, ${\displaystyle U}$, ${\displaystyle M}$ and ${\displaystyle N}$ given in Equation (3) are scouted. An example of the best response of type ${\displaystyle M}$ is given in Figures 5–7c and an example of the best response of type ${\displaystyle U}$ is given in Figures 6 and 7b.

$$\begin{array}{cc}\hfill p_2^U(p_1,a)& =p_1-t(L-b-a‘+ϵ)\hfill \end{array}$$

(3a)

$$\begin{array}{cc}\hfill p_2^M(p_1,a)& ={\displaystyle \frac{1}{2}}(p_1+t(L-a+b))\hfill \end{array}$$

(3b)

$$\begin{array}{cc}\hfill p_2^N(p_1,a)& =p_1+t(L-b-a-ϵ)\hfill \end{array}$$

(3c)

Table 2. Update of the follower in the (i,j) cell in the SHOT-b game.

suboutine SHOTBFOLLOW(BC,WPP,WXX,p2p,i,j,n)

double precision WPP(n,n),WXX(n,n)

integer BC(0:n+1); common/hot/rL,xx2

ux=0.d0;p2p=0.d0

DO jj=j-1,j+1;DO ii=i-1,i+1;ik=BC(ii);jh=BC(jj)

if(mod(ik+jh,2)==0)cycle

p2=WPP(ik,jh);uux=0.d0

DO jjj=j-1,j+1;DO iii=i-1,i+1;ik=BC(iii);jh=BC(jjj)

if(mod(ik+jh,2)==1)cycle

p1p=WPP(ik,jh);x1x=WXX(ik,jh)

call HOT(rL,p1p,p2,x1x,xx2,u1,u2)

uux=uux+u2

ENDDO;ENDDO

if(uux>ux)then;ux=uux;p2p=p2;endif

ENDDO;ENDDO

end

shows the Fortran code that implements the update of the follower’s price located in the (i, j) cell up to the p2p price, again in the SHOT-b game with ${\displaystyle t=1}$.

The Fortran codes that implement the update of the leader and of the follower in the SHOT-a game with ${\displaystyle t=1}$ are given in

Table A1. Code for the update of leader in the (i,j) cell in the SHOT-a game.

SUBROUTINE SHOTALEAD(BC,WPP,p1p,i,j,n)

double precision WPP(n,n);integer BC(0:n+1)

COMMON /HOT/ rL,rL2,r2L,xx1

ux=0.d0;p1p=WPP(i,j);fx2=rL2/1000.d0

DO jj=j-1,j+1;DO ii=i-1,i+1;ik=BC(ii);jh=BC(jj)

if(mod(ik+jh,2)==1)cycle;pp1=WPP(ik,jh)

p2x=0.d0;x2x=0.d0;u2x=0.d0

DO ix2=1,1001

x2=rL2+(ix2-1)*fx2;diffx=x2-xx1;sumx=x2+xx1

p2=pp1-diffx-0.001d0;if(p2x<0.d0) goto 1      !U

u2=p2*rL

if(u2>u2x)then;p2x=p2;x2x=x2;u2x=u2;endif

1       p2=(pp1+r2L-sumx)/2.d0                   !M

call HOT(rL,pp1,p2,xx1,x2,u1,u2)

if(u2>u2x)then;p2x=p2;x2x=x2;u2x=u2;endif

p2=pp1+diffx-0.001d0                  !N

call HOT(rL,pp1,p2,xx1,x2,u1,u2)

if(u2>u2x)then;p2x=p2;x2x=x2;u2x=u2;endif

ENDDO

call HOT(rL,pp1,p2x,xx1,x2x,u1,u2)

if(u1>ux)then;ux=u1;p1p=pp1;endif

ENDDO;ENDDO

END

and

Table A2. Follower update in the (i,j) cell in the SHOT-a game.

SUBROUTINE SHOTAFOLLOW(BC,WPP,WXX,p2p,x2x,i,j,n)

double precision WPP(n,n),WXX(n,n)

integer BC(0:n+1);COMMON /HOT/ rll,xx1

ux=0.d0;p2p=0.d0;x2x=0.d0

DO jj=j-1,j+1;DO ii=i-1,i+1;ik=BC(ii);jh=BC(jj)

if(mod(ik+jh,2)==0)cycle

p2=WPP(ik,jh);x2=WXX(ik,jh);uux=0.d0

DO jjj=j-1,j+1;DO iii=i-1,i+1;ik=BC(iii);jh=BC(jjj)

if(mod(ik+jh,2)==1)cycle

p1p=WPP(ik,jh)

call HOT(rll,p1p,p2,xx1,x2,u1,u2)

uux=uux+u2

ENDDO;ENDO

if(uux>ux)then;ux=uux;p2p=p2;x2x=x2;endif

ENDDO;ENDDO

END

in the Appendix A.

In the simulations of this work, ${\displaystyle N=200}$ and ${\displaystyle p_{max}=10}$. Only the model with ${\displaystyle t=1}$ will be considered, and the length of the market will be fixed to ${\displaystyle L=3.0}$ or to ${\displaystyle L=1.0}$. In the forthcoming figures, information regarding the leader player 1 will be in red, and the information regarding the follower player 2 will be in blue. A toy example of the initial iteration in the SHOT-b game is given in Figure A1 in the Appendix A.

4. Simulation of the Sequential Hotelling Game

4.1. The SHOT-b Game

4.1.1. Simulation Dynamics

Figure 4 deals with the dynamics in the simulation of the SHOT-b game with three values of ${\displaystyle b}$ up to ${\displaystyle T=20}$. In these three scenarios, the averages of the state variables quickly converge to a steady state, so that by ${\displaystyle T=20}$ (or before) the dynamics are virtually stabilized. This is so in the five simulations, i.e., five initial randomizations of prices and location of the follower, reported in the figure, which are almost coincident. In Figure 4a, the follower is located at the right end of the market segment (${\displaystyle b=0.0}$) uch that the leader exceeds the follower in the steady state, although to a very low extent. In contrast to this, in Figure 4b with the follower located at ${\displaystyle 3L/4}$ and in Figure 4c where the follower is located in the middle of the market segment (${\displaystyle b=1.5}$), the follower exceeds the leader in the steady state; this is particularly true in the latter case.

The patterns of ${\displaystyle p}$, ${\displaystyle (a,b)}$ and ${\displaystyle u}$ in one simulation of Figure 4c at ${\displaystyle T=20}$ are given in Figure A2 in the Appendix A.

Figure 5, Figure 6 and Figure 7 deal with the games in the steady states of the simulations of Figure 4. The layout of such games are shown in the far-left snapshots of the figures. The payoffs and demands of the games in which the price of one player is parameterized are shown in the central and far-right snapshots. These are obtained by sampling the ${\displaystyle [0,1.1(p+d_x)]}$${\displaystyle p}$-interval in 4000 equidistant points.

In Figure 5, it is ${\displaystyle k=0.589-0.000=0.589}$. In Figure 5b, it is shown that the best response to ${\displaystyle p_2=3.309}$ is not ${\displaystyle p_1=4.208}$ but ${\displaystyle p_1^M(p_2=3.309)=3.449\to }$${\displaystyle d_1=1.724\to u_1=5.496}$. Figure 5c proves that the best response to ${\displaystyle p_1=4.208}$ is ${\displaystyle p_2^M(p_1=4.208)=3.309=\to }$${\displaystyle d_2=1.665\to u_2=4.208·1.665=5.478}$.

In Figure 6b, it is shown that the best response to ${\displaystyle p_2=3.584}$ is not ${\displaystyle p_1=3.889}$ but ${\displaystyle p_1^U(p_2)=1.805=3.584-(2.250-0.471)-0.001}$. Figure 6c proves that the best response to ${\displaystyle p_1=3.889}$ is ${\displaystyle p_2^M(p_1=3.889)=3.584\to }$${\displaystyle d_2=1.792\to u_2=3.889·1.792=6.424}$.

In Figure 7b, it is shown that the best response to ${\displaystyle p_2=3.839}$ is not ${\displaystyle p_1=3.547}$ but ${\displaystyle p_1^M(p_2=3.389)=2.710=3.839-(1.500-1.129)-0.001}$. Figure 7c proves that the best response to ${\displaystyle p_1=3.547}$ is ${\displaystyle p_2^M(p_1=3.547)=3.838\to }$${\displaystyle d_2=1.919\to u_2=3.838·1.919=7.365}$.

The price values were the payoffs of the ${\displaystyle U}$ and ${\displaystyle M}$ responses equalize, i.e., ${\displaystyle u_i^U({\widehat{p}}_j)=u_i^M({\widehat{p}}_j),(i,j)=(1,2),(2,1)}$, are given in Equation (4)2. Thus, for example, in the scenario of Figure 7c it is ${\displaystyle {\widehat{p}}_1=9.000+0.371-1.500-4\sqrt{3.0·0.371})=3.651}$, very close to ${\displaystyle p_1=3.547}$. That is why ${\displaystyle u_2^U=7.357}$ is very close to ${\displaystyle u_2^M=7.365}$.

$${\widehat{p}}_1=(3L+a-b-4\sqrt{La})t,{\widehat{p}}_2=(3L-a+b-4\sqrt{Lb})t$$

(4)

No best response is of type ${\displaystyle N}$ in any the three (c)-scenarios of Figure 5, Figure 6 and Figure 7. This is because the players are not close enough to allow to the ${\displaystyle N}$ response to emerge as the best solution3.

4.1.2. ${\displaystyle a}$-Response Given ${\displaystyle b}$ and Prices

Figure 8 shows the features of the ${\displaystyle a}$-response in the HOT game given the prices achieved in the steady states of the simulations of Figure 4. In the three scenarios of Figure 8, the increase in ${\displaystyle a}$, i.e, movement to the center, induces an increase in the payoff of player 1 and a decrease in the payoff of player 24.

4.1.3. Parameterized b

Figure 9 deals with the simulation of the SHOT-b game with variable ${\displaystyle b}$. Figure 9a deals with the case of ${\displaystyle L=1.0}$, whereas Figure 9b deals with the case of ${\displaystyle L=3.0}$. In both scenarios, (${\displaystyle i}$) the leader exceeds the follower only with very low values of ${\displaystyle b}$, that is, when the follower is very far away from the market center, and (${\displaystyle ii}$) the maximum advantage of the follower is reached at ${\displaystyle b=L/2}$, that is, when the follower is located at the market center.

The average values of the (location, price) of the leader and that of the follower’s price shown in Figure 9 fit almost perfectly to the values in the SPE solution of the SHOT-b game given in Equation  (5), which induce ${\displaystyle d_1^{\star }=\sqrt{L{a}^{\star }}}$ 5, so that ${\displaystyle u_1^{\star }=\sqrt{L{a}^{\star }}\left(3L+a^{\star }-b-4\sqrt{L{a}^{\star }}\right)t}$ 6.

It is discovered that ${\displaystyle p_1^{\star }={\widehat{p}}_1(a^{\star },b)}$ and that the price in Equation (5c) is the ${\displaystyle M}$-response from ${\displaystyle b}$ to ${\displaystyle (a^{\star },p_1^{\star })}$, that is, ${\displaystyle p_2^{\star }=p_2^M(a^{\star },p_1^{\star })}$. Equation (5) generalize those provided in [10] for ${\displaystyle L=1.0}$. Incidentally, in the article [14], the ${\displaystyle L=1}$ scenario is also only taken into account with an additional, somehow spurious, variant in the notation: b is the distance to zero, not to one as is customary in the notation of the HOT game.

$$a^{\star }={\displaystyle \frac{1}{9}}\left(23L+3b-8\sqrt{L(7L+3b)}\right)$$

(5a)

$$p_1^{\star }=\left(3L+a^{\star }-b-4\sqrt{L{a}^{\star }}\right)t$$

(5b)

$$p_2^{\star }={\displaystyle \frac{1}{2}}\left(p_1^{\star }+t(L-a^{\star }+b)\right)$$

(5c)

According to Figure 9, the follower obtains the maximum payoff when he is located in the center of the market segment, in which case,

$$b^{\#}={\displaystyle \frac{L}{2}},a^{\#}={\displaystyle \frac{1}{9}}\left({\displaystyle \frac{49}{2}}-8\sqrt{{\displaystyle \frac{17}{2}}}\right)L=0.1307L$$

(6a)

$$p_1^{\#}=\left({\displaystyle \frac{5L}{2}}+a^{\#}-4\sqrt{L{a}^{\#}}\right)t=1.1846Lt$$

(6b)

$$p_2^{\#}={\displaystyle \frac{1}{2}}\left(p_1^{\#}+t({\displaystyle \frac{3L}{2}}-a^{\#})\right)=1.2769Lt$$

(6c)

The demands and payoffs associated to the solution in Equation (6) are7,

$$d_1^{\#}=0.3615L,u_1^{\#}=0.4282L^{2}t$$

(7a)

$$d_2^{\#}=0.6385L,u_2^{\#}=0.8153L^{2}t$$

(7b)

4.2. The SHOT-a Game

Figure 10 deals with the simulation of the SHOT-a game with variable ${\displaystyle a}$. Figure 10a deals with the case of ${\displaystyle L=1.0}$, whereas Figure 10b deals with the case of ${\displaystyle L=3.0}$. In both scenarios, (${\displaystyle i}$) the follower player 2 exceeds the leader, although the follower advantage decreases with increasing ${\displaystyle a}$ such that with high ${\displaystyle a}$ both players obtain a virtually equal payoff, (${\displaystyle ii}$) it is ${\displaystyle \overline{b}\simeq L/2}$ up to ${\displaystyle a}$ at least ${\displaystyle L/4}$, that is, the follower is located in the center of the market segment, provided that the leader is not close to the center, and (${\displaystyle iii}$) The aspect of the graphs changes at the critical value ${\displaystyle a_0}$ whose value is given in Equation (8) 8. Therefore, in Figure 10a, it is ${\displaystyle a_0=0.106}$, and in Figure 10b, it is ${\displaystyle a_0=0.318}$ .

$$a_0=\left({\displaystyle \frac{59}{2}}-24\sqrt{{\displaystyle \frac{3}{2}}}\right)L$$

(8)

In the two scenarios of Figure 10, if ${\displaystyle a<a_0}$, the average variables in the simulation fit very well the values in the SPE solution given in Equation  (9a). It is revealed that ${\displaystyle p_1^{\star }=\frac{1}{2}\left(3L+a-b^{\star }\right)t}$, and that ${\displaystyle p_2^{\star }=p_2^M(b^{\star },p_1^{\star })=\frac{1}{2}\left(p_1^{\star }+t(L-a+b^{\star })\right)}$.

$$b^{\star }={\displaystyle \frac{1}{2}}L,p_1^{\star }={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{5}{2}}L+a\right)t,p_2^{\star }={\displaystyle \frac{1}{4}}\left({\displaystyle \frac{11}{2}}L-a\right)t$$

(9a)

$$d_1^{\star }={\displaystyle \frac{1}{8}}\left({\displaystyle \frac{5}{2}}L+a\right),u_1^{\star }={\displaystyle \frac{1}{16}}{\left({\displaystyle \frac{5}{2}}L+a\right)}^{2}t$$

(9b)

$$d_2^{\star }={\displaystyle \frac{1}{8}}\left({\displaystyle \frac{11}{2}}L-a\right),u_2^{\star }={\displaystyle \frac{1}{32}}{\left({\displaystyle \frac{11}{2}}L-a\right)}^{2}t$$

(9c)

Figure 11 deals with the SHOT-a=0.0 game. Figure 11a shows the dynamics up to ${\displaystyle T=20}$, where the simulations quickly converge to virtually the same average steady-state variables. The game with the average values at ${\displaystyle T=20}$ of one simulation is shown in Figure 11b. Figure 11c shows the effect of the price response of player 2 in the scenario of the game of Figure 11b. Remarkably, the average value of the player 2 price reached through simulation, ${\displaystyle {\overline{p}}_2=4.154}$, is very close to ${\displaystyle p_2^M=4.115}$. Moreover, ${\displaystyle {\overline{u}}_2=u_2^M=8.468}$.

In the ${\displaystyle [a_0,L/4]}$ interval of Figure 10, the average variables in the simulation very well fit the values in the SPE solution given in Equation (10a). It is shown that ${\displaystyle p_1^{\star }={\widehat{p}}_1(b^{\star })=(3L+a-b^{\star }-4\sqrt{La})t}$ and that ${\displaystyle p_2^{\star }=p_2^M(p_1^{\star },b^{\star })}$, as in the ${\displaystyle [0,a_0]}$ interval 9.

$$b^{\star }={\displaystyle \frac{1}{2}}L,p_1^{\star }=\left({\displaystyle \frac{5}{2}}L+a-4\sqrt{La}\right)t,p_2^{\star }=2\left(L-\sqrt{La}\right)t$$

(10a)

$$d_1^{\star }=\sqrt{La},u_1^{\star }=\sqrt{La}\left({\displaystyle \frac{5}{2}}L+a-4\sqrt{La}\right)t$$

(10b)

$$d_2^{\star }=L-\sqrt{La},u_2^{\star }=2{\left(L-\sqrt{La}\right)}^{2}t$$

(10c)

Figure 12 deals with the simulation of the SHOT-a=L/4=0.75 game. Diverging from the scenario in Figure 11, in Figure 12 the average value of the player 2 price reached through simulation, ${\displaystyle {\overline{p}}_2=2.854}$, is not so close to ${\displaystyle p_2^M=2.993}$. In any case, ${\displaystyle {\overline{u}}_2=4.468}$ is not far from ${\displaystyle u_2^M=4.479}$. Incidentally, ${\displaystyle p_2^M=2.992}$ is very close to ${\displaystyle p_2^N=3.015}$ in Figure 12c.

The output of the lattice simulation becomes rather unclear with high ${\displaystyle a}$. Therefore, we will resort here to the direct implementation of the backwards induction principle. This has been performed by means of the Fortran code in

Table A3. Direct simulation of the SHOT-a game.

SUBROUTINE SHOTA-SCRATCH      integer, parameter::nsx=1000; double precision SHOTA(nsx+1,8)      commom/HOT/ rL,rL2; rix=rL2/nsx      plim=2.d0;if(rL==3.d0)plim=10.d0;pip=plim/nsx      DO ia=1,nsx+1;xa=(ia-1)*rix         uli=0.d0;pa=0.d0;xb=0.d0;pb=0.d0         DO ipi=1,nsx+1;plix=(ipi-1)*pip         xbx=0.d0;fop=0.d0;pli=0.d0;ufo=0.d0            DO ixb=1,nsx+1;bx=(ixb-1)*rix               x2=rll-bx;diffx=x2-xa               fopx=plix-diffx-0.001d0 !————-U—————               if(fopx<0.d0)goto 1               u2=fopx*rll               if(u2>ufo)then; xbx=bx;fop=fopx;pli=plix;ufo=u2; endif 1              fopx=(plix+rll-xa+bx)/2.d0 !——–M—————-               call XHOT(rll,plix,fopx,xa,x2,u1,u2)               if(u2>ufo)then; xbx=bx;fop=fopx;pli=plix;ufo=u2; endif               fopx=plix+diffx-0.001d0 !————-N—————-               call XHOT(rll,plix,fopx,xa,x2,u1,u2)               if(u2>ufo)then; xbx=bx;fop=fopx;pli=plix;ufo=u2; endif            ENDDO         call HOT(rL,pli,fop,xa,rL-xbx,u1,u2)         if(u1>uli)then; uli=u1;pa=pli;xb=xbx;pb=fop; endif         ENDDO         SHOTA(ia,1)=xa;SHOTA(ia,2)=pa;SHOTA(ia,3)=xb;SHOTA(ia,4)=pb      ENDDO      DO ia=1,nsx+1         xa=SHOTA(ia,1);pa=SHOTA(ia,2);xb=SHOTA(ia,3);pb=SHOTA(ia,4)         call XHOT(rL,pa,pb,xa,rL-xb,d1,d2,u1,u2)         SHOTA(ia,5)=d1;SHOTA(ia,6)=d2;SHOTA(ia,7)=u1;SHOTA(ia,8)=u2      ENDDO c————– print SHOTA————      END

in the Appendix A. This code produces the outputs in Figure 13, where player 1 obtains the maximum payoff at the location ${\displaystyle a^{\#}}$ already referred in Equation (6a) 10. Thus, the pair ${\displaystyle \left((a^{\#},p_1^{\#}),(b^{\#},p_2^{\#})\right)}$ whose formulas are given in Equation (6) is the SPE solution in the location–price SHOT game.

The demands in Equation (10) equalize at ${\displaystyle a=L/4}$ (${\displaystyle \leftarrow \sqrt{La}=L-\sqrt{La}}$). Further increase in ${\displaystyle a}$ up to ${\displaystyle a=3L/8}$ induces the price of the leader in the SPE solution to become of the form ${\displaystyle p_1^{\star }=\frac{L+b}{L-b}(L-b-a)}$ and the best response of the follower to become of the type ${\displaystyle N}$. Therefore, Equation (11) describe the SPE solution when ${\displaystyle a\in [L/4,3L/8]}$, where is ${\displaystyle b^{\star }=d_{1,2}^{\star }=L/2}$. At ${\displaystyle a=L/3}$, it is ${\displaystyle p_1^{\star }=L/2}$, ${\displaystyle p_2^{\star }=4L/3}$, and at ${\displaystyle a=3L/8}$, it is ${\displaystyle p_1^{\star }=3L/8}$, ${\displaystyle p_2^{\star }=L/2}$.

$$b^{\star }={\displaystyle \frac{L}{2}},p_1^{\star }=3\left({\displaystyle \frac{L}{2}}-a\right)t,p_2^{\star }=2\left(L-2a\right)t$$

(11a)

$$d_1^{\star }={\displaystyle \frac{L}{2}},u_1^{\star }=3\left({\displaystyle \frac{L}{2}}-a\right){\displaystyle \frac{L}{2}}t;d_2^{\star }={\displaystyle \frac{L}{2}},u_2^{\star }=2\left(L-2a\right){\displaystyle \frac{L}{2}}t$$

(11b)

Figure 14 deals with the SHOT-a game at ${\displaystyle a=0.85,L=3.0}$. Figure 14a shows that in the market segment ${\displaystyle [L/2,L]}$ the expenses regarding both players are equal such that the players in this segment are in principle indifferent as to which firm (player) to buy from. To overcome this indefiniteness, the Fortran code has been programmed so that consumers in this segment will buy from their closer firm, i.e., player 2.

In the ${\displaystyle [3L/8,L/2]}$${\displaystyle a}$-interval, ${\displaystyle b}$ equalizes ${\displaystyle p_2}$, with ${\displaystyle b^{\star }=p_2^{\star }=\frac{1}{2}(p_1^{\star }+L-a)}$11. Moreover, ${\displaystyle d_2^{\star }=b^{\star }=p_2^{\star }}$, with ${\displaystyle p_1^{\star }=2b^{\star }+a-L}$12, and as a result, Equation (12) describes the SPE solution in the ${\displaystyle [3L/8,L] }$${\displaystyle a}$-interval13.

$$b^{\star }=p_2^{\star }=(L-{\displaystyle \frac{1}{2}}\sqrt{2L(4a-L)})t,p_1^{\star }=(L+a-\sqrt{2L(4a-L)})t$$

(12a)

$$d_1^{\star }={\displaystyle \frac{1}{2}}\sqrt{2L(4a-L)},d_2^{\star }=L-{\displaystyle \frac{1}{2}}\sqrt{2L(4a-L)}$$

(12b)

Both prices continue to decrease with an increase in ${\displaystyle a>3L/8}$, which induces a decrease in both payoffs. At ${\displaystyle a=\frac{L}{2}}$ it is ${\displaystyle b^{\star }=p_2^{\star }=d_2^{\star }=(1-\frac{\sqrt{2}}{2})Lt}$,  ${\displaystyle p_1^{\star }=(\frac{3}{2}-\sqrt{2})Lt}$; therefore ${\displaystyle u_1^{\star }={(1-\frac{\sqrt{2}}{2})}^2{L}^{2}t}$,  ${\displaystyle u_1^{\star }=\frac{1}{2}(\frac{3}{2}-\sqrt{2})L^{2}t}$. It is ${\displaystyle {(1-\frac{\sqrt{2}}{2})}^2=0.086}$,  ${\displaystyle \frac{1}{2}(\frac{3}{2}-\sqrt{2})=0.043}$, so that ${\displaystyle u_1^{\star }>u_1^{\star }}$ at ${\displaystyle a=\frac{L}{2}}$ 14.

5. Quadratic Transportation Cost

In place of considering the linear transportation cost, it can be assumed that this cost is quadratic with respect to the distance, so that ${\displaystyle e_i=p+t{(s-x_i)}^2,i=1,2}$  [3]. In the Hotelling game with quadratic transportation cost (HOT2), the indifferent consumer is located at the ${\displaystyle \overline{s}}$ given by Equation (13) and the payoff functions (${\displaystyle u}$) are given in Equation (14).

$$\overline{s}=d_1={\displaystyle \frac{1}{2}}\left(s_x+{\displaystyle \frac{p_2-p_1}{t{d}_x}}\right),d_x=x_2-x_1$$

(13)

$$u_1=\left\{\begin{array}{ccc}0& \mathrm{if}& \overline{s}<0\\ d_1{p}_1& \mathrm{if}& 0\le \overline{s}\le L\\ L{p}_1& \mathrm{if}& \overline{s}>L\end{array}\right.,u_2=\left\{\begin{array}{ccc}L{p}_2& \mathrm{if}& \overline{s}<0\\ d_2{p}_2& \mathrm{if}& 0\le \overline{s}\le L\\ 0& \mathrm{if}& \overline{s}>L\end{array}\right.$$

(14)

In the HOT2 game, the undercutting solutions become ${\displaystyle p_1^U(p_2)=p_2-(2L-s_x)d_{x}t=p_2-(L-k)d_{x}t}$, ${\displaystyle p_2^U(p_1)=p_1-s_x{d}_{x}t=p_1-(L+k)d_{x}t}$15, while the ${\displaystyle M}$ solutions become ${\displaystyle p_1^M(p_2)=\frac{1}{2}(p_2+t(L+k)d_x)}$ and ${\displaystyle p_2^M(p_1)=\frac{1}{2}(p_1+t(L-k)d_x)}$. The intersection of such ${\displaystyle M}$ responses leads to prices in NE that are those given in Equation (2) multiplied by ${\displaystyle d_x}$. As a result, in the HOT2 game—opposite to what happens in the HOT game—the NE ${\displaystyle u_1^{*}}$ decreases with ${\displaystyle a}$ and ${\displaystyle u_2^{*}}$ decreases with ${\displaystyle b}$. This is mainly induced by the presence of the factor ${\displaystyle d_x=L-a-b}$ in the payoff

In the SHOT2 game, the prices in the SPE solution are given in Equation (15a) (see [12]). These prices induce the demands given in Equation (15b), and finally the payoffs given in Equation (15c), which equalize when ${\displaystyle k=0.314L}$16.

$$p_1^{\star }={\displaystyle \frac{1}{2}}(3L+k)d_{x}t,p_2^{\star }={\displaystyle \frac{1}{4}}(5L-k)d_{x}t$$

(15a)

$$d_2^{\star }={\displaystyle \frac{1}{8}}(3L+k),d_2^{\star }={\displaystyle \frac{1}{8}}(5L-k)$$

(15b)

$$u_1^{\star }={\displaystyle \frac{1}{16}}{(3L+k)}^2{d}_{x}t,u_2^{\star }={\displaystyle \frac{1}{32}}{(5L-k)}^2{d}_{x}t$$

(15c)

5.1. The SHOT2-b Game

In order to calculate the best response of the follower to a given price of the leader in the SHOT2-b game, we will resort to unsupervised simulation by calculating the response of the follower to a given ${\displaystyle p_1}$ in the ${\displaystyle [0,p_1+d_x]}$ interval across 1000 equidistant points to locate the best one. This is done so in

Table A4. Unsupervised updating of the leader in the (i,j) cell in the SHOT-b game with quadratic transportation cost.

subroutine RAWBLEAD(bc,WPP,WXX,p1p,x1x,i,j,n)

double precision WPP(n,n),WXX(n,n)

integer bc(0:n+1);common/hot/rll,xx2

ux=0.d0;p1p=WPP(i,j)

DO jj=j-1,j+1;DO ii=i-1,i+1;ik=bc(ii);jh=bc(jj)

if(mod(ik+jh,2)==1)cycle;pp1=WPP(ik,jh);xx1=WXX(ik,jh)

diffx=xx2-xx1

p2x=0.d0;u2x=0.d0;fp2=(pp1+diffx)/1000.d0

DO ipo=1,1001

p2=(ipo-1)*fp2

call HOT2(rll,pp1,p2,xx1,xx2,u1,u2)

if(u2>u2x)then;u2x=u2;p2x=p2;endif

ENDDO

call HOT2(rll,pp1,p2x,xx1,xx2,u1,u2)

if(u1>ux)then;ux=u1;p1p=pp1;x1x=xx1;endif

ENDDO;ENDDO

end

in the Appendix A, which shows the Fortran code that implements the update of the price and location of the leader in the cell (i,j) in such a raw way, where HOT2 implements the Hotelling game with quadratic transportation cost. This kind of brute-force simulation demands very high computer resources, much higher than those demanded by the code in Table A1. Incidentally, replacing HOT2 with HOT in Table A4, the subroutine will provide the unsupervised leader updating with linear transportation cost. More importantly, by activating in Table A4 the subroutines that implement the calculation in more sophisticated variants of the HOT game, one can address the simulation of the said game variants, in which mathematical analysis becomes almost impossible.

Figure 15 is the analogue to Figure 9 with quadratic transportation cost. In the ${\displaystyle L=1.0}$ scenario of Figure 15a, the follower player 2 exceeds the leader to roughly the same extent and the average value of ${\displaystyle a}$ is close to zero regardless of ${\displaystyle b}$. In the ${\displaystyle L=3.0}$ scenario of Figure 15b the main features of the ${\displaystyle L=1.0}$ scenario are preserved, it being notable that ${\displaystyle \overline{a}}$ is virtually zero regardless of ${\displaystyle b}$. Contrary to what happens with linear transportation cost in Figure 9, in the SHOT2-b game the payoff of follower player 2 decreases with the increase in ${\displaystyle b}$.

Taking ${\displaystyle a=0}$ in the prices in Equation (15a) (i.e., ${\displaystyle k=0-b=-b}$, ${\displaystyle d_x=L-b-0=L-b}$), the SPE solution of the SHOT2-b game becomes that in Equation (16), where it is ${\displaystyle p_1^{\star }=p_2^{\star }=8L^2/9}$ at ${\displaystyle b=L/3}$.

$$a^{\star }=0,p_1^{\star }={\displaystyle \frac{1}{2}}\left(3L-b\right)(L-b)t,p_2^{\star }={\displaystyle \frac{1}{4}}\left(5L+b\right)(L-b)t$$

(16a)

$$d_1^{\star }={\displaystyle \frac{1}{8}}\left(3L-b\right),d_2^{\star }={\displaystyle \frac{1}{8}}\left(5L+b\right)$$

(16b)

$$u_1^{\star }={\displaystyle \frac{1}{16}}{\left(3L-b\right)}^2(L-b)t,u_2^{\star }={\displaystyle \frac{1}{32}}{\left(5L+b\right)}^2(L-b)t$$

(16c)

The average values in the simulation of Figure 15 fit fairly well with those given in Equation  (16). Thus, for example, it is ${\displaystyle p_1^{\star }=p_2^{\star }=8/9}$ at ${\displaystyle a=1/3}$ in Figure 15a, and ${\displaystyle p_1^{\star }=p_2^{\star }=8.0}$ at ${\displaystyle a=1.0}$ in Figure 15b.

Figure 16 deals with the particular case of the SHOT2-${\displaystyle b=0.75}$ game from Figure 15b. Figure 16a, shows the dynamics in the simulation, therefore, it is the analogue to Figure 4b with quadratic transportation cost. Figure 16b shows the game with the average values of the variables reached at ${\displaystyle T=20}$. In such a game, it is as follows: ${\displaystyle d_x=2.250-0.026=2.224}$, ${\displaystyle k=0.026-0.750=-0.724}$. Figure 16c shows that the best response to ${\displaystyle p_1=9.232}$ is ${\displaystyle p_2^M(p_1=9.232)=8.738=\frac{1}{2}(9.223+(3.000-0.724)2.224)}$. Incidentally, the player 2 undercuts the player 1 in Figure 16c when ${\displaystyle p_2\le p_2^U(p_1=9.232)=4.170}$17.

5.2. The SHOT2-a Game

Figure 17 is the analogue to Figure 10 with quadratic transportation cost. An unexpected discontinuity of the first kind emerges circa ${\displaystyle a_0\simeq L/3}$. Before ${\displaystyle a_0}$, the average variables in the simulation fit in broad strokes to the solution in Equation (17a). This solution arises by making ${\displaystyle b=\frac{L}{2}}$ in Equation (15a) and is the solution given in Equation (9a) with the prices multiplied by ${\displaystyle d_x=\frac{L}{2}-a}$.

$$b^{\star }={\displaystyle \frac{L}{2}},p_1^{\star }={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{5}{2}}L+a\right)({\displaystyle \frac{1}{2}}L-a)t,p_2^{\star }={\displaystyle \frac{1}{4}}\left({\displaystyle \frac{11}{2}}L-a\right)({\displaystyle \frac{1}{2}}L-a)t$$

(17a)

$$d_1^{\star }={\displaystyle \frac{1}{8}}\left({\displaystyle \frac{5}{2}}L+a\right),d_2^{\star }={\displaystyle \frac{1}{8}}\left({\displaystyle \frac{11}{2}}L-a\right)$$

(17b)

$$u_1^{\star }={\displaystyle \frac{1}{16}}{\left({\displaystyle \frac{5}{2}}L+a\right)}^2({\displaystyle \frac{1}{2}}L-a)t,u_2^{\star }={\displaystyle \frac{1}{32}}{\left({\displaystyle \frac{11}{2}}L-a\right)}^2({\displaystyle \frac{1}{2}}L-a)t$$

(17c)

In order to improve the understanding of the effect of the variation of ${\displaystyle a}$ in the SHOT-a game, and quantify the discontinuity at ${\displaystyle a_0}$, we will resort to the code in Table A3 in the Appendix A, replacing the calls to SHOT with calls to SHOT2. This code produces the output shown in Figure 18, where the leader player 1 obtains his maximum payoff in Figure 18 at ${\displaystyle a=0.0}$. Thus, the SPE solution in the location–price SHOT2 game is that given in Equation (18a).

$$a^{\#}=0.0,p_1^{\#}={\displaystyle \frac{5}{8}}{L}^{2}t;b^{\#}={\displaystyle \frac{L}{2}},p_2^{\#}={\displaystyle \frac{11}{16}}{L}^{2}t$$

(18a)

$$d_1^{\#}={\displaystyle \frac{5}{16}}L,d_2^{\#}={\displaystyle \frac{11}{16}}L$$

(18b)

$$u_1^{\#}={\displaystyle \frac{25}{128}}{L}^{3}t,u_2^{\#}={\displaystyle \frac{121}{256}}{L}^{3}t$$

(18c)

The shift of the leader’s location to the left end of its market segment (${\displaystyle a^{\#}=0.0}$) in the SPE solution of the SHOT2 game is reminiscent of what happens in the HOT2 game, where the maximum differentiation principle applies, so that both players are drawn to the extremes of their market segments in the NE solution. Thus, it applies the opposite of the minimum differentiation principle, i.e., the tendency towards the center of the market, introduced in the seminal article by Hotelling. This is a rather surprising result, which leads some researchers in this field to invalidate the quadratic transportation cost in the HOT game.

The discontinuity at ${\displaystyle a_0}$ in Figure 18 is quantified as follows. In the scenario of Figure 18a at ${\displaystyle a=0.3290}$ it is ${\displaystyle p_1=0.242}$, ${\displaystyle b=0.500}$, ${\displaystyle p_2=0.222}$, and at ${\displaystyle a=0.3295}$ it is ${\displaystyle p_1=0.146}$, ${\displaystyle b=0.410}$. ${\displaystyle p_2=0.214}$. In the scenario of Figure 18b, at ${\displaystyle a=0.981}$ it is ${\displaystyle p_1=2.200}$, ${\displaystyle b=1.500}$, ${\displaystyle p_2=2.020}$, and at ${\displaystyle a=0.984}$ it is ${\displaystyle p_1=1.300}$, ${\displaystyle b=1.203}$, ${\displaystyle p_2=1.960}$. The discontinuity in the game variables at ${\displaystyle a_0}$ is transmitted mainly to the payoff of player 2 which decreases in such a way that at ${\displaystyle a_0}$ the payoffs of both players virtually equalize, and ${\displaystyle u_1}$ exceeds ${\displaystyle u_2}$ in the ${\displaystyle [a_0,L/2-ϵ]}$ interval. In any case, this is at a little extent and does not happen at ${\displaystyle a=L/2}$, where quite unexpectedly ${\displaystyle u_2>u_1}$, again a very little extent18.

Figure 19 deals with the SHOT2-a game at ${\displaystyle a=1.00,L=3.0}$. Figure 19a shows that, much as in Figure 14a, it is ${\displaystyle d_2=b}$, although expenses do not equalize in the ${\displaystyle (x_2,L]}$ market interval (which is not feasible with quadratic transportation cost). In Figure 19b it is ${\displaystyle p_2^M(p_1)=\frac{1}{2}(p_1+(L-k)d_x)=\frac{1}{2}(1.3+(3.0-1.0+1.2)0.8)=1.920}$. Note that in the game of Figure 19 it is ${\displaystyle u_1=u_2=2.304.}$

At ${\displaystyle a=1.500}$ in Figure 18b it is ${\displaystyle p_1=0.350}$, ${\displaystyle d_1=2.365}$, ${\displaystyle b=d_2=0.635}$, ${\displaystyle p_2=1.099}$, ${\displaystyle u_1=0.828}$, ${\displaystyle u_2=0.698}$ as shown in Figure 20. In Figure 20b ${\displaystyle u_2}$ becomes zero after the maximum when ${\displaystyle p_2=p_1+(L+b-a)(L-b-a)=p_1+(L/2+b)(L/2-b)=0.350+(1.500+0.635)(1.500-0.635)=2.198=2p_2^M}$.

In the SHOT2-a game with ${\displaystyle a\in (a_0,L/2]}$ it is ${\displaystyle p_2^M(p_1)\equiv p_2^{\star }=p_1^{\star }+{(L-b^{\star }-a)}^2}$. As a result, ${\displaystyle p_1^{\star }=(a+3b^{\star }-L)(L-b^{\star }-a)}$, ${\displaystyle b^{\star }=d_2^{\star }=\frac{1}{3}\left(2(L-a)-\sqrt{{(L-a)}^2-3p_1^{\star }}\right)}$19. Therefore, in the game of Figure 20, where ${\displaystyle a=L/2}$, it is ${\displaystyle p_2=p_1+{(\frac{L}{2}-b)}^2=0.350+{(1.500-0.635)}^2=1.099}$, and ${\displaystyle p_1=(3b-\frac{L}{2})(\frac{L}{2}-b)=(1.905-1.500)(1.500-0.635)=0.350}$, ${\displaystyle b=d_2=\frac{1}{3}\left(L-\sqrt{(L^2/4-3p_1}\right)13\left(3.000-\sqrt{(9/4-3·0.350}\right)=0.635}$.

6. Conclusions

Numerical simulation is a powerful tool for studying the sequential Hotelling game (SHOT) where the location of the leader (SHOT-a) or that of the follower (SHOT-b) is exogenously fixed. The main findings of this article are summarized in what follows.

• The subgame perfect equilibrium (SPE) solution exists in both scenarios regardless of the proximity of players in a finite market of length ${\displaystyle L}$.
• The follower overrates the leader as a general rule. The only exception applies to the SHOT-b game with very low ${\displaystyle b}$, that is, in which case the leader overrates the follower to a low extent.
• In the SHOT-b game, the advantage of the follower increases with ${\displaystyle b}$ in a linear-like way, reaching its maximum when the follower is located in the market center.
• In the SHOT-a game, the effect of the variation in ${\displaystyle a}$ is quite sophisticated, so the maximum payoff of the leader is reached at ${\displaystyle a=0.106L}$, which defines the SPE solution in the location–price SHOT game.
• In the SHOT game with quadratic transportation cost, the follower advantage notably increases compared to that found with linear transportation cost, and in the SPE solution in the location–price SHOT game, the leader shifts to his market end.

The study of the SHOT game with reservation cost [15,16] and of the SHOT game with elastic demand [14,15,17] is planned for a later study. The study of these sophisticated variants of the SHOT game will be addressed by implementing unsupervised or brute force simulation mechanisms which have already been tested in this work when dealing with the model with quadratic transportation cost.