# Location games with references

Gaëtan Fournier & Amaury Francou Aix Marseille Univ, CNRS, AMSE, Marseille,
France<EMAIL_ADDRESS>The project leading to this publication
has received funding from the french government under the “France 2030”
investment plan managed by the French National Research Agency (reference:
ANR-17-EURE-0020) and from Excellence Initiative of Aix-Marseille University -
A*MIDEX. ”This research was initiated during the student research internship
in AMSE of Amaury Francou<EMAIL_ADDRESS>
###### Abstract

We study a class of location games where players want to attract as many
resources as possible and pay a cost when deviating from an exogenous
reference location. This class of games includes political competitions
between policy-interested parties and firms’ costly horizontal
differentiation. We provide a complete analysis of the duopoly competition:
depending on the reference locations, we observe a unique equilibrium with or
without differentiation, or no equilibrium. We extend the analysis to a
competition between an arbitrary number of players and we show that there
exists at most one equilibrium which has a strong property: only the two most-
left and most-right players deviate from their reference locations.

_JEL Classification_ : C72, D43, L13, R30.

_Keywords_ : Location games, Spatial competition, Spatial voting theory,
Costly product differentiation.

##  1 Introduction

Spatial competitions arise when sellers maximize their clientele by competing
on the space of customers’ preferences or when political parties maximize
their electorate by competing in the space of political opinions, as
introduced in the seminal works of [Hotelling, 1990] and [Downs, 1957]. In the
most general version of these models, we can ignore the exact objective of the
agents and simply assume that they seek to maximize a resource through spatial
competition.

However, economic agents rarely have complete freedom when choosing their
locations: there is typically a reference situation, described by the
characteristics or the past choices of the agents or their predecessors. These
references are not necessarily controlled by the agents but have a significant
impact on the outcome of the competition and can not be ignored.

A first example one can think of is the situation where vendors choose the
location of their store. While the Hotelling-Downs model analyzes a
competition between sellers entering a new market, it is sometimes more
realistic to assume that they already have a historical address and that a
move is costly. The price of the move has no impact on the consumers, whose
economic behaviors depend only on the new locations, however it has a
strategic impact on the sellers’ decisions.

The general situation we study is the following: a finite number of players
compete for uniformly distributed resources on $[0,1]$. Each player $i$ has an
exogenous reference $r_{i}$, selects a location $x_{i}$ and pays a cost that
increases with the distance between $x_{i}$ and $r_{i}$. Resources are
attracted by the player with the closest location. The model, formally
detailed in section 2, can be applied in numerous fields of research. As an
illustration we now mention two motivating examples.

###  1.1 Motivating examples

Example 1: Political competition with policy-motivations.
In the classical Downs model, political parties only have office motivations:
they simply consider a policy as an opportunistic instrument, to be successful
in the election. In other words, they would say anything to seduce the
electorate. Therefore, they compete on the policy space by selecting platforms
$x_{i}\in[0,1]$ in order to maximize their electoral success. It is sometimes
more realistic to suppose that political parties also have policy motivations,
for example when they do not want to betray their sincere favorite policy
$r_{i}$. In this situation, parties have to compromise between two objectives:
maximizing their vote shares111The vote share might also represent the
probability to win a winner-takes-all election because voters have incomplete
information about political parties, see [Lindbeck and Weibull, 1987] for
example, or because parties have incomplete information about the distribution
of voters’ preferences [Patty, 2002]. and minimizing the distance between
their sincere opinions $r_{i}$ and the selected platforms $x_{i}$. Following
[Roemer, 2009], each motivation represents the objective of a faction of
political parties, the opportunists and the militants: while the opportunists
”desire to maximize the probability of the party’s victory”, the militants
”desire to propose a policy as close as possible to the party’s ideal point
and have little interest in winning the election”.

Example 2: Costly products’ differentiation.
Consider a market where firms want to maximize their clientele by selling a
product whose price is exogenously determined. This model applies for example
to newsstands, pharmacies or franchises of different types of services and
products. It also applies to the media markets or to any sector where
consumers are not charged with direct prices but through advertising.

We suppose that the good has a $1$-dimensional characteristic and that
consumers have symmetric single peaked preferences over this characteristic,
with a maximum uniformly distributed on the interval, so that consumers choose
to buy one unit of the product whose characteristic is the closest to their
ideal. Each firm $i$ has the exogenous expertise to produce a good with
characteristic $r_{i}$ and has the possibility to produce a good with
characteristic $x_{i}\neq r_{i}$ for a certain cost which increases with the
distance $d(r_{i},x_{i})$. The literature mentions this investment as the
”differentiation costs” (see for example [Eaton and Schmitt, 1994]). When the
characteristic is geographic, it may be interpreted as the cost of
transporting inputs bought from suppliers for example, otherwise it might be
interpreted as resulting from the process of modifying the standard product of
the firm, through research and development investments for example. The payoff
of firms is again composed of two terms: the first is the quantity of product
they sell, which is a good proxy for their profits as the price is fixed and
because we suppose that the production costs are small enough to give the
firms the incentives to sell as much as possible. The second term represents
the cost of the differentiation.

The literature review, in subsection 1.3, comes back to the above examples and
illustrates our contributions to the literature. We also briefly mention other
applications of our model.

###  1.2 Main results

We first study the duopoly competition. We provide a complete characterization
of the equilibrium for any pair of exogenous references
$(r_{1},r_{2})\in[0,1]^{2}$. An interesting question tackled by the literature
is to explain in which context do we observe differentiation or not: the
principle of minimal differentiation states that in many different spatial
competitions we observe similar equilibrium locations for both players.
Although this result is quite robust222The principle still holds in the case
of non-uniform distribution of consumers, in the case of sequential entry of
the players, in a winner takes all competition, or when there is uncertainty
about the preferences if players share the same prior., the principle has been
challenged by empirical evidences and theoretically by the introduction of
price competition (d’Aspremont et al. [1979]) or turnout in voting models
([Davis et al., 1970]) for example. In the current paper, we prove that an
equilibrium with or without differentiation can be observed depending on the
pair of references $(r_{1},r_{2})$. As stated formally in Proposition 1, there
are three different scenarios:

  1. 1.

If references are far from each other, there exists a unique equilibrium where
players differentiate.

  2. 2.

If references are close to each other but far from $\frac{1}{2}$, there is no
equilibrium.

  3. 3.

If references are both close to each other and close to $\frac{1}{2}$, there
exists a unique equilibrium where players do not differentiate
$(x_{1}^{*}=x_{2}^{*}=\frac{1}{2})$.

The intuition is the following: each player is subject to two possibly
conflicting forces, on the one hand each player wants to move towards his
opponent as such a move always increases the quantity of attracted resources,
but on the other hand they does not want to move too far from their reference
location. When the references are far from each other, each of the two players
can act as a local monopolist and find the optimal compromise between the two
forces. However, when the references are close to each other, the first order
conditions can not be satisfied without passing each other positions. If
references are close to $\frac{1}{2}$, this median location is chosen by both
players at equilibrium and the minimal differentiation principle is verified:
the unique equilibrium is $(\frac{1}{2},\frac{1}{2})$. Surprisingly,
equilibrium locations are not necessarily located in the interval between the
two references. Finally, if the references are close to each other but far
from $\frac{1}{2}$, the situation is unstable: players still want to be as
close as possible to each other, but if they select the same location
$x\neq\frac{1}{2}$ then each of them would benefit from an infinitesimal
deviation towards $\frac{1}{2}$ as he would attract more than half the
resources.

Our results quantitatively depend on the relative importance of the cost of
deviations from the references, which is represented by a parameter $c>0$. We
provide comparative statics to show that the larger $c$ is, the more often we
observe a differentiated equilibrium and the less often we observe the
equilibrium $(\frac{1}{2},\frac{1}{2})$. All together, we show that the
likeliness equilibrium existence is a non-monotonic function of the costs:
first decreasing then increasing with $c$. We explain this phenomenon and
illustrate graphically the different possible equilibria on the
$2$-dimensional space of $(r_{1},r_{2})\in[0,1]^{2}$. Finally, we investigate
the heterogeneous case where players have different parameters $c_{1}\neq
c_{2}$.

We then turn to the competition with more than two players. Among oligopolies,
the triopoly has a particular interest as this configuration is singular in
the literature of standard location games: it is the only configuration where
there exists no equilibrium. We prove that this non-existence result is robust
against the introduction of a small cost of deviation. We provide an explicit
upper bound for this robustness. If the costs increase above this threshold,
we prove that there exist three possible configurations depending on the
triplet of references: no equilibrium, a unique equilibrium where all players
differentiate or a unique equilibrium where two players do not differentiate.

We then consider the general case of $n$ players333The general case is the one
of $n\geq 5$ players. The less interesting case of four players slightly
differs and is briefly studied aside.. We prove that there exists either a
unique or no equilibrium depending on the reference locations. This result
differs from the spatial competition without references where there exists,
for $n>5$, a continuum of equilibria (see for example Eaton and Lipsey [1975]
and notice that the set is not very tractable as it involves $n^{2}$
equations). The introduction of the reference locations surprisingly
simplifies the equilibrium structure as it creates a tie-breaking rule. We
characterize the unique possible equilibrium and show a strong property: at
most four players select locations which are different from their references,
the two players with the most left or right references. A key argument for
this property is very intuitive: when a player changes his location within the
same interval between his neighbors, he gains some resources on one side but
looses the same quantity on the other side, so only the cost minimization
matters. We provide necessary and sufficient conditions under which the unique
equilibrium candidate is indeed an equilibrium, and we illustrate our result
in the simple case where references are regularly located on the unit
interval.

Finally, we implement a simple algorithm that computes the equilibrium
candidate for any references vector, and determine whether it is or not an
equilibrium. To make it possible, we show that despite the continuous action
space $[0,1]^{n}$, there only exist a finite number of possible best responses
for each player.

###  1.3 Literature review

In political economy, many papers are concerned with spatial competition.
While the original paper of [Downs, 1957] only considers office-motivated
candidates, [Wittman, 1973] considers a competition between two policy-
interested candidates and both papers gave birth to different literatures.444A
few papers tackle the question of the candidates’ motivations from an
empirical point of view, see for example [Fredriksson et al., 2011]. Between
these two extremes, some papers analyze the case of parties with hybrid
motivations, for example [Callander, 2008], [Saporiti, 2008] or [DRO, 2014].
The latter claims that ”although politicians might be more interested in
winning the elections, it seems reasonable to expect that policy
considerations will also enter into the candidate’s payoff function with some
weight”. The main difference with our paper is that the policy motivation of
candidates is measured by the distance between the ideal policy of the parties
and the policy implemented after the election. In the political interpretation
of our model, we do not consider a winner-takes-all election and therefore, we
do not specify which policy is implemented after the election. On the other
hand, we consider that players are political parties that have to compromise
between maximizing vote shares and minimizing the distance between the
preferred policy and the platform proposed by the party.
In this sense we refer to [Roemer, 2009]: the policy-interested candidates
introduced in the above papers are defined as the reformists while our policy-
interested candidates are defined as the militants. When the reformists
minimize the distance between their ideal and the implemented policy, the
militants minimize the distance between their ideal and the policy proposed by
the party.555Roemer also mention that ”political histories are replete with
descriptions of these three kind of party activists (the third kind being the
opportunists that are office-interested). For instance, Schorske (1993) calls
them, when describing the German Social Democratic Party: the party
bureaucrats, the trade union leadership and the radicals”. The solution
concept studied by Roemer differs from ours: his ”party-unanimity Nash
equilibrium” (PUNE) is an equilibrium in the game where a deviation is
profitable by the party only when it is unanimously preferred by all factions
of the party. The author provides some existence results and gives a Nash
bargaining interpretation of the solution concept. [Kartik and McAfee, 2007]
also consider a competition where some candidate are militants in the sense
that they do not want to propose a platform they do not believe in. In their
model, such candidates are preferred by voters and office-motivated candidates
use mixed strategies to pretend they also care about politics.

We now discuss papers concerned with the use of location games with references
in industrial organization. Our contribution in this field is to consider
differentiation costs paid by firms in order to produce a good which has a
different characteristic than the standard good. [Kishihara and Matsubayashi,
2020] consider a costly product re-positioning performed by two firms having
predetermined product characteristic base positions. The two firms compete on
both locations and prices, but the authors can only provide analytic results
for the equilibrium outcomes, as closed-form solutions can not be given.
[Correia-da Silva and Pinho, 2011] suppose that firms pay a quadratic cost to
differentiate their products from the standard product which is fixed to
$\frac{1}{2}$ for all agents. Firms compete on both location and price, and
the authors found that there is either a differentiated equilibrium or no
equilibrium at all. Although we do not take into account the impact of
locations on prices, we extend this analysis to the case where the standard
products are not necessarily fixed at $\frac{1}{2}$ and can in particular be
different. We also relax the restriction on the number of firms. We do not
believe that our analysis could keep this generality in a location-then-price
setting. Notice that [Eaton and Schmitt, 1994] also suppose that firms produce
goods which may be more or less costly to produce, according to whether their
specifications are distant from or close to the specifications of a basic
product. In a survey, [Brenner, 2010] mentions that ”a series of influential
papers have abstracted away from the choice of the price, and looked at
whether location equilibria exist, and how they can be characterized when the
location has no impact on the pricing”. We refer to this survey for the
literature concerned with fixed price location games and to [Loertscher and
Muehlheusser, 2011] for a broader discussion on markets where the hypothesis
of fixed price is particularly relevant.
We briefly mention papers concerned with other applications of our model. When
a social planner prefers the firms to produce a particular good for exogenous
reasons, firms are penalized for the distance between the good they produce
and a subsidized good. In this case, firms want to compromise between taking
strategic opportunities to increase their market shares and maximizing the
subsidies by following the recommendation of the planner. In [Lambertini,
1997], the author examines a duopoly in which he considers that firms can be
either taxed or subsidized depending on the choice of the characteristics,
which influence the equilibrium locations. Finally, let us mention that in the
literature of strategic advertising, when firms produce a good with a certain
characteristics but pretend, through advertisements, to produce a different
good in order to attract naive consumers, it is usually penalized by a cost
function. Such a cost typically increase with the difference between the two
products. For example, [Tremblay and Polasky, 2002] considers a duopoly
competition where the firms produce an identical product located at
$\frac{1}{2}$, but can generate a subjective product differentiation through
costly advertising.

##  2 Model and notations

We now formally define the location game in its normal form. A finite number
$n$ of players simultaneously select locations
$\textbf{x}=(x_{1},\dots,x_{n})$ in the action space $[0,1]^{n}$. In order to
explicit the players’ payoffs, we need to define $q_{i}(\textbf{x})$, the
quantity of resources that player $i$ attracts when the profile of locations
$\textbf{x}=(x_{1},\dots,x_{n})$ is played. It is the quantity of resources
$t$ that are closer to $x_{i}$ than to any other other $x_{j}$. In the case
where several players select the same location, they split their resources
equally. Only a negligible set of resources are at the same distance from two
different locations. Formally:

$q_{i}(x_{1},\dots,x_{n}):=\displaystyle\frac{\mu(\\{t\in[0,1],~{}|t-x_{i}|\leq|t-x_{j}|\text{
for every }j\neq i\\})}{Card(\\{x_{j}|x_{j}=x_{i}\\})}$

where $\mu$ is the Lebesgue measure (simply used to measure the lengths of
intervals).

Players are endowed with exogenous references $\textbf{r}=(r_{1},\dots,r_{n})$
and each player $i$ incurs as cost when choosing a location $x_{i}$ which is
different from his reference location $r_{i}$. This cost is given by
$\gamma(|x_{i}-r_{i}|)$ where $\gamma:[0,1]\rightarrow\mathbb{R}$ is a
continuous, strictly increasing and strictly convex function666The convexity
of the cost with respect to the distance is a common assumption: see for
example [d’Aspremont et al., 1979], where the author considers a the quadratic
cost functions to investigate the differentiation in a location-then-price
models and where linear costs induce technical difficulties or alternatively
[Villani, 2009] for the use of quadratic cost of distance in optimal
transport. such that $\gamma(0)=0$. We sometimes assume that
$\gamma(d)=cd^{2}$ where $c>0$, in order to obtain explicit results and to
provide a comparative static with respect to the parameter $c$. However, we
show that the main results hold without this simplifying assumption.

Player $i$’s payoff is then given by:

$g_{i}(\textbf{x},\textbf{r})=q_{i}(\textbf{x})-\gamma(|x_{i}-r_{i}|)$

It is useful to define the left (resp. right) neighbor of player $i$ as the
player that selects the largest (smallest) location which is strictly smaller
(larger) than $x_{i}$, if any, and the neighborhood of player $i$ as the
interval $(x_{i}^{\ell},x_{i}^{r})$ where $x_{i}^{\ell}$ is the left neighbor
of player $i$ if it exists and the boundary $0$ otherwise and $x_{i}^{r}$ is
the right neighbor of player $i$ if it exists and the boundary $1$ otherwise.
Finally, we say that a player is a left (resp. right) peripheral player when
he has no left (resp. right) neighbor.

##  3 The duopoly

In this section, we suppose for simplicity that $\gamma(d)=cd^{2}$ with $c>0$,
except in subsection 3.3 where we generalize the results.

###  3.1 Existence and characterization of equilibrium

######  Proposition 1.

The duopoly competition
Let $\textbf{r}=(r_{1},r_{2})\in[0,1]^{2}$ and suppose, without loss of
generality, that $r_{1}\leq r_{2}$.

  * •

If $r_{2}-\frac{1}{4c}\leq r_{1}+\frac{1}{4c}$, the unique possible
equilibrium is $(\frac{1}{2},\frac{1}{2})$.

It is an equilibrium if and only if
$\frac{1}{2}\in[r_{2}-\frac{1}{4c},r_{1}+\frac{1}{4c}]$.

  * •

If $r_{1}+\frac{1}{4c}<r_{2}-\frac{1}{4c}$, the unique possible equilibrium is
$(r_{1}+\frac{1}{4c},r_{2}-\frac{1}{4c})$.

It is an equilibrium if and only if
$\begin{cases}c(r_{2}-r_{1})^{2}+r_{1}+r_{2}-1-\frac{1}{4c}\geq
0~{}~{}~{}~{}~{}~{}(E1)\\\ c(r_{2}-r_{1})^{2}-(r_{1}+r_{2})+1-\frac{1}{4c}\geq
0~{}~{}~{}~{}(E2)\end{cases}$

The proof is postponed to subsection 6.2 and we now illustrate and comment the
above proposition.

Illustration and intuition: The figure below describes the equilibrium
structure with respect to the references $(r_{1},r_{2})\in[0,1]^{2}$.

Figure 1: The different equilibrium configurations with respect to
$(r_{1},r_{2})$, for $c=3$. In white there is no equilibrium, in orange
$(\frac{1}{2},\frac{1}{2})$ is the unique equilibrium, in blue
$(r_{1}+\frac{1}{4c},r_{2}-\frac{1}{4c})$ is the unique equilibrium.

As discussed in the introduction, we observe $3$ possible scenarios:

  1. 1.

In the orange square, references are such that the game admits a unique
equilibrium which is $(\frac{1}{2},\frac{1}{2})$ . As stated in Proposition 1,
this scenario happens when references are both close to each other:
$|r_{2}-r_{1}|\leq\frac{1}{2c}$ and close to the center:
$|r_{2}-\frac{1}{2}|\leq\frac{1}{4c}$ and
$|\frac{1}{2}-r_{1}|\leq\frac{1}{4c}$.

The proof of Proposition 1 combines two ingredients in this scenario. The
first ingredient is concerned with how players compromise between maximizing
resources and minimizing costs. On the one hand, when moving closer to his
opponent, the marginal gain of resources is constant and equal to
$\frac{1}{2}$, as the midpoint between the player and his neighbor moves two
times slower than the player. On the other hand the marginal cost of a
deviation further the references is increasing because the cost function
$\gamma(d)=cd^{2}$ is convex. Therefore, we find that there exists an optimal
deviation from the reference, which is given by a move of $\frac{1}{4c}$
towards the opponent (later named $\delta$ in the general case of a convex
function $\gamma$), as long as they don’t pass each other’s positions. If the
distance between the two references is closer than $\frac{1}{2c}$, then at
equilibrium players necessarily select the same location, otherwise at least
one player has a profitable deviation by moving closer to his opponent. The
second ingredient is formalized in claim 2 of Lemma 18:
$(\frac{1}{2},\frac{1}{2})$ is the only possible equilibrium where players
select the same location. Otherwise, each player has a profitable deviation by
moving marginally towards the center. Obviously, if references are to far from
$\frac{1}{2}$, the profile $(\frac{1}{2},\frac{1}{2})$ is not an equilibrium.
Because the constraints are stronger when $c$ increases, this first scenario
is less likely as the cost of deviating from the references becomes higher.

In this configuration, a counter-intuitive phenomenon can happen: it might be
that players select equilibrium locations that are strictly larger than both
references (or smaller). Take for example $c=3$ and $r=(0.42,0.45)$, then the
equilibrium is $(\frac{1}{2},\frac{1}{2})$.

  2. 2.

In the two symmetric blue areas, the game admits a unique equilibrium which is
$(r_{1}+\frac{1}{4c},r_{2}-\frac{1}{4c})$. In this case, references are far
enough from each other so that each agent acts as a local monopolist and
deviates from his reference towards his opponent by $\frac{1}{4c}$.

In this configuration, players do not have a profitable deviation within the
same neighborhood, that is on the same side of the opponent. Therefore, the
proof of Proposition 1 focuses on the cases where players have a profitable
deviation by relocating on the other side of their opponent.

Consider the situation where $r_{1}<x_{1}<x_{2}<r_{2}$. Player $1$’s payoff is
decreasing with $x_{1}$ on the whole interval $(x_{2},1]$ as it decreases the
resources he attracts and increases his costs. Therefore, the equilibrium
conditions are limited to one equation for Player $1$ not to deviate
marginally on the right of Player $2$777We prove that
$\displaystyle\lim_{\epsilon\downarrow 0}u_{1}(x_{2}+\epsilon,x_{2})\leq
u_{1}(x_{1},x_{2})$ and symmetrically for Player $2$. and one other condition
for Player $2$ not to deviate marginally to the left of Player $1$. These
equations are denoted $(E1)$ and $(E2)$ in Proposition 1. For instance, $(E1)$
can be written $c(r_{2}-r_{2})^{2}+(r_{1}+r_{2})\geq 1+\frac{1}{4c}$, it
states that it should not be the case that $r_{1}$ and $r_{2}$ are both close
to each other and small: indeed, it this case the left player has a profitable
deviation by deviating to the right of his opponent.

  3. 3.

In the two white areas, the game does not admit any equilibrium. The reason is
a repetition of the arguments above: the references are close to each other
but far from $\frac{1}{2}$ or because the references are very asymmetric (both
close to $0$ or to $1$) so that a player has a profitable deviation over his
opponent’s location. Note that this situation never occurs when references are
exactly symmetric $(r_{1}=1-r_{2})$.

The above figure is symmetric with respect to the linear transformation
$(x^{\prime},y^{\prime})=(y,x)$ as players are anonymous and
$(x^{\prime},y^{\prime})=(1-y,1-x)$ as the unit interval with uniform density
is symmetric.

Comments on the two illustrative examples mentioned in subsection 1.1.
The equilibrium $(\frac{1}{2},\frac{1}{2})$ is the unique equilibrium in the
game without references. In the field of political economics, it illustrates
the median voter theorem: at equilibrium both political parties select the
median voter’s opinion (here $\frac{1}{2}$ as voters are uniformly
distributed). When applied to this example, Proposition 1 explicitly narrows
the domain of validity of the median voter theorem. It shows how political
parties might differentiate when they have to compromise between office
motivations and policy interests.

In the second motivating example described in subsection 1.1, the principle of
minimal differentiation implies the production of two products with the same
characteristic. [d’Aspremont et al., 1979] challenges this principle by
introducing a different trade off between transportation costs and price
differentiation. It introduces the principle of maximal differentiation that
states on the contrary that two competitors will select opposite side of the
space they are competing in (in our case $0$ and $1$) in order to avoid a high
price competition. In our setting the price is fixed, we explain the
differentiation with only spatial arguments, the reference locations.
Proposition 1 shows that this ingredient is more subtle: depending on the
references, we may or not observe differentiation. Note that equilibria are
inefficient as the total clientele is fixed but players always pay a cost
because they select $x_{i}\neq r_{i}$ at the duopoly equilibrium. However, the
inefficiency does not necessarily increases with $c$, consider the case of a
differentiated equilibrium where the penalty paid by each player is
$c\left(\frac{1}{4c}\right)^{2}=\frac{1}{16c}$ which decreases with $c>0$.

###  3.2 Comparative statics

The trade-off between strategic opportunities and costs minimization relies on
the magnitude of the cost function. When $\gamma(d)=cd^{2}$, the parameter $c$
captures the importance of the references locations for players so we now
analyze how the equilibrium configurations varies with $c\in(0,+\infty)$.

######  Remark 2.

For a better interpretation of the numerical values taken by the parameter
$c$, we can consider the case of a differentiated equilibrium where players
select locations at a distance $\delta=\frac{1}{4c}$ from their references. In
the first motivating example, when supposing that $c=1$, we implicitly assume
that the compromise found by a political party between its policy interested
members and its office motivated members it that the party should deviate at a
distance equal $\frac{1}{4}$ from its reference towards a strategic
opportunity, so $25\%$ of the total political spectrum size. Taking $c=2.5$
leads to a compromise of a $10\%$ deviation.

The figure below describes the equilibrium configuration with respect to
$(r_{1},r_{2})\in[0,1]^{2}$ for increasing values of $c$.

Figure 2: The equilibrium configurations with respect to
$(r_{1},r_{2})\in[0,1]^{2}$, for $c\in\\{0.7,0.9,1.5,5,15\\}$. In white there
is no equilibrium, in orange $(\frac{1}{2},\frac{1}{2})$ is the unique
equilibrium, in blue $(r_{1}+\frac{1}{4c},r_{2}-\frac{1}{4c})$ is the unique
equilibrium.

We observe (and prove in Proposition 5 below) that when $c$ goes to $0$,
$(\frac{1}{2},\frac{1}{2})$ is the unique equilibrium for every profile
$(r_{1},r_{2})\in[0,1]^{2}$, as it is when $c=0$. On the other hand, when $c$
goes to infinity, $(r_{1}+\frac{1}{4c},r_{2}-\frac{1}{4c})$ is the unique
equilibrium for every profile $(r_{1},r_{2})$ such that $r_{1}\neq r_{2}$.

These two extreme cases indicates that the probability of existence of
equilibrium, when the profile of references is uniformly drawn on the square
$[0,1]^{2}$, is a non-monotonic function of $c$.888We compute the likeliness
of an equilibrium in a game where each reference would be drawn at random at
the beginning of the game, as it is the case for the favorite policies of
candidates with character in [Kartik and McAfee, 2007] for example. The
probability we compute is equal to the area of the blue and orange domains
together.

######  Proposition 3.

Suppose that $\textbf{r}=(r_{1},r_{2})$ is uniformly drawn on the square
$[0,1]^{2}$. The game admits an equilibrium with probability

$\mathbb{P}(c)=\min\left(1,1-\frac{(2+4c)^{\frac{3}{2}}-6c-5}{6c^{2}}\right)$

There exists $\theta\simeq 1.321$ such that $\mathbb{P}(c)$ is constantly
equal to 1 for $c\leq\frac{1}{2}$, decreasing for $c\in[\frac{1}{2},\theta]$
and increasing for $c\geq\theta$.

The proof of this proposition is given in 6.3. We prove there that $\theta$ is
a unique real solution of a third degree equation.

Illustration: In the following figure, the function $\mathbb{P}(c)$ is drawn
in blue. Simulations using the algorithm described in subsection 5.4 confirm
the theory with red dots.

Figure 3: Probability of existence of an equilibrium if $(r_{1},r_{2})\sim
U([0,1]^{2})$ with respect to $c\in[0,10]$

Interpretation:

  * •

For $c\leq\frac{1}{2}$, the first condition in Proposition 1 is always
satisfied:

$c\leq\frac{1}{2}\Rightarrow r_{2}-\frac{1}{4c}\leq r_{1}+\frac{1}{4c}\text{
for every }r_{1}\leq r_{2}\in[0,1]^{2}$

Therefore, $(\frac{1}{2},\frac{1}{2})$ is the unique equilibrium for any
references $(r_{1},r_{2})$. The equilibrium is the same as in the case $c=0$
because the cost function is too small to have any impact: the marginal cost
of deviating is always smaller than $\frac{1}{2}$, which is the marginal gain
of a strategic opportunity.

  * •

For $c\in[\frac{1}{2},\theta]$, the probability that an equilibrium exists
decreases with $c$. In this interval, the likeliness of the undifferentiated
equilibrium decreases, the likeliness of the differentiated equilibrium
increases and the first trend is stronger than the second one. We find that
$\theta\simeq 1.321$: according to Remark 2, this value is relatively small,
as it represents players that would opportunistically agree to deviate by a
distance of $\frac{1}{4\theta}\simeq 0.189$ from their references, in the unit
interval.

  * •

For $c\geq\theta$, the probability that an equilibrium exists increases with
$c$. In this interval, the undifferentiated equilibrium has almost
disappeared, so the increase of the likeliness of the differentiated
equilibrium makes the probability of existence larger.

######  Proposition 4.

Suppose that $\textbf{r}=(r_{1},r_{2})$ is uniformly drawn in the unit square.
The differentiated equilibrium $(r_{1}+\delta,r_{2}-\delta)$ is more likely
than the undifferentiated equilibrium $(\frac{1}{2},\frac{1}{2})$ if and only
if $c\geq\eta\simeq 1.17692$.

The proof of this results is somehow trivial as Proposition 1 immediately
implies that the probability that the unique equilibrium is
$(\frac{1}{2},\frac{1}{2})$ is $(\frac{1}{2c})^{2}=\frac{1}{4c^{2}}$. On the
other hand we easily compute the probability of the differentiated
equilibrium: $\mathbb{P}(c)-\frac{1}{4c^{2}}$ and the proposition follows. To
obtain the algebraic form of $\eta$, one needs to find the positive solution
to the equation $36c^{4}+8c^{3}-36c^{2}-24c-4=0$. Here we only provide the
numerical approximation. Again, according to Remark 2, the numerical value of
$\eta$ is relatively small, as it represents players that would
opportunistically agree to deviate by a distance of $\frac{1}{4\eta}\simeq
0.212$ from their references, in the unit interval.

In the current paper, we choose to analyze the equilibrium configuration as a
function of the pair $(r_{1},r_{2})$ and for a fixed parameter $c$. The
following proposition rephrases Proposition 1 when the pair $(r_{1},r_{2})$ is
fixed but the parameter $c$ varies.

######  Proposition 5.

Let $(r_{1},r_{2})\in[0,1]^{2}$. There exists $c_{0}>0$ such that for every
$c\in[0,c_{0}]$ the game admits $(\frac{1}{2},\frac{1}{2})$ as the unique
equilibrium of the game.
If $r_{1}\neq r_{2}$, there also exists $c_{1}$ such that for every
$c\in[c_{1},+\infty[$ the game admits $(r_{1}+\delta,r_{2}-\delta)$ as the
unique equilibrium of the game.

The previous proposition is illustrated in figure 2 where we can see that for
any pair of references, the equilibrium is undifferentiated for $c$ small
enough, and differentiated for $c$ large enough (and $r_{1}\neq r_{2}$). The
proof is straightforward: for $c$ small enough it is always the case that
$r_{2}-\frac{1}{4c}\leq\frac{1}{2}\leq r_{1}+\frac{1}{4c}$ so Proposition 6.2
states that there exists a unique equilibrium which is
$(\frac{1}{2},\frac{1}{2})$. On the other hand, if $c$ is large enough and
$r_{1}\neq r_{2}$, take without loss of generality $r_{1}<r_{2}$, then is it
always the case that $r_{1}+\frac{1}{4c}<r_{2}-\frac{1}{4c}$ and both
inequalities $(E1)$ and $(E2)$ hold, so Proposition 6.2 provides the existence
and uniqueness of the differentiated equilibrium.

This last proposition opens a mechanism design perspective: if the planner can
decide which parameter $c>0$ to implement, for example in the taxation setting
mentioned in the introduction, she basically decides whether there is an
equilibrium or not, and whether the equilibrium is differentiated or not.
Moreover, because the distance between players at a differentiated equilibrium
is given by $d(x_{2},x_{1})=r_{2}-r_{1}-\frac{1}{2c}$, she can also decide the
distance between firms in the interval $[0,r_{2}-r_{1})$.

###  3.3 General cost function $\gamma$

Proposition 1 can be generalized to the case where the cost function is given
by a strictly increasing and strictly convex function
$\gamma:[0,1]\rightarrow\mathbb{R}$. In this case, the following parameter
$\delta$ plays an important role.

######  Definition 6.

We denote $\delta$ the unique distance in $[0,1]$, if it exists, such that
$\gamma^{\prime}(\delta)=\frac{1}{2}$. If $\gamma^{\prime}(d)<\frac{1}{2}$ for
every $d\in[0,1]$ we set $\delta:=+\infty$ and if
$\gamma^{\prime}(d)>\frac{1}{2}$ for every $d\in[0,1]$ we set $\delta:=0$.

The uniqueness of $\delta$ is a consequence of the strict convexity of
$\gamma$. As noted in the discussion of proposition 1, the marginal gain when
moving closer to an opponent is constant equal to $\frac{1}{2}$, while the
marginal cost of a deviation further the reference is increasing as the cost
function $\gamma$ is convex. The parameter $\delta$ captures the turning point
where the second becomes larger than the first, therefore $\delta$ is the
maximal distance that a player beneficially deviates from its reference in the
same neighborhood. Note that in the particular case where $\gamma(d)=cd^{2}$
then $\delta=\frac{1}{4c}$.

######  Proposition 7.

Let $\textbf{r}=(r_{1},r_{2})\in[0,1]^{2}$. Suppose, without loss of
generality, that $r_{1}\leq r_{2}$. Then we have:

  * •

If $r_{2}-\delta\leq r_{1}+\delta$, the unique possible equilibrium is
$(\frac{1}{2},\frac{1}{2})$.

It is an equilibrium if and only if
$\frac{1}{2}\in[r_{2}-\delta,r_{1}+\delta]$.

  * •

If $r_{1}+\delta<r_{2}-\delta$, the unique possible equilibrium is
$(r_{1}+\delta,r_{2}-\delta)$.

It is an equilibrium if and only if
$\begin{cases}\gamma(r_{2}-r_{1}-\delta)-\gamma(\delta)+\frac{r_{1}+3r_{2}}{2}-1-\delta\geq
0~{}~{}~{}~{}(E1^{\prime})\\\
\gamma(r_{2}-r_{1}-\delta)-\gamma(\delta)-\frac{3r_{1}+r_{2}}{2}+1-\delta\geq
0~{}~{}~{}~{}(E2^{\prime})\end{cases}$

The proof of Proposition 7 strictly follows the proof of Proposition 1 and the
interpretation is unchanged. Remark that, although the unique equilibrium
locations only depend on $\delta$ (and not on the specific function $\gamma$),
the conditions for this candidate to be indeed an equilibrium depends on
$\gamma$, as we can see in the inequalities $(E1^{\prime})$ and
$(E2^{\prime})$.

###  3.4 Heterogeneous costs

We show in this subsection that Proposition 1 can be easily extended to the
case where players incur different cost functions $\gamma_{i}(d)=c_{i}d^{2}$.
We denote $\delta_{i}=\frac{1}{4c_{i}}$.

######  Proposition 8.

Let $r=(r_{1},r_{2})\in[0,1]^{2}$. Suppose, without loss of generality, that
$r_{1}\leq r_{2}$. Then we have :

  * •

If $r_{2}-\delta_{2}\leq r_{1}+\delta_{1}$, the unique possible equilibrium is
$(\frac{1}{2},\frac{1}{2})$.
It is an equilibrium if and only if
$\frac{1}{2}\in[r_{2}-\delta_{2},r_{2}+\delta_{2}]\cap[r_{1}-\delta_{1},r_{1}+\delta_{1}]$.

  * •

If $r_{1}+\delta_{1}<r_{2}-\delta_{2}$, the unique possible equilibrium is
$(r_{1}+\delta_{1},r_{2}-\delta_{2})$.
It is an equilibrium if and only if:
$\begin{cases}c_{1}(r_{2}-r_{1})^{2}-2c_{1}\delta_{2}(r_{2}-r_{1})+\frac{1}{2}(r_{1}+\delta_{1})+\frac{3}{2}(r_{2}-\delta_{2})+c_{1}(\delta_{2}^{2}-\delta_{1}^{2})-1\geq
0~{}~{}(E1^{\prime\prime})\\\
c_{2}(r_{2}-r_{1})^{2}-2c_{2}\delta_{1}(r_{2}-r_{1})-\frac{1}{2}(r_{2}-\delta_{2})-\frac{3}{2}(r_{1}+\delta_{1})-c_{2}(\delta_{2}^{2}-\delta_{1}^{2})+1\geq
0~{}~{}(E2^{\prime\prime})\end{cases}$

The proof is postponed to subsection 6.4.

Illustration: the above figures are obtained with $c_{1}=1$ and $c_{2}=2$ on
the left, and with $c_{1}=1$ and $c_{2}=10$ on the right.

Figure 4: The equilibrium structures with asymmetric costs:
$(c_{1},c_{2})=(1,2)$ on the left, and $(1,10)$ on the right. In white there
is no equilibrium, in orange $(\frac{1}{2},\frac{1}{2})$ is the unique
equilibrium, in blue $(r_{1}+\delta_{1},r_{2}-\delta_{2})$ is the unique
equilibrium.

Interpretation: The structure of equilibria remains unchanged: when references
are far from each other there is a differentiated equilibrium, if they are
close to each other and close to $\frac{1}{2}$, the unique equilibrium is
$(\frac{1}{2},\frac{1}{2})$. In this second case, the equilibrium conditions
are modified and do not define a square anymore but a rectangle. Finally if
they are close to each other but far from $\frac{1}{2}$ there is no
equilibrium.
The figures are still invariant to the linear transformation
$(x^{\prime},y^{\prime})=(1-x,1-y)$ as the unit interval with uniform density
is symmetric, but the figures are not symmetric with respect to the
transformations $(x^{\prime},y^{\prime})=(y,x)$ or
$(x^{\prime},y^{\prime})=(1-y,1-x)$ anymore, as it was the case when costs
where homogeneous, because players are not anonymous anymore.

##  4 The triopoly

###  4.1 Existence and characterization of equilibrium

The case where three players compete on the unit interval is particularly
interesting as it is the unique case where there is no pure equilibrium in the
game without cost (where $\gamma=0$). Indeed, it is easy to prove that at
equilibrium the two following contradictory statements hold:
(1) Peripheral players are paired,
(2) No more than two players share the same location.
An immediate consequence is that the triopoly does not admit any equilibrium.

Different papers tried to fill the gap, by introducing mixed equilibrium
(Shaked [1982]) or by relaxing the assumption that customers always buy to the
closest shop (De Palma et al. [1987]) for example.

In this section we show that the introduction of reference locations has an
impact on the existence of a triopoly equilibrium. More precisely: the non-
existence result is robust to the introduction of small reference costs. When
the costs increase above a certain threshold, there exists either a unique
equilibrium or no equilibrium, depending on the references. When the cost is
asymptotically large, there exists an equilibrium for almost every triplet of
references.

######  Proposition 9.

Suppose that $\gamma(d)=cd^{2}$ and consider the triopoly competition.
(1) If $c<\frac{5}{8}$, there is no equilibrium for any triplet of references
r.
(2) If $c\geq\frac{5}{8}$, there exists either a unique or no equilibrium
depending on the triplet of references.
(3) For every triplet of different references $r_{1}<r_{2}<r_{3}$, there
exists a unique equilibrium for $c$ large enough.

This proposition is proved in 6.5.2. In the proof we first characterize the
unique equilibrium candidate $\textbf{x}^{\star}$ in 6.5.1 that we now
describe, after introducing the useful notion of far-left and far-right
players.

######  Definition 10.

If $r_{1}+\frac{1}{4c}<r_{2}$ we say that Player $1$ is a far-left player. If
$r_{n}-\frac{1}{4c}>r_{n-1}$ we say that Player $n$ is a far-right player.
In the general case of strictly convex and increasing cost function $\gamma$,
we say that Player $1$ is far-left if $r_{1}+\delta<r_{2}$ and that Player $n$
is far right if $r_{n}-\delta>r_{n-1}$, where $\delta$ is defined in 6.

We emphasize that the notion of far player concerns a profile of references r
while the notion of peripheral player is a property of a profile of locations
x. As shown in Proposition 9 for three players, and later in Proposition 12
for the general case, a far-left or far-right player has a reference which is
so extreme that he never shares his equilibrium location with another player.
We now describe the equilibrium candidate in the triopoly competition (proof
in 6.5.1). Suppose without loss of generality that $r_{1}\leq r_{2}\leq
r_{3}$, then:

  * •

If Player $1$ is far-left and Player $3$ is far-right then the unique possible
equilibrium is
$\textbf{x}^{\star}=(r_{1}+\frac{1}{4c},r_{2},r_{3}-\frac{1}{4c})$.

  * •

If Player $1$ is not far-left but Player $3$ is far-right, the unique possible
equilibrium is
$\textbf{x}^{\star}=\left(\frac{r_{3}-1/4c}{3},\frac{r_{3}-1/4c}{3},r_{3}-\frac{1}{4c}\right)$,
and symmetrically, if Player $1$ is far-left but Player $3$ is not far-right,
the unique possible equilibrium is
$\textbf{x}^{\star}=\left(r_{1}+\frac{1}{4c},\frac{r_{1}+1/4c+2}{3},\frac{r_{1}+1/4c+2}{3}\right)$.

  * •

If neither Player $1$ nor Player $3$ are far-left or far-right, there is no
equilibrium.

The unique equilibrium candidate $\textbf{x}^{\star}$ is found by considering
the necessary condition that at equilibrium players do not have profitable
local deviation, that is a deviation within the same neighborhood. To fully
characterize the set of equilibria, one also has to consider deviations to a
different neighborhood. We postpone this discussion to section 5: we show
that, in the general case of $n$ players, to insure that no players has a
deviation in a different neighborhood, the vector of reference locations has
to satisfy a system of $n(n-1)$ equations. We now provide intuition and
illustration for Proposition 9.

The intuitive reason why the classical non-existence result in the triopoly
competition is robust against the introduction of a small cost of deviation
from a reference is the following:
\- If the references are close to $\frac{1}{2}$, the argument is the same as
in the case without costs: all three players want to locate together at the
center, but such a configuration is not stable as each player could deviate
marginally to his left or right and get half of the interval instead of
sharing a third.
\- Therefore, the most favorable case for existence of an equilibrium is the
case where $(r_{1},r_{2},r_{3})=(0,\frac{1}{2},1)$. The description of the
unique equilibrium candidate above gives that $x_{1}=\frac{1}{4c}$,
$x_{2}=\frac{1}{2}$ and $x_{3}=1-\frac{1}{4c}$, and we find that if
$c\leq\frac{5}{8}$, Player $2$’s payoff is $\frac{1}{5}$ which makes a
deviation to the left of Player $1$ (or to the right of Player $2$)
profitable.

Illustration: The graph below is concerned with the existence of equilibrium
in the triopoly competition.

Figure 5: Probability that the triopoly competition admits an equilibrium when
$\textbf{r}\sim\mathscr{U}([0,1]^{3})$ and $\gamma(d)=cd^{2}$ for
$c\in(0,10)$.

The graph illustrates the fact that for $c$ small enough, the probability that
an equilibrium exists is zero (Proposition 9 is stronger as it also shows that
there does not exist a negligible set of equilibria neither). Although the
probability of existence goes to $1$ as $c\rightarrow+\infty$, it increases
slowly: half of the profiles provide an equilibrium only when $c\simeq 15.35$.
According to Remark 2, this numerical value of $c$ is relatively large, as it
represents players that would opportunistically agree to deviate only by a
distance of $\frac{1}{4c}\simeq 0.016$ from their references, in the unit
interval.

As an illustration, we now analyze the triopoly competition in two particular
cases.

###  4.2 Illustrative example 1: symmetric references profiles

We consider the particular case where the references profiles are symmetric.
Therefore, we suppose that $(r_{1},r_{2},r_{3})=(r_{1},\frac{1}{2},1-r_{1})$
with $r_{1}\in[0,\frac{1}{2})$.

######  Proposition 11.

Suppose that $\gamma(d)=cd^{2}$. The triopoly competition with references
$\textbf{r}=(r_{1},\frac{1}{2},1-r_{1})$ admits a unique possible equilibrium
which is
$\textbf{x}^{\star}=(r_{1}+\frac{1}{4c},\frac{1}{2},1-r_{1}-\frac{1}{4c})$.
This is an equilibrium if and only if
$r_{1}\leq\phi(c):=\frac{3+2c-2\sqrt{4+2c}}{4c}$.

The proof is postponed to subsection 6.6.

This result illustrates Proposition 9 as $\phi$ is an increasing function: if
$c$ is small enough, there exists no equilibrium. We can compute that for
$c\leq\frac{2\sqrt{2}-1}{2}\simeq 0.914$ there exists no equilibrium for any
symmetric profile of location. It improves the general threshold of
$\frac{5}{8}$ provided for the general case. We also observe that $\phi(c)$
goes to $\frac{1}{2}$ when $c$ goes to $+\infty$, so that the game admits a
unique equilibrium for every $r_{1}\in[0,\frac{1}{2})$ if $c$ is large enough.

###  4.3 Illustrative example 2: one reference is fixed to $\frac{1}{2}$

The following figure illustrates the equilibrium characterized in subsection
4.1 in the case where a reference is fixed to $\frac{1}{2}$ and
$\gamma(d)=10d^{2}$.

Figure 6: The equilibrium structure with respect to
$(r_{1},r_{2})\in[0,1]^{2}$ when $r_{3}=\frac{1}{2}$. In blue: there exists a
unique equilibrium where all players select different locations, in orange:
there exists a unique equilibrium where two players select the same location,
in white there is no equilibrium.

When both $r_{1}$ and $r_{2}$ are far from $\frac{1}{2}$, the figure show 4
symmetrical sub-figures:
\- the bottom-left sub-figure illustrates the case where $r_{1}\leq r_{2}\leq
r_{3}=\frac{1}{2}$. This sub-figure is quite similar to the figure 1 obtained
in the duopoly competition. This is explained by the fact that the third
player locates at $r_{3}=\frac{1}{2}-\frac{1}{4c}$ and that the two players
compete on the sub-interval $[0,x_{3})$. A deviation to the right of $x_{3}$
would be too costly. As in the duopoly competition, the orange rectangle
represents the cases where the two players select the same location, which is
here $\frac{1}{6}-\frac{1}{12c}$, i.e. the location where they attract the
same quantity of resources from the left and from the right when the third
player is located at $\frac{1}{2}-\frac{1}{4c}$ (because claim 2 of Lemma 18
applies).
\- take now the top left sub-figure, concerned with the case where $r_{1}\leq
r_{3}=\frac{1}{2}\leq r_{2}$. In this case, we find that the equilibrium is
differentiated,
$\textbf{x}=(r_{1}+\frac{1}{4c},\frac{1}{2},r_{2}-\frac{1}{4c})$ and the
condition for this profile to be an equilibrium is that there is no profitable
deviation of Player $2$ or $3$ to the left of Player $1$, and no profitable
deviation of Player $1$ or $3$ to the left of Player $2$. These conditions
involve quadratic equation that are similar to $(E1)$ and $(E2)$ in
Proposition 6.2.
The top-right and bottom-right sub-figures are symmetric to the previous ones.
Indeed, the figure is symmetric with respect to the linear transformation
$(x^{\prime},y^{\prime})=(y,x)$ as players are anonymous and
$(x^{\prime},y^{\prime})=(1-y,1-x)$ as the unit interval with uniform density
is symmetric.

##  5 The general case of $n$ players

In this section, we only suppose that the cost function $\gamma$ is a $C^{1}$,
strictly convex and strictly increasing function. We first prove that for any
vector of references the game admits at most one equilibrium and we
characterize the unique equilibrium candidate and its properties. We then
consider the sufficient equilibrium conditions and we prove that these
(infinitely many) conditions can be reduced to a list of $n(n-1)$ equations.
Finally, we illustrate our result with the particular case of regularly
located references.

###  5.1 Existence, uniqueness and characterization of the equilibrium

The next proposition deals with the case of $n\geq 5$ players. The duopoly and
triopoly are studied in section 3 and 4. The case where $n=4$ is also
particular because it is possible that all players are peripheral, which is
not possible when $n\geq 5$ as proved in Claim 1 of Proposition 18. For the
sake of completeness, the case of $4$ player is provided in subsection 5.3 and
poses no conceptual difficulties.

Proposition 12 exhibits the unique equilibrium candidate and Proposition 14
provides necessary and sufficient conditions for this candidate to be an
equilibrium. The parameter $\delta$ is defined in 6 and the notions of far-
left and far-right players are introduced in 10.

######  Proposition 12.

Let $n\geq 5$ and $r_{1}\leq r_{2}\leq\dots\leq r_{n}$. The unique possible
equilibrium $\textbf{x}^{\star}$ of the game with references r is the
following:

  * •

If Player $1$ is a far-left player, he is the only left-peripheral player and
locates at $x_{1}^{\star}=r_{1}+\delta$. Otherwise, both Player $1$ and Player
$2$ are the left-peripheral players and they locate at $\frac{r_{3}}{3}$.

  * •

Symmetrically, if Player $n$ is a far-right player, he is the only right-
peripheral player and locates at $x_{n}^{\star}=r_{n}-\delta$. Otherwise, both
Player $n$ and Player $n-1$ are the right-peripheral players and they locate
at $\frac{2+r_{n-2}}{3}$.

  * •

All non peripheral players locate at their references: $x_{i}^{\star}=r_{i}$.

The proof is given in subsection 6.7.1. We now discuss the interesting
properties of the equilibrium candidate:
(1) the equilibrium, when it exists, is unique. This property differs from the
analysis of location games without references, where there exists for $n>6$ a
continuum of equilibria (a polyhedron of dimension $n-5$). The reason for
uniqueness in the current paper is that the cost of deviating from the
reference location is a tie-breaking rule: while the quantity of resources
that a player attracts does not depend on the exact position of a player in a
fixed neighborhood, the total payoff is not constant as the costs are strictly
increasing with the distance.
(2) at equilibrium, at most $4$ players select a location which is different
from their references: the two most left and most right players. Therefore,
when studying the game with many players, only the border effects have to be
looked at, i.e. the behavior of players $1$ and $2$ (symmetrically of players
$n-1$ and $n$). All the interior players locate at their references.
(3) the border effects are the same for any $n\geq 3$: either player $1$’s
reference is far enough to player $2$’s reference and then player $1$ acts as
a monopolist and satisfies his first order conditions, or the $2$ most left
references are close enough and players select the same location. In this case
the choice of this location is dictated by 18, claim 2: they must be located
at $\frac{1}{3}$ of the interval $[0,r_{3}]$. The argument is symmetric on the
right.

We illustrate Proposition 12 with a numerical example.

######  Example 13.

Suppose that $\mathbb{r}=(0.10,0.13,0.42,0.52,0.75,0.95)\in[0,1]^{6}$ and
$\gamma(d)=5d^{2}$. Player 1 is not far-left since
$r_{2}-r_{1}\leq\delta=0.05$, but Player $6$ is far-right since
$r_{6}-r_{5}>\delta$. The profile $\textbf{x}^{\star}$ is therefore such that
$x_{1}=x_{2}=r_{3}/3$ and such that $x_{6}=r_{6}-\delta$. All other players
selects $x_{i}=r_{i}$. Thus,
$\mathbb{x}^{\star}=(\frac{r_{3}}{3},\frac{r_{3}}{3},r_{3},r_{4},r_{5},r_{6}-\delta)=(0.14,0.14,0.42,0.52,0.75,0.90)$
is the unique equilibrium candidate.

$|$$|$$|$$|$$|$$|$$0$$r_{1}$$r_{2}$$r_{3}$$r_{4}$$r_{5}$$r_{6}$$1$$x_{1}$$x_{2}$$x_{3}$$x_{4}$$x_{5}$$x_{6}$

######  Proposition 14.

The equilibrium candidate described in Proposition 12 is an equilibrium if and
only if the three following conditions hold:
(1) If Player 1 is not far-left then $\frac{r_{3}}{3}\in[r_{2},r_{1}+\delta]$.
(2) Symmetrically, if Player $n$ is not far-right then
$\frac{2+r_{n-2}}{3}\in[r_{n}-\delta,r_{n-1}]$.
(3) $\Delta_{i}^{j}(\textbf{x}^{\star})\geq 0$ for every $i\neq
j\in\\{1,\dots,n\\}$, where $\Delta_{i}^{j}(\textbf{x})$ denote the loss of
player $i$ when he deviates infinitesimally close to Player $j$’s location
when x is played. More precisely:

$\Delta_{i}^{j}(\textbf{x})=\begin{cases}\lim\limits_{\begin{subarray}{c}\epsilon\to
0\\\
\epsilon>0\end{subarray}}u_{i}(\textbf{x})-u_{i}(x_{j}+\epsilon,\textbf{x}_{-i})\text{
if }x_{i}<x_{j}\\\ \lim\limits_{\begin{subarray}{c}\epsilon\to 0\\\
\epsilon>0\end{subarray}}u_{i}(\textbf{x})-u_{i}(x_{j}-\epsilon,\textbf{x}_{-i})\text{
if }x_{i}>x_{j}\\\ 0\text{ if }x_{i}=x_{j}\end{cases}$

Conditions $(1)$ and $(2)$ are concerned with local deviations of peripheral
players when they are paired. If player $1$ is not a far-left player, then the
unique possible equilibrium described in Proposition 12 is such that
$x_{1}=x_{2}=\frac{r_{3}}{3}$. In the proof, we show that if
$\frac{r_{3}}{3}>r_{1}+\delta$ then Player $1$ has a profitable deviation:
$\frac{r_{3}}{3}$ is too far from his reference (at a distance larger than
$\delta$, which the maximal distance he can deviate from $r_{1}$ as a
peripheral player). On the other hand, if $\frac{r_{3}}{3}<r_{2}$ then Player
$2$ would improve his payoff by playing $\frac{r_{3}}{3}+\epsilon$ for
$\epsilon>0$ small enough: the quantity of attracted resources is the same but
the reference cost is smaller.
Condition $(3)$ is concerned with global deviation: if
$\Delta_{i}^{j}(\textbf{{x}}^{\star})<0$ then player $i$ has a profitable
deviation by selecting a location infinitesimally close to player $j$: the
market opportunity is larger than the reference cost of such a deviation.
Among the infinity of possible deviations, we show that if these $n-1$
deviations are not profitable, then the profile is an equilibrium.

######  Example 15.

We come back to example 13. The equilibrium candidates is
$\mathbb{x}^{\star}=(\frac{r_{3}}{3},\frac{r_{3}}{3},r_{3},r_{4},r_{5},r_{6}-\delta)=(0.14,0.14,0.42,0.52,0.75,0.90)$.
It satisfies condition $(1)$ as
$\frac{r_{3}}{3}=0.14\in[0.13,0.15]=[r_{2},r_{1}+\delta]$. Condition $(2)$ is
empty as player $6$ is far-right. Condition $(3)$ translates to:
$(\Delta_{1}^{j})_{j\neq 1}=(0,~{}0.594,~{}0.899,~{}2.1695,~{}3.232)$
$(\Delta_{2}^{j})_{j\neq 2}=(0,~{}0.51,~{}0.785,~{}1.9865,~{}3.004)$
$(\Delta_{3}^{j})_{j\neq 3}=(0.442,~{}0.442,~{}0.3395,~{}0.6595,~{}1.242)$
$(\Delta_{4}^{j})_{j\neq 4}=(0.747,~{}0.747,~{}0.075,~{}0.3545,~{}0.787)$
$(\Delta_{5}^{j})_{j\neq 5}=(1.9105,~{}1.9105,~{}0.5945,~{}0.4045,~{}0.2025)$
$(\Delta_{6}^{j})_{j\neq 6}=(3.303,~{}3.303,~{}1.427,~{}1.037,~{}0.2475)$
we see that $\Delta_{i}^{j}\geq 0$ for every $i\neq j$ so the sufficient
conditions are satisfied and $\mathbb{x}^{\star}$ is an equilibrium.

###  5.2 Illustrative example: regular location of references

In this subsection, we consider the case where references are regularly
distributed on $[0,1]$ to illustrate our results.

######  Proposition 16.

In the game with $n\geq 5$ players where $\gamma(d)=cd^{2}$ and
$r=\left(\frac{1}{n+1},\frac{2}{n+1},...,\frac{n}{n+1}\right)$, we have that:

$\textbf{x}^{\star}=\left(\frac{1}{n+1}+\frac{1}{4c},\frac{2}{n+1},...,\frac{n}{n+1}-\frac{1}{4c}\right)$
is an equilibrium if and only if $c\geq\frac{1+\sqrt{6}}{4}(n+1)$.

Otherwise there is no equilibrium.

The proposition is proved in 6.8. The unique equilibrium candidate is such
that only players $1$ and $n$ are peripheral. The intuition for the result is
the following: when $\textbf{x}^{\star}$ is played, the peripheral players are
players who attract the more resources while Players $2$ and $n-1$ are players
who attract the less resources. For this profile to be an equilibrium, we need
$c$ to be large enough so that Players $2$ does not have a profitable
deviation to deviate to the left of Player $1$. As $n$ becomes large, the
distance between players decreases and the constraint on $c$ is stronger.

###  5.3 The case where $n=4$

For the sake of completeness, we here give the equilibrium configuration in
the case of $n=4$ players.

######  Proposition 17.

Let $\textbf{r}=(r_{1},r_{2},r_{3},r_{4})\in[0,1]^{4}$. Suppose, without loss
of generality, that $r_{1}\leq r_{2}\leq r_{3}\leq r_{4}$. Then we have:

  * •

If Player $1$ is a far-left player and Player $4$ is a far-right player then
the unique possible equilibrium is
$\textbf{x}^{\star}=(r_{1}+\delta,r_{2},r_{3},r_{4}-\delta)$.

  * •

If Player $1$ is not a far-left player but Player $4$ is a far-right player,
the unique possible equilibrium is
$\textbf{x}^{\star}=(\frac{r_{3}}{3},\frac{r_{3}}{3},r_{3},r_{4}-\delta)$.

  * •

Symmetrically, if Player $1$ is a far-left player but Player $4$ is not a far-
right player, the unique possible equilibrium is
$\textbf{x}^{\star}=(r_{1}+\delta,r_{2},\frac{r_{2}+2}{3},\frac{r_{2}+2}{3})$.

  * •

If neither Player $1$ nor Player $4$ are far left or far-right, there is no
equilibrium.

The necessary and sufficient conditions given in Proposition 14, for the
equilibrium candidate to be an equilibrium, also apply for $n=4$.

This proposition is given here for the sake of completeness. The results
slightly differ as it might be the case that no player select $x_{i}=r_{i}$.
However, its proof poses no conceptual difficulties and relies on a case-by-
case analysis using the same arguments as the general case of $n\geq 5$
players.

###  5.4 Algorithm

Figures 1, 2, 3, 4, 5, 6 or similar figures with different parameters can be
reproduced using the algorithms described in this section and available at
https://github.com/FournierAMSE
/Location-games-with-references.

Each program is concerned with a particular case of $2$, $3$, $4$ or $n\geq 5$
players. Each program typically contains the following functions:

  * •

equilibrium(r,c): For the given reputation vector r and $c>0$, this function
verifies whether the unique equilibrium candidate characterized in Proposition
12 (for the case $n\geq 5$) is an equilibrium. To do so, it checks whether the
candidate satisfies each necessary and sufficient conditions provided in
Proposition 14.

  * •

probability_NE(nb_draws,c): This function estimates the probability of
existence of an equilibrium by performing several random draws of reputation
profiles r and computing the ratio of equilibrium to the number of draws
(NE/nb_draws).

  * •

NE_plot(nb_draws,c) : This function performs several random draws of
$(r_{1},r_{2})$ and plots the couples for which there exists an equilibrium.
When plotting these figures with $n>2$, we need to fix $n-2$ reputations to
obtain a $2$-dimensional figure.

##  6 Proofs

For simplicity, we use the vocabulary from the second motivating example
presented in the introduction: resources are customers, players are sellers,
the consumers buying to a given seller are its clientele. We first introduce
some useful notations:

  * •

For a profile $\textbf{x}=(x_{1},\dots,x_{n})$ of locations in $[0,1]^{n}$ and
a point $y\in[0,1]$, we denote $\operatorname{Card}(y,\textbf{x})$ the number
of players who selected location $y$ when the profile x is played.
$\operatorname{Card}(y,\textbf{x})=0$ when $y\notin\\{x_{1},\dots,x_{n}\\}$.
If $\operatorname{Card}(y,\textbf{x})=2$ we say that $2$ players are paired in
$y$.

  * •

We denote by $q_{i}(\textbf{x})$ the quantity of customers attracted by Player
$i$. These consumers are called his clientele. With the notations introduced
at the end of section 2, we have:

$q_{i}(\textbf{x}):=\frac{x_{i}^{r}-x_{i}^{\ell}}{2\operatorname{Card}(x_{i},\textbf{x})}$

It will sometimes be useful to consider $q_{i}^{\ell}(x)$ and $q_{i}^{r}(x)$
the quantity of customers attracted by player $i$ from the left and from the
right:

$q_{i}^{\ell}(\textbf{x}):=\frac{x_{i}-x_{i}^{\ell}}{2\operatorname{Card}(x_{i},\textbf{x})}$
$q_{i}^{r}(\textbf{x}):=\frac{x_{i}^{r}-x_{i}}{2\operatorname{Card}(x_{i},\textbf{x})}$

We obviously have the relation:
$q_{i}(\textbf{x})=q_{i}^{\ell}(\textbf{x})+q_{i}^{r}(\textbf{x})$.

The proof section intensively use the following equilibrum properties.

###  6.1 Some equilibrium properties

In the following, $\gamma$ is a continuous, strictly increasing and strictly
convex function such that $\gamma(0)=0$.

######  Lemma 18.

Suppose that in the game with $n$ players, references r and cost function
$\gamma$, an equilibrium x is played:

  1. 1.

Players are either single or paired: for any player $i$ we have

$Card(x_{i},\textbf{x})\leq 2$

  2. 2.

If two players are paired, the quantity of consumers buying to their locations
from the left is equal to the quantity of consumers buying from the right:

$Card(x_{i},\textbf{x})=2\Rightarrow
q_{i}^{\ell}(\textbf{x})=q_{i}^{r}(\textbf{x})$

  3. 3.

The quantity of consumers that a non-peripheral player attracts does not
change when he deviates to a location between the same neighbors (also when he
is paired). Formally: $y\mapsto q_{i}(y,\textbf{x}^{-i})$ is constant with
$y\in(x_{i}^{\ell},x_{i}^{r})$ for a non-peripheral player $i$.

  4. 4.

The quantity of consumers that a paired, left-peripheral and not right-
peripheral player attracts does not change when he deviates to his right, as
long as he doesn’t go over his right neighbor: $y\mapsto
q_{i}(y,\textbf{x}^{-i})$ is a constant function on the interval
$[x_{i},x_{i}^{r})$. The same statement holds for a right-peripheral player
deviating to his left.

  5. 5.

A left-peripheral players which is not right-peripheral is located on the
right of his reference: $r_{i}\leq x_{i}$ for $i$ left and not right-
peripheral (resp. $r_{j}\geq x_{j}$ for $j$ right and not left-peripheral).

  6. 6.

If the reference profile is sorted, then equilibrium is also sorted:
$r_{1}\leq\dots\leq r_{n}\Rightarrow x_{1}\leq\dots\leq x_{n}$.

###### Proof.

Here we choose to avoid formalism when possible as the results are intuitive.
(1) Suppose that three players select the same location. Each of them gets a
third of the clientele. If one of these players makes an infinitesimal move to
the left, he loses consumers from the right but does not share consumers from
the right anymore. Either left of right consumers is larger than a third of
the total clientele, so either an infinitesimal deviation to the left or to
the right improves the quantity of attracted consumers. Because the mentioned
deviations are infinitesimal and the cost function is continuous, we described
a profitable deviation. The same argument applies with any number of players
$n\geq 3$.
(2) Suppose that two players select the same location and that they attract
more consumers from the left ($q^{\ell}>q^{r}$). If a player makes an
infinitesimal move to the left, he looses the right consumers but does not
share the left consumers anymore. Again, because $\gamma$ is continuous, we
described a profitable deviation.
(3) The statement is obvious if the player is single, as his clientele only
depends on the distance between his left and right neighbors, which is
constant. In the case where the player is paired, he attracts half the
consumers buying from the left and half the consumers buying from the right.
At equilibrium these two quantities are equal (Claim 2 above). A deviation to
the left would change his right neighbor but not the quantity of attracted
consumers.
(4) Suppose that a player is paired, left-peripheral and not right-peripheral.
Again, a deviation to the right would change his left neighbor but not the
total quantity of attracted consumers. This statement does not hold for a
deviation to the left where he attract less consumers.
(5) If $x_{i}<r_{i}$ for $i$ left and not right-peripheral player then player
$i$ has a profitable deviation to his right. It also holds when $i$ is paired
due to Claim 4 above.
(6) Suppose that there exists $i<j$ such that $x_{j}<x_{i}$. We have either
$x_{i}\leq r_{j}$ or $r_{j}<x_{i}$. Suppose first that $x_{i}\leq r_{j}$ so
that $x_{j}<x_{i}\leq r_{j}$. In this situation Player $j$ may be non-
peripheral (paired or not) or left-peripheral (paired or not).
\- If Player $j$ is non-peripheral (paired or not), Claim 3 provides a
profitable deviation within the same neighborhood, towards his reference.
\- If Player $j$ is left-peripheral and paired, Claim 4 provides a profitable
deviation to his right.
\- If Player $j$ is left-peripheral and not paired, a small deviation to his
right increases his clientele and decreases his cost.
Suppose now that $r_{j}<x_{i}$, so that $r_{i}\leq r_{j}<x_{i}$ and
$x_{j}<x_{i}$. In this situation Player $i$ may be non-peripheral (paired or
not) or right-peripheral (paired or not).
\- If Player $i$ is non-peripheral (paired or not), Claim 3 above gives us
that he would have a profitable local deviation towards his reference.
\- If Player $i$ is right-peripheral (paired or not), he has a profitable
deviation towards his reference as it lowers his cost while increasing (or
maintaining if paired) his quantity of attracted resources. ∎

An immediate consequence of the previous lemma is the following strong
equilibrium property:

######  Lemma 19.

Suppose that in the game with $n$ players, references r and cost function
$\gamma$, an equilibrium x is played: if Player $i$ is non-peripheral then
$x_{i}=r_{i}$.

###### Proof.

Suppose that $x_{i}\neq r_{i}$ for a non-peripheral player $i$. Claim 3 of
Lemma 18 tells that $q_{i}$ is constant in the interval
$(x_{i}^{\ell},x_{i}^{r})$ (also when $i$ is paired). Therefore, a deviation
towards his reference within the same neighborhood is profitable as it reduces
the costs.
∎

###  6.2 Proof of Proposition 1.

Consider the duopoly competition with $\gamma(d)=cd^{2}$. We suppose, without
loss of generality that $r_{1}\leq r_{2}$.

We first investigate the differentiated equilibrium.
Suppose that $r_{1}+\frac{1}{4c}<r_{2}-\frac{1}{4c}$ and that
$\textbf{x}^{\star}=(x_{1},x_{2})$ is an equilibrium. Lemma 18 (claim 6) gives
that $x_{1}\leq x_{2}$. We now show that $x_{1}<x_{2}$. The same Lemma (claim
2) proves that if $x_{1}=x_{2}$ then $x_{1}=x_{2}=\frac{1}{2}$. Suppose ad
absurdum that $\textbf{x}^{\star}=(\frac{1}{2},\frac{1}{2})$ is an
equilibrium.

  * •

If $r_{1}+\frac{1}{4c}<\frac{1}{2}$, consider Player 1 deviating towards
$r_{1}<\frac{1}{2}$. The equilibrium condition gives that
$g_{1}((\frac{1}{2}-\epsilon,\frac{1}{2}),\textbf{r})-g_{1}(\textbf{x}^{\star},\textbf{r})=2c\epsilon(\frac{1}{2}-r_{1})-c\epsilon^{2}-\frac{\epsilon}{2}\leq
0$. Dividing by $\epsilon>0$ and taking $\epsilon\rightarrow 0$ leads to
$\frac{1}{2}\leq\frac{1}{4c}+r_{1}$ which is in contradiction with our
hypothesis.

  * •

Symmetrically, if $\frac{1}{2}\leq r_{1}+\frac{1}{4c}$, consider Player 2
deviating towards $r_{2}$. We have:
$g_{2}((\frac{1}{2},\frac{1}{2}+\epsilon),\textbf{r})-g_{2}(\textbf{x}^{\star},\textbf{r})=2c\epsilon(r_{2}-\frac{1}{2})-c\epsilon^{2}-\frac{\epsilon}{2}\leq
0$. Dividing by $\epsilon>0$ and taking $\epsilon\rightarrow 0$ leads to
$r_{2}-\frac{1}{4c}\leq\frac{1}{2}$. Because
$r_{1}+\frac{1}{4c}<r_{2}-\frac{1}{4c}$, the previous inequality contradicts
with our hypothesis.

We proved that $x_{1}<x_{2}$. In this case, the payoff functions are given by
$g_{1}(x_{1},x_{2})=\frac{x_{1}+x_{2}}{2}-c(x_{1}-r_{1})^{2}$ and
$g_{2}(x_{1},x_{2})=1-\frac{x_{1}+x_{2}}{2}-c(r_{2}-x_{2})^{2}$. The first
order conditions give the optimal location for each player since their payoff
functions are concave. We obtain $x_{1}=r_{1}+\frac{1}{4c}$ and
$x_{2}=r_{2}-\frac{1}{4c}$. We have proved that if
$r_{1}+\frac{1}{4c}<r_{2}-\frac{1}{4c}$ then
$(r_{1}+\frac{1}{4c},r_{2}-\frac{1}{4c})$ is the unique equilibrium candidate.

To conclude that this profile is indeed an equilibrium we have to check that
players have no incentive to deviate on the other side of their opponent
(which would change the expression of the payoff function as $x_{1}>x_{2}$).
We now prove that such deviations are not profitable when inequalities $(E1)$
and $(E2)$ are satisfied.

The first of the two inequalities below translates the fact to Player 1 has no
incentive to deviate to location $x_{2}+\epsilon$ as $\epsilon$ goes to zero
positively (this criteria is both necessary and sufficient since his payoff
would decrease with $\epsilon>0$). Symmetrically, the second inequality
translates the fact to player 2 has no incentive to deviate to location
$x_{1}-\epsilon$ as $\epsilon$ goes to zero positively.

$g_{1}(\textbf{x}^{\star},\textbf{r})=\frac{r_{1}+r_{2}}{2}-\frac{1}{16c}\geq
1-(r_{2}-\frac{1}{4c})-c(r_{2}-\frac{1}{4c}-r_{1})^{2}$

and

$g_{2}(\textbf{x}^{\star},\textbf{r})=1-\frac{r_{1}+r_{2}}{2}-\frac{1}{16c}\geq(r_{1}+\frac{1}{4c})-c(r_{2}-\frac{1}{4c}-r_{1})^{2}$

Those two inequalities simplify to:

$c(r_{2}-r_{1})^{2}+r_{1}+r_{2}-1-\frac{1}{4c}\geq 0$
$c(r_{2}-r_{1})^{2}-(r_{1}+r_{2})+1-\frac{1}{4c}\geq 0$

which are $(E1)$ and $(E2)$ in Proposition 1.
Note that we also proved that player 1 has no incentive to deviate to
$x_{1}=x_{2}$. Indeed, we have that
$u_{2}(x_{2},x_{2})=\frac{1}{2}-c(x_{2}-r_{2})^{2}=\displaystyle\lim_{\epsilon\downarrow
0}\frac{u_{2}(x_{2}-\epsilon,x_{2})+u_{2}(x_{2}+\epsilon,x_{2})}{2}$ and both
terms in the numerator are smaller than $u_{2}(x_{1},x_{2})$ as we prove that
is it not profitable for Player 1 to deviate to $x_{2}-\epsilon$ nor to
$x_{2}+\epsilon$ for any $\epsilon>0$.

We now investigate the undifferentiated equilibrium.

Suppose now that $r_{2}-\frac{1}{4c}\leq r_{1}+\frac{1}{4c}$ and that
$\textbf{x}^{\star}=(x_{1},x_{2})$ is an equilibrium.

To prove that $(x_{1},x_{2})=(\frac{1}{2},\frac{1}{2})$, it is sufficient to
show that $x_{1}=x_{2}$ (Lemma 18, claim 2).

Suppose, ad absurdum, that $x_{1}<x_{2}$. In this case the payoffs are given
by $g_{1}(x_{1},x_{2})=\frac{x_{1}+x_{2}}{2}-c(x_{1}-r_{1})^{2}$ and
$g_{2}(x_{1},x_{2})=1-\frac{x_{1}+x_{2}}{2}-c(r_{2}-x_{2})^{2}$. The
equilibrium conditions give that $\frac{\partial
g_{1}(x_{1}+\epsilon,x_{2})}{\partial\epsilon}<0$ for $\epsilon$ small enough,
which simplifies to $\frac{1}{4c}+r_{1}\leq x_{1}$ when $\epsilon\rightarrow
0$. It also gives that $\frac{\partial
g_{2}(x_{1},x_{2}-\epsilon)}{\partial\epsilon}<0$ for $\epsilon$ small enough
which simplifies to $x_{2}\leq r_{2}-\frac{1}{4c}$ when $\epsilon\rightarrow
0$. We would then have $\frac{1}{4c}+r_{1}\leq x_{1}<x_{2}\leq
r_{2}-\frac{1}{4c}$, which contradicts our hypothesis.

We proved that if $r_{2}-\frac{1}{4c}\leq r_{1}+\frac{1}{4c}$, the unique
equilibrium candidate is $(\frac{1}{2},\frac{1}{2})$. It remains to prove that
if $r_{2}-\frac{1}{4c}\leq\frac{1}{2}\leq r_{1}+\frac{1}{4c}$ then
$(\frac{1}{2},\frac{1}{2})$ is an equilibrium.

Suppose that $r_{2}-\frac{1}{4c}\leq\frac{1}{2}\leq r_{1}+\frac{1}{4c}$.
Suppose first that $r_{1}\leq\frac{1}{2}$.
First, notice that player 1 has no profitable deviations in the interval
$(\frac{1}{2},1]$ as it decreases his clientele and increases his costs.
Furthermore, any deviations in $[r_{1},\frac{1}{2}]$ would give Player $1$ a
payoff
$g_{1}((\frac{1}{2}-\epsilon,\frac{1}{2}),\textbf{r})=g_{1}(\textbf{x},\textbf{r})+2c\epsilon(\frac{1}{2}-r_{1})-c\epsilon^{2}-\frac{\epsilon}{2}$
(with $0<\epsilon<\frac{1}{2}-r_{1}$). Because $\frac{1}{2}-\frac{1}{4c}\leq
r_{1}$, we have
$2c\epsilon(\frac{1}{2}-r_{1})-c\epsilon^{2}-\frac{\epsilon}{2}\leq 0$ and the
deviation is not profitable. Finally, any deviation in the interval
$[0,r_{1})$ gives a payoff strictly smaller than a deviation in $r_{1}$ which
is not profitable. The same arguments hold for $r_{1}>\frac{1}{2}$ and for
Player $2$ so we proved that $(\frac{1}{2},\frac{1}{2})$ is an equilibrium.

###  6.3 Proof of Proposition 3

We compute the area of the set $\mathscr{E}=\\{\textbf{r}\in[0,1]^{2}\text{
for which the duopoly competition admits an equilibrium}\\}$. The computation
is made easier after a rotation of the axis:
$(x,y)=\left(\frac{\sqrt{2}}{2}(r_{1}-r_{2}),\frac{\sqrt{2}}{2}(r_{1}+r_{2})\right)$
and when we consider the non-equilibrium area noted $\mathscr{N}$. By
symmetry, we only compute $\mathscr{B}$, the bottom part of $\mathscr{N}$,
below the line $y=\frac{\sqrt{2}}{2}$. We deduce
$\mathscr{E}=1-\mathscr{N}=1-2\mathscr{B}$.

$|$$-$$0$$r_{1}$$r_{2}$$x$$y$$x_{1}$$x_{0}$$-x_{1}$$-x_{0}$$\mathscr{B}$$\mathscr{B}$
Figure 7: The equilibrium area $\mathscr{E}$ in hatched, the non-equilibrium
area $\mathscr{E}$ in white. $\mathscr{B}$ is the bottom part of
$\mathscr{N}$.

To compute $\mathscr{B}$, we compute the integral of the parabola between
$-x_{0}$ and $x_{0}$ and we subtract the missing parts.

$\mathscr{B}=\int_{-x_{0}}^{x_{0}}(-\sqrt{2}cx^{2}+\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{8c})\,dx-\Bigg{(}2\times\frac{x_{0}^{2}}{2}\Bigg{)}-\Bigg{(}\int_{-x_{1}}^{x_{1}}(-\sqrt{2}cx^{2}+\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{8c})\,dx-2\bigg{(}x_{1}(\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{4c})+\frac{x_{1}^{2}}{2}\bigg{)}\Bigg{)}$

which gives
$\mathscr{A}=1-2\mathscr{B}=1-\frac{(2+4c)^{\frac{3}{2}}-6c-5}{6c^{2}}$. We
finally have
$\mathbb{P}(c)=\min\left(1,\mathscr{A}\right)=\min\left(1,1-\frac{(2+4c)^{\frac{3}{2}}-6c-5}{6c^{2}}\right)$.

We now prove that $c\mapsto\mathbb{P}(c)$ is constant on $[0,\frac{1}{2}]$,
decreasing on $[\frac{1}{2},\theta]$ and increasing on $[\theta,+\infty[$ for
$\theta\simeq 1.1321$.

First, for $c\in(0,\frac{1}{2}]$, we obtain that $(\frac{1}{2},\frac{1}{2})$
is an equilibrium for every profile of reference so $\mathscr{P}(c)=1$.
Furthermore, for $c>\frac{1}{2}$, we find that $c\rightarrow\mathbb{P}(c)$ is
a smooth function and that $\frac{d\mathbb{P}(c)}{dc}$ has the same sign as
$f(c)=\sqrt{4c+2}(c+2)-(3c+5)$. We have
$f^{\prime}(c)=\sqrt{2}\sqrt{2c+1}+\frac{\sqrt{2}(c+2)}{\sqrt{2c+1}}-3$ and
$f^{\prime\prime}(c)=\frac{3c\sqrt{2}}{(2c+1)^{\frac{3}{2}}}>0$. Because
$f^{\prime}$ is increasing and because $f^{\prime}(\frac{1}{2})=\frac{3}{2}>0$
we obtain that $f^{\prime}$ is positive and therefore that $f$ is increasing.

Because $f(\frac{1}{2})=-\frac{3}{2}$ and that
$\displaystyle\lim_{c\rightarrow+\infty}f(c)=+\infty$, the intermediate value
theorem provides the existence of $\theta$ such that $f(x)\leq 0$ when
$x\leq\theta$ and $f(x)\geq 0$ when $x\geq\theta$.

Solving $\frac{d\mathbb{P}(c)}{dc}=0$ gives a polynomial equation of degree
$3$ that provides a unique real solution
$\theta=\frac{1}{12}\left(3\sqrt[3]{4\sqrt{26}+73}+\sqrt[3]{1971-108\sqrt{26}}-9\right)\simeq
1.321$.

###  6.4 Proof of Proposition 8.

The proof is a straightforward generalization of Proposition 1: the quantity
$\frac{1}{4c}$ that appears with homogeneous costs is replaced by either
$\delta_{1}=\frac{1}{4c_{1}}$ or $\delta_{2}=\frac{1}{4c_{2}}$. When repeating
the proof, there are two minor modifications:
(1) in the case where $r_{2}-\delta_{2}\leq r_{1}+\delta_{1}$: in the
homogeneous case, the necessary condition for $(\frac{1}{2},\frac{1}{2})$ to
be an equilibrium is that $\frac{1}{2}\in[r_{2}-\delta,r_{1}+\delta]$ and this
condition was sufficient to obtain that
$\frac{1}{2}\in[r_{1}-\delta,r_{1}+\delta]\cap[r_{2}-\delta,r_{2}+\delta]$.
This is not the case anymore with heterogeneous costs, so the necessary
condition is now written with the two conditions
$\frac{1}{2}\in[r_{1}-\delta_{2},r_{1}+\delta_{2}]$ and
$\frac{1}{2}\in[r_{2}-\delta_{2},r_{2}+\delta_{2}]$.
(2) in the case where $r_{1}+\delta_{1}<r_{2}-\delta_{2}$: similarly to the
homogeneous case, to insure that Player 1 has no incentive to deviate to the
right of Player 2, and reciprocally that Player 2 has no incentive to deviate
to the left of Player 1, we write the two conditions on the references
$r_{1},r_{2}$. Because the costs functions have changed, these conditions are
now provided by inequalities $(E1^{\prime\prime})$ and $(E2^{\prime\prime})$.

###  6.5 The triopoly

####  6.5.1 The equilibrium candidate.

Let $\textbf{r}=(r_{1},r_{2},r_{3})\in[0,1]^{3}$ and $\gamma(d)=cd^{2}$.
Without loss of generality we suppose that that $r_{1}\leq r_{2}\leq r_{3}$.
Let $\textbf{x}^{*}=(x_{1},x_{2},x_{3})$ be an equilibrium of the game. Remark
that Claim 6 of Lemma 18 gives that $x_{1}\leq x_{2}\leq x_{3}$.

We start by providing a list of triopoly equilibrium properties and we then
conclude on the equilibrium structure.
We show that if Player $1$ is not a far-left left player, he is necessarily
paired with Player $2$ at equilibrium. Suppose ad absurdum that
$r_{1}+\delta\geq r_{2}$ and that $x_{1}<x_{2}$ ($\delta$ is defined in 6 and
is equal to $\frac{1}{4c}$ in this context). Player $1$’s payoff is
$g_{1}(\textbf{x}^{\star},\textbf{r})=\frac{x_{1}+x_{2}}{2}-c(x_{1}-r_{1})^{2}$.
Because this function is concave and because its derivative is null in
$r_{1}+\delta$, it is then increasing on $[0,r_{1}+\delta]$. We now prove that
$x_{2}\leq r_{1}+\delta$, which implies that Player $1$’s payoff is strictly
increasing on $[0,x_{2})$ and contradicts $x_{1}<x_{2}$. Suppose first that
$x_{2}<x_{3}$, Lemma 19 gives that $x_{2}=r_{2}$ so that $x_{2}\leq
r_{1}+\delta$. Suppose now that $x_{2}=x_{3}$, both players are right-
peripheral. We necessarily have $x_{3}=x_{2}\leq r_{2}$ (Lemma 18, claim 5)
and therefore $x_{2}\leq r_{1}+\delta$. We proved that if Player $1$ is not
far-left he is necessarily paired with Player $2$. The same arguments hold for
Player $3$ if he is not far-right.

A direct consequence is that if neither Player $1$ nor Player $3$ are far-left
and far-right, there is no equilibrium as three players can not share the same
location (Lemma 18, claim 1).

We secondly show that, if a peripheral player is far-left, he is not paired.
The equilibrium conditions imply that any deviation $\epsilon$ of Player $1$
towards his reference is not profitable, so that
$g_{1}((x_{1}-\epsilon,x_{2},x_{3}),\textbf{r})-g_{1}((x_{1},x_{2},x_{3}),\textbf{r})=-\frac{\epsilon}{2}+2c(x_{1}-r_{1})\epsilon-c\epsilon^{2}\leq
0$. Dividing by $\epsilon>0$ and taking $\epsilon\rightarrow 0$ gives that
that $x_{1}\leq r_{1}+\frac{1}{4c}$. Moreover, we have $r_{1}\leq r_{2}\leq
x_{1}=x_{2}$ (Claim 5 of the same Lemma) which gives a contradiction with our
hypothesis that Player 1 is far left. The same arguments arise for the
symmetric case of a far-right player.

Because Player $1$ is not paired when he is far-left, we show that
$x_{1}=r_{1}+\delta$. Suppose that $r_{1}+\delta<r_{2}$. In this situation
Player $1$’s payoff is
$g_{1}(\textbf{x}^{\star},\textbf{r})=\frac{x_{1}+x_{2}}{2}-c(x_{1}-r_{1})^{2}$.
The first and second order conditions gives us that his payoff is increasing
with $x_{1}\in[0,r_{1}+\delta]$ and decreasing for
$x_{1}\in[r_{1}+\delta,x_{2})$. His payoff is maximized in
$x_{1}=r_{1}+\delta$, which is the only position possible at equilibrium in
$[0,x_{2})$. The same arguments arise for the symmetric case of a far-right
player.

In the equilibrium situation where $x_{1}<x_{2}<x_{3}$, Lemma 19 gives that
$x_{2}=r_{2}$. Finally, Claim 2 of Lemma 18 gives that if Players $1$ and $2$
are paired $x_{1}=x_{2}=\frac{x_{3}}{3}$.

We can now conclude on the structure of the triopoly equilibrium:

  * •

First, if both Player $1$ and Player $3$ are far-left and far-right, then
$x_{1}=r_{1}+\delta$, $x_{2}=r_{2}$, $x_{3}=r_{3}-\delta$.

  * •

If Player $1$ is not far-left but Player $3$ is far-right, Player $1$ and $2$
are paired and Player $3$ is alone on $x_{3}=r_{3}-\delta$. Player $1$ and $2$
are positioned on $\frac{x_{3}}{3}=\frac{r_{3}-\delta}{3}$. The symmetric
situations leads to $x_{1}=r_{1}+\delta$ and
$x_{2}=x_{3}=\frac{r_{1}+\delta+2}{3}$.

  * •

If both Player $1$ and Player $3$ are not far-left nor far-right, then there
is no equilibrium.

####  6.5.2 Proof of Proposition 9

Suppose first that $c<\frac{5}{8}$, that is $\delta=\frac{1}{4c}>\frac{2}{5}$.
We first prove that there exists no equilibrium with a far-left and a far-
right players. Indeed, in this case: $x_{1}=r_{1}+\delta>\frac{2}{5}$ and
$x_{3}=r_{3}-\delta<\frac{3}{5}$. Therefore, Player $2$’s payoff is strictly
less than $\frac{1}{10}$ and deviation to $x_{1}-\epsilon$ for $\epsilon$
small enough is profitable as Player $2$’s deviation payoff is at at least
$\frac{2}{5}-c(\frac{1}{5})^{2}>\frac{2}{5}-\frac{1}{30}>\frac{1}{5}$.

We now prove that there exists no equilibrium with a far-left player and no
far-right player (the other case is symmetric). In this case,
$x_{1}=r_{1}+\delta>\frac{2}{5}$. According to the equilibrium candidate
described in 6.5.1, Player $2$ and $3$ share the location
$\frac{r_{1}+\delta+2}{3}$. Player $2$’s payoff decreases with $r_{1}$ and is
at most $\frac{1}{5}$. We now prove that Player $2$ has a profitable deviation
to the left of Player $1$. In this location, Player $2$ attracts a quantity of
consumers arbitrary close to $r_{1}+\delta\geq\delta>\frac{2}{5}$. Because the
cost function is convex the deviation is the least profitable when $r_{1}=0$
and $r_{2}=1$ and the deviation increases the costs by at most
$c(\frac{3}{5})^{2}-c(\frac{1}{5})^{2}<\frac{1}{5}$. When Player $2$ deviates
to the left of Player $1$, his payoff change is strictly larger than
$\frac{2}{5}-\frac{1}{5}-\frac{1}{5}=0$, and is therefore profitable. The
exact same argument applies to Player $3$. We proved that there exists no
equilibrium.

Suppose that $c\geq\frac{5}{8}$. There exits at most one equilibrium, as
proved in the previous subsection where we exhibit the unique equilibrium
candidate.

As long as $r_{1}<r_{2}<r_{3}$, there exists $c$ large enough so that Player
$1$ and $3$ are far left and far right. The unique equilibrium candidate is
therefore $(r_{1}+\frac{1}{4c},r_{2},r_{3}-\frac{1}{4c})$. For $c$ large
enough, the cost of any deviation for any player is strictly larger than $1$
which the maximal payoff a player can obtain in the game, so deviations are
not profitable.

###  6.6 Proof of Proposition 11

Consider the triopoly competition where
$\textbf{r}=(r_{1},\frac{1}{2},1-r_{1})$ and $\gamma(d)=cd^{2}$.

If Player $1$ is not far-left then by symmetry Player $3$ is not far-right and
Proposition 6.5.1 gives that there is no equilibrium. At equilibrium, we must
have Player $1$ and $3$ being respectively far-left and far-right.

According to 6.5.1, the unique possible equilibrium is
$(r_{1}+\delta,\frac{1}{2},1-r_{1}-\delta)$. We start by providing necessary
conditions. Because Player $2$ has no profitable deviation to the left of
Player $1$, we have:

$\frac{1-r_{1}-\delta-r_{1}-\delta}{2}\geq
r_{1}+\delta-c(\frac{1}{2}-r_{1}-\delta)^{2}$

which simplifies to the second order equation

$c^{2}r_{1}^{2}-r_{1}(c^{2}+\frac{3}{2}c)+(\frac{c^{2}}{4}+\frac{c}{4}-\frac{7}{16})\geq
0$

For the equilibrium candidate to be an equilibrium, $r_{1}$ should satisfy
$r_{1}\notin\left[\frac{3+2c-2\sqrt{2c+4}}{4c},\frac{3+2c+2\sqrt{2c+4}}{4c}\right]$.
Because $\frac{3+2c+2\sqrt{2c+4}}{4c}>\frac{1}{2}$ while $r_{1}<\frac{1}{2}$,
we can simplify the condition to
$r_{1}\leq\frac{3+2c-2\sqrt{2c+4}}{4c}=:\phi(c)$.

Now suppose that
$\textbf{x}=(r_{1}+\frac{1}{4c},\frac{1}{2},1-r_{1}-\frac{1}{4c})$ and that
$r_{1}\in[0,\phi(c)]$. We prove that x is an equilibrium.

First notice that players have no profitable deviation within their
neighborhoods. Indeed, the first and second order conditions provide that the
payoff is maximized when $x_{1}=r_{1}+\delta$ and $x_{3}=r_{3}-\delta$, and
Claim $3$ Lemma 18 gives that Player $2$ prefers to locate at $r_{2}$ in this
neighborhood.

Moreover, players have no deviations to a different neighborhood: a deviation
of Player $1$ to $x\in(\frac{1}{2},1]$ provides a smaller payoff than a
deviation to the symmetric location $1-x\in[0,\frac{1}{2})$ which is not a
profitable deviation, as it belongs to the neighborhood as $x_{1}$. It is not
profitable to deviate to $x_{2}$ neither as such a deviation is less
profitable than a deviation to $x_{2}-\epsilon$ for $\epsilon$ small enough,
which is not profitable. Following the above computations, Player $2$ has no
incentive to deviate to the left of Player $1$ or to the right of $3$ as long
as $r_{1}\leq\phi(c)$. Finally, Player $2$ has no incentive to deviate to
$x_{1}$ as neither $x_{1}-\epsilon$ nor $x_{1}+\epsilon$ are profitable
deviations as $\epsilon\rightarrow 0$. The same arguments hold for $x_{3}$.

###  6.7 Proof of section 5

####  6.7.1 Proof of Proposition 12

Suppose that x is an equilibrium in the game with references r and suppose,
without loss of generality, that $r_{1}\leq\dots\leq r_{n}$.

As a consequence of Lemma 18 (Claim 6), Players $1$ and $n$ are peripheral.
Players $2$ and $n-1$ can be peripheral or non-peripheral. Because at most $2$
players can share the same location at equilibrium (Claim 1, Proposition 18),
players $3$, $4$, …, $n-2$ are non-peripheral. For these players, Lemma 19
claims that $x_{i}=r_{i}$.

Suppose that Player $1$ is a far-left player, i.e. that $r_{1}+\delta<r_{2}$.
We prove that $x_{1}<x_{2}$. Suppose, ad absurdum, that $x_{1}=x_{2}$. Using
Lemma 18, we have $q_{1}^{\ell}(x)=q_{1}^{r}(x)$, and because their right
neighbor is the non-peripheral player $3$ located at $x_{3}=r_{3}$, we have
$x_{1}=x_{2}=\frac{r_{3}}{3}$.

On the one hand, an infinitesimal deviation to his left is not profitable to
Player $1$, so we have that
$\gamma^{\prime}(\frac{r_{3}}{3}-r_{1})\leq\frac{1}{2}$. On the other hand we
have $r_{1}+\delta<r_{2}\leq\frac{r_{3}}{3}$, where the second inequality
comes from the fact that Player $2$ has no profitable infinitesimal deviation
to his right. Because $\gamma^{\prime}$ is strictly increasing, we have
$\gamma^{\prime}(\frac{r_{3}}{3}-r_{1})>\gamma^{\prime}(\delta)=\frac{1}{2}$
and we face a contradiction. Player $1$ is not paired with Player $2$.

Because Player $1$ is peripheral and single, his payoff is given by
$u_{i}(\textbf{x})=\frac{x_{1}+x_{2}}{2}-\gamma(x_{1}-r_{1})$ in the interval
$[0,x_{2})$ and the first and second order conditions give that his payoff is
maximized in $x_{1}=r_{1}+\delta$. The argument is symmetric for Player $n$.

Suppose now that Player $1$ is not a far left-player, i.e. that
$x_{1}+\delta\geq r_{2}$. If $x_{1}<x_{2}$, his payoff is given by
$u_{i}(\textbf{x})=\frac{x_{1}+x_{2}}{2}-\gamma(x_{1}-r_{1})$ and is therefore
strictly increasing with $x_{1}\in[0,x_{2})$. We conclude that Players $1$ and
$2$ are paired. This is only possible if $x_{1}=x_{2}=\frac{r_{3}}{3}$ (Lemma
18, claim 2). The argument is symmetric for players $n-1$ and $n$.

####  6.7.2 Proof of Proposition 14

We first show that conditions $(1)$, $(2)$ and $(3)$ are necessary for the
described profile to be an equilibrium.

Suppose that condition $(1)$ doesn’t hold, i.e. Player $1$ is a far-left
player but $\frac{r_{3}}{3}\notin[r_{2},r_{1}+\delta]$.
\- If $\frac{r_{3}}{3}<r_{2}$, then player $2$ has a profitable deviation to
his right: the quantity that Player $2$ attracts is constant on the interval
$[\frac{r_{3}}{3},r_{3})$ (Claim 4, Proposition 18) but his reference cost
decreases on the interval $[\frac{r_{3}}{3},r_{2}]$.
\- If $\frac{r_{3}}{3}>r_{1}+\delta$, then Player $1$ has a profitable
deviation to his left: by making an infinitesimal move to his left, he obtains
a marginal loss in the quantity of consumers which is equal to $\frac{1}{2}$
while the marginal diminution of his reference cost is
$\gamma^{\prime}(x_{1}-r_{1})=\gamma^{\prime}(\frac{r_{3}}{3}-r_{1})>\gamma^{\prime}(r_{1}+\delta-
r_{1})=\gamma^{\prime}(\delta)=\frac{1}{2}$.

The case where condition $(2)$ doesn’t hold is symmetric.

Suppose that condition $(3)$ doesn’t hold, i.e. there exists $i$, $j$ such
that $\Delta_{i}^{j}(\textbf{x}^{*})<0$. Then, by definition of
$\Delta_{i}^{j}(\textbf{x}^{*})$, Player $i$ has a profitable deviation by
playing $x_{j}+\epsilon$ for $\epsilon$ small enough if $i<j$, or by playing
$x_{j}-\epsilon$ for $\epsilon$ small enough if $i>j$.

We now prove that conditions $(1)$, $(2)$ and $(3)$ are sufficient. We suppose
that $\textbf{x}^{*}$ is the equilibrium candidate described in Proposition
12, that the three conditions hold and we prove that $x^{*}$ is an
equilibrium.

Take any player $i\in\\{3,\dots,n-2\\}$. He has no profitable deviation within
his neighborhood $[x_{i}^{l},x_{i}^{r}]$ where his payoff is maximized at
$x_{i}=r_{i}$.
We now prove that player $i$ has no profitable deviation towards a different
neighborhood. Consider any player $j$ such that $x_{j}<x_{i}$. The deviation
payoff of Player $i$ increases in the interval $(x_{j}^{\ell},x_{j})$ as the
quantity of attracted consumers is constant and the reference cost is
decreasing. However, condition $(3)$ gives that
$u_{i}(\textbf{x})\geq\lim\limits_{\begin{subarray}{c}\epsilon\to 0\\\
\epsilon>0\end{subarray}}u_{i}(x_{j}-\epsilon,\textbf{x}_{-i})$, which proves
that there is no profitable deviation in the interval $(x_{j}^{\ell},x_{j})$.
Note that a deviation to $x_{j}$ is not profitable neither as it gives Player
$i$ a payoff $u_{i}(x_{j},x_{-i})=\lim\limits_{\begin{subarray}{c}\epsilon\to
0\\\
\epsilon>0\end{subarray}}\frac{u_{i}(x_{j}+\epsilon,\textbf{x}_{-i})+u_{i}(x_{j}-\epsilon,\textbf{x}_{-i})}{2}$,
both terms in the numerator being smaller than $u_{i}(\textbf{x})$ due to
condition $(3)$. Because the same argument applies to every player $j$ such
that $x_{j}<x_{i}$, we proved that Player $i$ has no incentive to deviate to
any location in $[0,x_{i}^{\ell}]$. A symmetric argument applies with any
player $j$ such that $x_{j}>x_{i}$ and we proved that Player $i$ has no
profitable deviation towards a different neighborhood.

If Player $1$ is far-left and $\textbf{x}^{*}$ is played, Player $2$ is not
peripheral so he has no profitable deviation. Suppose now that Player $1$ is
not far-left, so that Player $2$ is peripheral. We prove that he has no
profitable deviation neither. Condition (1) gives
$x_{2}=\frac{r_{3}}{3}\in[r_{2},r_{1}+\delta]$.
\- Player $2$ has no profitable deviations within $(x_{2},x_{3}]$ due to Lemma
18, Claim 4.
\- Player $2$ has no profitable deviations within $[r_{2},x_{2})$ neither: for
any deviation $x_{2}-\epsilon$ in this interval, we have
$u_{2}(x_{2}-\epsilon,\textbf{x}_{-i})-u_{2}(\textbf{x}))=-\frac{\epsilon}{2}-c\epsilon^{2}+2c\epsilon(\frac{r_{3}}{3}-r_{2})$.
Yet, since $\frac{r_{3}}{3}\leq r_{1}+\delta\leq r_{2}+\delta$, we have
$-\frac{1}{2}+2c(\frac{r_{3}}{3}-r_{2})\leq 0$ and therefore
$-\frac{\epsilon}{2}-c\epsilon^{2}+2c\epsilon(\frac{r_{3}}{3}-r_{2})<0$. The
deviation is not profitable.
\- Player $2$ has no profitable deviations within $[0,r_{2})$ as they are even
less profitable than a deviation in $[r_{2},x_{2}^{*})$.
We conclude that Player $2$ doesn’t have any profitable deviation in his
neighborhood $[0,x_{3})$.
Moreover, the same argument than before applies for global deviation:
condition (3) implies that any deviation in a different neighborhood is not
profitable neither. The same arguments hold regarding Player $n-1$.

If Player $1$ is far-left, he is the only peripheral player. He has no local
deviations, as his payoff is maximum in $[0,x_{2})$ for
$x_{1}=r_{1}+\delta=x_{1}^{*}$, due to the first and second order conditions.
Again, conditions $(3)$ implies that any deviation in a different neighborhood
is not profitable.
If Player $1$ is far-left, he is paired with Player $2$. In this case, the
same arguments used for Player $2$ above applies and Player $1$ has no
profitable deviation. The same applies to Player $n$.

###  6.8 Proof of Proposition 16

We consider the case of $n\geq 5$ players where $\gamma(d)=cd^{2}$ and
$\mathbb{r}=(\frac{1}{n+1},\frac{2}{n+1},...,\frac{n}{n+1})$.

Suppose that x is an equilibrium. Using Proposition 12, we have
$x_{i}=r_{i}=\frac{i}{n+1}$ for every $i\in\\{3,\dots,n-2\\}$. If Player $1$
is not far-left, Proposition 14 gives that
$\frac{r_{3}}{3}=\frac{1}{n+1}\in[r_{2},r_{1}+\delta]=[\frac{2}{n+1},\frac{1}{n+1}+\delta]$,
which is not the case here. Therefore, there is no equilibrium when Player $1$
is not far-left, which is the case when $c<\frac{n+1}{4}$.

The existence of an equilibrium implies $c\geq\frac{n+1}{4}$. In this case
players $1$ and $n$ are respectively far-left and far-right. The unique
equilibrium candidate is
$\mathbb{x^{\star}}=(r_{1}+\delta,r_{2},r_{3},...,r_{n-1},r_{n}-\delta)$. A
necessary condition for this profile to be an equilibrium is that Player $2$
does not profitably deviate on the left of Player $1$, that is:

$r_{1}+\delta-c(r_{2}-r_{1}-\delta)^{2}\leq\frac{r_{3}-r_{1}-\delta}{2}$

which simplifies to $c\geq\frac{1+\sqrt{6}}{4}(n+1)$. Note that this
constraints is harder to satisfy than the constraint $c\geq\frac{n+1}{4}$.

We now prove that
$\mathbb{x^{\star}}=(r_{1}+\delta,r_{2},r_{3},...,r_{n-1},r_{n}-\delta)$ is an
equilibrium when $c\geq\frac{1+\sqrt{6}}{4}(n+1)$. Repeating standard
arguments, no player has an incentive to deviate within his neighborhood. We
now prove that Players have no profitable deviation to a different
neighborhood.

Any player $i\in\\{3,\dots,n-2\\}$ has a payoff equal to $\frac{1}{n+1}$.
Therefore, it is not profitable to deviate in the interval between two non-
peripheral players. Moreover, because $c\geq\frac{1+\sqrt{6}}{4}(n+1)$, Player
$2$ does not benefit from deviating to the t of Player $i$. Such a deviation
is even less profitable for a player $i\in\\{3,\dots,n-2\\}$. The same
argument applies for a deviation on the right of player $n$.

Players $2$ and $n-1$ have no incentives to relocate respectively on the left
of Player $1$ or on the right of Player $n$ because
$c\geq\frac{1+\sqrt{6}}{4}(n+1)$. Any other deviation is even less profitable
as the interval are smaller and the reference costs are higher.

Players $1$ and $n$ have no incentives to deviate anywhere as any deviation
both reduces their clientele and increases their references costs.

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