RETURN OF THE ICE CREAM MEN
A Combined Game Theoretical, Agent- Based Approach to Discrete Hotelling
Games
Nabi Abudaldah, Wim Heijman, Pieter Heringa, Pierre v. Mouche
January 8, 2013
Abstract
We discuss a well-known spatial problem in which two vendors compete for market share on a
one dimensional plane. This problem is so far especially popular in game theory. However, with
game theoretical analysis it soon becomes extremely tedious and complicated to study more
sophisticated versions of the problem (e.g. in a two-dimensional space or with more vendors). In
this article we show how a basic analytic version of the model can be studied as a discrete game
and provide its main characteristics such as Nash equilibria. We then explain how the same
problem can be solved using an agent-based model. We show that the basic analytic model
yields the same results as the agent-based model. Moreover, the agent-based model can easily
be adapted to include more complex assumptions, and we analyse some hypotheses to show
this. This not only contributes to the scientific debate on the use of agent-based modelling for
problems in game theory, it can also be of use in the development of location models for realworld applications.
Keywords: Game Theory, Agent-Based Modelling, Hotelling’s Law, Hotelling Games
1
1.
Introduction
For over eighty years, many people have been intrigued by one single article of professor
Harold Hotelling. Maybe the most fascinating thing about Hotelling's model is its simple
appearance at first sight. Still, it touches upon so many issues, that authors publish new
variations of Hotelling's model until today. However, the elegance and simple nature of
Hotelling’s model also limit its value for real world applications. Real-world location questions
often involve two dimensional spaces, multiple actors, a heterogeneous demand over space, and
(policy) restrictions on sites where firms are able to locate. It is important to extend the models
with such more sophisticated specifications, as earlier work has shown that even minor changes
in the assumptions of the model sometimes leads to enormous changes in its outcomes (Eiselt
and Laporte 1989). Including such specifications into Hotelling’s model becomes analytically
challenging and laborious using conventional analytical methods. Modern simulation techniques
combined with a strongly increased computational power offer new perspectives for such realworld problems.
There is an on-going discussion on the possibilities and limitations of game theory and (agentbased) simulation (see Moss 2001; Balzer et al. 2001). However, it is hard to compare analytic
game theory and agent-based modelling because they are based on different assumptions. In this
article we show how agent-based modelling can be used to analyse problems for which usually
game theoretical models are used. Moreover, we show how agent-based modelling can be easily
employed to simulate more complicated (and probably more realistic) situations. This is a
promising approach both for researchers looking for extended, flexible models and for analysts
and policy makers in search of answers to complicated location problems. However, a limitation
is that contrary to game theory no general valid theorems can be proven as fundamentally only a
finite number of instances can be tested.
The remainder of this article is organised as follows: Section 2 gives a brief overview of the
different assumptions underlying different variations of Hotelling's model. Section 3 analyses
the equilibria of a discrete Hotelling game with given and equal prices. The purpose of this
exercise is twofold: first, it is not self-evident that discrete games have the same equilibria as
their continuous counterparts. Second, the discrete game can be used to validate the outcomes of
agent-based (simulation) models. In Section 4 we present the outcomes of such an agent-based
model and compare them with the findings of the discrete game model. We also extend the
agent-based model with multiple players and two dimensions. In Section 5 we present our
conclusions and discuss implications for future research.
2
Throughout this article, we use various terms interchangeably. Firm and seller are defined as
perfect equivalents of each other. The same goes for consumer, customer and buyer.
2.
Literature overview
Let us first turn to the model as it was described by Hotelling in his seminal article on this topic
(Hotelling 1929). The buyers of a certain commodity are distributed uniformly along a line with
length l. On the line are the two businesses A and B, with respective distances a and b from the
two extremes of the line. All buyers bring their bought commodities home at a cost of c per unit
of distance. Hotelling makes some important assumptions, according to him without loss of
generality. He assumes that (1) the cost of production of both firms is zero; (2) one unit quantity
of the commodity is sold in each unit of time, per unit of length of the line (the demand is
therefore at the extreme of inelasticity); (3) no buyer has a preference for any of the sellers, all
existing preferences in reality are symbolized by the transportation price (and hence reflected in
the 'final' price that is paid by the customer). Hotelling himself did not mention another
important assumption he makes, namely that the transportation costs per unit product are taken
to be independent of the number of units transported (Lerner and Singer 1937). The finding by
Hotelling that (under certain assumptions) two firms on a line will tend to locate as close
together as possible at the centre of the line, became generally known as the Hotelling Law, or
the principle of minimum differentiation (Eaton and Lipsey 1975). Hotellings findings are later
disputed in a stream of literature started by Vickrey (1964) and attracting more attention after
the formalisation of the arguments by d’Aspremont et al. (1979).
Stevens (1961) was the first to suggest that Hotelling’s law can be treated as a game theoretical
problem. Since then, most contributors to the literature on this law have combined the main
insights and propositions of Hotelling with later knowledge on equilibria in game theory. The
easiest way to view Hotelling's model as a game, is to strictly divide his model into two pieces,
and make a game for each: a pricing game and a location game. In the Hotelling Pricing Game,
firms can set prices, given certain locations (locations are fixed). In the Hotelling Location
Game, firms can choose a location, but prices are fixed (Rasmusen 2007).
Many amendments can be made to the original model of Hotelling. There is indeed a very wide
body of literature in which adaptations of the model are examined, with a large variety of
different outcomes. For a selective but useful overview, see Gabszewicz and Thisse (1992),
Graitson (1982), Miron (2010) or Kilkenny and Thisse (1999).
Apart from the assumptions Hotelling himself mentioned, there are a few assumptions that hold
for almost all game-theoretical variations on the basic model: most authors (implicitly) assume
3
rational players, a symmetric game (i.e. a player is indifferent for being any of the players), and
complete information (i.e. all players know all payoffs and strategies available to all players).
To obtain some oversight, authors that provide an overview of earlier literature usually
categorize models according to the main assumptions made. Most important are the:

number of players or firms;

decision variables of these players (location or price);

order of the game (including whether or not the game is sequential);

physical form of the market;

economic form of the market (cooperation, oligopoly or pure competition);

elasticity of demand;

distribution of consumers;
possibility for new entrants to enter the market.A similar model in continuous space is analysed
by Puu (2002). The assumption of fixed and equal prices can be a realistic one for various real
world cases. One could think of situations where authorities have regulated the prices, or the
firms themselves have agreed to fix the prices. Another possibility is that earlier in the
production chain (e.g. at the manufacturer) prices are chosen, and that the retailers have to stick
to the agreed prices. Simplifying a bit, one could say that the two ice cream sellers on the beach
are replaced with two newspaper vendors at a main street. They sell the same newspaper, at the
very same price, but from different locations along the street.
It can also be relevant in cases where the locations do not represent geographical places. For
example for models on elections for political parties; only the positioning is considered, prices
are irrelevant (Hotelling 1929; Smithies 1941; Anderson et al. 1992).
The assumption of a uniform distribution of consumers is rather common in location models,
probably because of its convenience for analysis (e.g. Tabuchi and Thisse 1995).
Studies on a two-dimensional or higher-dimensional space are scarce, most probably because
the analysis is complex, laborious and tedious. Eiselt (2011) points out a few directions for
future research. He mentions the possibility of extensions to more dimensions, but immediately
adds that it can be doubted if this is a promising niche, as the first results have shown it to be
very difficult, even for pure locational models. Some have for example come up with circular or
triangular markets to loosen the strict assumption of a linear market (Lerner and Singer 1937;
Vickrey 1964; Eaton and Lipsey 1975; Gupta et al. 2004). However, there are a few analyses
with a two-dimensional space. There is a well-known conjecture of Eaton and Lipsey (1975)
that for more than two firms in an two-dimensional space, an equilibrium with zero conjectural
4
variation is unlikely to exist. However, Okabe and Suzuki (1987) show with Voronoi-diagrams
that for very large numbers of firms the configuration has stable equilibria in the inner area of
square regions. Tabuchi (1994), Irmen and Thisse (1998) and Veendorp and Majeed (1995) all
show that (under specific conditions) firms in a bounded two-dimensional space minimize
differentiation on one dimension, and maximize it on the other. Discrete models, even in a onedimensional space, are also scarce. An overview of the few existing studies can be found in
Serra and Revelle (1994); there is some more recent material in Prisner (2011). In the next
section we will show how such a discrete game can be written in strategic form, and we will
analyse its outcomes.
3.
Game-Theoretical Analysis
In the first instance, we analyse a model with the following characteristics:
1. Fixed and equal (mill) prices;
2. Uniform distribution of customers;
3. Variable spatial elasticity of demand (factor w);
4. One dimensional, discrete space;
5. Two players.
Later on in this article we show that an agent-based version of this model can be easily adapted
to more complex characteristics: we modify characteristic number 4 and 5 to a two-dimensional
space and multiple players. Such extensions are of great importance for real world applications.
Consider the following game in strategic form, to be called standard discrete Hotelling game
with two players, denoted by 1 and 2. Each player i has the same action set Xi={0,…,n}; here n
is a positive integer. And the payoff functions f1,f2 : X1  X2  ℝ are given by
5
So, for instance in case n = 10,
A possible abstract interpretation of a standard discrete Hotelling game is the following.
Consider the n+1 points {0,1,…,n} on the real line, to be referred to as vertices; here n is a
positive integer. The two players simultaneously and independently choose a vertex; player i
chooses vertex xi. The payoff of player i, fi (x1,x2), is the number of vertices that is the closed to
him; if a vertex has equal distance to both players, then this vertex contributes only ½ to each
player. Of course, this abstract notion allows for various economic interpretations (as discussed
in the previous section). Below we present various fundamental observations based on three
simple propositions.
Proposition 1:
1. The discrete Hotelling game is symmetric, i.e. f2 (x1,x2) = f1 (x2,x1) for all (x1,x2).
2. f1 + f2 = n + 1, thus the discrete Hotelling game is a constant-sum game.
Proof:
1. We see that:
2. This immediately follows from the formulas for f1 and f2. Q.e.d.
As a standard discrete Hotelling game is a constant-sum game, all strategy profiles are fully
cooperative, i.e. maximise the total payoff function f1 + f2 and therefore are (strongly) pareto
efficient. As a standard discrete Hotelling game is constant-sum, it is strictly competitive, that is
for all strategy profiles (a1,a2) and (b1,b2) it holds that:
6
As the game is strictly competitive, it follows that:
Proposition 2:
1. For each player i and for all Nash equilibria (a1,a2) and (b1,b2) it holds
that f i (a1,a2) = f i (b1,b2).
2. The set of Nash equilibria is a Cartesian product.
3. Either all Nash equilibria are not strict, or all Nash equilibria are strict1.
As a standard discrete Hotelling game is symmetric, the game is fair in the sense that it does not
matter for a player whether he is player 1 or 2. As the game is symmetric, it follows that:
Proposition 3:
If (n1,n2) is a Nash equilibrium, then also (n1,n2) is a Nash equilibrium.
Proof:
Suppose now we have a discrete Hotelling game with a Nash-equilibrium. Proposition 2(1)
guarantees that the payoff of player 1 is in each Nash-equilibrium the same; denote this value by
v. The above makes it possible to determine v quickly: let (n1,n2) be a Nash-equilibrium. By
Proposition 3, also (n2,n1) is a Nash equilibrium. By Proposition 1(1), f2(n1,n2) = f1(n2,n1). By
Proposition 2(1), f1(n1,n2) = f1(n2,n1). Now n+1 = f1(n1,n2) + f2(n1,n2) = 2f1(n1,n2) and therefore:
Discrete Hotelling games do not seem to be well-known in the literature. Some literature does
exist, see for example Prisner (2011) and references therein. In fact, the standard discrete
Hotelling game is a special case of a Voronoi game. In a Voronoi game with two players, both
players simultaneously choose a vertex. The payoff for a player is the number of vertices closer
to that player then to the other one, plus half of the number of vertices with equal distance. It is
known that each Voronoi game has at least one Nash equilibrium (Prisner, 2011). For our case,
it is straightforward to prove the following theorem:
Theorem 1
Each standard discrete Hotelling game has a symmetric Nash equilibrium. Even:
1
A Nash equilibrium is strict if at this equilibrium, for each player a change of his strategy leads to a
lower payoff.
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1. In case n is even, Nash equilibrium is at:
There is no other Nash equilibrium.
2. In case n is odd, Nash equilibria are at:
There are no other Nash equilibria.
The above standard discrete Hotelling game can be extended in at least three ways; the game
should (1) include more than two players, (2) vertices further away should contribute less to
payoffs and (3) vertices should be arranged according to a different geometry. We will now
consider such a game, but we restrict ourselves to hypercubes. The formal definition of such a
game should be clear. Only the distance factor needs explanation. We do this here explicitly in
the case of two players and the 1-dimensional geometry.
Consider the strategy profile (x1,x2). Then the exact same vertices as in the earlier considered
standard discrete Hotelling game contribute to the payoff of a player: these are the vertices that
are closest to him and, if it exists, a vertex that has equal distance to both players. Take such a
vertex. If it is at distance d, then it contributes wd to the payoff if it is not a shared vertex, and
otherwise it contributes wd/2; here w[0,1]. So in the standard discrete Hotelling game, w = 1.
We call w the distance factor. The so defined game is symmetric, i.e. f2(x1,x2) = f1(x2,x1).
The reader is invited to check that for n = 4 and w = ½ the game is the following bi-matrix game
(with row (column) i + 1 for the action i of player 1 (2)):
Its set of Nash equilibria is:
Formally, for positive integers m, n, N and w[0,1] denoted by E(m,n,N,w) represents the set of
Nash equilibria for a general discrete Hotelling game. N denotes the number of players and w
the distance factor. Symbol m refers to the m-dimensional case of a hypercube with (n+1)m
vertices. So m = 1 and N = 2 concerns the case studied earlier. As far as we know such a game
has never been studied before.
8
The analysis of this variant of the discrete Hotelling game is not straightforward at all (if
w{0,1}). Probably it can be proved by hand that for each w there exists a Nash equilibrium.
However, this seems to be much more tedious. One of the questions for further research is
whether there is an abstract argument that guarantees the existence of an equilibrium. Quite
obvious is:
Proposition 4:
If N ≤ (n+1)m, i.e. in case there are not more players than vertices, then E(m,n,N,0) ≠ . Even
then E(m,n,N,0) is the set of all multi-strategies (x1,…,xN) with #{x1,…,xN} = N, i.e. with all
numbers x1,…,xN different.
However, less obvious is:
Proposition 5:
E(1,1,N,w) ≠ 
Proof:
This holds as in this case we have to do with a symmetric game in strategic form where the
strategy set of each player has exactly two elements (Cheng et al., 2004). Q.e.d.
Again obvious is:
Proposition 6:
If m = 1 and w = 0, then each strategy profile (x1,x2) with x1 ≠ x2 is fully cooperative.
Finally we like to make some conjectures. They will be tested in the next section as hypotheses.
Here are the conjectures:
I.
E(1,n,2,w) ≠ , i.e. if there are two players, then in the one-dimensional case there
always exists a Nash equilibrium.
II.
E(1,n,2,w) ≤ 2 if and w{0,½,1}, i.e. if there are two players and w{0,½,1}, then in
the one-dimensional case there is a unique Nash equilibrium (and its permuted
counterpart).
III.
If N ≥ 3, then E(m,n,N,w) ≠  for w > 0 small enough.
IV.
In the one-dimensional case with two players and w{0,1}, the strategy profile
is fully cooperative.
9
4.
Agent-Based Model
The conjectures made above can be tested as hypotheses in an agent-based model. Before we
proceed to do so, we briefly explain how one can build the model of Hotelling within an agentbased framework. Agent-based models are computer simulation models of heterogeneous agents
that communicate and interact with each other and with their environment according to certain
rules (Gilbert 2008). By explicitly formulating the rules by which Hotelling’s agents operate, we
are able to build an agent-based equivalent of Hotelling’s model.
There are two types of agents in Hotelling’s model that interact with each other; buyers and
sellers. Both agents have a position within a one or two dimensional discrete and finite
Euclidian grid-space. Every grid-cell houses exactly one buyer. Sellers are free to move to any
grid-cell if they wish to relocate themselves. The agent-based simulation runs for any number of
discrete time cycles (ticks). Every tick, sellers evaluate all possible positions to situation and sell
their ice-cream to buyers. Buyers at their turn make a choice to consume some units of icecream from the closest seller relative to their position. When two sellers are at the exact same
distance, the buyer will choose to consume half of its demand from one seller and half of its
demand from the other sellers. This same decision rule holds for any number of sellers. We use
the NetLogo agent-based modelling toolkit developed by Wilensky (1999), which is freely
available for academic purposes2.
In our NetLogo implementation, small circles represent buyers and bigger circles represent
sellers. Buyers are anonymous while sellers have unique ID numbers, which are shown at the
lower-right of their symbols. At every tick one seller gets the turn to choose an optimal location
for itself and buyers will buy from all sellers given the new configuration. Then the turn is given
to the next seller to relocate itself to an optimal location and again the buyers buy from the
sellers given the new spatial configuration. This is repeated indefinitely until the end-state is
reached, in which none of the sellers has used its turn to relocate to a new location. We tested
four hypotheses by simulating the model with different parameter sets.
Hypothesis 1: For the parameter set given below, a Nash-equilibrium exists; even standard
best response dynamics leads to a Nash-equilibrium regardless of the starting
position of the vendors.
Parameters
N=2
2 See http://www.wageningenur.nl/en/Expertise-Services/Chair-groups/Social-Sciences/AgriculturalEconomics-and-Rural-Policy-Group/Publications.htm
10
n+1 = 20
w = { 0.0, 0.1, 0.2 … 1.0 }
Test
Test every starting position with w running from 0 to 1 in steps of 0.1 and
observe whether the simulation is in Nash-equilibrium or not.
Outcome
At w = 0, any starting position is a Nash-equilibrium, except when both vendors
are at the same position. At any other value of w, a Nash-equilibrium is reached.
Thus hypothesis 1 is corroborated.
Hypothesis 2: If w decreases, the average distance between the two vendors in the Nash
equilibria increases. The average distance is computed as the sum of all
distances between the vendors in all possible Nash-equilibria divided by the
number of possible Nash-equilibria.
Parameters
N=2
n+1 = 20
w = { 0.0, 0.1, … ,1.0 }
Test
Run the model and observe the distance between the two vendors at the state of
Nash-equilibrium.
Outcome
There is a clear trend between parameter w and the distance between the two
vendors as seen in the above figure. We repeated the simulation 100 times with
N = 2 and w = { 0.0, 0.05, 0.10, …, 1.0 } and n + 1 = 20. After a total of 2100
runs we ran a regression model dist=β0+β1w+ε and found a significant negative
contribution of parameter w to distance (dist) with an R2 of 0.637.
At w = 0, every starting position of the vendors, when not being at the same
11
position, is a Nash-equilibrium. When w is high, the distance at Nashequilibrium between the vendors is small and vice versa. Thus Hypothesis 2 is
accepted.
Hypothesis 3:
For 0 < w < 1, in a cooperative setting, the two vendors aiming at maximum
total revenue will locate as close as possible to ¼ and ¾ of total space available.
Parameters
N=2
n = 20
w = {0.1, 0.2, … ,0.9}
The vendors in this model are adjusted to not only optimize their own
individual revenue, but to optimize the sum of revenues of all vendors.
Test
Place the sellers on random positions every run. Run the model for 25 ticks and
note the last position of both sellers. Repeat 100 times.
Outcome
After running the model 225 times in total (25 times each setting of 10 ticks)
we observe that the position of sellers is indeed centered around ¼ and ¾ of the
space as shown in the above figure. Thus Hypothesis 3 is corroborated.
Hypothesis 4:
In a two-dimensional space, for a small enough w, there exists at least one
Nash-equilibrium; even standard best response dynamics leads to a Nash
12
Equilibrium, regardless the starting position of the two vendors.
Parameters
N = { 3, 4, 5, 6 }
n+1 = 10 x 10
w = { 0.0, 0.1, … ,1.0 }
Test
Run the model 10 times per parameter set and observe whether the simulation
ends in a Nash-equilibrium or not.
Outcome
In Table 1, we set out parameters w and N (vendors) against the model outcome
(nash). In the table we see the number of times in which the model ended in a
Nash-equilibrium at every parameter setting (w = {0.0 … 1.0} and vendors = {
3 … 6 }). At every number of vendors at lower w setting (w < 0.5), we observe
the simulation ending in a Nash-equilibrium. Vice versa, we observe at every
number of vendors at higher w setting (w > 0.9), that the model does not reach
Nash-equilibrium. For the intermediate values of w (0.6 ≤ w ≤ 0.9) there is no
certain outcome. Where N = 3, 5 and 6, the number of Nash-equilibria equals 0
for 0.8 ≤ w ≤ 1.0. For N = 4 the number of Nash-equilibria equals 0 only for w
= 1.0.
Table 1: Number of runs ending in Nash-equilibrium
Parameter w
Vendors (N)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
3 10 10 10 10 10 10
7
6
0
0
0
4 10 10 10 10 10 10 10
7
9
8
0
5 10 10 10 10 10
8
3
1
0
0
0
6 10 10 10 10 10
8
10
0
0
0
0
In Table 1, we observe Nash-equilibria at smaller w settings, therefore
Hypothesis 4 is accepted.
5.
Conclusions
In this article we wondered whether Agent-Based Modelling (ABM) could bring further
progress to the analysis of Hotellings Law. As ABM is a discrete approach we first adapted the
standard continuous game theoretic approach to a discrete framework. This led to a number of
proven propositions and 4 conjectures that were corroborated by way of testing of 4 connected
13
hypotheses in an ABM-framework. Especially the extension from one to two dimensions and
the variation in the spatial demand elasticity appeared to be difficult to solve using a discrete
analytical game theoretical approach. In this situation ABM proved to be a useful approach for
testing the essential hypotheses.
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