Figures and data in Spatial competition with interacting agents | International Journal of Microsimulation

Abstract

Using agent-based modeling, we generalize Hotelling’s model of spatial competition with more than two firms in a two-dimensional space. Firms choose both price and location to maximize profits. The principle of minimum differentiation does not hold in general. Local duopolies emerge from the interaction between firms. Firms do not spread uniformly across the two-dimensional space, nor do they all charge the same price. Firms in more competitive locations charge lower prices and generate less profit.

This article contains 12 figures and 2 tables.

Figures

**Figure 1.** Minimum distance to nearest when firms choose locations but prices are fixed.

**Figure 2.** Maximum distance to nearest firm when firms choose locations but prices are fixed.

**Figure 3.** Minimum distance to nearest firm when firms choose both prices and locations.

**Figure 4.** Maximum distance to nearest firm when firms choose both prices and locations.

**Figure 5.** Proportion of firms that form local duopoly pairs (less than 3 units distance from one other firm at the end of the simulation run).

**Figure 6.** Characteristic locations with two, three and seven firms when firms choose locations but prices are fixed.

**Figure 7.** Characteristic locations with two, three and seven firms when firms choose both prices and locations.

**Figure 8.** Price versus distance to nearest competitor when firms choose prices but locations are fixed (left panel) and when firms choose both locations and prices (right panel).

**Figure 9.** Profit versus price when firms choose prices but locations are fixed (left panel) and when firms choose both locations and prices (right panel).

**Figure 10.** Sensitivity analysis for the distance between nearest firms (top panel), the fraction of firms involved in local duopolies (middle panel), and the average price (bottom panle), with respect to firms’ sampling percentage (what fraction of consumers each firm surveys about potential patronage during decision-making).

**Figure 11.** Sensitivity analysis for the distance between nearest firms (top panel), the fraction of firms involved in local duopolies (middle panel), and the average price (bottom panel), with respect to customer rationality (the percent chance that each customer will choose to patronize their minimal cost firm, instead of choosing randomly among all firms).

**Figure 12.** Average price versus grid size for 3 firms that are choosing both location and price.

Tables

Algorithm 1. Pseudocode for the agent-based model.

Initialization:

customer agents placed on each lattice point of 41x41 grid.

for each firm F_i ∈ F1, F2, …, F_N:

place firm Fi on a random lattice point.

set initial price for Fi ← 10

Each time step:

let M (the set of movement actions) = |Up, Down, Left, Right}

let P (the set of price change actions) = |+1,-1}

for each firm F_i:

F_i selects m_i ∈ M s.t. calculatedProfitAfter(m_i) is maximized

F_i selects p_i ∈ P s.t. calculatedProfitAfter(p_i) is maximized

for each firm F_i:

F_i takes actions mi and pi

for each customer Cj:

C_j selects firm F_min s.t. price(F_min) + transportationCost(F_min, C_j) is minimized.

C_j purchases one unit of the good from F_min

Table 1. Selling prices for the firms shown in Figure 7.

Region	A	B	C	D	E	F	G	H	I	J	K	L
Price charged	1	1	8	2	2	2	2	3	3	2	3	10

Notes: Firms positioned farther apart from others (e.g., C and L) charge higher prices than firms in competitive local duopolies, because they are able to raise prices without losing as many customers as firms who have nearby competitors.