Question

Consider a location game with five regions on the beach in which a vendor can locate. The regions are arranged on a straight line, as in the original game discussed in the text. Instead of there being two vendors, as with the original game, suppose there are three vendors who simultaneously and independently select on which of the five regions to locate. There are thirty consumers in each of the five regions; each consumer will walk to the nearest vendor and purchase a soda, generating a \(\$ 1.00\) profit for the vendor. Assume that if some consumers are the same distance from the two or three nearest vendors, then these consumers are split equally between these vendors. (a) Can you rationalize the strategy of locating in region 1 ? (b) If your answer to part (a) is "yes," describe a belief that makes locating at region 1 a best response. If your answer is "no," find a strategy that strictly dominates playing strategy 1 .

Short answer

The viable move for a vendor in region 1 depends on competition's distribution; otherwise, mid regions offer more strategic advantage.

Step-by-step solution

Understanding the Game

In this scenario, there are five distinct regions labeled as 1 to 5 in a linear arrangement, with three vendors choosing where to set up shop. Each of these five regions hosts 30 consumers. Each vendor will aim to place themselves in a position such that the maximum number of consumers can easily reach them—a closer proximity typically ensures higher consumer count.

Analyzing Strategy of Locating in Region 1

To think about the strategy of choosing region 1, consider the location of region 1 as one endpoint of the linear arrangement. The consumers from region 1 will go to the closest vendor if it is located in their region. However, region 1's consumers would also visit region 2 if the vendor is closer or equally close (as per the equal split rule between equidistant vendors). If a vendor positions in region 1 while others are randomly placed, the strategy relies heavily on the latter's locations.

Evaluating Best Response for Locating in Region 1

For vendor position in region 1 to be considered a 'best response', assume a belief where at least one other vendor will occupy regions 3, 4, or 5, ensuring that the consumers, left unattended in these farther regions, do not inherently result in a competitive disadvantage against vendors placed at mid or farther areas. The idea is that with one vendor in region 1, and competitive others spaced towards the mid to end range, the market is best equalized.

Considering belief scenarios

Regions 2 acts as a balance—vendors positioned optimally around it will enable strategic advantage opening all ends from the middle. Consumers in the region are split between equal-distance vendors, so anticipating movement, vendor 1 positioned here assumes one or more vendors will either equally prioritize the middle or be hemmed to mid-far regions, enabling equally sharing customers at equidistantly split boundaries.

Identifying Dominating Strategy

If choosing region 1 does not seem optimal based on other plausible vendor locations, check if being centrally located either at region 2 or 3 offers full access and higher immediate consumer density. Thus, centrally placed vendors benefit from equidistant consumer distribution.

Key concepts

Location Games

Location games are strategic interactions where players (like vendors) choose positions in a competitive market. The goal is to maximize reach to consumers or resources. In the given problem, this game takes place on a linear beach with five regions. Vendors aim to select positions to attract the highest number of consumers. This involves understanding not only the geographical layout but also anticipating the competitors' moves.

Location games are common in market competition scenarios. They require vendors to think about proximity to consumers and competitors' locations. Choosing well can provide a competitive edge. The particularity here is that consumers move to the nearest vendor, so vendors must understand and predict movements to optimize their placement. Vendors also must consider how far consumers are willing to travel and how split decisions among equidistant options might impact them.

Strategy Selection

In any game involving multiple players making simultaneous decisions, strategy selection is crucial. Here, each vendor independently chooses a region on the beach to set up their shop. The decision directly affects the potential to maximize profits based on consumer access.

Vendors must deliberate their location considering the behavior of other vendors and consumer movement. Each position has its benefits and trade-offs. Regions closer to many consumers may seem attractive but can also invite more competitors. Thus, selecting a strategy requires analyzing competitor tendencies and consumer distribution, then optimizing one’s position to leverage these factors. A good strategy takes into account the movement of competitors and consumers equally, ensuring a higher likelihood of success.

• Choose a spot that maximizes your consumer catchment.
• Consider possible competitor locations.
• Think about consumer behaviors and their willingness to move.

Nash Equilibrium

Nash Equilibrium in location games involves each vendor choosing a location where any deviation from that chosen strategy, given everyone else sticks to their strategies, does not result in higher profits.

In this location game, achieving Nash Equilibrium means finding a setup where no vendor benefits from changing their region after the locations are set. It's about balance; all vendors optimally placed such that collective profits are stabilized.

• All vendors choosing the best strategy for their own gain.
• Competitors can’t do better by unilaterally changing their strategy.
• Market conditions and consumer distribution remain stable.

Nash Equilibrium is critical because it guides vendors toward strategies that are best for maintaining competitive stability. Achieving this helps vendors avoid continuous repositioning or over-competition, simplifying strategy decisions.

Consumer Distribution

Consumer distribution refers to how consumers are spread across the five regions of the beach, with 30 consumers per region in this game. Understanding distribution is key to predicting consumer behavior and maximizing vendor profit.

The challenge lies in drawing consumers from their initial locations to the vendor’s location. Regions with dense consumer numbers are attractive, but vendors must consider distance and how evenly consumers can be split among nearby vendors. If consumers are strategically placed, they march towards closer vendors, splitting equally among equally distant options. Vendor positioning should thus take into account:

• Consumer concentration in each region.
• Proximity advantages to cluster multiple regions’ consumers.
• Potential equal splits with other vendors.

By understanding consumer distribution patterns, vendors can optimize their locations to capture the maximum number of consumers possible efficiently.