Game Theory: Hotelling Model of Spatial Competition

Picture this: you’re walking along a kilometre-long track on the beach, looking for an ice-cream stand on a warm day. After the first 100 metres, there’s nothing, after 200, there’s nothing, after 300, there’s nothing. But after you’ve walked 200 more metres, you are suddenly met with more ice-cream stands than you’ve seen in the past decade. So many competitors set up next to each other, but why?

It seems illogical at first, because setting up in your own location away from everyone else might appear to be most profitable. For example, if you set up 300 metres North along the track (assume you may walk in a North or South direction along the track) instead of next to everyone else at the center, then everyone travelling North would surely buy from you: because they wouldn’t be bothered walking another 200 metres to center where everyone else is! 

However, what about the people walking from the South? After walking 500 metres they will come to the plethora of ice-cream stands, so almost certainly they won’t walk the extra 200 metres to you. This might cause you to replicate the same idea but 300 metres South: but the same problem arises, just with the “Northern” originated customers now. 

Now assume that the number of people along the kilometre-long track are uniformly distributed, that is there are an identical number of people in any spot. 

As such, to maximise the number of ice-creams sold (assuming price is fixed and competitive to your other competitors on the beach), you want to have the closest proximity to the greatest number of beach-goers. 

It might seem fair if you and a competitor set up equidistant from the center, meaning that each of you would get ½ of customers each, but if you quickly shift your stand closer centrally, your customer base will increase due to a larger number of customers on your side.

If the entry points are at A and B (1st Diagram), standing 250m from the closest entry point yields equivalent market share.

However, as shown in diagram 2, moving closer to the center 500m from each entry point will yield a larger market share for that vendor as all customers to the left (towards C) will come there, as well as half the customers between the two vendors located 375m from C and 250m from D respectively. 

The optimal strategy, therefore, would be to place your stand in the center, which your competitor may also do to result in ½ of market share each.

At this stage, ceteris paribus, it is impossible to deviate from your central position and be better off. As such, the locations of you and your competitor are now in Nash Equilibrium, meaning that you cannot change your “strategy” to get more customers assuming your competitor stays in the center.

More broadly, if you set up your stand in location x₁ metres from the closest entry point and the second sets up at x₂ metres from the same entry point for 0 ≥ x₁  ≥ x₂  ≥ 1000 (on a 1 kilometre long track), assuming you and your competitor have negligible product differentiation, you will get:

of the customers and your competitor

If you set x₁ = x₂, both expressions will be the same. But to optimise your strategy you will push x₁ towards 500 (location of the center) to attain an inequality in your favour. This means that unless x₁ = x₂ = 500, there is always a way for either one of you or your competitor to generate more sales through moving closer to the center point (500m from each entry).

Now assume that instead of just two vendors (you and your competitor), there are now n  N>2 competitors along the one kilometre track. The question now becomes much harder! 

It may be tempting to think that each of you can stand (1/n+1)th along the track to generate equal market share, but like we saw before, if any vendor shifts their cart towards the center by some amount p <1/(n+1)th of the total distance (1000m), their market share increases.

Thus, the answer must surely be that all of you accumulate at the center, most likely getting (1/n)th market share due to price not being a differentiating factor.

Whilst this is the most commonly seen strategy (everyone at the center), the system is technically not at Nash Equilibrium given that some vendors will be “sandwiched” in the middle of the two vendors on the outside, giving them the incentive to move and thus creating an endless loop! 

References

1. William Spaniel (2013)Game Theory 101 (#40): Hotelling's Game and the Median Voter Theorem. 17 July. Available at: https://www.youtube.com/watch?v=THVrl_2Mu1A (Accessed: 30 September 2025).

2. Thomas Nechyba (2022)27.2.The Hotelling Model. 22 July. Available at: https://www.youtube.com/watch?v=QoGOv-Hh17k (Accessed: 30 September 2025).

3. William Spaniel (2013)Game Theory 101 (#39): Hotelling's Game. 17 July. Available at: https://www.youtube.com/watch?v=jILgxeNBK_8 (Accessed: 30 September 2025).