Solving Sequential Games II

Where to set up shop?


In my previous blog I introduced the extensive form of games that are used in illustrating sequential games (order matters). These games are solved using backwards induction. Today we are going to analyze a scenario of where a restaurant should open up shop taking into consideration a competitor looking to open shop as well.

Pieology and Blaze Pizza are competing for world domination. It’s hard to miss these locations in town as they are popping up everywhere. Due to their long term strategy for aggressive expansion, Pieology is looking to open a location at a new shopping center. However, Blaze Pizza wants to capture market share in the new shopping center as well. The owners of the shopping center are willing to give Pieology first choice for the exact location (first-mover advantage). Which location should Pieology choose?

Sequential Pie

Observe the game tree above. Pielogy has the option of either choosing the West, South, or East side of the shopping center. Once Pieology makes there choice of location, Blaze Pizza will come in and establish a new location based on Pieology’s choice. Using backwards induction we can solve for the optimal choice for Pieology to maximize profit. If Pieology chooses West, Blaze Pizza can select either West, South, or East. Blaze Pizza will prefer to set up a location at the East side of the shopping center since 3 (East) > 2 (South) > 1(West).

Sequential Pie 2

Now let’s assume Pieology chooses to build the new location on the South side of the shopping center. Blaze Pizza will evaluate their potential profits based on the possible locations. Blaze Pizza’s preference is to build a new location on the East side of the shopping center due to 3 (East) > 2 (West) > 1 (South).

Sequential Pie 3

Similar analysis can be used to determine Blaze Pizza’s optimal choice if Pieology chooses the East side of the shopping center. Based on the possible profits, Blaze Pizza will select the West side of the shopping center since 3 (West) > 2 (South) > 1 (East).

Sequential Pie 4

Based on backwards induction what would be the optimal choice for Pieology?

If Pielogy chooses West: (West, East) = (4,3)

If Pieology chooses South: (South, East) = (3,3)

If Pieology chooses East: (East, West) = (5,3)

As you can see Pieology will always have higher profits than Blaze Pizza due to their first-mover advantage. However, their is an optimal location for Pieology. Based on the payouts (profits), Pieology will choose to establish their new location on the East side of the shopping center since 5 (East) > 4 (West) > 3 (South).

Sequential Pie 5

To optimize profits for both businesses, it is best that they establish locations furthest from each other at the new shopping center. We will continue to look at sequential games in the next blog which will include more examples as we continue to solve these types of games. I look forward to your feedback in the comments below.

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