# A Game Design

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A Game Design
Two supermarkets site in a town on a certain distance. Town residents may usually go to the nearest supermarket, but they can choose the other supermarket because it not takes long trip to reach the farther supermarket.
Now they will sell a new kind of yogurt that estimate 200 unit will be sold per week in the town. They must decide the price. There are 3 prices they can choose: \$10, \$14, \$18.
If they sell the yogurt for a same price, they will sell the same unit. If one supermarket sells \$4 more than the other supermarket does, half of the near residents will go to the other supermarket to buy the yogurt. If one supermarket sells \$8 more, the other supermarket will sell the entire unit.
So we will have the chart below, it shows the revenue the supermarkets can get in different price situation:
Supermarket 2
Supermarket 2

Supermarket 1
Supermarket 1
Price/unit(\$) | 10 | 14 | 18 | 10 | 1000,1000 | 1500,700 | 2000,0 | 14 | 700,1500 | 1400,1400 | 2100,900 | 18 | 0,2000 | 900,2100 | 1800,1800 |

In order to get higher revenue, supermarkets want to choose a proper price. But they will not know the price the other supermarket set and the price will be not changed during the week.
In the situation, to predict possible price decision from competitor and derive strategy, supermarkets can get chart below:
Supermarket 2
Supermarket 2

Supermarket 1
Supermarket 1
Price/unit(\$) | 10 | 14 | 18 | 10 | 1000,1000 | 1500,700 | 2000,0 | 14 | 700,1500 | 1400,1400 | 2100,900 | 18 | 0,2000 | 900,2100 | 1800,1800 |
So we can see when they all set the price as \$10, they can get Nash equilibrium.
Neither of them has a dominant strategy. Dominated strategy of supermarket1 is \$18, and of supermarket 2 is \$18 too.
The nature of conflict make both of them want to get the largest revenue which is 2100 while the other gets only

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