Saturday, July 7, 2012

CTY: Are Game Theory and Economics really that similar?

This is my second week at CTY studying Game Theory and Economics. For today, I decided to show a combination between the two.  Let's say two companies are producing Q goods overall, q1 from the first company and q2 from the second. The equation for their relation of production is: Pq1 = 200q1 - 2Qq1 Let's see what happens if we simplify.  Pq1 = -2q1^2 + (200 - 2q2)q1 This is a quadratic equation, which we have looked at several times. Since the leading coefficient is negative, there is a point that is the highest point on the graph which is the vertex.  To solve for the x-coordinate (q1's production), you would be doing -b/2a with a being the first coefficient and b being the second.  If you do this, you get: -(200 - 2q2)/2(-2) 2q2 - 200/-4 q2 - 100/-2 100 - q2/2 Basically, this number is the best your company can do with this equation. For q2, the same logic applies.  100 - q1/2 To find q1's actual number, we assume that q2 will play rationally and follow this formula. That means we can substitute this formula in for q2 in the q1 problem. This gives us: q1 = 100 - q2/2 q1 = 100 - (100 - q1/2)/2 4q1 = 200 - 100 + q1 3q1 = 100 q1 = 33.3... In other words, q1 and q2 should both always do 33.3... to get the best outcome for them. The game theory is their choosing of the strategy and the economics is its application. I found it cool how this relation is so clear in this problem even though they seem so different.  Bonus: Here is another fun problem that we learned last week.  There is a kingdom ruled by two tyrants: the king and the dragon. They are normally in good terms with each other until now. The townspeople are split between the two, and they decide that the only fair way to decide a leader is to fight to the death.  The dragon suggests a fire breathing contest and the king suggests a juggling contest, but that wasn't fair. The king then proposed this idea: There are ten numbered wells in our kingdom. Each well has a poison in the water that can only be treated by the poison in a higher numbered well. If the poison isn't treated in an hour, they are dead. One day, we will both serve each other a glass of water, and then go our separate ways. After an hour, we will see who is alive.  The dragon does not know where well #10 is and the king does, and the king is to smart to not catch a following dragon. However, the dragon accepts the king's challenge, and after the hour, the king is somehow dead and the dragon is left to rule the kingdom.  How did the dragon do it? As usual, I will post the answer in a month.

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