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Nash Equalibrium

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1. In a symmetric model, team 1 and team 2 would invest money until they reach Nash equilibrium and their profits are maximized. Since this model is a symmetric model, both teams have the same incentive to win and therefore at equilibrium it can be assumed that win% for team 1 will equal the win% for team 2. Knowing these givens and the equation given, profits for team 1 can be found as followed: π1 = Vw1 – t1 → V(2-g)/4 π1 = (500)(2-.5)/4 π1 = 187.5
The profit can then be plugged back into the initial equation to determine team 1’s investment: π1 = Vw1 – t1
The win percentages for both teams will be .5 because .5 + .5 = 100% in a symmetric model 187.5 = 500(.5) – t1 t1= 62.5
Since this is a symmetric model, it is assumed that the investments and profits for team 1 will equal team 2’s investments and profits. π1 = π2 187.5 = π2 t1 = t2 62.5 = t2
2. For model 1, profits are larger when g=1/2 because the investments made by both teams are lower and therefore π1 = Vw1 – t1 will result in higher profits. The reason team 1 and team 2’s investments will be lower is because there is less incentive to invest when the advantage for investing goes down, as it does with a lower sensitivity parameter. A lower g (sensitivity parameter) means that teams need to investment less to yield a higher win%. The following graph shows the difference in profits for g=1 and g= ½: 3. To calculate the Nash equilibrium for an asymmetric model win % and team investment need to be calculated first because they depend on the sensitivity of each team’s market outcome: w1 = Vg1 / Vg1 + Vg2 w1 = 500.5 / 500.5 +300.5 w1 = .563508
Next is team 1’s investment found with the following formula: t1 = (g)(V1)((Vg1Vg2)/(Vg1+Vg2)2) t1 = (.5)(500)((500.5*300.5)/( 500.5+300.5)2) t1 = 61.4917
Profits can now be found for team

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