In the competition to attract athletes and produce champion?teams, universities increased their spending on college athletics four times faster than overall university spending from 2001 through 2003. Schools have poured money into athletic programs even though studies show that this practice does not increase winning rates or alumni donations? ("Review &? Outlook,"Wall Street? Journal,May? 27, 2005,? W15). ?Nonetheless, suppose that money does matter in producing championships. University A spends mA on its football team and University B spends mB. The fraction of the time that University A wins is wA=mA/(mA+mB)?, and the fraction of time that B wins is wB=?(1?wA?)=mB/(mA+mB). Suppose that each university wants to maximize its profit from having sports teams. The expected profit of University? i, i=?A, ?B, is ?i=vi(mi/mA+mB?), where vi is the value to University i of winning a game. Show that if vA=vB?, in the Nash equilibrium each school wins? one-half of its games. If vA=vB=?v, then each school will win? one-half of its games because each school will spend mi=.............on athletics.

In the competition to attract athletes and produce champion teams, universities increased their spending on college athletics four times faster than overall university spending from 2001 through 2003. Schools have poured money into athletic programs even though studies show that this practice does not increase winning ratus or alumni donations ("Review & Outlook," Wall Street Journal, May 27. 2005, W15). Nonetheless, suppose that money does matter in producing championships. University A spends ma on its football team and University B spends mp. The fraction of the time that University A wins is w." ma + ma and the fraction of Sme that ma B wins is wg = (1 -WA)"- Suppose that each university wants to maximize its profit from having sports teams, The expected profit of University , im A, B, is ma + me imA + me ". where V, is the value to University i of winning a game. Show that if VA " Vp. in the Nash equilibrium each school wins one-half of its games. If v. Ve =v. then each school will win one-half of its games because each school will spend my |on athletics. (Proparty format your expression using the fools in the palette. Hover over fools to see keyboard shortcuts. E g., a fraction can be created with the / character)