Game Theory: Unusual Question #3 and #4

[fnord_too]

Reducing the equation:

C = R + .25*(1-R) = .75R+.25

((1+R)/2)/((1+C)/2) = (1+R)/(1+C) = (1+R)/(1.25+.75R)


What we want is the derivative of this whole nasty function. I appologize in advance for the verbosity and pedestrian nature of my calculus, I went into discrete math because I was no great fan of the continuous variety.

Lets call 1+R U and 1.25+.75R V

so we want d/dR 0 + .75R - .375 + (1-R)*E(raise)
= 0 + .75 + d/dR E(Raise) - d/dR R*E(raise) = .75 + some other stuff to be computed presently

d/dR E(Raise) = d/dR .75R+.25 +(.75 - .75R)*2*((U/V)-(V/U))
= .75 + d/dR 1.5*(U/V - V/U) - .75R * (U/V)-(V/U)

= .75 + d/dR 1.5 U/V - 1.5 V/U - .75RU/V + .75RV/U (mommy make the hurting stop)

d/dR U/V = (V dU - U dV) / (V^2)

dU = 1, dV = .75, V^2 = .5625*R^2+.9375R + 1.5625

d/dR U/V = (.75R + 1.25 - .75(1+R))/(.5625*R^2+.9375R + 1.5625)
= .5/(.5625*R^2+.9375R + 1.5625)

d/dR V/U = (if I got this right):
(.75*(1+R) - .75R-1.25)/r^2 +2R +1
= -.5/(R+1)^2

d/dR RU/V = d/dR R+R^2/V = ((V*(2R+1)) + (U*(1.5R + 1.25))/(.5625*R^2+.9375R + 1.5625)
The top simplifies to .5 R

for d/dR RV/U we get -.5R/(R+1)^2 (and yes in retrospect I see the easy way to have done that!)

now where the hell am I? I can't finish grinding this out right now. If someone has mathmatica or the likes handy, please maximize the E(x) function and let me know where R is. Alternately if someone really likes Calc, do this by hand. Alternately, I will do it later, but after about an hour or so, mostly doing my work in a text editor (yuk) I am ready to leave this for a while.