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Unformatted text preview: Indiﬀerence Surfaces for Lotteries
A
L TEX ﬁle: LotteryIndifSurfnball — Daniel A. Graham <daniel.graham@duke.edu>, June 22, 2005 In: Needs ["Graphics‘ContourPlot3D‘"]
Needs ["Graphics‘ImplicitPlot‘"]
Consider an individual whose vonNeuman utility function is equal to the natural log of the payoﬀ In: u [w_] :=Log [1 + w]
where w is measured in millions of dollars and let (x,y,p) denote a lottery that pays $x with probability
p and $y with probability 1p. It follows that this person, let’s call him Nat Log, would be indiﬀerent
between a certain prize of $2 and any lottery (x,y,p) for which In:
Out: pu [x] + 1 − p u y ==u [2]
p Log [1 + x] + 1 − p Log 1 + y == Log [3]
or, equivalently, for which In:
Out: indiﬀ = pu [x] + 1 − p u y − u [2]
−Log [3] + p Log [1 + x] + 1 − p Log 1 + y
is equal to zero. This surface in x,y,p space can be plotted as follows where the origin is in the lower
left hand corner, the x and y axes form the two edges of the bottom of the shallow box and the p axis
forms the third (vertical) edge. In: ContourPlot3D indiﬀ, {x, 0, 4} , y , 0, 4 , p , 0, 1 , PlotPoints → {5, 5} , PlotLabel → "Indiﬀerence Surface" ; Indifference Surface Representative crosssections correspoding to diﬀerent values of p are illustrated below.
In: ImplicitPlot Table indiﬀ == 0/.p → i, {i, 0, 1, .2} , {x, 0, 4} , y , 0, 4 , PlotPoints → 50, PlotStyle → Table [Hue [i] , {i, 0 Page 1 of 2 y
4 3 2 1 x
1 2 3 4 Note that each is (weakly) convex. The horizontal line corresponds to p=0  when there is no chance
of winning x  and the vertical line to p=1  when there is no chance of winning y. In between, the
indiﬀerence curves "twist" clockwise as p increases from 0 to 1. Each of these curves is, of course, a
crosssection of the threedimensional indiﬀerence surface illustrated before. Page 2 of 2 ...
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 Spring '09
 Physics

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