Economics 206
Spring 2007 (Prof. G. Loury)
Solution of Racing Problem in Assignment 7
[Racing Game] Let
V
be the value of the prize, and let
C
(
x
)
be the cost to a player of taking
a step toward the
fi
nish line of size
x
. A player gets to move every second period, so if
δ
is
the perperiod discount factor, then
β
=
δ
2
is the discount factor that applies to any value
which a player expects to receive on his next turn. Consider the singleplayer problem of
optimally approaching the
fi
nish line from distance
X
:
φ
(
X
) =
Max
(
x
n
,N
)
{
β
N
V
−
N
X
n
=0
β
n
C
(
x
n
)

N
X
n
=0
x
n
≥
X
}
,
where
N
≥
0
is an integer, and
n
∈
{
0
,
1
, ..., N
}
The expression above embodies a player’s
choice of how many steps to take to reach the
fi
nish line (
N
+ 1
), and of how big to make
each step (
x
n
). The value function
φ
(
X
)
gives the net return to either player of having a
“free run” to the
fi
nish line without competition from the other player.
Now, consider the following recursion:
C
(
X
0
)
=
V,
and for
k
≥
0 :
C
(
X
k
+1
−
X
k
)
=
βφ
(
X
k
)
.
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 Spring '07
 G.LOURY
 Economics, Microeconomics, Recursion, finish line, Prof. G. Loury

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