Mike Tymn, in his National Master's News column (Issue 275, July 2001, p.6,)
argues that the Boston Marathon awards structure discriminates against males
and older competitors. He bases his argument on an analysis of the
distribution of women (and young competitors) in the race relative to
quality of performance. The gist of it is that women are paid better than
they deserve based on the quality of their performances relative to world and
course records. Older competitors are paid less than they deserve based upon
age grading criteria. Moreover, the distribution of awards is not equitable
in terms of the overall numbers in each division.
Whether or not Tymn's contentions about Boston are true, and, more generally,
which types of award structure are most equitable, practical, enticing, or
would meet some other criterion, are all interesting topics for debate, but
I shall not pursue them here. Rather, I shall discuss the implications of
an awards structure which is quite different from the typical ones, those
based upon measuring performances relative to the most outstanding one.
Suppose, on the contrary, that in each age/sex category, race management
agrees to pay contestants based upon their performance relative to the AVERAGE
performance in that category. Thus, an average or below performer would
receive nothing, while those who excel relative to their peers would receive
more or less according to how much better than average they performed.
Elementary statistics provides a natural way to measure the amount by which a
given runner excels relative to average. The measure, called the Z-score,
is defined as z = (x - m)/s, where x is a given runner's time, m is the
average of all times in the category, and s is the standard deviation of
those times. The more negative z, the better the performance.
Since it unwieldy to work with negative quantities, it is more natural here
to define z as (m - x)/s, and we will do so. For example, suppose a runner
ran of time of 1000 seconds in a race where the average performer in his
division ran 1050 seconds. If the standard deviation of all performances in
the division were 80 seconds, then the given athlete would receive a "score" of
(1050 - 1000)/80 = 0.625.
How much should a runner be paid for a given z? It's up to the race director,
of course. Let us assume there is some function P(z) (P for prize,) chosen
by the race director, which gives the dollar amount to be paid to a runner
whose score in the race is z. Presumably, performances which are average or
worse should receive no prize, so P(z) should be zero for z less than or equal
to zero. (Average or worse performances correspond to non-positive values of
z.) On the other hand, better than average performances should be paid more
or less according to how much better than average they are, i.e, according
to how large the z score is. For example, if P(z) = z, or more generally some
constant multiple of z, then a performance which is twice as good as another -
as measured by z - would be awarded twice as much. If a race director wanted
to pay an accelerating premium for good performances, he might choose P(z) to
be z-squared or even some higher power of z.
How much could a race director expect to pay out if he were to adopt such a
scheme? The answer would depend upon how race performances are
distributed relative to each other. (The real quantity of interest would be
the expected payout per runner since the prizes would presumably be financed
by entry fees.) Thus, to answer the question we would need to calculate
E(P(z)), where E denotes expected value. If performances are normally
distributed, then z has a standard normal distribution and it is straightforward
to calculate the expected value given a particular function P. For example,
if P(z) is z-squared (z^2), then the expected value is 0.5. Since this
represents a fixed cost per runner, such an awards scheme could easily be
financed by charging an appropriate entry fee.
Suppose now that a race advertises a particular payout scheme (i.e, a P(z).)
In addition, suppose the runners in a particular age/sex division get together
before the race and conspire to rig the finish time distribution so as to
bilk the race director out of as much money as possible, agreeing to split
the proceeds among themselves after the race. How much could they earn this
way, and what should be their optimal strategy?
The answer depends on how quickly P(z) grows with z. In particular, the
function z^2 represents a critical case: here it turns out that the runners'
individual earnings can never exceed 1, and that 1 can only be attained in
the limit as the number of runners tends to infinity. To achieve this, the
field should select one runner to run a very fast time, and the rest should
all tie with an extremely slow time. The exceptional runner's very high
z score earns him a big payoff, which he then splits with his co-conspiritors.
Nevertheless, despite the conspiracy, the cost per runner is fixed and the
race can finance it with entry fees.
If the function P grows faster, the same stategy as defined in the last
paragraph leads to arbitrarily large payouts (as the number of runners
becomes large.) Unless the race puts a cap on entries, such collusion will
lead to bankruptcy. (Race directors take note!)
If P grows more slowly than z^2, the situation is quite different. Suppose,
for example, that P grows only linearly with z. Faced with such an awards
structure, the best a runner can do is to convince all but one other
individual in his division not to run the race. The two runners then run
unequal times, earning z scores of +1 and -1. Then they split the unit sized
award, winning 0.5 each. This is the best they can do. (The proofs of this
and the other assertions above make interesting exercises.)