Can you solve the basketball riddle? - Dan Katz
You’ve spent months creating
a basketball-playing robot,
the Dunk-O-Matic,
and you’re excited to demonstrate it
at the prestigious Sportecha Conference.
Until you read an advertisement:
“See the Dunk-O-Matic face human players
and automatically adjust its skill
to create a fair game for every opponent!”
That's not what you were told to create.
You designed a robot that shoots baskets,
sometimes successfully and sometimes not,
taking turns with a human opponent.
No one said anything about teaching it
to adjust its performance.
Maybe the CEO skimmed an article
about AI and overpromised,
setting you up for public embarrassment.
Luckily, you installed a feature
where given any probability q,
you can adjust the robot to have that
probability of success on each attempt.
You swiftly gather information,
and jackpot:
your team has a dossier on all
potential demo participants,
including the probability each has
of making baskets.
In each match, the human shoots first,
then the robot, then the human again,
and so on until someone makes
the first successful basket and wins.
You can remotely adjust the Dunk-O-Matic’s
probability between opponents.
What should that probability
be for each opponent,
so that the human has a 50% chance
of winning each match?
Pause here to figure it out yourself.
Answer in 3
Answer in 2
Answer in 1
You might guess that q
should be equal to p.
But that ignores the advantage
of going first.
Suppose p and q are both 100%.
Even though the competitors
are equally skilled,
the first player always wins.
So a deeper analysis is required.
One approach involves adding up every
chance the human has to win,
using geometric series.
A geometric series is an infinite
sum of numbers,
where each number is the previous number
multiplied by a common ratio.
Two facts about geometric series
are useful here.
First, if the common ratio r
of a geometric series
has absolute value less than 1,
the series has a finite total.
And second, if the first number
in the series is a,
that total is: a divided by 1-minus-r.
How does this help us calibrate our robot?
Remember that the human has probability p
of making a basket.
Since they go first, they have probability
p of winning on the first try.
What’s the probability that they win
on the second try?
That attempt only happens
if both players miss.
The probability of a miss is 1 minus
the probability of a success,
so the miss probabilities are
1-minus-p and 1-minus-q.
The chance of both happening
is the product of those values.
So the probability of two failures
and then a human success
is p times (1-minus-p) times (1-minus-q).
Winning on the third try requires
another round of misses,
so that chance is p multiplied
by the double-miss probability twice.
If we add all the possible probabilities
of a human win,
the total is the sum
of a geometric series.
Since the first number in the series is p,
and the ratio is this product
that’s less than 1,
the sum will be
(p divided by 1) minus the ratio.
We want this sum to be 1/2.
Using some algebra to solve for q,
we find that q should equal p
divided by 1-minus-p.
If p is greater than 50%,
q would need to be bigger than 1,
which can’t happen.
In that case, a fair game is impossible,
because the human has a better-than-50%
chance of winning immediately.
The robot's total probability is also
the total of a geometric series.
How does this series compare
to the human’s?
To win, the robot needs some number
of double misses,
then a human failure
followed by a robot success.
If q equals p over 1-minus-p,
(1-minus-p) times q is p.
For our choice of q, not only do these
series have the same sum,
but they’re the same series!
We could bypass geometric series
by starting with this reasoning.
The robot’s chances of winning in the
first round is (1-minus-p) times q,
and so if we want that chance to match
the human’s first-round chance,
we want it to equal p,
making q: p over 1-minus-p.
More rounds may occur,
but before each round,
the competitors are tied,
so everything effectively restarts.
If they have the same odds of winning
in the first round,
they also will in the second round,
and so on.
The demonstration goes perfectly,
but while you didn't want
to embarrass yourself,
you also didn’t want
to deceive the public.
Taking the stage, you explain
your company’s false promises
and your hastily ad-libbed solution.
Thankfully, the ensuing bad press
is directed at your employers,
and it turns out the
presentation volunteers
own a more employee-friendly
robotics company.
After some tedious intellectual
property litigation,
you find yourself at a healthier workplace
with a regular spot on a
pickup basketball team.
