Imagine someone walks up to you and hands you
a tiny, a really tiny slip of paper. Like maybe
the size of a fortune cookie insert? Yeah, exactly
like that. Barely bigger than a fortune cookie
insert. And they tell you with just absolute
deadpan sincerity that written on this scrap
is the fundamental mathematical algorithm behind
every single decision you have ever made. Right.
And not just that, but every decision you will
ever make. You probably assume it was a joke.
Right? I mean, I would. Oh, absolutely. It sounds
like science fiction. It really does. But today,
we're essentially holding that exact metaphorical
slip of paper. Welcome to the deep dive, by the
way. Today's deep dive is based on what might
actually be our smallest source text ever. It
is incredibly brief. Yeah, it's literally just
a remarkably short Wikipedia stub about this
concept in decision theory. It's called the complete
class theorem. We're talking about, I don't know,
a handful of really dense sentences. But the
sheer density of the information, I mean, that
is what makes it so compelling. We are dealing
with an intensely concentrated nugget of mathematical
statistics here. In academia, you sometimes find
these tiny unassuming theorems that just quietly
act as the load -bearer. pillars for entire fields,
like behavioral economics or logic. And that
is exactly what we have here. It really is fascinating.
I have to admit, when I first looked at the source
material, I was pretty skeptical. I was like,
how much can we really extract from this? Right,
it doesn't look like much. But then as you parse
the actual phrasing, you realize this text is
delivering this huge universal ultimatum about
the actual mechanics of human choice. So our
mission today for you, the listener, is to take
this highly compressed statistical puzzle and
carefully expand it, word by word. Expand it
and really look at the implications. Exactly.
We're going to examine the ultimate filter for
making the optimal choice and see how it actually
maps onto our everyday lives. I think that is
the right approach. We really need to dismantle
the machinery of this theorem piece by piece,
because once you actually grasp the underlying
mechanisms, the specific mathematical constraints
the text is outlining, it just forces a total
reevaluation of what you even consider rational
thought. OK, let's unpack this before we get
into the really heavy statistical claims. The
text establishes a sort of baseline. Right. It
has to set the stage. Yeah. It gives us this
strict requirement for what makes a decision
mathematically valid in the first place. And
it introduces a specific term. It calls it an
admissible decision rule. Yes. And the way the
text defines admissibility is critical because
it does so entirely by negation. Meaning it tells
us what it isn't. Exactly. It does not tell you
what a perfect decision looks like. It only tells
you what an invalid decision looks like. It states
that if you are using an inadmissible rule than,
quote, there is a rule that is sometimes better
and never worse. Wow. OK. So if we invert that
logic, an admissible decision rule is one where
absolutely no alternative exists that is sometimes
better and never worse. Precisely. That phrase,
sometimes better and never worse, that's the
threshold for keeping a choice on the table at
all. I want to ground this for a second. Let's
look at a classic commuter dilemma. OK. Let's
hear it. Say you're driving to work, and you
have two routes. Route A is the highway. Based
on all available data, it takes exactly 20 minutes
every single day, regardless of conditions. And
it's like a magical highway, but sure. Right,
work with me here. Route B is the back roads.
It usually takes 20 minutes, but if you get stuck
behind a delivery truck or something, it takes
30 minutes. We have to be careful there, though.
For that analogy to really hold up mathematically,
we have to assume strict ceteris paribus. Which
means? all other things being equal. Right. Yes,
exactly. Because if route B offers a stunning
scenic view that you really enjoy, then the utility
of that drive changes. It might not be strictly
worse to take 30 minutes if you're enjoying the
scenery. OK, fair point. But if we are isolating
purely for time and fuel efficiency, if getting
there fast is the only goal, then yes, your analogy
works perfectly. Right. So assuming we only care
about getting to the office as fast as possible
in that tightly constrained scenario, route A
is our sometimes better, never worse option.
Because the highway is never going to take 30
minutes, so it is never worse than the back roads.
But it is sometimes better, because on the days
there happens to be a truck on the back roads,
the highway beats it by 10 minutes. Exactly.
So if you choose the back roads, knowing this
information, you are choosing a mathematically
dominated strategy. You're picking a rule where
a strictly superior alternative just plainly
exists. So the theorem is basically saying that
the first step of decision theory is just eliminating
the obviously dumb choices. That's a great way
to put it. It really is. That concept of dominance
is the foundational sorting mechanism in decision
theory. By eliminating those dominated, sometimes
worse options, it leaves us with our admissible
class of choices. It just clears the junk out.
It sweeps the board clean of mathematically unjustifiable
actions. But while that sounds obvious when we
talk about a highway, it is merely the setup.
The complete class theorem uses this baseline
of admissibility to launch its actual, far more
aggressive premise. Which brings us to the core
of the text, because once we have isolated those
admissible decisions, once we've thrown out the
back roads and kept only the strategies that
aren't mathematically self -defeating, the theorem
asks a huge question. What do all of these surviving
valid decisions have in common? And the answer
the text provides is absolute. Yeah, it really
is. It says they are all fundamentally Bayesian.
What's fascinating here is the sheer uncompromising
nature of that claim. The theorem mathematically
proves that all admissible decision rules are
equivalent to the Bayesian decision rule for,
as the text specifically words it, some utility
function and some prior distribution. Wow. There
are no exceptions in the math. every single one.
I want to really dig into those two specific
terms, utility function and prior distribution,
because they seem to form the engine of this
entire framework. I mean, if every good choice
we make is DASION, we have to understand what
we're actually calculating in our heads. Well,
let's start with the utility function. In statistics,
your utility function is the mathematical representation
of your goals, your risk tolerance, and the specific
payoffs you are chasing. So it's like a quantifiable
metric of what you value in any given scenario.
Exactly. It is what you want out of the situation.
OK. And the prior distribution. That is your
pre -existing probability map. It encompasses
the historical data, the baseline odds, and essentially
the state of the world as you understand it,
right up until the very moment you make your
choice. OK. I want to take those two mechanisms
and apply them to something way less abstract
than theoretical statistics. Sure. Let's look
at a highly relatable scenario. Say you are trying
to decide between going to a local pizza place
or a popular taco truck for dinner. And classic
dilemma. Right. If we apply the theorem here,
your utility function is essentially your craving,
maybe combined with your caloric needs and your
budget. OK. Let's say we quantify it. Your utility
for eating a great size of pizza tonight is 100
units of satisfaction. But the taco truck represents,
say, 80 units. Oh. So pizza is the clear winner
on utility. But you cannot just make the decision
based on utility alone because the outcomes are
not guaranteed. Because life happens. Right.
And that is exactly where the prior distribution
enters the calculation. Your prior distribution
is your knowledge of the environment. So you
know from past experience that the pizza place
is clear across town. At 7 p .m. there is an
80 percent chance of gridlock traffic. which
obviously degrades your utility. Sitting in traffic
is definitely negative utility. Yeah, huge penalty.
But the taco truck is just down the street, meaning
there's only a 5 % chance of a delay. Now, let's
look at the actual mechanics of the Bayesian
update here, rather than just stating the final
result. To make an admissible choice, you are
calculating the expected utility. Okay, how does
that work? You multiply the raw utility of the
pizza, those hundred units you mentioned, by
the probability of actually securing it without
that massive negative utility penalty of sitting
in traffic for an hour. Got it. So, if the traffic
penalty reduces the net utility of the pizza
to, let's say, 40 units, but the reliable taco
truck offers a solid expected utility of 75 units,
then the mathematically admissible choice is
the tacos. Even though I wanted the pizza more
in a vacuum. Exactly, because we don't live in
a vacuum. But what happens when the environment
shifts? Like, let's say you're walking out the
door fully intending to get tacos because it's
the optimal admissible choice. But you check
your phone and see a notification that a massive
water main break just flooded the street where
the taco truck usually parks. Ah. Well, that
is the very essence of a Bayesian update. Your
prior distribution has just been radically altered
by new evidence. The probability of the taco
truck being accessible just plummeted from 95
percent down to, say, 10 percent. Which fundamentally
rewrites the expected utility. Suddenly, the
previously admissible taco choice becomes mathematically
inferior. Yes. If you ignore that notification
and just to the taco truck anyway, you are no
longer operating optimally. You are choosing
an inadmissible rule. Unless of course your utility
function is highly idiosyncratic. What do you
mean? Well, if you genuinely value the physical
exercise of walking to a flooded street more
than you value actually eating food, then the
choice remains admissible for your specific utility
function. OK, that's wild to think about. But
it actually explains a lot of seemingly irrational
human behavior. When people make choices that
look totally foolish to outsiders, it is often
because their internal utility function heavily
weights variables that others simply cannot see.
Like what? Like pride or spite. or just a desire
for novelty. They are still making an admissible
choice based on their math, we just don't have
access to their numbers. So what does this all
mean? We're looking at a framework here that
leaves absolutely no room for mystical thinking.
The text's conclusion is structured as this rigid
dichotomy. It states, quote, Thus, for every
decision rule, either the rule may be reformulated
as a Bayesian procedure, or there is a rule that
is sometimes better and never worse. There is
no third option. Right. If we connect this to
the bigger picture, you begin to realize why
this theorem is so disruptive to how we view
ourselves. Because humans deeply romanticize
intuition. Oh, heavily. We love the narrative
of the brilliant executive who ignores all the
data and just goes with their gut. Or the master
chess player who makes a move based on a feeling.
We treat intuition as this magical third category
of cognition that somehow transcends logic. But
this little Wikipedia stub effectively outlaws
that third category. It traps us in a binary.
You are either performing a Bayesian calculation
or you are making an error. Exactly. It completely
demystifies intuition. If an executive makes
a gut call that ultimately succeeds and that
decision was truly optimal and admissible, the
complete class theorem dictates that they did
not bypass logic at all. They just didn't realize
they were doing math. Right. Instead... Their
brain executed a rapid subconscious Bayesian
procedure. Their quote unquote gut feeling was
actually just a highly compressed prior distribution.
It's a massive internal database built over decades
of industry experience, micro -observations,
and pattern recognition. They ran the expected
utility math instantly. They just didn't realize
they were doing algebra. Their conscious mind
experienced the output of the equation as a feeling.
Precisely. Take a professional quarterback reading
a complex defense at the line of scrimmage. They
have, what, three seconds to make a choice? Yeah,
barely that. They aren't guessing. They are accessing
thousands of hours of film study, which is their
prior distribution, and instantly calculating
the probability of a blitz against the utility
of a specific passing route. So when they throw
an interception? When they throw an interception,
it is not because their magic failed. It is because
their prior distribution was incomplete. Or they
failed to properly update their probabilities
when a linebacker shifted at the last second.
They just made an inadmissible choice. It is
a profoundly humbling way to view human intellect.
It suggests that our absolute greatest moments
of genius are just instances where our internal
mathematical models align perfectly with reality.
It really is. But, you know, accepting that every
valid choice is inherently Bayesian creates a
massive problem for statisticians. How do you
actually prove that mathematically when human
variables are infinite? Right. That is the wall
you hit. And this is where the source text introduces
its only specific academic example. Yes, the
text shifts from the abstract theorem to the
practical constraints required to actually prove
it. It brings up the work of Thomas Shelburne
Ferguson from his 1994 text, Mathematical Statistics,
a decision theoretic approach. Yeah, this stub
highlights a very specific theorem from Ferguson.
It reads, quote, if the sample space is closed
and the parameter space is finite, then the class
of Bayes rules is complete. We really need to
elevate the discussion here, because these aren't
just casual terms. No, they are incredibly strict
boundaries. Ferguson is outlining the exact conditions
required for the math to be ironclad. When he
demands a closed sample space, he means that
the set of all possible outcomes must be strictly
bounded and known in advance. So no surprises.
Exactly. You cannot have unexpected black swan
events. Right. You are essentially demanding
a ceiling on reality. You have to know every
single state of nature that could possibly occur
before you even make your decision. Yes. And
the second constraint, demanding that the parameter
space is finite, means you must be dealing with
a limited, countable number of variables. This
raises an important question about the limits
of mathematical models. Why did Ferguson have
to artificially bind the space to prove the theorem?
Why couldn't he just prove it for everything?
Because without a finite parameter space, you
run straight into the problem of computational
intractability. In an unbounded environment,
the number of prior probabilities you have to
constantly update expands exponentially. The
math just explodes. Here's where it gets really
interesting. because real life is almost never
a closed sample space with finite parameters.
Never. Operating under Ferguson's constraints
is like trying to calculate the optimal route
to work, but doing so under the assumption that
the roads will always remain exactly as they
are. Right. But in reality, the city planners
are actively adding new roads, changing the speed
limits, and physically moving your office building
while you are currently driving. The parameter
space is constantly, endlessly expanding. That
is an excellent way to conceptualize it. In a
closed finite system like a board game or maybe
a highly controlled clinical trial, the complete
class theorem holds perfectly. You can map every
single variable. Surely in chess. Exactly. But
as you transition into macroeconomics or global
politics or honestly even just navigating a messy
interpersonal conflict, you are introducing infinite
parameters and open -ended sample spaces. The
pure pristine application of the theorem becomes
impossible to execute in real time. Even if the
underlying logic remains completely sound, you
just simply cannot compute an infinite number
of variables. Which is exactly why advanced decision
theory struggles to create perfectly predictive
models for everyday human behavior. The theory
is complete, but our capacity to measure the
parameters is profoundly incomplete. Wow. So,
to summarize the intellectual journey we have
taken today, and remember, this is all based
on just a few short sentences from a stub. We
started with the concept of admissibility. learning
that the very first step of rationality is simply
eliminating choices where a strictly better alternative
exists. Sweeping the board clean. Right. And
from there, we discovered that every single optimal
valid decision that survives that filter is fundamentally
a Bayesian calculation. We are constantly weighing
our utility functions against our prior knowledge.
We learned that there is no magical third option.
You are either updating your probabilities based
on evidence to maximize your outcomes, or you
are actively leaving a better option on the table.
And we saw how statisticians like Sergusson had
to artificially freeze reality into these closed,
finite boxes just to mathematically prove the
completeness of these rules, which really highlights
the massive gap between pure math and the absolute
chaos of the real world. It is a comprehensive
framework for understanding choice. It really
changes how you see things. It does. It really
does. But before we finish this deep dive, there
is one final easily overlooked detail in the
source text that I want you, the listener, to
consider. Ah, yes. It is buried in a tiny parenthetical
aside in the very first paragraph. And it just...
beautifully reframes this entire struggle between
perfect math and imperfect reality. You're talking
about the caveat about the sequence. Yes. The
text states that an admissible rule is equivalent
to a Bayesian rule for a specific utility in
prior distribution and then it quietly adds in
parentheses or for the limit of a sequence of
prior distributions. That one phrase mathematically
acknowledges the passage of time. Exactly. By
introducing the limit of a sequence, the theorem
acknowledges that our prior knowledge is not
just some static snapshot. It is a sequence.
It's a continuous dynamic flow of data that is
constantly updating with every single micro interaction
we have with the world. And in calculus, a limit
represents a value that a sequence approaches
as the inputs approach infinity. It is a horizon
line. You can get closer and closer, but you
never actually touch it. So I really want you
to ponder what that parenthetical actually implies
for human rationality. If making the ultimate
mathematically perfect choice requires us to
chase the theoretical limit of an infinitely
updating sequence of prior assumptions, at what
point does our biology force us to fail? We are
bounded by time and cognitive capacity. We literally
cannot compute to infinity. Precisely. At what
point does our human inability to process an
infinite sequence force us into making mathematically
inadmissible choices simply because we just ran
out of time to run the Bayesian update? We have
to act eventually. Right, we have to choose a
career or buy a house or pick a partner now.
Even though the sequence of data is still flowing,
we are forced to be imperfect. Not because the
theorem is flawed, but because the clock runs
out before we can reach the limit. It suggests
that true mathematical optimality isn't a destination
we can ever actually reach. It is merely an asymptote
we continually strive toward in the dark. And
to think, we've found that profound realization
hidden in a tiny Wikipedia scub, just a handful
of sentences outlining the ultimate algorithm
of choice. Keep that horizon line in mind the
next time you are trying to make a difficult
decision. You are just a biological, Bayesian
engine doing your absolute best before the time
runs out.