### 2017 Madison Math Joke Competition – results

 Participants from left: Gina Stuessy, Ben West, Ronak Mehta, Josh Jacobson, Michael Schirle, Jen Birstler, Mark Yerrington

The 2017 Madison math Joke/vegan potluck competition (hosted by Health eFilings) was a great success. Participants competed in six categories:
1. Best pickup line
2. Best pun
3. Best easily understood joke
4. Best song/poem/artistic presentation
5. Best under proclaim rules (one person should find the joke funny, but after that as few people as possible should find it funny)
6. Best composite food + joke score
By popular demand, here are the jokes given by our participants. Note that not everyone gave me a written copy of their jokes, so not every joke verbally presented is listed in this document.

# Pun

## Ronak

A frequentist was arrested for trying to use the sample mean as a robust estimator, but got off because they had no priors!

## Ben (winner)

The case history of Mr. A Triangle, presented by the staff of Gray-Sloan Memorial Hospital
Mr. A Triangle presented to urgent care with pain in his neck. Due to the acute nature of his case, he was immediately transferred to the ER. Unfortunately, the ER had an unusually high patient volume that day, with a long line stretching out the door. Due to the obtuseness of Dr Doug Ross, the staff failed to notice parallels between his case and others.

A Triangle's neck pain was originally diagnosed as a tear in the rhomboids muscles or perhaps the trapezius, but a full patient history revealed A Triangles isosceles nature and attention shifted to the scalene and deltoid muscles.

Fortunately, Dr Meredith Gray happened to be walking by, and she noticed that A Triangles internal angles did not sum to 180°. Further investigation revealed that the patient had been exercising on an elliptical machine.

Internal angle misalignment due to elliptical surfaces is such a rare condition that it was not even recognized until its description in the mid-1800s by Dr János Bolyai. A Triangle was known for hyperbolic statements, to such an extent that some even considered him to be a lune, but eventually the staff was convinced that he needed to be transferred via medevac plane to the symmetry ward.

Dr Callie Torres was the intern in charge of patient intake on the symmetry ward that day. It was her third rotation after having received her MD and, since she specialized in triangular medicine, the third rotation was her final one. Callie was sometimes jealous of her colleagues in other specialties who had more rotations available, particularly those who could glide right through medical school, but upon reflection she felt she had made the right choice: her colleagues in the tiling specialties consistently had to deal with translators, after all.

Dr Torres brought A Triangles case history to the group, but the group was tense due to unresolved issues between Dr Torres and her former love interest Dr O'Malley. Dr Torres had told Dr O'Malley that, while their relationship was solid, it was purely platonic and, since the hospital already had five solid platonic relationships, the other staff members had trouble distinguishing this one.

The group recommended an angular bisection to return A Triangles scalene muscle to normal shape, but Dr Torres had attended a Greek school and was therefore more familiar with dissection than bisection. The patient was referred to Dr Derek Shepherd, who discovered that the disease vectors were still unknown, despite treatment guidelines requiring a known determinant. Dr Shepherd wished to discover the vector product, but as he had lost the use of his arms in an airplane accident he could not apply the right hand rule.

A Triangle was therefore sent to surgery under the supervision of Dr Maranda Bailey, who immediately determined that not a bisection but a trisection was called for. Unfortunately, upon administration of anesthesia A Triangle began to code, and only a ruler and compass were available. As trisection was impossible, Dr Bailey proceeded with successive bisection.

A Triangle was discharged two days later, with a small angular irregularity. As the difference from a true trisection was arbitrarily small, we feel that discussion of any differences is nonconstructive and irrational.

# Easily Understood

## Ben and Gina (winner)

Abbott and Costello do linear algebra
[On phone]
Izzy: Hello?
Ben: Hey Izzy!
Izzy: Hey, what’s up? You excited for the baseball season this year?
Ben: Yeah, that’s why I was calling… do you know who our coach is?
Izzy: Yeah, I think it’s Miss Gina, the math teacher?
Ben: Oh, okay. I had thought it was Mr. Gomez!
Izzy: No, I’m pretty sure it’s Miss Gina.
Ben: Okay, the math teacher. I’ll go find her because I have some questions.

Ben: Miss Gina?
Gnia: Yes Ben?
Ben: I was wondering if I could ask you about the bases?
Gina: Aye, certainly.
Ben: Could you tell me third base?
Gina: K
Gina: Look, you are in a three-dimensional space right?
Ben: Sure…
Gina: All right, then you have three basis units, and the third one is K
Ben: No, not basis, bases
Gina: Fine, you have three bases, and the third one is K
Ben: K?
Gina: Third-base!
Ben: No, like in baseball you have these different bases...
Gina: It doesn't matter whether you have a ball or something else, the bases are still the same.
Ben: But if you have a baseball diamond…
Gina: Look, you can have a ball, or a diamond, or any shape that you want, but a linear
transformation will bring the third base back to K.
Ben: Fine, then do you know first base?
Gina: I
Ben: Okay, will you tell me?
Gina: I!
Ben: Go ahead and tell me, then!
Gina: Aye, I said I!
Ben: K…
Gina: Third-base!
Ben: Oh, so can you tell me third base?
Gina: K.
Ben: Look, it seems improbable to me that you don't understand what I mean by base.
Gina: Oh, improbable. Bayes… You mean like Thomas Bayes?
Ben: Is Thomas a base?
Gina: Well, sure. Thomas Bayes.
Ben: Thomas base?
Gina: Thomas Bayes.
Ben: Oh great, Thomas base! What base is he?
Gina: I'm not sure. I assume he was at least the second Bayes after his father… It's a pretty common
name...
Ben: So he's the second base?
Gina: I mean, I'm not sure how many Bayes there have been. I would guess that he's at least the
second, but he could be the third Bayes.
Ben: So he’s third base?
Gina: Third-base?
Ben: Yeah.
Gina: K.
Ben: K?
Gina: K.
Gina: Aye
Ben: I?
Gina: First base.
Ben: First base?
Gina: I.
Ben: K…
Gina: Third-base!

## Ronak

How many Mobius strips does it take to change a lightbulb?
None, a baby doesn’t wear lightbulbs!

# Songs/Poem/artistic presentation

## Unbiased Estimation - Jen (winner)

When we met, I was on a drunkard’s walk.
With the Markov chains on my wrists, I was a slave to repeat my mistakes almost surely.
I didn’t know what to expect - my life was highly variable until you normalized these Cauchy curves.
Now, everyday I know what to expect - it’s mu.

At our first moment, we integrated and at our second moment, we integrated again.
These moments have such high value to me, I naturally log all of them in this poem to you.

I used to think I was bounded but you showed me life has no limits.
You told me I’d cross positive determinants that would help me solve this system.
I knew your predictions were credible, even prior to our conjugation.

We’ll always be each other's Bayes.
My maximum likelihood is your posterior.
It’s okay that your p-value is small, you’re still significant to me.
Honestly, though, it’s sufficiently large.

Limerick #1 - Ronak

There once was a number named pi,
Who thought fractions could only be lies,
For such is the case,
In the real metric space,
That irrationals are dense on the line!

Limerick #2 - Ronak

There once was a sequence for me,
That was about a third less than a three,
But over the limit,
Five-thirds it did summit,
For their value approached that of an e!

# Proclaim Rules

## Ben (winner)

As a young man, Abraham Lincoln worked as a rail splitter.  He was notoriously thrifty,  and therefore purchased the smallest plot of land which would allow him to split rails: a rail splitting field.

### Summary

It is possible that our universe is infinite in both time and space. We might therefore reasonably consider the following question: given some sequences $u = (u_1, u_2,\dots)$ and $u' = (u_1’, u_2’,\dots)$ (where each $u_t$ represents the welfare of persons living at time $t$), how can we tell if $u$ is morally preferable to $u’$?

It has been demonstrated that there is no “reasonable” ethical algorithm which can compare any two such sequences. Therefore, we want to look for subsets of sequences which can be compared, and (perhaps retro-justified) arguments for why these subsets are the only ones which practically matter.

Adam Jonsson has published a preprint of what seems to me to be the first legitimate such ethical system. He considers the following: suppose at any time $t$ we are choosing between a finite set of options. We have an infinite number of times in which we make a choice (giving us an infinite sequence), but at each time step we have only finitely many choices. (Formally, he considers Markov Decision Processes.) He has shown that an ethical algorithm he calls “limit-discounted utilitarianism” (LDU) can compare any two such sequences, and moreover the outcome of LDU agrees with our ethical intuitions.

This is the first time that (to my knowledge), we have some justification for thinking that a certain algorithm is all we will "practically" need when comparing infinite utility streams.

### Limit-discounted Utilitarianism (LDU)

Given $u = (u_1, u_2,\dots)$ and $u' = (u_1’, u_2’,\dots)$ it seems reasonable to say $u\geq u’$ if
$$\sum_{t = 0} ^ {\infty} (u_t - u_t’) \geq 0$$
Of course, the problem is that this series may not converge and then it’s unclear which sequence is preferable. A classic example is the choice between $(0, 1, 0, 1,\dots)$ and $(1, 0, 1, 0,\dots)$. (See the example below.)

LDU handles this by using Abel summation. Here is a rough explanation of how that works.

Intuitively, we might consider adding a discount factor $0< \delta< 1$ like this:
$$\sum_{t = 0} ^ {\infty} \delta ^ t (u_t - u_t’)$$
This modified series may converge even though the original one doesn’t. Of course, this convergence is at the cost of us caring more about people who are born earlier, which might not endear us to our children.

Therefore, we can take the limit case:
$$\liminf_{\delta\to 1 ^ -} \sum_{t = 0} ^ {\infty} \delta ^ t (u_t - u_t’)$$
This modified summand is what’s used for LDU.

LDU has a number of desirable properties, which are summarized on page 7 of this paper by Jonsson and Voorneveld. I won’t go into them much here other than to say that LDU generally extends our intuitions about what should happen in the finite case to the infinite one.

#### Example

Suppose we want to compare $u = (1, 0, 1, 0,\dots)$ and $u' = (0, 1, 0, 1,\dots)$. Let's take the standard series:
\begin{align} \sum_{i = 0} ^\infty (u_i - u_i') & = (1-0) + (0-1) + (1-0) + (0-1) +\dots\\ & = 1-1+1-1+\dots\\ & =\sum_{i = 0} ^\infty(-1) ^ i \end{align}
This is Grandi’s series, which famously does not converge under the usual definitions of convergence.

LDU though will place in a discount term $\delta$ to get:
$$\sum_{i = 0} ^\infty (-1) ^ i\delta ^ i =\sum_{i = 0} ^\infty (-\delta) ^ i$$
It is clear that this is simply a geometric series, and we can find its value using the standard formula for geometric series:
$$\sum_{i = 0} ^\infty (-\delta) ^ i = \frac {1} {1+\delta}$$
Taking the limit:
$$\liminf_{\delta\to 1 ^ -}\frac {1} {1+\delta} = 1/2$$
Therefore, the Abel sum of this series is one half, and, since $1/2 > 0$, we have determined that $(1, 0, 1, 0,\dots)$ is better than (morally preferable to) $(0, 1, 0, 1,\dots)$.

This seems kind of intuitive: as you add more and more terms, the value of the series oscillates between zero and one, so in some sense the limit of the series is one half.

### Markov Decision Processes (MDP)

Markov Decision Processes, according to Wikipedia, are:
At each time step, the process is in some state $s$, and the decision maker may choose any action $a$ that is available in state $s$.  The process responds at the next time step by randomly moving into a new state $s'$, and giving the decision maker a corresponding reward $R_a(s,s')$.
The probability that the process moves into its new state $s'$ is influenced by the chosen action.  Specifically, it is given by the state transition function $P_a(s,s')$.  Thus, the next state $s'$ depends on the current state $s$ and the decision maker's action $a$.
At each time step the decision-maker chooses between a finite number of options, which causes the universe to (probabilistically) move into one of a finite number of states, giving the decision-maker a (finite) payoff. By repeating this process an infinite number of times, we can construct a sequence $u_1, u_2,\dots$ where $u_t$ is the payoff at time $t$.

The set of all sequences generated by a decision-maker who follows a single, time independent, (i.e. stationary) policy is what is considered by Jonsson. Crucially, he shows that LDU is able to compare any two streams generated by a stationary Markov decision process. [1]

### Why This Matters

My immediate objection upon reading this paper was “of course if you limit us to only finitely many choices then the problem is soluble – the entire problem only occurs because we want to examine infinite things!”

After having thought about it more though, I think this is an important step forward, and MDPs represent an importantly large class of decision processes.

Even though the universe may be infinite in time and space, in any time interval there is plausibly only finitely many states I could be in, e.g. perhaps because there are only finitely many neurons in my brain.

(Someone who knows more about physics than I might be able to comment on a stronger argument: if locality holds, then perhaps it is a law of nature that only finitely many things can affect us within a finite time window?)

Sequences generated by MDPs are therefore plausibly the only set of sequences a decision-maker may need to practically consider.

### Outstanding Issues

My biggest outstanding concern with modeling our decisions with an MDP is that the payoffs have to remain constant. It seems likely that, as we learn more, we will discover that certain states are more or less valuable than we had previously thought. E.g. we may learn that insects are more conscious than previously expected, and therefore insect suffering affects our payoffs more highly than we had originally thought. It seems like maybe one could have a “meta-MDP” which somehow models this, but I’m not familiar enough with the area to say for sure.

A more theoretical question is: what sequences can be generated via MDPs? My hope is that one day someone will show LDU (or a similarly intuitive algorithm) can compare any two computable sequences, but I don’t think that this is that proof.

Lastly, we have the standard problems of infinitarian fanaticism and paralysis. E.g. even if our current best model of the universe predicted that MDP was exactly correct, there would still be some positive probability that it was wrong and then our “meta-decision procedure” is unclear.

### Conclusion

Overall, I don't think that this completely solves the questions with comparing infinite utility streams, but it's a large step forward. Previous algorithms like the overtaking criterion had fairly "obvious" incomparable streams, with no real justification for why those streams would not be encountered by a decision-maker. LDU is not complete, but we at least have some reason to think that it may be all we "practically" need.

I would like to thank Adam Jonsson for discussing this with me. I have done my best to represent LDU, but any errors in the above are mine. Notably, the justification for why MDP's are all we need to consider is entirely mine, and I'm not sure what Adam thinks about it.

1. This is not explicitly stated in Jonsson's paper, but it follows from the proof of theorem 1. Jonsson confirmed this in email discussions with me.