# Research Notes

## Random Unpolished Thoughts

This page is for storing half-baked ideas in one place so that I don't just forget about them. If you are reading this, don't expect coherent models or profound insights. Expect something stylistically more along the lines of a long-form twitter ramble.

Competitive Equilibrium with altruism? with one-sided altruism?

### April 3rd: Zoom Call with David

• Ordinal Vs. Cardinal Pref and Meng/Shapley etc.
• Cardinal Happiness
• Parent's understanding young kid's preferences better than the kid does.
• Utility vs Wellbeing
• Candorcet Winners

AI content curation is annoying. With a flawed system, I can learn its flaws and adapt them to suit my needs. With AI curation, the system is anti-inductive, and I can never learn how it works, so if the AI's goals don't match mine, there is nothing I can do.

How to count to 100 stacks. Fragility and stacks.
Huge variablity in externalities from production. If industry level taxes are uniform across producers, then it is possible that the average emissions will rise.? World in data protein graph.

Preferences are rankings over the entire space of allocations. Not competing rankings for each axis. Common refrain on online that animal welfare is important because of scale. Even if chickens are 1% as morally noteworthy as humans, there are tons of chickens. But I would argue that morality is not monotonically increasing in chicken numbers.I would be sad if there are no happy chickens. I would be slightly more sad if there is one chicken and it is sad. But fill the universe with happy chickens. That's horriffic. Fill the universe with happy humans, that's great.
Information. If information can be muffled, then a coarser typespace is what we're going to see in equilibrium. Delayed signals
Joe Roots's extension of Gibbersatherwates. Similar extension of Arrows?

In five cucumbers, can we find a counterexample to the greedy strategy? This is just me getting nerdsniped.

For simplicity, assume zero sum version. Two players. Number of cucumbers doesn't matter.

With one card left, lead player wins iff their card is $$\leq$$ the follower's

There are two cards left, you are the follower. You have cards X and Y, with $$X \geq Y$$. Lead player plays Z. If Z > X, you must play Y. Otherwise, you have a choice of X or Y, and you should play X because playing Y dominates playing X in the final round. Therefore, the game is solved after the lead plays in the penultimate round.

There are two cards left, you are the lead. You have cards A and B, with $$A > B$$. Opponent has cards X and Y, with $$X \geq Y$$. Now what do? If you play a card greater than X, the opponent will play Y then follow in the final round with X. Otherwise the opponent will play X and lead in the final round with Y. Let's look at all the possibilities:

• Case 1: $$A > B > X \geq Y$$. You will lose either way.
• Case 2: $$A > B = X \geq Y$$. Play A, and the opponent plays plays Y; You lead with B, opponent loses. You play
• Case 3: $$A > B X \geq Y$$
• Case 4: $$A > B X \geq Y$$
• Case 5: $$A > B X \geq Y$$

There are two tricks left. You are the lead, with cards $$A$$ and $$a$$, where $$A > a$$ (If A=a, your choice doesn't matter). Your opponent has cards $$B$$ and $$b$$. You don't know what the values of $$B$$ and $$b$$ are, but without loss of generality, have $$B \geq b$$. Your opponent will play optimally, so the following is true:

• If the card you play is strictly greater than B, the opponent will play b, you will take the trick and lead the final round. And then in the final round, your opponent will play B.
• If you played A, you win when a ≤ B.
• If you played a, you win when A ≤ B.
• Otherwise, if the card you play is less than or equal to B, the opponent will play B, take the trick, and lead the final round with b.
• If you played A, you win when a < b.
• If you played a, you win when A < b.

But if you played a and a > B, it can't be that A ≤ B. And if A < B, then you win when playing A if a < b.

So therefore A is a winning move iff (A > B AND a ≤ B) OR (A ≤ B AND a < b). This condition is equal to the union of mutually exclusive events: (a < b) OR (A > B AND b ≤ a ≤ B). We can then find the condition for when A loses by applying De Morgan's Law. NOT((a < b) OR (A > B AND b ≤ a ≤ B)) = NOT(a < b) AND NOT(A > B AND b ≤ a ≤ B) = (a ≥ b) AND (A ≤ B OR b > a OR a > B). This simplifies to ...((a ≥ b AND A ≤ B) OR (a > B))

Likewise, a is a winning move iff (a > B AND A ≤ B) OR (a ≤ B AND A < b). Note that (a > B AND A ≤ B) is impossible. Note also that (A < b) implies (a ≤ B). Thus we can simplify, and say that a is a winning move iff (A < b). And a is a losing move iff (A ≥ b)

Putting this together, we get the following results:

• Either card wins when ((a < b) OR (A > B AND b ≤ a ≤ B)) AND (A < b). This simplifies to (A < b).
• A wins, a loses: ((a < b) OR (A > B AND b ≤ a ≤ B)) AND (A ≥ b) simplifies to: (a < b AND A ≥ b) OR (A > B AND b ≤ a ≤ B)... (a < b AND B ≥ A ≥ b) OR (A > B AND a ≤ B)
• a wins, A loses: ((a ≥ b AND A ≤ B) OR (a > B)) AND (A < b)...((a ≥ b AND A < b))...FALSE
• T
• Either card loses: ((a ≥ b AND A ≤ B) OR (a > B)) AND (A ≥ b)...

Now remember, you don't know the actual values of b and B. In Game Theory terms, your information set is determined only by the values of a and A, and your choice of strategy must depend only on that information. (In an actual game, the sequence of cards already played would also determine your information set.)

Which card is better? The cases where you win either way or lose either way are irrelevant to this decisions. What matters is the values of the probabilities: A is better when a is better when Pr((a ≥ b AND A < b)) > Pr((a < b AND A ≥ b) OR (A > B AND b ≤ a ≤ B))

Can we find a case where it is better to play your low card? We would need that Pr((A > B AND a ≤ B)OR(a < b AND A = B)) < Pr().

Suppose A cant equal B and a cant equal b. Then need Pr((A > B AND a < B)) < Pr((a > b AND A < B)). Say A = 8, a = 7, and all other 7s,8s are in discard. Otherwise, dscard is even. Play the 8, So actually nvm, always better for penultimate lead to play high you are following in third to last round. what do? Lead next round will win iff (a < b) OR (A > B AND b ≤ a ≤ B). Lead will lose iff ((a ≥ b AND A ≤ B) OR (a > B)). Is there a situation where you will lose iff you lead? They just played a 7. You can beat them with 6,7,10. They have two cards, four in deck. Pool of six. That pool consists of 5,6,6,6,7,?.

### Rent Control

Here's a pickle. All the experts agree that rent control is a bad idea. Makes housing quality worse, increases time spent driving, increases housing costs in the long run etc. etc. But cities keep enacting rent control. Why?

• Maybe there is some sort of principal agent type problem where city councils are incentivized to enact bad policies which sound good to the average voter?
• Maybe its just a matter of existing house owners engaging in rent-seeking behavior to restrict the supply of housing and make their own property have inflated value?
• Maybe it's just that lawmakers only value the wellbeing of current residents and not potential residents?
• Maybe economists are all wrong, and price signals are a martian conspiracy?

Here's the thought that struck me. Maybe it's a time-consistency problem. Wherein enacting rent control has short term benefits, but long term harms. And lawmakers care more about the short term either because of political re-election type stuff, or simply because forecasting long-term effects is more difficult than hearing about the immediate problems facing the people who come into city council meetings.

In Econ 101, we show students that a binding price ceiling has an ambiguous effect on consumer surplus. In a price-control-induced shortage, we can no longer use the price mechanism to allocate the good, and so we can't say which of the buyers will actually get to buy. If the people with the highest valuation of the good end up buying it, consumer surplus might decrease. But if the people with a lower valuation on the good end up buying it, then consumer surplus goes down. Many people are hurt for the benefit of the few.

Sitting on the toilet and thinking about this, my intution is that it takes time to switch housing. So after rent control is enacted, in the short run it's the people already living there that get the houses, and people are actually better off. But then the value that people place on living somewhere changes. You could have kids being born or moving out. You could get job offers in different locations. And so if you take any set of potential residents, the distribution of the values that they place on the housing will drift over time towards the overall population's distribution.

So here's a silly little toy example to play around with this intuition:

• There are 4 houses in the city. No depreciation or construction. Four houses forever.
• There are 12 people, and each person values a house in the city at either 11 dollars per period (urbanite) or 9 dollars per period (countryfolk). There are initially 4 urbanites and 8 countryfolk, and the urbanites start with the houses.
• Each house holds exactly one person, and any people without one of these 5 houses will have to live outside the city. (In the suburbs or in Texas or whatever.)
• The value a person gets from living in the city is equal to the difference between their value and the price. The value of living outside the city is 0. Assume that the price of outside housing is exactly equal to how much poeple value it.
• Life circumstances change, and so at the end of each period, one urbanite will become countryfolk and one countryfolk will become an urbanite, at random.
• Also at the end of a period, anyone who wants to leave the city can, and then vacancies can be filled by someone moving in. If there are multiple people who want to move in, then pick one at random.

So first, let's look at what happens if the price is set to 10 dollars. Each urbanite with a house earns 1 unit of surplus, so there are 4 units of surplus total. At the end of the period, one of the urbanites becomes a countryfolk, while one of the countryfolk becomes an urbanite. Because the newly turned countrylover now is only willing to pay 9 dollars to live in the city, they move out, and then the new urbanite is the only person who wants to move in, and does so.

INSERT IMAGES HERE

And even if we start with all four houses occupied by countryfolk, they'll move out at the end of the first period, and so we will end up with all four houses occupied by the people willing to pay more. Every period, 4 total surplus.

Now lower the price to 9. In this first period, where the urbanites all live in the city, each resident is getting 2 units of surplus, for a total of 8. But at the end of the period, when one of the resident urbanites turns into a countryfolk, they will choose to stay in their house. And next period, the total surplus will be only 6. Then after that, we could either have the resident countryfolk switch back to being an urbanite, or have another one of the resident urbanites switch to being a countryfolk. In the long run, we'll only have one third of residents actually being urbanites.

TODO: Insert picture, do little simulation to find expected path of surplus over time.

Now obviously, this example doesn't actually have a price mechanism in the first place, so doesn't look at why prices get high. And the fact that rent control discourages the building of new houses is a big part of why it causes long term harm. I mean, the price controls above don't even reduce the housing supply. Also moving is costly, so there's a friction there... Thoughts for another time.