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?
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.
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:
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:
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:
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,?.
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?
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:
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.