I’m in the middle of the third book in Robert Caro’s biography of Lyndon Johnson. In brief, Caro’s thesis is that (1) Lyndon Johnson cares only about power, and (2) Lyndon Johnson is spectacularly skilled at politics. Moreover, (2) holds in a strong sense: Johnson is not simply skilled at politics, but far more skilled than nearly everyone around him. As a result, Johnson’s life is an example of asymmetric play in a theoretically symmetric game, and a beautiful illustration of how such asymmetric play is equivalent to the game itself having asymmetric rules.

I’ve used Thomas Wang’s integer hash functions for years for various purposes. Using techniques invented by Bob Jenkins for general hashing (e.g., hashes of strings), Wang derived several hash specialized for fixed size integer input. His 64-bit version is

uint64_t hash(uint64_t key) {
  key = (~key) + (key << 21); // key = (key << 21) - key - 1;
  key = key ^ (key >> 24);
  key = (key + (key << 3)) + (key << 8); // key * 265
  key = key ^ (key >> 14);
  key = (key + (key << 2)) + (key << 4); // key * 21
  key = key ^ (key >> 28);
  key = key + (key << 31);
  return key;
}

Key properties include avalanche (changing any input bit changes about half of the output bits) and invertibility. Recently I wanted to make explicit use of the inverse, in order to verify that zero would never arise as the hash of a given set of inputs. This property would allow me to initialize the hash table in question (which takes up several gigabytes) with zeros and avoid an explicit occupied bit on each entry. Thus, I needed inverse_hash(0).

Consider the following simplified version of Texas holdem, with two players Alice and Bob:

1. Alice and Bob are each dealt two private cards.

2. Alice posts a small blind of 1, Bob posts a big blind of 2.

3. Alice either folds, calls, or raises by any amount $\ge 2$.

4. Bob either calls or folds.

5. Five more shared cards are dealt, and the winner is determined as usual.

6. Both Alice and Bob have infinite stack sizes, so only expected value matters.

Consider the following poker-like game, played with two players: Alice and Bob.  Bob posts a blind of 1.  Both players are dealt a single, continuous hand chosen uniformly at random from $[0,1]$.  Alice can fold, call, or raise any amount $b \gt 0$ (calling means $b = 0$).  Bob either calls or folds.

My original plan was to work out the Nash equilibrium for this game, and therefore derive interesting smooth curves describing the optimal way for Alice to bluff and generally obscure her hand.  It didn’t quite work out that way, but the result is still interesting.