### circulating denominations (part 4)

… and wallet distributions.

This is part of the Toronto visit series.

“Do you have change for $5?” “I can only give you one loonie and two lizes” “What?” Dumps coins on counter. “Oh…” (Canada has no bills under$5 and circulates the $1 and$2 coins.)

Before playing with Canadian money, I had thought that a $2 denomination, whether coin or bill, would be a great idea. But the problem I encountered here was that I was just unable to get very many$1 coins when the $2 coin was also widely circulating. This makes sense, because each transaction at most ends up giving you one additional$1 coin if done optimally. But if you had to always pay odd dollar-amount fees like the $3 streetcar fares, then you need many$1 coins which you don’t have. Compare this to the US system, where you get lots of $1 bills from daily transactions — up to four$1 bills in a transaction ($0-$4 in change). It surprised me that the latter situation is more flexible, because I did not take into account the dynamic effects that repeated transactions have.

### resolving the St. Petersburg paradox

The St. Petersburg paradox is based on one of those gambling games where the usual model of using expected gain to decide whether to play the game gives a counter-intuitive result.

In the simplest of examples, you pay some entry fee to play the game, $1 is put in a pot by a counterparty, then a coin is repeatedly flipped and the pot is doubled on every coin flip by the counterparty, until “tail” comes up. You receive the money in the pot. The expected gain of this game is infinite, regardless of the initial entry fee. So it would seem that one should always play the game, regardless of the amount demanded as entry fee. But, as the article points out, “few of us would pay even$25 to enter such a game.”

### A little Markovian problem

Here it is:

A has a fair coin and B has a fair coin. They flip coins together, but only keep track of their own sequences of heads and tails. A stops if the sequence “HHT” appears. B stops if the sequence “HTH” appears. Which player is more likely to stop first?