# riceissa comments on Meetup Notes: Ole Peters on ergodicity

• (I’ve only spent several hours thinking about this, so I’m not confident in what I say below. I think Ole Peters is saying something interesting, although he might not be phrasing things in the best way.)

Time-average wealth maximization and utility=log(wealth) give the same answers for multiplicative dynamics, but for additive dynamics they can prescribe different strategies. For example, consider a game where the player starts out with $30, and a coin is flipped. If heads, the player gains$15, and if tails, the player loses $11. This is an additive process since the winnings are added to the total wealth, rather than calculated as a percentage of the player’s wealth (as in the 1.5x/​0.6x game). Time-average wealth maximization asks whether , and takes the bet. The agent with utility=log(wealth) asks whether , and refuses the bet. What happens when this game is repeatedly played? That depends on what happens when a player reaches negative wealth. If debt is allowed, the time-average wealth maximizer racks up a lot of money in almost all worlds, whereas the utility=log(wealth) agent stays at$30 because it refuses the bet each time. If debt is not allowed, and instead the player “dies” or is refused the game once they hit negative wealth, then with probability at least 18, the time-average wealth maximizer dies (if it gets tails on the first three tosses), but when it doesn’t manage to die, it still racks up a lot of money.

In a world where this was the “game of life”, the utility=log(wealth) organisms would soon be out-competed by the time-average wealth maximizers that happened to survive the early rounds. So the organisms that tend to evolve in this environment will have utility linear in wealth.

So I understand Ole Peters to be saying that time-average wealth maximization adapts to the game being played, in the sense that organisms which follow its prescriptions will tend to out-compete other kinds of organisms.