I had an idea for a different way to evaluate meta-options. A meta-option behaves like a call option where the price equals the current value of the equity and the strike price equals the cash salary you'd be able to get instead.[1]

If I compare an equity package worth $100K per year versus a counterfactual cash salary of $80K and assume a volatility of 70% (my research suggests that small companies have a volatility around 70–100%), the call option for the equity that vests in the first year is worth $38K, and the call option for the equity that vests in the 4th year is worth $62K (which is equivalent to a 13% annual return). So on average, a meta-option on a 4-year equity package is worth somewhere in the ballpark of a 20% annual return.

[1] This is kind of wrong because with a normal stock option you don't have to pay the strike until you exercise, but with an employee meta-option, you have to give up your counterfactual salary as soon as you start working, and you don't vest for the first year so you have to give up a full year of cash salary no matter what. If you have monthly vesting, the fact that you have to pay at the beginning of the month instead of the end doesn't matter much.

I disagree-voted to indicate that I did not donate my mana because of this post (I use Manifold sometimes but I have only a trivial amount of mana)

I feel your pain. I hope the amount of upvotes and hearts you're getting helps you feel better, but I know brains don't always work that way (mine doesn't).

I believe this sort of thing doesn't get much attention from EAs because there's not really a strong case for it being a global priority in the same way that existential risk from AI is.

It's really hard to judge whether a life is net positive. I'm not even sure when my own life is net positive—sometimes if I'm going through a difficult moment, as a mental exercise I ask myself, "if the rest of my life felt exactly like this, would I want to keep living?" And it's genuinely pretty hard to tell. Sometimes it's obvious, like right at this moment my life is definitely net positive, but when I'm feeling bad, it's hard to say where the threshold is. If I can't even identify the threshold for myself, I doubt I can identify it in farm animals.

If I had to guess, I'd say the threshold is something like

• if the animals spend most of their time outdoors, their lives are net positive
• if they spend most of their time indoors (in crowded factory farm conditions, even if "free range"), their lives are net negative

it seems important for my own decision making and for standing on solid ground while talking with others about animal suffering.

To this point, I think the most important things are

1. whatever the threshold is, factory-farmed animals clearly don't meet it
2. 99% of animals people eat are factory-farmed (in spite of people's insistence that they only eat meat from their uncle's farm where all of the animals are treated like their own children etc)

The use of quantitative impact estimates by EAs can mislead audiences into overestimating the quality of quantitative empirical evidence supporting these estimates.

In my experience, this is not a winnable battle. Regardless of how many times you repeat that your quantitative estimates are based on limited evidence / embed a lot of assumptions / have high margins of error / etc., people will say you're taking your estimates too seriously.

Thanks for linking your paper! I'll check it out. It sounds pretty good from the abstract.

Do you have some estimate of the cost-effectiveness of helping slaughterhouse workers as compared to, say, cage-free campaigns?

I came up with a few problems that pose challenges for ergodicity economics (EE).

First I need to explicitly define what we're talking about. I will take the definition of the ergodic property as Peters defines it:

The expected value of the observable is a constant (independent of time), and the finite-time average of the observable converges to this constant with probability one as the averaging time tends to infinity.

More precisely, it must satisfy

$\underset{T\to \infty }{lim}\frac{1}{T}\underset{0}{\overset{T}{\int }}f\left(W\right(t\left)\right)dt=\int f\left(W\right)P\left(W\right)dW$

where W(t) is wealth at time t, f(W) is a transformation function that produces the "observable", and the second integral is taking the expected value of wealth across all bet outcomes at a single point in time. (Peters specifically talked about wealth, but W(t) could be a function describing anything we care about.)

According to EE, a rational agent ought to maximize the expected value of some observable f(W) such that that observable satisfies the ergodic property.

Problem of choosing a transformation function

According to EE, making decisions in a non-ergodic system requires applying a transformation function to make it ergodic. For example, if given a series of bets with multiplicative payout, those bets are non-ergodic, but you can transform them with $f\left(W\right(t),W(t+1\left)\right)=\log \left(\frac{W(t+1)}{W\left(t\right)}\right)=\log \left(W\right(t+1\left)\right)-\log \left(W\right(t\left)\right)$, and the output of $f$ now satisfies the ergodic property.

(Peters seems confused here because in The ergodicity problem in economics he defines the transformation function f as a single-variable function, but in Evaluating gambles using dynamics, he uses a two-dimensional function of wealth at two adjacent time steps. I will continue to follow his second construction where f is a function of two variables, but it appears his definition of ergodicity is under-specified or possibly contradictory.)

The problem: There are infinitely many transformation functions that satisfy the ergodic property.

The function "f(x) = 0 for all x" is ergodic: its EV is constant wrt time (because the EV is 0), and the finite-time average converges to the EV (b/c the finite-time average is 0). There is nothing in EE that says f(x) = 0 is not a good function to optimize over, and EE has no way of saying that (eg) maximizing geometric growth rate is better than maximizing f(x) = 0.

Obviously there are infinitely many constant functions with the ergodic property. You can also always construct an ergodic piecewise function for any given bet ("if the bet outcome is X, the payoff is A; if the bet outcome is Y, the payoff is B; ...")

Peters does specifically claim that

1. A rational agent faced with an additive bet (e.g.: 50% chance of winning $2, 50% chance of losing $1) ought to maximize the expected value of $f\left(W\right(t),W(t+1\left)\right)=W(t+1)-W\left(t\right)$
2. A rational agent faced with a multiplicative bet (e.g.: 50% chance of a 10% return, 50% chance of a –5% return) ought to maximize the expected value of $f\left(W\right(t),W(t+1\left)\right)=\log \left(W\right(t+1\left)\right)-\log \left(W\right(t\left)\right)$

These assumptions are not directly entailed by the foundations of EE, but I will take them as given. They're certainly more reasonable than f(x) = 0.

Problem of incomparable bets

Consider two bets:

Bet A: 50% chance of winning $2, 50% chance of losing $1

Bet B: 99% chance of 100x'ing your money, 1% chance of losing 0.0001% of your money

EE cannot say which of these bets is better. It doesn't evaluate them using the same units: Bet A is evaluated in dollars, Bet B is evaluated in growth rate. I claim Bet B is clearly better.

There is no transformation function that satisfies Peters' requirement of maximizing geometric growth rate for multiplicative bets (Bet B) while also being ergodic for additive bets (bet A). Maximizing growth rate specifically requires using the exact function $f\left(W\right(t),W(t+1\left)\right)=\log \left(\frac{W(t+1)}{W\left(t\right)}\right)$, which does not satisfy ergodicity for additive bets (expected value is not constant wrt t).

In fact, multiplicative bets cannot be compared to any other type of bet, because $\log \left(\frac{W(t+1)}{W\left(t\right)}\right)$ is only ergodic when W(t) grows at a constant long-run exponential rate.

More generally, I believe any two bets are incomparable if they require different transformation functions to produce ergodicity, although I haven't proven this.

Problem of risk

This is relevant to Paul Samuelson's article that I linked earlier. EE presumes that all rational agents have identical appetite for risk. For example, in a multiplicative bet, EE says all agents must bet to maximize expected log wealth, regardless of their personal risk tolerance. This defies common sense—surely some people should take on more risk and others should take on less risk? Standard finance theory says that people should change their allocation to stocks vs. bonds based on their risk tolerance; EE says everyone in the world should have the same stock/bond allocation.

Problem of multiplicative-additive bets

Consider a bet:

Bet A: 50% chance of doubling your money, 50% chance of losing $1

More generally, consider the class of bets:

50% chance of multiplying your money by a, 50% chance of losing b dollars

Call these multiplicative-additive bets.

EE does not allow for the existence of any non-constant evaluation function for multiplicative-additive bets. In other words, EE has no way to evaluate these bets.

Proof.

Consider bet A above. By the first clause of the ergodic property, the transformation must satisfy (for some wealth value $x$)

$\frac{1}{2}f(x,2x)+\frac{1}{2}f(x,x-1)=k$

for some constant $k$. This equation says $f$ must have a constant expected value.

Now consider what happens at x = -1. There we have $\frac{1}{2}f(-1,-2)+\frac{1}{2}f(-1,-2)=k$ and therefore $f(-1,-2)=k$.

That is, f(-1, -2) must equal the expected value of f for any x.

we can generalize this to all multiplicative-additive bets to show that the transformation function must be a constant function.

Consider the class of all multiplicative-additive bets. The transformation function must satisfy

$\frac{1}{2}f(x,ax)+\frac{1}{2}f(x,x-b)=k_{ab}$

for some constants a, b which define the bet (in Bet A, a = 2 and b = 1). (Note: It is not required that a and be be positive.)

The transformation function $f(x,y)$ must equal $k_{ab}$ when $x=\frac{b}{1-a}$. To see this, observe that $ax=\frac{ab}{1-a}$ and $x-b=\frac{b}{1-a}-\frac{b(1-a)}{1-a}=\frac{ab}{1-a}$, so $f(x,ax)=f(x,x-b)=f(\frac{b}{1-a},\frac{ab}{1-a})$.

For any pair $a,b$ defining a particular additive-multiplicative bet, it must be the case that $f(\frac{b}{1-a},\frac{ab}{1-a})$ is a constant (and, specifically, it equals the expected value of the transformation function with parameters $a,b$).

Next I will show that, for (almost) any $x,y$ pair in $f(x,y)$, there exists some pair $a,b$ that forces $f(x,y)$ to be a constant.

Solving for $a,b$ in terms of $x,y$, we get $a=\frac{y}{x},b=x-y$. This is well-defined for all pairs of real numbers except where $x=0$. For any pair of values $x\ne 0,y$ we care to choose, there is some bet parameterized by $a=\frac{y}{x},b=x-y$ such that $f(x,y)$ is a constant.

Therefore, if there is a function $f(x,y)$ that is ergodic for all additive-multiplicative bets, then that function must be constant everywhere (except on the line $x=0$, i.e., where your starting wealth is 0, which isn't relevant to this model anyway). A constant-everywhere function says that every multiplicative-additive bet is equally good.

EDIT: I thought about this some more and I think there's a way to define a reasonable ergodic function over a subset of multiplicative-additive bets, namely where a > 1 and b > 0. Let

$f(x,y)=\log \left(y/x\right)\text{if}x<y$ $f(x,y)=y-x\text{if}x>y$

This gives $k=\frac{1}{2}\log \left(a\right)-\frac{1}{2}b$ which is the average value, and I think it's also the long-run expected value but I'm not sure about the math, this function is impossible to define using the single-variable definition of the ergodic property so I'm not sure what to do with that.