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
limT→∞1TT∫0f(W(t))dt=∫f(W)P(W)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(W(t),W(t+1))=log(W(t+1)W(t))=log(W(t+1))−log(W(t)), 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
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(W(t),W(t+1))=W(t+1)−W(t)
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(W(t),W(t+1))=log(W(t+1))−log(W(t))
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(W(t),W(t+1))=log(W(t+1)W(t)), 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(W(t+1)W(t)) 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)
12f(x,2x)+12f(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 12f(−1,−2)+12f(−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
12f(x,ax)+12f(x,x−b)=kab
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 kab when x=b1−a. To see this, observe that ax=ab1−a and x−b=b1−a−b(1−a)1−a=ab1−a, so f(x,ax)=f(x,x−b)=f(b1−a,ab1−a).
For any pair a,b defining a particular additive-multiplicative bet, it must be the case that f(b1−a,ab1−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=yx,b=x−y. This is well-defined for all pairs of real numbers except where x=0. For any pair of values x≠0,y we care to choose, there is some bet parameterized by a=yx,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(y/x)if x<yf(x,y)=y−xif x>y
This gives k=12log(a)−12b 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.
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:
More precisely, it must satisfy
limT→∞1TT∫0f(W(t))dt=∫f(W)P(W)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(W(t),W(t+1))=log(W(t+1)W(t))=log(W(t+1))−log(W(t)), 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
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(W(t),W(t+1))=W(t+1)−W(t)
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(W(t),W(t+1))=log(W(t+1))−log(W(t))
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:
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(W(t),W(t+1))=log(W(t+1)W(t)), 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(W(t+1)W(t)) 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:
More generally, consider the class of bets:
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)
12f(x,2x)+12f(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 12f(−1,−2)+12f(−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
12f(x,ax)+12f(x,x−b)=kab
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 kab when x=b1−a. To see this, observe that ax=ab1−a and x−b=b1−a−b(1−a)1−a=ab1−a, so f(x,ax)=f(x,x−b)=f(b1−a,ab1−a).
For any pair a,b defining a particular additive-multiplicative bet, it must be the case that f(b1−a,ab1−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=yx,b=x−y. This is well-defined for all pairs of real numbers except where x=0. For any pair of values x≠0,y we care to choose, there is some bet parameterized by a=yx,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(y/x)if x<y f(x,y)=y−xif x>y
This gives k=12log(a)−12b 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.