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Four agents are observing infinite streams of outcomes in ${\{S,F\}}$. None of them knows the future outcomes and as good Bayesianists they represent their beliefs about unknowns as probability distributions:

• Agent 1 believes that outcomes are i.i.d. with probability ${1/2}$ of success.
• Agent 2 believes that outcomes are i.i.d. with probability ${\theta}$ of success. She does not know ${\theta}$; She believes that ${\theta}$ is either ${2/3}$ or ${1/3}$, and attaches probability ${1/2}$ to each possibility.
• Agent 3 believes that outcomes follow a markov process: every day’s outcome equals yesterday’s outcome with probability ${3/4}$.
• Agent 4 believes that outcomes follow a markov process: every day’s outcome equals yesterday’s outcome with probability ${\theta}$. She does not know ${\theta}$; Her belief about ${\theta}$ is the uniform distribution over ${[0,1]}$.

I denote by ${\mu_1,\dots,\mu_4\in\Delta\left(\{S,F\}^\mathbb{N}\right)}$ the agents’ beliefs about future outcomes.

We have an intuition that Agents 2 and 4 are in a different situations from Agents 1 and 3, in the sense that are uncertain about some fundamental properties of the stochastic process they are facing. I will say that they have structural uncertainty’. The purpose of this post is to formalize this intuition. More explicitly, I am looking for a property of a belief ${\mu}$ over ${\Omega}$ that will distinguish between beliefs that reflect some structural uncertainty and beliefs that don’t. This property is ergodicity.

Definition 1 Let ${\zeta_0,\zeta_1,\dots}$ be a stationary process with values in some finite set ${A}$ of outcomes. The process is ergodic if for every block ${\bar a=(a_0,\dots,a_{k})}$ of outcomes it holds that

$\displaystyle \begin{array}{rcl} \lim_{n\rightarrow\infty}&\frac{\#\left\{0\le t < n|\zeta_t=a_0,\dots,\zeta_{t+k}=a_{k}\right\}}{n}=\\&\mathop{\mathbb P}(\zeta_0=a_0,\dots,\zeta_{k}=a_{k})~\text{a.s}.\end{array}$

A belief ${\mu\in \Delta(A^\mathbb{N})}$ is ergodic if it is the distribution of an ergodic process

Before I explain the definition let me write the ergodicity condition for the special case of the block ${\bar a=(a)}$ for some ${a\in A}$ (this is a block of size 1):

$\displaystyle \lim_{n\rightarrow\infty}\frac{\#\left\{0\le t < n|\zeta_t=a\right\}}{n}=\mathop{\mathbb P}(\zeta_0=a)~\text{a.s}.\ \ \ \ \ (1)$

In the right side of (1) we have the (subjective) probability that on day ${0}$ we will see the outcome ${a}$. Because of stationarity this is also the belief that we will see the outcome ${a}$ on every other day. In the left side of (1) we have no probabilities at all. What is written there is the frequency of appearances of the outcome ${a}$ in the realized sequence. This frequency is objective and has nothing to do with our beliefs. Therefore, the probabilities that a Bayesian agent with ergodic belief attaches to observing some outcome is a number that can be measured from the process: just observe it long enough and check the frequency in which this outcome appears. In a way, for ergodic processes the frequentist and subjective interpretations of probability coincide, but there are legitimate caveats to this statement, which I am not gonna delve into because my subject matter is not the meaning of probability. For my purpose it’s enough that ergodicity captures the intuition we have about the four agents I started with: Agents 1 and 3 both give probability ${1/2}$ to success in each day. This means that if they are sold a lottery ticket that gives a prize if there is a success at day, say, 172, they both price this lottery ticket the same way. However, Agent 1 is certain that in the long run the frequency of success will be ${1/2}$. Agent 2 is certain that it will be either ${2/3}$ or ${1/3}$. In fancy words, ${\mu_1}$ is ergodic and ${\mu_2}$ is not.

So, ergodic processes capture our intuition of processes without structural uncertainty’. What about situations with uncertainty ? What mathematical creature captures this uncertainty ? Agent 2’s uncertainty seems to be captured by some probability distribution over two ergodic processes — the process “i.i.d. ${2/3}$” and the process “i.i.d. ${1/3}$”. Agent 2 is uncertain which of these processes he is facing. Agent 4’s uncertainty is captured by some probability distribution over a continuum of markov (ergodic) processes. This is a general phenomena:

Theorem 2 (The ergodic decomposition theorem) Let ${\mathcal{E}}$ be the set of ergodic distributions over ${A^\mathbb{N}}$. Then for every stationary belief ${\mu\in\Delta(A^\mathbb{N})}$ there exists a unique distribution ${\lambda}$ over ${\mathcal{E}}$ such that ${\mu=\int \theta~\lambda(\text{d}\theta)}$.

The probability distribution ${\lambda}$ captures uncertainty about the structure of the process. In the case that ${\mu}$ is an ergodic processes ${\lambda}$ is degenerated and there is no structural uncertainty.

Two words of caution: First, my definition of ergodic processes is not the one you will see in textbooks. The equivalence to the textbook definition is an immediate consequence of the so called ergodic theorem, which is a generalization of the law of large numbers for ergodic processes. Second, my use of the word `uncertainty’ is not universally accepted. The term traces back at least to Frank Knight, who made the distinction between risk or “measurable uncertainty” and what is now called “Knightian uncertainty” which cannot be measured. Since Knight wrote in English and not in Mathematish I don’t know what he meant, but modern decision theorists, mesmerized by the Ellsberg Paradox, usually interpret risk as a Bayesian situation and Knightian uncertainty, or “ambiguity”, as a situation which falls outside the Bayesian paradigm. So if I understand correctly they will view the situations of these four agents mentioned above as situations of risk only without uncertainty. The way in which I use “structural uncertainty” was used in several theory papers. See this paper of Jonathan and Nabil. And this and the paper which I am advertising in these posts, about disappearance of uncertainty over time. (I am sure there are more.)

To be continued…

I will have more to say about the Stony Brook conference, but first a word about David Blackwell, who passed away last week. We game theorists know Blackwell for several seminal contributions. Blackwell’s approachability theorem is at the heart of Aumann and Maschler’s result about repeated games with incomplete information which Eilon mentions below, and also of the calibration results which I mentioned in my presentation in Stony Brook (Alas, I was too nervous and forgot to mention Blackwell as I intended too). Blackwell’s theory of comparison of experiments has been influential in the game-theoretic study of value of information, and Olivier presented a two-person game analogue for Blackwell’s theorem in his talk. Another seminal contribution of Blackwell, together with Lester Dubins, is the theorem about merging of opinions, which is the major tool in the Ehuds’ theory of Bayesian learning in repeated games. And then there are his contributions to the theory of infinite games with Borel payoffs (now known as Blackwell games) and Blackwell and Fergurson’s solution to the Big Match game.