Common Knowledge of Rationality + Consistent Alignment of Beliefs = Common Priors

Heap and Varoufakis summarize the last assumption of game theory's axiomatization of rational behavior in the "consistent alignment of beliefs" axiom: no instrumentally rational person can expect another likewise rational person who has the same information to develop different thought processes.

This is usually justified by the Harsanyi doctrine: when two rational people examine the same information, they must draw the same inferences, and independently come to the same conclusion.

Robert Aumann fiercely defended this principle in his article "Agree to Disagree" (1976) and his earlier article "Subjectivity and Correlation in Randomized Strategies" (1974).

Aumann argues, if you assess it is going to rain tomorrow with 75% probability and I assess it will rain tomorrow with 33% probability, then we must have different information and we should update our probabilities accordingly until we converge on some shared probability estimate. That is, through dialogue, we (as rational actors) will arrive at a conclusion we both agree upon.

When we combine "consistent alignment of beliefs" with the common knowledge of rationality, we end up with common priors (i.e., a source of beliefs). The connection is this: if you know you are rational and you know your adversary is rational and (using consistent alignment of beliefs) you know your thoughts about what your adversary might be doing have the same origin as your thoughts about your own actions along the same line as your adversary's thoughts, THEN you adversary's actions will never surprise you. Beliefs are consistently aligned in the sense, if you actually were able to know your adversary's plans, you wouldn't want to alter your beliefs about those plans. Conversely, if your adversary knew about your planned actions, then your adversary wouldn't want to alter their beliefs they hold about your prospective actions which underpin their planning about their future actions.

Observe this dialogue needs to happen in "real" (i.e., historical) time and not in "logical time" (in the sense of the length of a logical derivation of hypothetical dialogue). Without such actual dialogue, there's no need to come to any agreement. Scott Aaronson has shown (arXiv:cs/0406061) such dialogue can be done in finite time and, in some sense, "efficiently".

One of the problems with this, the inference of common priors from the premises on the Common Knowledge of Rationality coupled to the consistent alignment of beliefs argues the dialogue occurs in "logical time".

The problem with this is for "one shot games", where interactions between the players occur only once and in the absence of communication, there is literally no opportunity for such dialogue.

Prior Beliefs

We need some "initial beliefs" for our rational actors to have, so as to avoid an infinite regress in reciprocal expectation of actions pursued. We saw how rational actors update their beliefs via Bayesian updates, but we need some "initial prior" to start the process. Without common priors, we can get senseless results.

But the choice of prior probability distributions in Bayesian analysis can impact the posterior distribution considerably. The field of "Reference Priors" uses information theory to measure how the choice of prior distribution affects the posterior probability. The choice of priors has a rich history and while it is true "objective" (or "noninformative") priors have "minimal impact" on the posterior, but that is not the same as "zero impact". Noninformative priors can lead to improper posterior, which is dangerous. How we choose a prior seems to be a hotly contested topic (does the choice of priors "matter"? What is an appropriate way to do it?) which Andrew Gelman has written extensively on.

Even if we restrict ourselves to only "stable" priors, I'm not sure this is much progress.

Revenge of the Nerds German Philosophers

One thing which the German philosophers Kant and Hegel pondered was the self-conscious reflection of human reason upon itself. Can our reasoning faculty turn on itself and, if it can, what can it infer? Phrased more relevantly, when reason knowingly encounters itself in a game, does this tell us anything about what reason should expect of itself?

Hegel's Phenomonology of Spirit (or more generally, his philosophy of Spirit) addresses this train of thought (and more). Further Hegel takes Reason reflecting on reason as it reflects on itself as part of the restlessness which drives history. Outside of history, for Hegel, there are no answers for the question of what one's reason demands of others' reason. History provides a changing set of answers.

Also worth mentioning is that game theory uses "reason" akin to Hume's usage in his famous passage We speak not strictly and philosophically when we talk of the combat of passion and reason. Reason is, and ought only to be the slave of passions, and can never pretend to any other office than to serve and obey them. Reason is a tool to help achieve the ends of subjective passions. Hegel rejoins in his lectures on the History of Philosophy, in chapter 2 on Hume in particular, In itself reason thus has no criterion whereby the antagonism between individual desires, and between itself and the desires, may be settled. Thus everything appears in the form of an irrational existence devoid of thought; the implicitly true and right is not in thought, but in the form of an instinct, a desire.

Kant's Critique of Pure Reason via his Transcendental Dialectic investigates Reason's excesses. For other Kantian repudiations of game theoretic "reason", see O'Neil's Constructions of Reason (1989), e.g., page 27 et seq.

Conclusion

So we finally have answered the question posed so long ago: beliefs are formed by taking into account common knowledge of rationality coupled to consistent alignments of beliefs. This bootstraps a rational actor's belief system by considering that actor's rational adversary's beliefs which have already solved the riddle of what is the original actor's belief system.

And if that sounds circular...that's because it is...

References

• Shaun Hargreaves Heap and Yanis Varoufakis, Game Theory: A Critical Introduction. Second ed., Routledge. (This is the axiomatization scheme I am following.)
• John Searle, Rationality in Action. MIT Press, 2001. (This provides a different set of axioms for rational behaviour, equivalent to the axioms of game theory, and discusses implicit assumptions & its flaws.)
• S. Morris, "The Common Prior Assumption in Economic Theory". Economics and Philosophy 11 (1995) 227–253. Eprint.
• John Harsanyi, "Games with Incomplete Information Played by 'Bayesian' Players: Part 1, The Basic Model". Management Science 14, 3 (1967) 159–182. Eprint
• Robert J. Aumann, "Agreeing to Disagree" (PDF). The Annals of Statistics 4, 6 (1976) 1236–1239. doi:10.1214/aos/1176343654.
• Scott Aaronson, Common Knowledge and Aumann’s Agreement Theorem [blogpost]
• Scott Aaronson, "The Complexity of Agreement". Proceedings of ACM STOC (2005) pp. 634–643, eprint arXiv:cs/0406061

Common Knowledge of Rationality

Game theorists sought a solution to initial belief formation by treating beliefs as purely subjective assessments of a situation or matter, which could degenerate into permitting almost any belief to (and thereby action from) instrumentally rationally actors. The resolution to this was another axiom: the common knowledge of rationality.

As a rational actor, it would be prudent to stipulate your adversaries are rational actors themselves. Consequently, they would also stipulate you are rational, too. This Common Knowledge of rationality turns out to be a tacit axiom of game theory.

As an example of this, in the episode "Peak Performance" in "Start Trek: The Next Generation", the android Commander Data analyses Commander Riker's strategic abilities in a memorable scene:

DATA: I have several examples of Commander Riker's battle technique. At the Academy, he calculated a sensory blind spot on a Tholian vessel and hid within it during a battle simulation. And as a lieutenant aboard the Potemkin, his solution to a crisis was to shut down all power, and hang over a planet's magnetic poles, thus confusing his opponent's sensors.

TROI: And from these specifics, what general conclusion can you extrapolate?

DATA: Only twenty-one percent of the time does he rely upon traditional tactics. So, the Captain must be prepared for unusual cunning. Counsellor, Commander Riker will assume we have made this analysis, and knowing that we know his methods, he will alter them. But, knowing that we know that he knows that we know, he might choose to return to his usual pattern.

Data would continue in this manner ad infinitum had he not been interrupted, and it is precisely what the common knowledge of rationality states. We could formally generalize this thus:

1. each person is instrumentally rational
2. each person knows (1)
3. each person knows (2)
4. each person knows (3)
5. ...and so on ad infinitum.

How does this help? By itself, it has a fundamental problem which Heap and Varoufakis illustrate in the following example.

Suppose you have a desire to be "fashionable" when deciding what clothes to wear. But this requires taking into account that other people want to be "fashionable" too. So you need to take into account what clothes they will wear, when deciding what clothes you will wear (in order to realize your desire to be fashionable).

However, other people want to be "fashionable" too, and they will select what to wear based on the expectations of what other people (including you) will wear.

So you need to account that what clothes they will wear depends on what they think you will wear, which affects what clothes you are planning to wear. But other fashionistas, knowing this, will adjust what they wear accordingly. Knowing that you know that they know you know, you now can adjust accordingly.

And so on. This process doesn't really stop, unless we add another assumption. The belief formation doesn't "settle down" without adding an assumption about common priors, which is what Bernheim's "Rationalizable Strategic Behavior" (1984) and Pearce's "Rationalizable Strategic Behavior and the Problem of Perfection" (1984) do.

[History: D. Lewis introduced the concept of "common knowledge" when analyzing a philosophical problem in the book Conventions (1969), but Robert Aumann imported the concept to economics in his 1976 paper "Agreeing to disagree" (Annals of Statistics 4 (1976) pp. 1236–1239).]

References

• Shaun Hargreaves Heap and Yanis Varoufakis, Game Theory: A Critical Introduction. Second ed., Routledge. (This is the axiomatization scheme I am following.)
• John Searle, Rationality in Action. MIT Press, 2001. (This provides a different set of axioms for rational behaviour, equivalent to the axioms of game theory, and discusses implicit assumptions & its flaws.)
• John Geanakoplos, "Common Knowledge". Journal of Economic Perspectives 6, 4 (1992) pp. 53–82
• John Geanakoplos, "Common Knowledge". Chapter 40 in Handbook of Game Theory with Economic Applications vol 2 (eds. R.J. Aumann and S. Hart), North Holland 1994, pp. 1437–1496. SemanticScholar.
• Pierre Lescanne, "Mechanizing Common Knowledge Logic using COQ". Annals of Mathematics and Artificial Intelligence 48 1-2 (2006) pp 15–43, eprint.
• Pierre Lescanne, "Common knowledge logic in a higher order proof assistant?" arXiv:0712.3147

Good Statistical Writing

(Disclaimer: these are preliminary, "stream of consciousness" thoughts. I may completely change my mind after further thought.)

When I think of good writing in journalism, I usually think about a profile in The New Yorker: a piece that is entertaining, informative, and factual.

But how to write "good statistical writing"? I have found a page from UCLA [archived] about statistical writing, but appears to be geared for presenting research to a more academic (or at least statistically competent) audience.

FiveThirtyEight doubtless is one of the biggest clearinghouse of statistical writing. But in lieu of compelling writing, their data journalism (for elections, at least) seems to amount to a cacophony of plots between scattered text. I'm not sure how I feel about their writing, but I don't want to cozy up next to the fireplace with it (unlike, say, The New Yorker's profiles).

Problem: two stories spliced into one. I suppose one problem might be that statistical writing necessitates telling two disjoint stories concurrently: the narrative, and "showing one's work" for the statistics. For this reason, it reminds me of Edward Gibbon who wrote with one style for the main text of his Decline and Fall of the Roman Empire, and another style in his footnotes where he "showed his work" (wrestling with sources, comparing differences, weighing evidence, etc.).

Would this suffice for statistical writing oriented towards the lay audience?

The amount of writing necessary to "show my work" in statistics would make the footnotes rather lengthy and daunting. I doubt it would be an adequate or even proper way to write.

I have been tempted to relegate "showing my work" to R Markdown organized in a Git repository structured in a way mimicking Jekyll's directory layout, and showing the main results in this blog. What does this mean?

The git repo has a /_posts/ subdirectory containing "posts". These are the scratch work corresponding to the posts appearing here, on PoliticalArithmetic, in R markdown.

This would separate the two stories into two texts. I'm uncertain if this solution sidesteps the entire problem, as though answering the problem How to handle statistics in writing? with the unsatisfying retort Don't.

Implicit Problem: What is good writing? This, I think, is harder to answer. But it is worth asking, if one wants to write "good writing". Like art, I don't know what defined good writing, but I know good writing when I see it.

At the same time, it seems undesirable to merely have writing coquetted with statistical jargon and numbers.

In this sense, from a literary or (in the Aristotlean usage) rhetorical perspective, statistics is a new "literary device" that is not being adequately handled as such.

"Good statistical writing" has to handle statistics as a literary device to be deployed in "good writing".

Perhaps the key to good statistical writing is the key to all writing: telling a good story. Statistics is another tool in the literary toolbox means towards that end, a teamplayer rather than a star.