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