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

Criticisms of Instrumental Rationality

As a follow up to the post Agents are Instrumentally Rational, I thought it would be good to discuss a number of criticisms to instrumental rationality. It's healthy to do so, since political actors are not cold, calculating automatons. In elections, voters do not vote rationally. Politicians may or may not behave rationally. If political actors are (gasp) human, perhaps examining the flaws of rationality will illuminate aspects to better model scenarios. And unlike economists, we are trying to fit theory to reality.

There is a growing literature testing predictions "instrumental rationality" makes. Alarmingly, the vast majority of this literature finds actors are not instrumentally rational.

The basic problem seems to be, expected utility models actors as computers trying to optimize some quantity. But the human brain is more like a sophisticated "pattern recognition machine", and there are builtin "short circuits" to avoid heavy computations but tend to produce false-positives. (This is useful for doing things which do not need heavy computations; it is a feature, sometimes a bug.) This is the pioneering work of Daniel Kahneman and Amos Tversky. There is a wonderful book Thinking: Fast and Slow, by Kahneman himself, summarizing the research.

There are two points of particular concern I'll mention here. In another post, I will go on a philosophical spelunking on where this particular notion of "rationality" comes from (Hume) and the criticisms it has faced in the past.

Allais Paradox

Lets try testing the framework of preferences over prospects, specifically the independence axiom. You have to make one choice between two alternative lotteries:

Lottery 1:
Win $2500 with probability 33%
Win $2400 with probability 66%
Win nothing with 1% probability

Lottery 2: win $2400 with certainty.

Once you made that choice, you need to choose between two more lotteries:

Lottery 3:
Win $2500 with probability 33%
Win nothing with probability 67%

Lottery 4:
Win $2400 with probability 34%
Win nothing with probability 66%

Which would you prefer? What does expected utility suggest?

The expected winnings from Lottery 1 would be $2409, whereas the expected winnings from Lottery 2 is $2400. The instrumentally rational individual would pick Lottery 1, even though most people empirically choose Lottery 2.

Similarly, the expected winnings from lottery 3 is $825, whereas the expected winnings for lottery 4 is $816. But again, people choose lottery 4 over lottery 3.

The Allais Paradox is the observation that, in experiments, volunteer behavior directly contradicts rational behavior. Sugden's Rational Choice: A Survey of Contributions from Economics and Philosophy reviews the literature on this topic.

The conflict is with the axiom of independence. If we use an alternative axiomatization for rational behavior ("Savage's axioms"), the Allais paradox contradicts the "sure-thing principle".

Source of Beliefs

So, how does a rational actor "acquire" beliefs? (An instructive exercise for the reader, harking back to Socrates, is to consider how the reader "acquires" beliefs.)

For some political actors, it doesn't really matter. I'm pretty certain Senator Ted Cruz has beliefs on almost everything, and as far as how he acquired them, well, it doesn't matter.

For other types of political actors, like your "everyday voters", it does matter. Voter belief is actually a hotly debated topic: to what degree voters are "rational actors", where they acquire their party identification or how they develop affinity for a candidate, these are all hot topics.

There is universal agreement in the literature rational agents update their beliefs using Bayes inference. We should recall from probability that Bayes' theorem is a generalization of contrapositive in logic. Heuristically, what happens is we have some mathematical model of the world using random variables and parameters (denoted θ in the literature, possibly a vector of parameters). Some event E occurs, and we use adjust our model's parameters based on the event occurring.

[I don't have the space to describe the details (though I should in some future blog post), the interested reader is encouraged to read John Kruschke's Doing Bayesian Data Analysis for details.]

But where do the initial estimates for prior distributions come from? Where do the "initial beliefs" emerge? There are two answers to this query.

First answer: from the search for information itself. People do not sit around waiting for "information" to fall into their laps. No! Rational actors must actively pursue information. The original "prejudices" (initial beliefs) are adjusted as more information is actively obtained.

When does a rational actor cease seeking information? Economists answer, with either a sigh or a smile, the rational actor will stop when the utility of the information gained equals the cost of the search for that information (with the cost evaluated in utility terms). As long as there is more utility gleaned from seeking than it costs to seek, a rational actor will keep searching. This is a rather cute, self-consistent solution.

But it begs the question: how does an actor know how to evaluate the utility of the new information prior to obtaining it? Perhaps our actor has formulated expectations about the value of additional information. How did our actor acquire that expectation of the value of information?

A waggish defender might say, "By acquiring information about the value of information up to the point where the marginal benefits of this (second-order) information were equal to the costs." This solution really degenerates into an infinite regress, since we can ask the same question of how the actor knows value of this second-order information.

There are two ways to stop the infinite regress:

1. Something additional is needed. This concedes the instrumental rationality paradigm is incomplete.
2. The only alternate would be to assume that the individual knows the benefits the actor can expect from a little more search because the actor knows the full information set. But then there is no problem: the actor knows everything already!

I wonder if we might not be more generous with the waggish defender, and try to bootstrap some interpolated polynomial from first-order, second-order, ..., N-order costs of information? My intuition suggests the answer to be "In general, no; only for a few special edge cases can this bootstrap occur coherently."

A last remark: this discussion reminds me of Meno's paradox (not to be confused with Zeno's paradox). In Plato's dialogue, Meno, Meno asks Socrates And how will you inquire into a thing when you are wholly ignorant of what it is? Even if you happen to bump right into it, how will you know it is the thing you didn't know? [80d1-4] Socrates reformulates it thus [A] man cannot search either for what he knows or for what he does not know[.] He cannot search for what he knows—since he knows it, there is no need to search—nor for what he does not know, for he does not know what to look for. [80e] Or, phrased more relevantly for our discussion, how can a rational actor actively pursue information about a matter which the actor is completely ignorant of?

I'm sure the rejoinder would be, "Rational actors seek information precisely how Socrates sought answers to questions on matters he professed ignorance about." But it still dodges the question.

Second answer: Beliefs as purely subjective assessments. This is following Savage's The foundations of statistics (1954), where beliefs are purely subjective assessments. They are what they are, and only revealed ex post by the choices people make.

This avoids a lot of the problems plaguing the first answer. Unfortunately, we have some experimental evidence casting doubt on the consistency of such subjective assessments and more generally on the probabilistic representations of uncertainty; most famous of which is the Ellsberg paradox.

Game theory has been pursuing the line of reasoning Savage provides. But then this may license any kind of action, rendering instrumental rationality nearly vacuous. Game theorists have sought to prevent this "purely subjective assessments" turning against itself [i.e., letting "anything" be a solution to describe rational behavior] by supplementing instrumental rationality with the assumption of the common knowledge of rationality. This leads to weak solutions to game theoretic problems apparently called Rationalizability, not to be confused with the psychological mechanism of "Rationalizing" (i.e., lying to one's self to feel better).

References

• Shaun Hargreaves Heap and Yanis Varoufakis, Game Theory: A Critical Introduction. Second ed., Routledge.
• John Searle, Rationality in Action. MIT Press, 2001.

Savage Axioms

• L. J. Savage, The foundations of statistics. John Wiley & Sons Inc., New York (1954).
• Judea Pearl, The Sure-Thing Principle. UCLA Cognitive Systems Laboratory, Technical Report R-466 (2016).
• Dov Samet, "The sure-thing principle and independence of irrelevant knowledge". Israel Institute of Business Research, 2008.
• Edi Karni, Savages’ Subjective Expected Utility Model. Lecture notes, 2005.

Animals and Rational Behaviour

Having introduced a notion of Instrumental Rationality, perhaps a good question to ask is "What is an example of a rational agent which is not human?"

It might seem that animals behave rationally at times; for example, worker bees give up their own production in favor of other offspring of the queen, surely this must give some benefit to the donor. This idea (generalizing the bee situation) is known as Hamilton's Rule, and it doesn't really work. For example, it is neither testable nor does it make predictions.

With the bees, the altruist (i.e., worker bee) cooperates by giving a benefit b to the recipient (another offspring) at a cost c to itself. Both b and c are measured in terms of fitness, specifically the expected number of offspring. Naively one might expect b > c to suffice, but Hamilton's major insight was that relatedness ("degree of kinship") r between donor and recipient must enter into the equation, giving us Hamilton's rule br > c.

Game theorists are overjoyed to hear this can be derived from utility maximization, and one might expect it to have a status similar to Newton's laws in physics. However, Nowak, Tarnita, and Wilson have argued that Hamilton's rule almost never holds. In short, simple game theoretic models here fail to describe the biological situation.

Decades ago, evolutionary biologists would have treated Hamilton's rule as an "iron law". That no longer seems to be the case. For more on this, see de Vladar and Szathmáry's "Beyond Hamilton's Rule".

Agents are Instrumentally Rational

Game Plan: We'll introduce the notion of "instrumental rationality" as an ordering of alternatives with some technical condition. Then we'll discuss measures of "preference" via utility functions. Then we conclude by discussing maximizing utility under uncertainty.

Loosely put, individuals who are instrumentally rational have preferences over various "things" (e.g., baby-back ribs are preferred to chicken, and chicken is preferred to bread). Such individuals are deemed "rational" for picking actions which satisfy those preferences. The only constraint is that preferences are ordered in some suitably "weakly coherent" way (e.g., ribs are still preferred to bread if there is no chicken).

The convention is to call these "things" preferred as "Alternatives".

Definition 1. Let an actor be choosing between countably many possible different alternatives x₁, x₂, x₃, …. An actor is called Instrumentally Rational if the actor has preferences satisfying the following conditions:

1. Reflexivity: No alternative x_i is less preferred than itself.
2. Completeness: For any two alternatives x_i and x_j, either (1) x_i is strictly preferred over x_j, (2) x_j is strictly preferred over x_i, or (3) the actor is indifferent between the two alternatives.
3. Transitivity: For any alternatives x_i, x_j, x_k, if x_i is no less desired than x_j, and if x_j is no less desired than x_k, then x_i is no less desired than x_k.
4. Continuity: For any alternatives x_i, x_j, x_k, if x_i is (strictly) preferred to x_j, and if x_j is (strictly) preferred to x_k, then there exists some "composite" of x_i and x_k (call it y) which is equally as desired as x_j.

Remark 1 (On Continuity). There are two ways to interpret the continuity axiom. The first perspective is to think of y as a "basket" containing "bits" of x_i and "bits" of x_k. For example, if x_i is "18 ribs", x_j is "half a roasted chicken", and x_k is "10 rolls", then there is some composite ("9 ribs and 5 rolls") which is equally as desirable as half a chicken.

The other perspective is to think of y as a lottery, where the actor obtains x_i with probability p (0 < p < 1) and x_k with probability 1 − p. The continuity axiom then says there is some p for which the actor is indifferent between the lottery y and the alternative x_j.

Remark 2 (Ordering, Utility Functions). The first three axioms taken together implies the actor has a well-defined preference ordering (in the mathematical sense). When the continuity axiom is added, the preference ordering may be "represented" by a utility function (i.e., a function assigning to each alternative x_i some real number U(x_i) reflecting the "utility" or "desire" for that alternative). An actor making choices to satisfy his or her preference ordering can be viewed "as if" maximizing his or her utility function.

Now, discussions of "utility" of an alternative should not be confused with the philosophy of Utilitarianism. A utility function just assigns some numbers such that the ordering induced by it is the same as the actor's preference relation. That is to say, U(x_i) > U(x_j) if and only if x_i is strictly preferred to x_j. The numbers represented by U(x_i) are measured in utils, which is Agent-dependent and measures that Agent's preference for the given alternative.

Ordinal Utilities, Cardinal Utilities, Maximizing Expected Utility

Definition 2. If we assign utility "arbitrarily" but in a manner consistent with the preference ordering (e.g., for any alternatives X and Y such that X is preferred to Y, we assign the utilities such that U(X) > U(Y) but otherwise the quantities remain arbitrary), then we call such utility the Ordinal Utility.

Here we must stress again there are two important points of assigning utility in a manner which captures only the preference ordering (and nothing else).

First, ordinal utility does not describe the agent's "intensity of desire" for an alternative. The "strength of preference" is not captured by this notion. So how much more I want ribs than chicken is not adequately described by this notion, just the fact that I really want ribs right now (and not chicken, much less bread).

Second, ordinal utility cannot be compared "across agents". The ordinal utility I assign to a full rack of baby-back ribs cannot be compared to anyone else's ordinal utility for, say, Lasagna. We can only compare my ordinal utility for baby-back ribs against my ordinal utility for Lasagna.

Dealing with Uncertainty

My local BBQ joint smokes 1 pig per day, and when it's all sold, there's no more. If I am hungry, should I go before the lunch rush or afterwards?

Here we must talk of Prospects, outcomes and their associated probabilities.

For our particular situation, there is a decision I must make (go before the lunch rush or after) and two outcomes (there is food left, or they ran out of food). One prospect is given by the possible outcomes to a given choice of going before the lunch rush (go before lunch AND they have food, go before lunch AND no more food; p, 1 − p). The other prospect is given by the decision to go after the lunch rush (go after lunch AND they have food, go after lunch AND no more food; q, 1 − q).

Observe, each decision has different possible outcomes, but the probabilities for the outcomes on a given decision must sum to 100%: something must happen when I take a decision.

We now need to consider preference ordering over prospects.

Definition 3. Suppose a person must choose between actions with uncertain outcomes, in the sense that: each action has various possible outcomes associated with it, each with some probability. We call this action a Prospect and represent it by a pairing of the possible outcomes with their respective probabilities (y₁, y₂, ...; p₁, p₂, ...) where the outcome y_i occurs with probability p_i, and the probabilities sum to 1 = p₁ + p₂ + ... (since there must be an outcome to the action).

Remark. There is a "nested structure" to prospects, in the sense that y_i might be an "atomic outcome" (e.g., "there will be food", "there will be no food", "it will rain", "the world will end", etc.) or another prospect (imagine "I flip a coin; if it is heads, then I do this action, but if it is tails then I do some other action").

An actor's Preferences over Prospects are called Consistent if the preference satisfies axioms (1), (2), and (3) of Definition 1, and:

4. Continuity: Consider three prospects y_i, y_j, and y_k, and suppose the first is preferred to the second and the second is preferred to the third. Then there exists some probability p such that the prospect (y_i, y_k; p, 1 − p) is equally as preferable to y_j (compare to the second interpretation forwarded in Remark 1).
5. Preference increasing with probability: If y_i is preferred to y_j, letting y_m = (y_i, y_j; p₁, 1 − p₁) and y_n = (y_i, y_j; p₂, 1 − p₂), then y_m is preferred to y_n only if p₁ > p₂.
6. Independence: For any three prospects y_i, y_j, and y_k, if y_i is preferred to y_j, then there exists a probability p such that the prospect (y_i, y_j; p, 1 − p) is no less desired than (y_i, y_k; p, 1 − p)

Given this notion of "consistent preferences over uncertain prospects", how can we develop a notion of instrumental rationality?

Maximizing Expected Utility

The first step is to introduce the notion of Cardinal Utility, which assigns to a given outcome y_i the intensity for an agent's preference for that outcome u(y_i).

The second step is to consider the Expected Utility of a Prospect y = (y₁, y₂, ...; p₁, p₂, ...) as the sum E_u[y] = u(y₁)p₁ + u(y₂)p₂ + ..., which is the expected value of the "random variable".

Now, an agent with cardinal utility u(-) is considered instrumentally rational if it picks the action whose prospect has the maximum expected utility.

Example 1. If I go to my favorite BBQ restaurant before the lunch rush, the prospect looks like ("get food", "no food"; 0.95, 0.05). If I leave after the lunch rush, the prospect looks like ("get food", "no food"; 0.1, 0.9).

My cardinal utility function looks like u(get food) = 10, u(no food) = −30.

The expected utility for going before the lunch rush is then

E[before]	=	u(get food)×0.95 + u(no food)×0.05
	=	10×0.95 − 30×0.05
	=	9.5 − 1.5
	=	8

The expected utility for going after the lunch rush is then

E[after]	=	u(get food)×0.1 + u(no food)×0.9
	=	10×0.1 − 30×0.9
	=	1 − 27
	=	−26

Since 8 > −26, it is rational to go before the lunch rush to try to get food.

Next time, we'll discuss flaws with this notion of instrumental rationality, both logical and empirical.

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.)