Common knowledge, rationality, and reasonableness in game theory

Common knowledge, rationality and reasonableness in introductory game theory

Beliefs, knowledge, and rationality

Looking beyond beliefs, preferences, and the structure of games, we discuss the concepts of common knowledge and rationality, and its consequences for decision-making.

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Common beliefs and knowledge
Towards a solution
Rationality and sensibility

Common beliefs and knowledge

In a previous post, we introduced readers to two foundations of game theory. Firstly, beliefs and preferences are central to our perceptions and the way we behave. Secondly, many problems are strategically interdependent. That is, the payoff from your actions depends not only on your choices, but on the choices that other people take too.

So, not only is what you and others believe important, but what they believe you believe, and what you believe they believe that you believe, and so on. This is the idea behind common beliefs or common knowledge – a significant but unspoken property of social interactions and organization.

“We say that an event or fact is common knowledge among a group of people if everyone knows it, everyone knows that everyone knows it, everyone knows that everyone knows that everyone knows it, and so on.”
Michael Suk-Young Chwe. (2001). Rational Ritual: Culture, Coordination, and Common Knowledge.

Beliefs matter

Put another way, your beliefs or knowledge of someone else’s beliefs or knowledge can affect the way you behave. Teenagers, for example, adopt new fashion trends not only because they think it’s cool, but because they think others think it’s cool. And other teens adopt the new fashions because they think that others think it’s cool too.

Green-eyed logicians

In introductory courses, we typically assume that specification and rules of the game are fixed and common knowledge to all players. Real life, of course, is often murkier. Players can modify the game or abandon it altogether. While we can always add more branches to a tree, or more rows to a table, the challenge for theorists is to simplify the model for usability. This may seem a little abstract right now. But as we’ll see later, common knowledge has important consequences for decision-making and problem-solving in strategic situations. (To whet your appetite, TED-Ed’s Green-eyed Logician’s Puzzle is one, albeit extreme, example of common knowledge in action.)

Towards a solution

Previously, we also discussed how to specify strategic situations. That is, to describe the players, choices, information and payoffs in a sequential or simultaneous game format. We’re now one step closer to solving the game. To do so, we have to understand what actions players will take.

One reliable approach is to eliminate nonsensical or inferior strategies. Elimination is an attractive first step since it simplifies the problem and reveals the actions we ought to focus on. To eliminate strategies, we have to make assumptions about player behavior, and assumptions about how players think about the behavior of others. The critical assumption in game theory and much of economics, of course, is rationality.

Rationality and sensibility

Simply put, rational players select actions to maximize their own payoff or best interest, given their options, beliefs, and preferences. Remember also that people differ in their preferences and beliefs. What is rational for one individual is not necessarily rational to another. More on this later.

Common knowledge of rationality

Another important assumption is that rationality is common knowledge. If, for example, players A and B avoid inferior strategies, then they are rational. And if Player A knows that Player B avoids irrational strategies (and vice versa), and that Player A knows that Player B knows that Player A avoids irrational strategies (and vice versa), and so on, then there is common knowledge of rationality.

While this sounds contrived, it’s a common property in strategic situations. In a game of chess, players will choose moves to improve their winning chances. The players also know that their opponent will do the same. And that their opponent knows this about them as well. And it is this mix of rationality, common knowledge, and preferences (e.g., the desire to win the game), that enables players to formulate beliefs about each other’s likely behavior.

Theorists also consider the possibility when common knowledge is limited. Advanced courses in game theory, for example, discuss the role of rationalizability. We omit the nuances here for brevity and simplicity. The assumptions above should allow us to explore the most interesting parts of game theory without getting bogged down in technicalities. Keep in mind, however, that economists, mathematicians, and philosophers have a lot more to say about rationality and common knowledge, and its consequences for decision-making.

Skill and sense

Introductory courses in game theory make the implicit assumption that players are of reasonable sense and skill. That is, players are free from myopia and human error. This simplification helps us to grapple with the complexities of decision-making. It’s important to remember, however, that this is a simplifying assumption. Chess illustrates, for example, that nobody can play a perfect game. There are far too many permutations. And people are fallible. Of course, advanced topics in game theory can allow for similar concessions. You need only specify the game correctly (and have an appetite for complexity).

“How can I win the game of tick-tack-toe? It is well known that I cannot, if I assume (in keeping with the convention of game theory) that my opponent understands the game perfectly.”
Garrett, Hardin. (1968). The Tragedy of the Commons.

Rationality and reasonableness

Finally, it’s vital not to conflate rationality with reasonableness. Some people are driven by self-interest, others by altruism and a sense of community. Acting on emotions doesn’t make you irrational. If people are consumed by wrath or pride, by want or envy, we need only specify their preferences and payoffs to reflect this. Put another way, someone is rational when they pursue actions that are consistent with their worldview. You might think it unreasonable or nonsensical, but that doesn’t necessarily make it irrational.

“Two people can both be rational and yet evaluate the same outcome quite differently. People don’t just look at the dollars. They’re motivated by many things — pride, fairness, jealousy, spite, vengefulness, altruism, and charity are just a few of the possibilities. … Simply dismissing someone as irrational closes the mind. Much better is to work harder at seeing the world as the other person sees it. … In fact, if you try to impose your rationality on others, who’s the one who is really being irrational?”
Adam Brandenburger, & Barry Nalebuff. (1996). Co-opetition.