The rationale for rationality, part II.

We last spoke of rationality by looking at economic rationality and specifically looking at a solution concept in game theory, which supposedly is adopted by rational agents in order to 'solve' a game. It turned out that economic rationality lead to a rather queer outcome which many would deem irrational. Perhaps this can allow us to say that economic rationality is necessary but far from sufficient in specifying the conditions of what is meant by 'rationality.' The game we looked at, the Prisoners' Dilemma, had a much more efficient outcome than the one achieved - maybe articulating our reasoning for why that outcome is rational could help us in providing a robust account of rationality.

The aspect of rationality that I am going to introduce is epistemic rationality, which means rationality pertaining to knowledge and beliefs. Typically in economics the notion of rationality doesn't need to be expanded to include epistemic rationality because economics is largely about efficiency and maximizing payoffs. That and describing epistemic rationality in math would be difficult, at least for me and my oft misunderstanding undergraduate mind! Regardless, I think it's a crucial notion which can't be ignored, especially in game theory, which is a sub-discipline of economics.

So, epistemic rationality deals with forming good judgments/beliefs (interchangeable terms) from a body of evidence. You can think of beliefs as a function of evidence actually. We're given evidence - facts about the physical world, memories, pre-held beliefs - and then we use rules of inference to form judgments. In your everyday intermediate-college-student economic analysis, this type of rationality just isn't necessary. However, in game theory I think that this notion of rationality is necessary.

In the Prisoners' Dilemma, we've seen how decision principles meant to fulfill the purpose of economic rationality can lead to irrational outcomes. This is also apparent in other games, such as centipede games. In those games, players have the option to take either one coin from a finite sum of coins or to take two coins, which ends the game. The players alternate turns. Economic rationality would have the first player take two coins immediately and thus end the game, and the solution concept employed here is backward induction.

A two-player game (players A and B) in extensive form

Backward induction entails ex ante reasoning here (before the game starts) and it goes like this. If I'm player 1 and my only concern is garnering the greatest payoff I can (as stipulated by economic rationality), then I should see that my greatest payoff will be from choosing to take two coins at the end of the game. However, I will also see that player 2 is aware of this, and player 2 doesn't want a pay off of '2' when she could get '3' on the preceding turn by taking two coins. Player 2 will preempt me. This reasoning carries back all the way to my first move, where I should supposedly take two coins to maximize my payoff. Now, be aware the centipede games usually include 101 coins total, so the difference between the payoffs at the end and beginning of the game are much greater.

Now, everybody who is anybody knows that taking two coins on the first move is a bad idea! If this were a longer game then anyone would agree that the players would 'cooperate' until the end and get much bigger payoffs! Thus, again, we have a disparity between economic rationality and the intuitively rational strategy. This is where I think that the need for epistemic rationality becomes explicitly clear.

If it's assumed, as it often is with the centipede game or Prisoners' Dilemma, that both players are perfectly rational, then an extension of the notion of rationality to include epistemic rules of inference will lead to an entirely different outcome which is in line with our intuitions. Look at it this way, both players want to maximize their payoffs, which is something a perfectly rational agent would do according to economists, and that seems pretty right. However, both players also have a common knowledge between them. This entails that both players know that the other is perfectly rational, the other knows every detail of the game, and all of that other mutha' jazz. Remember this and take the following into consideration. It has been argued that rules of inference lead ultimately to a strong thesis (dubbed the 'Uniqueness Thesis) which says that a given "body of evidence justifies at most one proposition out of a competing set of propositions" (Feldman 2007). In other words, this thesis claims that there is at most only one rational thing to believe given a certain set of evidence. Of course, perfectly rational agents in the centipede game have the same body of evidence, and if our notion of rationality is extended to include epistemic rationality, then each player will also come to the same conclusion according to the Uniqueness Thesis. Each player, knowing the other is perfectly rational (as stipulated by the assumption of common knowledge) will know that both players will then come to believe the same thing. Knowing this, each player will know that whatever he is thinking, the other will also be thinking the same. Thus, player 1 may either believe that the best strategy is to take two coins immediately or that he should cooperate until the end. Thanks to his well-justified belief that the other player will be thinking the same as he, then he should clearly one-coin it 'til the end because that will garner of a much greater payoff. Ta-da! Taking 'rational' to include epistemic rationality also leads to an outcome in line with our intuitions. Very similar reasoning can be applied to the Prisoners' Dilemma, which will then lead to the more efficient strategy profile of (Cooperate, Cooperate).

If this reasoning sounds good, then this gives good reason to think that both payoff maximization and adherence to rules of inference are necessary conditions for being rational. The question now is if the conjunction of these is also sufficient, and that will be something best left for another post!