In
economics
Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and intera ...
and
game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
, a participant is considered to have superrationality (or renormalized rationality) if they have
perfect rationality
The term ''Homo economicus'', or economic man, is the portrayal of humans as agents who are consistently rational and narrowly self-interested, and who pursue their subjectively defined ends optimally. It is a word play on ''Homo sapiens'', u ...
(and thus maximize their
utility
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosopher ...
) but assume that all other players are superrational too and that a superrational individual will always come up with the same strategy as any other superrational thinker when facing the same problem. Applying this definition, a superrational player playing against a superrational opponent in a
prisoner's dilemma
The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...
will cooperate while a rationally self-interested player would defect.
This
decision rule
In decision theory, a decision rule is a function which maps an observation to an appropriate action. Decision rules play an important role in the theory of statistics and economics, and are closely related to the concept of a strategy (game theory ...
is not a mainstream model within the
game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
and was suggested by
Douglas Hofstadter
Douglas Richard Hofstadter (born February 15, 1945) is an American scholar of cognitive science, physics, and comparative literature whose research includes concepts such as the sense of self in relation to the external world, consciousness, an ...
in his article, series, and book ''
Metamagical Themas
''Metamagical Themas'' is an eclectic collection of articles that Douglas Hofstadter wrote for the popular science magazine ''Scientific American'' during the early 1980s. The anthology was published in 1985 by Basic Books.
The volume is subst ...
''
[ – reprinted in: ] as an alternative type of rational
decision making
In psychology, decision-making (also spelled decision making and decisionmaking) is regarded as the cognitive process resulting in the selection of a belief or a course of action among several possible alternative options. It could be either rati ...
different from the widely accepted
game-theoretic
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applic ...
one. Superrationality is a form of
Immanuel Kant
Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher and one of the central Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works in epistemology, metaphysics, ethics, and ...
's
categorical imperative
The categorical imperative (german: kategorischer Imperativ) is the central philosophical concept in the deontological moral philosophy of Immanuel Kant. Introduced in Kant's 1785 ''Groundwork of the Metaphysic of Morals'', it is a way of evalu ...
,
and is closely related to the concept of
Kantian equilibrium proposed by the economist and
analytic Marxist John Roemer
John E. Roemer (; born February 1, 1945 in Washington, D.C., to Ruth Roemer and Milton Roemer, namesake of Roemer's law) is an American economist and political scientist. He is the Elizabeth S. and A. Varick Stout Professor of Political Scienc ...
.
Hofstadter provided this definition: "Superrational thinkers, by recursive definition, include in their calculations the fact that they are in a group of superrational thinkers."
This is equivalent to reasoning as if everyone in the group obeys Kant's categorical imperative: "one should take those actions and only those actions that one would advocate all others take as well."
Unlike the supposed "
reciprocating human", the superrational thinker will not always play the equilibrium that maximizes the total social utility and is thus not a
philanthropist
Philanthropy is a form of altruism that consists of "private initiatives, for the Public good (economics), public good, focusing on quality of life". Philanthropy contrasts with business initiatives, which are private initiatives for private goo ...
.
Prisoner's dilemma
The idea of superrationality is that two logical thinkers analyzing the same problem will think of the same correct answer. For example, if two people are both good at math and both have been given the same complicated problem to do, both will get the same right answer. In math, knowing that the two answers are going to be the same doesn't change the value of the problem, but in the game theory, knowing that the answer will be the same might change the answer itself.
The
prisoner's dilemma
The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...
is usually framed in terms of jail sentences for criminals, but it can be stated equally well with cash prizes instead. Two players are each given the choice to cooperate (C) or to defect (D). The players choose without knowing what the other is going to do. If both cooperate, each will get $100. If they both defect, they each get $1. If one cooperates and the other defects, then the defecting player gets $200, while the cooperating player gets nothing.
The four outcomes and the payoff to each player are listed below.
One valid way for the players to reason is as follows:
# Assuming the other player defects, if I cooperate I get nothing and if I defect I get a dollar.
# Assuming the other player cooperates, I get $100 if I cooperate and $200 if I defect.
# So whatever the other player does, my payoff is increased by defecting, if only by one dollar.
The conclusion is that the rational thing to do is to defect. This type of reasoning defines game-theoretic rationality and two game-theoretic rational players playing this game both defect and receive a dollar each.
Superrationality is an alternative method of reasoning. First, it is assumed that the answer to a symmetric problem will be the same for all the superrational players. Thus the sameness is taken into account ''before'' knowing what the strategy will be. The strategy is found by maximizing the payoff to each player, assuming that they all use the same strategy. Since the superrational player knows that the other superrational player will do the same thing, whatever that might be, there are only two choices for two superrational players. Both will cooperate or both will defect depending on the value of the superrational answer. Thus the two superrational players will both cooperate since this answer maximizes their payoff. Two superrational players playing this game will each walk away with $100.
Note that a superrational player playing against a game-theoretic rational player will defect, since the strategy only assumes that the superrational players will agree.
Although standard game theory assumes common knowledge of rationality, it does so in a different way. The game-theoretic analysis maximizes payoffs by allowing each player to change strategies independently of the others, even though in the end, it assumes that the answer in a symmetric game will be the same for all. This is the definition of a game-theoretic
Nash equilibrium
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...
, which defines a stable strategy as one where no player can improve the payoffs by unilaterally changing course. The superrational equilibrium in a symmetric game is one where all the players' strategies are forced to be the same before the maximization step. (There is no agreed-upon extension of the concept of superrationality to asymmetric games.)
Some argue that superrationality implies a kind of
magical thinking
Magical thinking, or superstitious thinking, is the belief that unrelated events are causally connected despite the absence of any plausible causal link between them, particularly as a result of supernatural effects. Examples include the idea that ...
in which each player supposes that their decision to cooperate will cause the other player to cooperate, even though there is no communication. Hofstadter points out that the concept of "choice" doesn't apply when the player's goal is to figure something out, and that the decision does not cause the other player to cooperate, but rather the same logic leads to the same answer independent of communication or cause and effect. This debate is over whether it is reasonable for human beings to act in a superrational manner, not over what superrationality means, and is similar to arguments about whether it is reasonable for humans to act in a 'rational' manner, as described by game theory (wherein they can figure out what other players will or have done by asking themselves, what would I do if I was them, and applying
backward induction
Backward induction is the process of reasoning backwards in time, from the end of a problem or situation, to determine a sequence of optimal actions. It proceeds by examining the last point at which a decision is to be made and then identifying wha ...
and
iterated elimination of dominated strategies).
Probabilistic strategies
For simplicity, the foregoing account of superrationality ignored
mixed strategies: the possibility that the best choice could be to flip a coin, or more generally to choose different outcomes with some probability. In the
prisoner's dilemma
The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...
, it is superrational to cooperate with probability 1 even when mixed strategies are admitted, because the average payoff when one player cooperates and the other defects are the same as when both cooperate and so defecting increases the risk of both defecting, which decreases the expected payout. But in some cases, the superrational strategy is mixed.
For example, if the payoffs in are as follows:
:: CC – $100/$100
:: CD – $0/$1,000,000
:: DC – $1,000,000/$0
:: DD – $1/$1
So that defecting has a huge reward, the superrational strategy is defecting with a probability of 499,900/999,899 or a little over 49.995%. As the reward increases to infinity, the probability only approaches 1/2 further, and the losses for adopting the simpler strategy of 1/2 (which are already minimal) approach 0. In a less extreme example, if the payoff for one cooperator and one defector was $400 and $0, respectively, the superrational mixed strategy world be defecting with probability 100/299 or about 1/3.
In similar situations with more players, using a randomising device can be essential. One example discussed by Hofstadter is the
platonia dilemma In the platonia dilemma introduced in Douglas Hofstadter's book ''Metamagical Themas'', an eccentric trillionaire gathers 20 people together, and tells them that if one and only one of them sends them a telegram (reverse charges) by noon the next d ...
: an eccentric trillionaire contacts 20 people, and tells them that if one and only one of them send him or her a telegram (assumed to cost nothing) by noon the next day, that person will receive a billion dollars. If they receive more than one telegram or none at all, no one will get any money, and communication between players is forbidden. In this situation, the superrational thing to do (if it is known that all 20 are superrational) is to send a telegram with probability p=1/20—that is, each recipient essentially rolls a
20-sided die and only sends a telegram if it comes up "1". This maximizes the probability that exactly one telegram is received.
Notice though that this is not the solution in the conventional game-theoretical analysis. Twenty game-theoretically rational players would each send in telegrams and therefore receive nothing. This is because sending telegrams is the
dominant strategy
In game theory, strategic dominance (commonly called simply dominance) occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. Many simple games can be solved using dominance. The ...
; if an individual player sends telegrams they have a chance of receiving money, but if they send no telegrams they cannot get anything. (If all telegrams were guaranteed to arrive, they would only send one, and no one would expect to get any money).
See also
*
Evidential decision theory Evidential decision theory (EDT) is a school of thought within decision theory which states that, when a rational agent is confronted with a set of possible actions, one should select the action with the highest ''news value'', that is, the action w ...
*
Instrumental convergence
Instrumental convergence is the hypothetical tendency for most sufficiently intelligent beings (both human and non-human) to pursue similar sub-goals, even if their ultimate goals are quite different. More precisely, agents (beings with agency) m ...
*
Moral realism
Moral realism (also ethical realism) is the position that ethical sentences express propositions that refer to objective features of the world (that is, features independent of subjective opinion), some of which may be true to the extent that they ...
*
Newcomb's problem
In philosophy and mathematics, Newcomb's paradox, also known as Newcomb's problem, is a thought experiment involving a game between two players, one of whom is able to predict the future.
Newcomb's paradox was created by William Newcomb of the ...
*
Perfect rationality
The term ''Homo economicus'', or economic man, is the portrayal of humans as agents who are consistently rational and narrowly self-interested, and who pursue their subjectively defined ends optimally. It is a word play on ''Homo sapiens'', u ...
*
Prisoner's dilemma
The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...
References
{{Douglas Hofstadter
Game theory
Behavioral economics
Rational choice theory