Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 1 / 50 What is Sociology? What is Sociology?
Mathematical Sociology, Agent-Based Modeling and Artificial Societies
Dirk Helbing and Research Team
Chair of Sociology, in particular of Modeling and Simulation
Septemer 22, 2008
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 1 / 50 What is Sociology? What is Sociology? What is Sociology? Sociology focusses on such phenomena as interactions and exchanges between individuals at the “micro-level”, and group dynamics, group development, and crowds at the “macro-level”. Sociologists are interested in the individual, but primarily within the context of larger social structures and processes, such as social roles, race and class, and socialization. They use a combination of qualitative research designs and highly quantitative methods, such as procedures for sampling and surveys. Sociologists in this area are interested in a variety of demographic, social, and cultural phenomena. Some of their major research fields are the emergence of cooperation, social inequality, deviance, conflicts, and social change, norms, roles and socialization.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 2 / 50 Theoretical Approach Auguste Comte Theoretical Approach
Auguste Comte (1798–1857) is the “father” of sociology and proposed a rational (“positivistic”) approach to the study of society, based on observation and experiment. As key concepts he distinguished social statics (social institutions and their relationships, ...) and social dynamics (processes of social change as “natural” breakdowns and rearrangements of social structures). In the beginning, he called his approach “social physics”, but later he used the term “sociology” (meaning knowledge of society). He considered sociology to be the queen of sciences. His approach should also provide a practical basis on which to create a better society, e.g. more stable social order.
Source: Smelser
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 3 / 50 Theoretical Approach Max Weber
Max Weber (1864–1920) proposed to elaborate “ideal types” as abstract statements of the essential, though often exaggerated characteristics of social phenomena, in order to sharpen comparisons. For example, he explored religions by contrasting the ‘ideal Protestant’ with the ‘ideal Jew’, ‘ideal Hindu’, and ‘ideal Buddhist’, knowing that these models did not precisely descibe any actual individuals. The ideal types serve to be contrasted with the actual, empirical reality. “Ideal” in this context, by the way, does not mean “good” or “best”, but something like “stylized” or “abstracted”. Weber also advocated to become involved in politics outside the classroom, while striving for scientific neutrality in the professional/scientific work.
Source: Macionis and Plummer
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 4 / 50 Theoretical Approach Talcott Parsons
Talcott Parsons (1902–1979) advocated using “analytical realisms” to build sociological theory. Theory in sociology should be a generalized, coherent system of abstract concepts, which do not correspond to concrete phenomena, but reflect the important features of the social world without being overwhelmed by empirical details. That is, concepts should abstract from empirical reality (with all its diversity and confusion) common analytical elements, in order to isolate phenomena from their embeddedness in the complex relations constituting social reality. Initially, an elaborate classification and categorization of social phenomena is needed that reflects significant features of social phenomena.
Source: Turner
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 5 / 50 Organisation of this Course Table of Content Organisation of this Course
1 22. Sept.: Cognitive Dissonance/Decision Making (Dirk Helbing) 2 29. Sept.: Social Segregation (Wenjian Yu) 3 06. Okt.: Social Inequality (Sergi Lozano) 4 13. Okt.: Game Theory and Social Cooperation (Heiko Rauhut) 5 20. Okt.: Friendship Network Formation (Albert Diaz-Guilera) 6 27. Okt.: Group Dynamics (Sergi Rauhut) 7 3. Nov.: Crowd Behavior (Anders Johansson) 8 10. Nov.: Market Dynamics (Albert Diaz-Guilera) 9 17. Nov.: Conflicts, Wars and Violence (Lubos Buzna) 10 24. Nov.: Deviance and Crime (Heiko Rauhut) 11 1. Dez.: Innovation Dynamics (Albert Diaz-Guilera) 12 8. Dez.: Emergence of Conventions and Norms (Dirk Helbing) 13 15. Dez.: Language and Culture, afterwards Exam (Albert Diaz-Guilera) Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 6 / 50 Organisation of this Course Table of Content
Chapter 1 Cognitive Dissonance and Decision Making
Dirk Helbing www.soms.ethz.ch [email protected]
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 7 / 50 Modeling Behavior Introductory Remark Modeling Behavior
A good predictor for future behavior is past behavior. However, attempts to describe human behavior as behavioral responses to environmental stimuli had different degrees of success. The possibility to condition reflexes encouraged the mechanistic idea of a deterministic stimulus-response relationship (inspired by physics and biology). However, the description of reasoned or planned action turned out to be more complicated (requiring to distinguish voluntary behavior from the previously mentioned kinds of behavior). The related stimulus-response theories are based on the concept of attitude and had to be more and more generalized.
Source of the following transparencies, if not otherwise stated: Fishbein/Ajzen
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 8 / 50 Modeling Behavior Classical Conditioning Classical Conditioning Paradigm (Pavlov, 1890s)
Unconditioned Stimulus (UCS) Unconditioned Response (UCR) Conditioned Stimulus (CS) Conditioned Response (CR)
When the UCS preceeds the CS, little learning is observed; the strongest learning effect occurs when the CS precedes the UCS by a short time interval such as 0.5 seconds (causality matters). Food and drinks, e.g., are primary reinforcers, conditioned stimuli are secondary reinforcers. Classical Conditioning [VIDEO] Smart Cat [VIDEO]
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 9 / 50 Modeling Behavior Operant Conditioning Instrumental or Operant Conditioning (Skinner,
1938) Reinforcement learning Principle of reward and punishment Frequency of reinforcement, temporal relation, and magnitude of feedback are relevant Similar stimuli may cause the same or similar Abbildung: Skinner Box responses (association-based generalization) Operant Conditioning+“Behaviorism”[VIDEO]
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 10 / 50 Modeling Behavior Response Consistency Response Consistency Stimulus-Response Consistency A person consistently performs the same response or set of particular responses in the presence of a given stimulus object.
Response-Response Consistency The individual is predisposed toward performing a certain class of behaviors, i.e. the responses elicited by one object are consistent with one another.
Evaluative Consistency Knowledge of the person’s attitude does not permit prediction of any specific behavior, but he or she is predisposed to a certain degree of favorability in the behavior toward the stimulus object.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 11 / 50 Modeling Behavior Theory of Reasoned Action Theory of Reasoned Action
1 Attitude: A person’s favorable or unfavorable evaluation of an object → emotions, affect 2 Belief: The cognitive information (opinion) one has about an object → view/interpretation of the world 3 Behavioral Intention: The person’s likelihood to perform the behavior. 4 Behavior: The observable (“overt”) act of the subject.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 12 / 50 Attitudes and Behavior Definitions Attitudes Attitude is viewed as a latent affective variable that guides individual behavior. Attitude may be defined as a 1 learned 2 predisposition 3 to respond in a consistently favorable or unfavorable manner with respect to a given object.
One distinguishes 3 types of consistency: 1 stimulus-response consistency 2 response-response consistency 3 evaluative consistency
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 13 / 50 Attitudes and Behavior Measurement of Attitudes Measurement of Attitudes
Fishbein and Ajzen (1972) found more than 500 different operations designed to measure attitude. These operations include standard attitude scales; indices across various verbal items; single statements of feelings, opinions, knowledge, or intentions; observation of one or more overt behaviors; and physiological measures. As affect is the most essential part of the attitude concept, a polar evaluative dimension (scale) is essential in the measurement of attitudes. A well-known method to measure attitudes are semantic differentials.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 14 / 50 Attitudes and Behavior Semantic Differentials Semantic Differentials (Osgood/Tannenbaum, 1957)
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 15 / 50 Attitudes and Behavior The 3 Main Factors The 3 Main Factors
A factor analysis results in the following 3 main factors: evaluation, activity, and control
fair - strong - active ( respect, admiration, benevolent strength ) I
strong - passive ( no evaluation )
unfair - strong - actie ( fear, anger, powerful II fair - weak - passive e!ective evil ) ( well-wishing but ine!ective, milk - toast - like )
III weak - active ( no evaluation )
unfair - weak - passive ( cowardice, disgust, repugnance )
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 16 / 50 Attitudes and Behavior Stimulus Complexes Stimulus Complexes When two or more stimuli are combined (at the same time), one speaks of a stimulus complex. The overall attitude toward the stimulus object is then calculated as
n m X X A = biei or A = pjuj , i=1 j=1
where ei is the evaluation of the object’s attribute i, n the number of relevant attributes, and bi ≥ 0 the strength of belief regarding attribute i, i.e. the subjective probability that the object has this attribute. The second formula corresponds to the expectancy value theory, where pj represents the subjective probability (“expectancy”) that the behavior following from the attitude will have the consequence j, which is valuated with the utility uj.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 17 / 50 Attitudes and Behavior Appetence-Aversion Conflict Appetence-Aversion Conflict
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 18 / 50 Theories of Attitude Change Attitude Change and Learning Attitude Change and Learning
Attitudes are learned. They result from past experience. Once established, an attitude may influence the formation of new beliefs. Similarly, performance of a particular behavior may lead to new beliefs about the object, which may in turn influence the attitude. Changes in beliefs can initiate changes in any other variable (attitudes, intentions, and behaviors). Therefore, exposing a person to new information can trigger a chain of effects. In persuasive communication, communicator credibility (and, therefore, reputation) is important, otherwise new information may be doubted and rejected.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 19 / 50 Theories of Attitude Change Congruity Principle Congruity Principle (Osgood and Tannenbaum, 1955)
The “mediating reaction characteristic of each [stimuli] shifts toward congruence with that characteristic of the other, the magnitude of the shift being inversely proportional to the intensities of the interacting reactions”, i.e. the more polarized an evaluation, the less it will shift toward the other. The point of resolution is predicted as
Pn |ei| A = i=1 biei with bi = Pn . j=1 |ej|
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 20 / 50 Theories of Attitude Change Balance Theory Balance Theory (Heider, 1946)
The theory concerns a person’s perceptions of triads involving the following three elements: the focal person (a), another person (b), and an object or event (x). A balanced state is said to exist if a has similar attitudes toward the two other elements of a triad, i.e. aLb and aLx (“associative” coupling), or aLb and aLx (“dissociative” coupling), where L represents liking and L represents dislike relationships. There is a tendency that unbalanced states will be replaced by balanced ones (where the “economic principle” favors a minimum number of attitude changes). If a change is not possible, the state of imbalance will produce tension (comparable to “frustrated states” in “spin glasses”).
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 21 / 50 Theories of Attitude Change Transitions Transitions to Balanced States
x x x x
a b a b a b a b
x x x x
a b a b a b a b
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 22 / 50 Theories of Attitude Change Dissonance Theory Dissonance Theory (Festinger, 1957) Dissonance due to inconsistencies between attitude and behavior causes arousal/an emotional state of discomfort, which tends to change the dissonant attitude (“rationalization”). Specifically, dissonance theory considers the relations between cognitive elements (beliefs or attitudes or self-perceptions of behaviors). “Consonance”, “dissonance”, and “irrelevance” are used to describe three kinds of relations that may exist between any two cognitive elements x and y. x and y are dissonant, if not-x follows from y. The magnitute D of dissonance increases with the importance of the elements to the person: Pn d=1 Id D = Pn Pm , d=1 Id + c=1 Ic
where Id = bd|ed| is the importance of dissonant element d and Ic = bc|ed| is the importance of consonant element c.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 23 / 50 Theories of Attitude Change Reduction of Dissonance Reduction of Dissonance Cognitive dissonance may result from: 1 decision making, 2 forced compliance, 3 voluntary or involuntary exposure to dissonant information, or 4 disagreement with other persons. To reduce the magnitude of dissonance, a person may add new cognitive elements that are consonant with the element in question (i.e. actively seek new consonant information and try to avoid exposure to dissonant information). A typical effect is the increase of one’s evaluation of a chosen alternative (“rationalization”). However, a person may also reduce the importance of elements in a dissonant relation. The greater the magnitude of dissonance, the greater the expected change in belief → Ash experiment
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 24 / 50 Theories of Attitude Change Comparison Comparison In balance theory, links may be either beliefs or attitudes, while congruity theory considers only p’s attitudes toward o and x, and dissonance theory deals only with inconsistency between beliefs. The distinction between imbalance and dissonance corresponds closely to the distinction between affective and cognitive inconsistency. Dissonance is created only to the extent that the person feels he or she had the freedom of choice. Whereas balance theory is concerned with perceived relations between o and x, the congruity principle deals with “objective” or stated relations, i.e. assertions. A state of congruence exists when the evaluations of two objects are equally intense, i.e. balanced configurations are incongruous, unless the evaluations of both objects are equally polarized. As a given assertion may not be believed, the congruity principle requires a correction for incredulity.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 25 / 50 Theories of Attitude Change Theory of Planned Behavior Theory of Planned Behavior (Ajzen, 1985)
Attitude
Subjective Intention Behavior Norms
PBC
Attitude toward performing a given behavior is related to the person’s beliefs that performing the behavior will lead to certain consequences (“anticipation”). The greater the “perceived behavioral control” (PBC), the more positive the behavioral intention and the more likely the actual performance of the behavior. Source: Leone/Perugini/Ercolani Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 26 / 50 Theory of Choice Decision Concepts Decision Concepts
EUT = Expected Utility Theory
Source: Slides of Lennart Schalk, after Gigerenzer et al., 1999
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 27 / 50 Theory of Choice Bounded Rationality Bounded Rationality Rational choice theory assumes that humans can be reasonably approximated or described as “rational” entities, who would never do anything violating their preferences. The concept of bounded rationality revises this assumption to account for the fact that perfectly rational decisions are often not feasible in practice due to the finite computational resources available for making them.
Source: Extracts from Wikipedia Note that limited memory and processing capacity does not exclude that people make optimal or close-to optimal choices, if the limitation of resources is considered. However, such a concept implies different behaviors than assuming hyperrational agents with unlimited capacities, as done by the concept of “homo economicus”. Ernst Fehr distinguishes the ”homo sociologicus” following prevailing social norms (see “Social Action”) and “homo reciprocans” driven by the tendency to reciprocate cooperation (see “Game Theory”). Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 28 / 50 Theory of Choice Satisficing Satisficing (Herbert Simon, 1956) The word satisfice was coined by Herbert Simon as a portmanteau of “satisfy” and “suffice”. In decision making, satisficing explains the tendency to select the first option that meets a given need or select the option that seems to address most needs rather than the “optimal” solution. Satisficing occurs in consensus building when the group looks towards a solution everyone can agree on even if it may not be the best. In economics, satisficing is a behavior which attempts to achieve at least some minimum level of a particular variable, but which does not necessarily maximize its value. In many circumstances, the individual might be uncertain about what constitutes a satisfactory outcome. In this case, the individual can only evaluate outcomes on the basis of their probability of being satisfactory. If the individual chooses that outcome which has the maximum chance of being satisfactory, then this individual’s behavior is theoretically indistinguishable from that of an optimizing individual under certain conditions.
Source: Extracts from Wikipedia
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 29 / 50 Theory of Choice Fast and Frugal Heuristics Fast and Frugal Heuristics (Gigerenzer, 1996)
According to Gigerenzer, humans take decisions often in an intuitive way, using fast and frugal heuristics. (“Frugal” means something like simple, economic, in the sense of resource-saving.) Such heuristic decision rules can perform better than common methods from multivariate statistics, because successful decision-making requires to ignore irrelevant information. It is not true that more information, more decision time, and more options lead to better decisions. Therefore, the related research question is: “When is more better, and when is less more?”
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 30 / 50 Theory of Choice Examples Examples of Fast and Frugal Heuristics Recognition Heuristics “If you recognize one of two objects, but not the other, then infer that the recognized object has the higher value with respect to the considered criterion and choose it.”
Take-the-Best Heuristics “Compare all alternatives in question with respect to the most important criterium and choose the alternative(s) which match(es) the criterium best. If several alternatives are judged to be best, compare them with respect to the second most important criterium, etc. In this way, exclude inferior options, until a single one remains, and choose it.”
Source: After extracts from Wikipedia Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 31 / 50 Theory of Choice Examples Further Examples of Fast and Frugal Heuristics
Minimalist Heuristics “Evaluate the alternatives in question with respect to the most important criterium and select the best. If there are several equally valuated best alternatives, choose any of them.”
Guessing Heuristics “If you don’t have any criteria to distinguish between two alternatives, guess which is the better one, or flip a coin.”
Tallying Heuristics “Weight all decison criteria equally and select the alternative which reaches the highest evaluation.”
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 32 / 50 Theory of Choice Adaptive Toolbox Adaptive Toolbox
There are many fast and frugal heuristics. An individual applies the one which appears to be adapted best to the decision problem (note the similarities to the use of heuristics in Operation Research, in particular if NP-hard, i.e. numerically complex problems must be solved) Source: Transparencies of Lennart Schalk, after Todd and Gigerenzer (2000), p. 740
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 33 / 50 Rational Choice Theory Choice under Uncertainty Choice under Uncertainty This case concerns situations with a lack of information. This is often modeled by adding a Gumbel-/Weibull-distributed random utility j to the known part uj = u(xj) of the utility, which is a quantitative measure of the attitude A towards/preference for xj.
Why a Gumbel distribution? The maximum of two Gumbel-distributed values is again Gumbel distributed.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 34 / 50 Rational Choice Theory Multinomial Logit Model Multinomial Logit Model (McFadden, 1973)
Among two alternatives, the one is taken for which the sum of the known and random part of the utility is larger. The probability p(xj) of chosing the action with outcome xj is then given by the multinomial logit model eu(xj)/σ p(xj) = . P u(xi)/σ i e Here, σ is a measure for the uncertainty of information. In the limit σ → 0, the alternative with the highest utility is chosen with certainty. For u(xj)/σ = ln f(xj) (“Weber-Fechner law”), where f(xj) is the relative frequency of a successful outcome xj, we obtain the law of effect: p(xj) = f(xj).
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 35 / 50 Rational Choice Theory Choice without Risk and Uncertainty Choice without Risk and Uncertainty: Preference Order We start with the classical theory of choice under certainty, where each agent has sufficient information about the available set of actions with perfectly predictable consequences. That is, there is a function that maps each action ai into a specific outcome xi.
The notation xi xj (“weak preference”) shall reflect that outcome xj is not preferred to xi. If and only if xi xj and not xj xi, we speak of strict preference, xi xj. If xi xj and xj xi, xi is indifferent to xj, xi = xj. An agent’s behavior is rational, if he or she selects an outcome that is valued as least as much as any other (i.e. if not any other action could reach a higher valued outcome). To generate predictions about choice behavior, we require the notion of weak preferences and weak ordering defined in the following.
Source of this and the following transparencies (apart from Choice under Uncertainty): McCarty/Meirowitz Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 36 / 50 Rational Choice Theory Definitions Definitions Completeness
Confronted with any two options xi and xj, a person can determine whether he or she does not prefer options xi to option xj, does not prefer xj to xi, or does not prefer either. When preferences satisfy this property, they are complete, which means there is no noncomparability problem.
The maximal set for a weak preference relation “” on a choice set X is the largest subset M(, X) ⊂ X with xi xj for all xi, xj ∈ M(, X).
Reflexiveness
means that a person does not prefer an outcome xi (in the maximal set) to itself, i.e. xi xi for any outcome xi.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 37 / 50 Rational Choice Theory Definitions Further Definitions Transitivity
Confronted with three options xi, xj, and xk, if a person does not prefer xj to xi and does not prefer xk to xj, then he or she must not prefer xk to xi, i.e. xi xj and xj xk imply xi xk for all xi, xj, xk. Preferences satisfying this property are transitive.
Acyclicity
is an alternative concept meaning that we have x1 xn for any finite set of outcomes {x1,..., xn} for which xi xi+1 for all i < n.
The properties of completeness, reflexivity, and transitivity (or completeness, reflexivity, and acylicity) together define weak ordering.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 38 / 50 Rational Choice Theory Utility Theory Utility Theory
Rather than basing rational choice on binary preferences and maximal sets, preferences can be represented by a real-valued utility function u(xi).
Definition: For all elements xi, xj of the maximal set, the utility function must have the property that u(xi) ≥ u(xj) if and only if xi xj.
Then, u(xi) > u(xj) implies xi xj, and u(xi) = u(xj) implies xi = xj. Note that the utility function is not necessarily unique, i.e. there may be several functions fulfilling the above definition.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 39 / 50 Rational Choice Theory Choice under Risk Choice under Risk
If actions ai and outcomes oj are linked probabilistically rather than deterministically (via a unique relationship), one speaks of choice under risk. Then, the probability that outcome xj follows action ai is represented by J X pij ≥ 0 with pij = 1 . j=1
The vector ~pi = (pi1,..., piJ) is called the distribution or lottery of outcomes associated with action ai. We are interested to represent people’s preferences between two lotteries ~pi and ~pj. Independence
means that, for any three lotteries ~pi,~pj,~pk, we have ~pi ~pj (lottery ~pi is weakly preferred to ~pj) if and only if [α~pi + (1 − α)~pk] [α~pj + (1 − α)~pk] for any value α ∈ (0, 1).
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 40 / 50 Rational Choice Theory Expected Utility Function Expected Utility Function (von Neumann/Morgenstern, 1944)
Under certain conditions, there exists a real-valued utility function u(xj) that assigns a number uj = u(xj) to each outcome xj of an action ai such that ~pi ~pj if and only if EU(~pi) ≥ EU(~pj) with the expected utility function J X EU(~pi) = piju(xj) . j=1
That is, given the existence of such a utility function, the preference for a lottery can be measured as a weighted average of the utilities u(xj) of the outcomes xj, where the weights are the occurence probabilities pij of the respective outcomes.
Assigning utilities to the most and least preferred outcomes, x1 and xJ, the utilities of any other outcome xj can be determined as the expected utility pu(x1) + (1 − p)u(xJ) of the lottery that is indifferent compared to having outcome xj with certainty. Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 41 / 50 Rational Choice Theory Risk Preferences Risk Preferences
Let us now assume that u(xj) represents the monetary value of outcome xj. Then, a person is called risk averse, if he or she prefers the certain outcome x = px1 + (1 − p)x2 over the lottery returning outcome x1 with probability p1 = p and the outcome x2 with probability p2 = (1 − p), i.e. if
u(x) = u(px1 + (1 − p)x2) > pu(x1) + (1 − p)u(x2) .
If the “>” sign is replaced by a “<” sign, correspondingly behaving agents are called risk acceptant or risk seeking. Otherwise, if an “=” sign applies, agents are said to behave risk neutral. Risk aversion corresponds to a strictly concave utility function u(x) (e.g. a root function), risk acceptance to a strictly convex utility function, and risk neurality corresponds to the function u(x) = x.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 42 / 50 Prospect Theory Allais Paradox Allais Paradox Lottery a: 33% chance of 2,500$, 66% chance of 2,400$, 1% chance of 0$ Lottery b: 2,400$ for sure 82% of test persons choose lottery b, despite lower expected utility.
Lottery c: 33% chance of 2,500$, 67% chance of 0$ Lottery d: 34% chance of 2,400$, 66% chance of 0$ 83% of test persons choose lottery c, which questions risk-avoiding behavior.
Both experiments together show that subjects often make choices inconsistent with the independence axiom of expected utility theory.
Source of this and the following transparencies, if not otherwise stated: McCarty/Meirowitz
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 43 / 50 Prospect Theory Kahnemann’s and Tversky’s Findings Kahnemann’s and Tversky’s Findings (1979)
The independence axiom is violated in particular when all lotteries are far from sure things:
Lottery a: 45% chance of 6,000$, 55% chance of 0$ Lottery b: 90% chance of 3,000$, 10% chance of 0$ Lottery c: 0.1% chance of 6,000$, 99.9% chance of 0$ Lottery d: 0.2% chance of 3,000$, 99.8% chance of 0$
Test persons prefer lottery b over a and c over d. Because the large payoffs in lotteries c and d have negligible probabilities, subjects prefer the one with the bigger prize. When both probabilities are reasonably high, subjects are inclined to take the one that is relatively more certain.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 44 / 50 Prospect Theory Losses are Different from Gains Losses are Different from Gains
The previously stated risk-averseness does not hold when lotteries are over losses rather than gains:
Lottery a: 80% chance of -4,000$, 20% chance of 0$ Lottery b: -3,000$ for sure Lottery c: 20% chance of -4,000$, 80% chance of 0$ Lottery d: 25% chance of -3,000$, 75% chance of 0$
A preference for certainty would speak for a preference of b over a and c over d. However, test persons prefer a over b and d over c. Therefore, people are risk averse with respect to gains, but accept risks with respect to losses.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 45 / 50 Prospect Theory Framing Framing Suppose a person has been given 1,000$ and is offered Lottery a: 50% chance of an additional 1,000$, 50% chance of 0$ Lottery b: 500$ for sure For comparison, suppose the person has been given 2,000$ and is offered Lottery c: 50% chance of losing 1,000$, 50% chance of 0$ Lottery d: Loss of 500$ for sure Test persons prefer b over a, but c over d. Hence, the framing of the lotteries matters.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 46 / 50 Prospect Theory Kahnemann’s and Tversky’s Theory Kahnemann’s and Tversky’s Theory (1979): Editing Phase 1 Coding/Framing: People evaluate gains and losses separately. In a first step, the reference point is determined. 2 Combination: People combine probabilities associated with identical outcomes. 3 Segregation: People identify and separate the riskless components of a choice. 4 Cancellation: People ignore the common elements of two lotteries. 5 Simplification: People round probabilities and drop unlikely outcomes from their considerations. 6 Detection of dominance: People drop any lottery that is first-order stochastically dominated from their consideration.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 47 / 50 Prospect Theory Kahnemann’s and Tversky’s Theory Kahnemann’s and Tversky’s Theory (1979): Evaluation Phase Definition: Let p be the probability of x and q the probability of y, while a payoff of 0 or nothing happens with probability 1 − p − q. Prospects are then defined as strictly positive if x, y > 0 and p + q = 1, strictly negative, if x, y < 0 and p + q = 1, while regular in all other cases.
For a regular prospect, agents maximize
V(x, p; y, q) = π(p)v(x) + π(q)v(y) ,
where v(x) and v(y) with v(0) = 0 are the values of each outcome, while π(p) and π(q) with π(0) = 0 and π(1) = 1 are weights based on the outcome probabilities.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 48 / 50 Prospect Theory Kahnemann’s and Tversky’s Theory Strictly Positive or Strictly Negative Prospects For strictly positive or strictly negative prospects such as x > y > 0 or x < y < 0, where p + q = 1, agents maximize
V(x, p; y, q) = v(y) + π(p)[v(x) − v(y)] .
In this way, the risk-free component v(y) is separated from the risky component v(x) − v(y).
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 49 / 50 Time Preferences
Time Preferences Agents value current utility more than future utility. This observation is classically described by a discount factor δ: Utilities that are t time periods in the future are discounted by a factor δt (“geometric discounting”, corresponding to exponential interest rates).
However, empirical evidence is more in favor of hyperbolic discounting, which assumes that at time 0 agents discount the utility at time t by a factor
h(t) = (1 + αt)−γ/α
with parameters α, γ > 0. Note that α ≈ 0, otherwise hyperbolic discounting weights the future heavily, which implies the problem of time consistency.
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/ Mathematical Sociology Dirk Helbing and Research Team Zurich Septemer 22, 2008 50 / 50 Bibliography
Bibliography
Main source: M. Fishbein and I. Ajzen (1975) Belief, attitude, intention, and behavior: An introduction to theory and research (Addison-Wesley, Reading, MA) L. Leone, M. Perugini, and A. P. Ercolani (1999) A comparison of three models of attitude-behavior relationships in the stuying behavior domain, Eur. J. Soc. Psychology 29, 161–189. Simon, H. A. (1956) Rational choice and the structure of the environment. Psychological Review 63(2), 129-138. Simon, H. A. (1990) Invariants of human behavior. Annual Review of Psychology 41, 1-19. Tversky, A., and Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science 185, 1124-1131. Gigerenzer, G. and Goldstein, D. G. (1996) Reasoning the fast and frugal way: Models of bounded rationality. Psychological Review 103(4), 650-669. Gigerenzer, G., Todd, P. M., and The ABC Research Group (1999). Simple Heuristics That Make Us Smart (Oxford University Press, New York).
Chair of Sociology, in particular of Modeling and Simulation http://www.soms.ethz.ch/