I n t r o d u c t i o n t o
P s y c h o l o g i c a l D e c i s i o n T h e o r y
Introduction to Mathematical Psychology and Data Analysis
on the occasion of the
European Mathematical Psychology Meeting,
Prof.em. Dr. Hans Christoph Micko
Am Radeland 28, D-21244 Buchholz i.d.Nordheide, Germany
Tel/Fax: (++49)(+531)360500 Email: firstname.lastname@example.org
Mathematical psychologists attempt to formulate their theories in a
formal (mathematical) language.
- Precision and consequently easier refutability of theories.
- Comparability of goodness of fit of theories (e.g. by maximum likelihood methods).
In research on human decision making the situation is different.
The field started in mathematical economics with the development of formalised prescriptive theories for rational decision making (mainly by v. Neumann and Morgenstern in mid 19. century).
Psychology of decision making started with attempts to use those normative theories as descriptive theories of human decision making.
In view of many inconsistencies of human decision making the theories had to be modified and nowadays, while still serving as a background, tend to be discarded as a descriptive theories.
From the necessary modifications much was learned about the peculiarities of human decision making, particularly about how a decision problem is (as opposed to should be) formulated before a choice is made, i.e. how choice alternatives, utilities and probabilities of outcomes, decision stages etc. are identified or estimated.
Here we shall look at such modifications and the resulting insights
for the psychology of decision making. As we shall see, human decision making just does not follow rationality and laws of logic to the extent required by mathematical theories.
Prescriptive or Normative Decision theory considers three typical situations:
1.) Decisions under Certainty:
The choice determines the outcome with certainty (e.g. purchase of a good or brand), i.e. there is a one to one relationship between choice alternatives ai and outcomes oi, ai oi, aj oj.
2.) Decisions under Risk (Gambles):
A choice alternative can have more than one outcome ai (oim) depending on uncontrollable states of nature sim. The probabilities pim = p(sim) of the outcomes are known to the decision maker.
3.) Decisions under Uncertainty:
A choice alternative can have more than one outcome. The probabilities of the outcomes are unknown (e.g. choice of a strategy against an enemy who may select his counter strategy later).
In fact, the probabilities of outcomes or uncontrollable states of nature are neither completely certain nor completely uncertain. The properties of a car after 5 years of use are uncertain at the time of purchase and sometimes more or less accurate estimates of unknown outcome probabilities may be made, e.g. estimates of choice probabilities for various counter strategies.
Classical theories assume that the choice alternatives, goals (e.g. profit maximisation) and possible outcomes of a choice are specified.
That is a simplification which enables the study of principles of rational decision making but can at best be more or less approximated in everyday decisions,
pT (ai) = pA (ai) . pT (A)
1.) D e c i s i o n s u n d e r C e r t a i n t y
Rational Decision rule: Choose the alternative yielding the most preferred outcome.
That rule tacitly employs a binary preference relation ( > ) among the alternatives which has the following mathematical properties: It is
- irreflexive (i.e. ai > ai) with ai > aj denoting: ai is preferred to aj and ai > aj denoting: ai is not preferred to aj
- asymmetric (i.e. ai > aj implies aj > ai)
- transitive (i.e. ai > aj and aj > ak imply ai > ak) and
- connected (i.e. either ai > aj or aj > ai).
Such a relation defines a strong order (strict rank order) and permits a natural number i, j, to be assigned to every alternative ai, aj, such that i > j if ai > aj. That isomorphic mapping (a, >) (N, >) of the ordered set of alternatives on a set of ordinal numbers indicates the alternative which is most preferred, second most preferred, third most preferred....least preferred. The numbers may be regarded as measures of the utility of the alternatives on an ordinal scale.
(Please distinguish" > : is preferred to" and " = : is equivalent to"
from " > : is greater than" and " = : is equal to".
The signs are somewhat difficult to discriminate, but context helps.)
Utility is a theoretical variable inferred from observable preferences choices or -judgements. Since preference is considered to be the behavioural consequence of higher utility, the theories of choice behaviour and utility measurement are closely related.
A strict order of preference among alternatives is not always found, because as a rule the preference relation among alternatives is
- not connected (i.e ai > aj and aj > ai hold)
since decision makers are frequently indifferent between some of the outcomes which may be called equivalent to them.
This suggests the supplementation of the preference relation by an indifference relation (=) which is:
- reflexive (i.e. ai = ai)
- symmetric (i.e. ai = aj implies aj = ai) and
- transitive (i.e. ai = aj and aj = ak imply ai = ak).
The indifference and preference relation together define a weak order
(a,=,>), i.e. a strong order between equivalence classes of alternatives, which can be mapped on a number system (N,=,>) such that the same resp. different natural numbers are assigned to alternatives within the same resp. different equivalence classes.
Even the requirements of a weak order are to strict for a description of human preferences. In particular, the transitivity of the indifference relation is usually violated. This can be easily explained by noting that the differences between the utilities of the possible alternatives of a choice may be too small to be noticed. Consider the famous example of three cups of coffee o1, o2, o3 containing slightly increasing amounts of sugar. The difference in concentration can be made so small that they remain unnoticed and lead to indifference judgements between adjacent cups but large enough to be noticed for the pair
o1, o3 leading to a preference judgement. Such a "weak order" with overlapping equivalence classes and consequently intransitive indifference relation is called a strong partial order (or semi-order, if a just noticeable difference can be defined on the utility scale). An intransitive indifference relation between two outcomes no longer indicates equal values on the utility scale but an unnoticeably small difference only.
Inconsistencies due to small utility differences may and do result in violations of the asymmetry of preferences as well, particularly if a preference judgement is enforced (i.e. equivalence judgement prohibited). In that case we find even under constant environmental and motivational conditions sometimes ai > aj and sometimes aj > ai.
Therefore, we may consider choice probabilities p(ai > aj): For an exactly equivalent or even identical pair ai, aj we may expect
p(ai > ai) = p(ai > ai ) = p(aj > aj) = .5. As, say, the utility difference of the pair ak , al increases, the probability p(ak > al) of choosing ak over al increases towards 1 (or decreases towards 0, if u(al) > u(ak)).
Clearly, a necessary, but not sufficient, precondition for the utility difference to increase is an increasing dissimilarity of the pair ai, aj.
We shall make use of that relationship later.
Accordingly, the reliability of a preference judgement can be regarded as a behavioural indicator of the size of small utility differences | ai - aj | and may be used to construct an interval scale of utility which reflects not only the rank order of utilities but also the sizes of their differences.
If choice alternatives resp. outcomes vary in one dimension only, like sweetness of coffee, medium as well as large utility differences can be expected to result in reliable preference judgements p(ai > aj) = 1 or 0. Therefore, choice alternatives resp. outcomes of sufficiently small utility differences are needed for constructing an interval scale from unreliable preference judgements.
1a) Luce´s_Choice Axiom.
A powerful theoretical basis for constructing even a ratio scale (with meaningful zero-origin, interpretable utility ratios and differences but arbitrary measurement unit) is the famous Choice Axiom by D. Luce. It assumes independence of choice probability ratios from irrelevant alternatives. Formally the axiom states, with pX (Y) denoting the probability of selecting the subset Y X from the set X of alternatives:
and consequently for any A T, e.g. A = (ai, aj):
Luce´s plausible and powerful choice axiom permits the construction even of a ratio scale of utilities by defining the odds of choosing one alternative over another as the ratio of their utilities.
For choice probabilities three transitivity properties can bee defined:
If p(ai > aj) > .5 and p(aj > ak) > .5, then
Strong stochastic transitivity: p((ai > (ak) > Max(p((ai > (aj), p((aj > (ak)).
Medium stochastic transitivity: p(ai > ak) > Min(p(ai > aj), p(aj > ak)).
Weak stochastic transitivity: p(ai >ak) > .5.
Luce´s choice axiom requires strong stochastic transitivity, as can be seen from the multiplication of p(ai > aj)/p(aj >ai) = u(ai)/u(aj) > 1 by p(aj >ak)/p(ak > aj) = u(aj)/u(ak) > 1, which yields the product
u(ai)/u(ak) = p(ai >ak)/p(ak >ai) which is larger than its two factors.
Strong stochastic transitivity is a severe requirement following from Luce´s quite plausible axiom. Frequently, however, it is not met by human preferences.
1b) Multi-Attribute Decision Theory.
(here for decisions under certainty only)
Even well distinguishable alternatives may yield inconsistent preference judgements particularly if they vary simultaneously on several attributes. Consider e.g. the famous choice alternatives:
a1: journey to Rome
a2: journey to Paris
a3: journey to Paris + 10.- EURO
Obviously some or many (of us) will be more or less indifferent between Rome and Paris and judge a1 = a2 as well as a1 = a3, but most of these will prefer a2 > a2, thus violating the transitivity of the equivalence relation. That inconsistency appears to be due to a similarity effect i.e. the focussing of attention on the larger difference in local geography when comparing a2 with a1, and on the only small difference in money when comparing a2 and a2.
The rational decision rule of selecting the most preferred decision alternative breaks down in the case of the alternatives varying along several attributes 1,..,l,..,L (e.g. effectiveness, beauty, durability, cheapness, etc. of a buy), because usually different alternatives will, be most preferred with respect to different attributes (or with respect to different decision goals). The decision maker must somehow combine specific preferences with respect to different attributes to an overall preference, i.e. he must look for a function
u(ai) = f(u1 (ai),..,u1 (ai),..,uL (ai))
determining an overall utility from the specific utilities with respect to different attributes.
The Linear Compensatory Rule
u(ai) = w1 . u1 (ai)
assigns an importance weight wl to every attribute (resp. decision goal) 1,..,l,..,L and computes the weighted sum of the specific utilities.
The linear compensatory rule, when applied to subjects´ numerical judgements of specific and overall utilities predicts the latter surprisingly well from the former. When applying it to judgements of specific utilities of different experts (e.g. economist and ecologists) their weighted sums agree better than their judgements of overall utility.
The rule is taught in decision counselling and training: 1. define goals or desired attributes of outcomes, 2. assign importance weights to attributes 3. assign specific utilities to outcomes, 4. take weighted sum, 5. compare resulting overall utilities with intuitive judgements thereof, 6. if both agree, select alternative with highest overall utility.
The linear compensatory rule requires the measurement of specific utilities on interval scales of common unit for all attributes, and ratio scale measurements for the importance weights, since weaker (e.g. ordinal) scales would produce any result depending on the arbitrary choice of measurement units and origin. Subjects´ numerical estimates of specific utilities and importance weights are assumed to represent measurements on the required scales but that assumption is untestable.
and if overall preferences are B > F and D > H then A > I must hold.
| || || ||Price|| |
| || ||Cheap||Medium||Expensive|
Conjoint measurement is used widely in market research to study the effect of changing specific properties of products on their status in the hierarchy of overall preferences. Since all utilities are measured on interval scales, the size of improvements with respect to various attributes (e.g. price reductions vs. quality improvements) in different regions (e.g. low price vs. medium price) on the size of overall utility improvements can be compared.
Of course, the above consistency conditions may not hold for some set of preferences. But if simple preferences are inconsistent, more demanding utility ratings must be doubted even more and should not be accepted because of lack of consistency tests.
The Conjunctive Rule
u(oi) = Min(u1(oi),.., u1 (oi),.., uL (oi)
equates overall utility to the worst of the specific utilities all of them measurded on a common scale. It is used in exams, if bad marks (in important subjects) cannot be compensated for by good marks in other fields, or if e.g. risks of health and life are are regarded as negative properties of alternatives that cannot be compensated.(Approximation: u(oi) = w1 . log(u1 (oi)) because of overweighting of small (or large negative) specific utilities).
The opposite Disjunctive Rule
u(oi) = Max(u1(oi),..,u1 (oi),..,uL (oi))
equating overall utility with the strongest quality of an alternative determines e.g. the survival in an expert world and is sometimes used in the comparison of candidates for academic positions, fellowships etc. (Approximation: u(oi) = w1 . e^u1 (oi)).
The Lexicographic Rule requires the ranking of attributes (goals) with respect to importance. Alternativeai is preferred over aj if it is (sufficiently) preferred with respect to the most important attribute or vice versa. If neither is preferred, the alternatives are compared with respect to the second, third ,...., least most important attribute until one of them is (sufficiently) preferred.
The rule makes sense but may yield intransitive preferences: Consider birthday presents from Germany, Spain and Portugal, ranked by you in that order with respect to your more important attribute "money value" and in reverse order with respect to your less important attribute "style". If the differences in money value are small you may violate transitivity by preferring the German over the Portuguese present because of its just sufficiently higher money value but otherwise the Portuguese over the Spanish and the Spanish over the German present because of superior style.
The Elimination by Aspects Rule considers desirable aspects of the alternatives which are conceived as dichotomous (= either present or absent).
Importance weights w1 are assigned to the aspects. Then in step 1 all aspects are ignored that are common to all alternatives and, thus, do not discriminate between them. In step 2 some remaining aspect l is selected with probability proportional to its importance weight p1 = w1 / w1. In step 3 all alternatives are eliminated that do not possess aspect l. These three steps are repeated with the set of remaining alternatives ... and so on until one alternative remains.
Given the importance weights, the probabilities of selecting each alternative can be computed. They obey some testable properties, e.g. medium stochastic transitivity, but not Luce´s independence from irrelevant alternatives. The rule predicts the above similarity effect, because for many of us journeys to Rome and Paris have, besides many common, about equally many and on the average equally important unique desirable aspects. Therefore, the chances of picking a unique desirable aspect of Rome or Paris, when applying the elimination by aspect rule, are about the same, and the one additional, at most moderately important, aspect of receiving 10 Euro is not likely to be picked as decisive from the many unique aspects. The consequence is indifference between Rome and Paris or rather a probability of p =.5 for preferring one journey over the other . On the other hand, there is only one unique aspect discriminating a journey to Paris and a journey to Paris + 10 Euro. It will be picked with probability p = 1 and result in the certain choice of the second alternative.
2) D e c i s i o n s u n d e r R i s k (G a m b l e s).
We discuss choices between gambles at some length, in spite of the fact that human decisions with known probabilities other than one or zero are rare. We do so, because it may tell us something about subjective estimates of probabilities and about the subjective utility of (different amounts of) money. Our considerations are based on the assumption that humans behave reasonably and equate the utility of a gamble to estimates of its expected utility which would be obtained on the average if the gamble were played many times. Estimates of subjective probability and utility may be subject to random error about, but not to systematic bias from, the values predicted by an appropriate theory of choices among gambles.
Gambles can have, and mostly have, more than one outcome, say outcomes 1,..,i,j,..,n, usually associated with different monetary gains or losses i.e. positive or negative utilities. In economics utility, ui, is regarded as proportional or equal to monetary value, vi, which can be measured on a ratio scale permitting a meaningful interpretation of differences, sums and ratios. The probabilities of occurrence of the outcomes pi, pj, are known and measured even on an absolute scale. Therefore, the expected value of a gamble pi. vi can be computed, i.e. the monetary value which would be obtained on the average if the gamble were played many times. The expected values of gambles measured on ratio scales with the same measurement unit (i.e. in the same "currency") can be compared, including those of "gambles", in which some outcome i occurs with probability pi = 1, yielding the value vi, and all others occur with probability pj = 0.
Gambles can be described by listing the probabilities and monetary values of their n outcomes, e.g. in the following way:
// p1, v1 / p2, v2 /..../ pi, vi /..../ pn, vn //.
A gamble with two outcomes can be abbreviated:
// p1, v1 / (1-p1) v2 // = // p1, v1,, v2 //,
and even further, if the value of one outcome is zero, say v2 = 0:
// p1, v1 / (1-p1), 0 // = // p1, v1,, 0 // = // p1, v1 //.
If we consider a "gamble" which has a sure outcome we simplify the denotation further to
// 1, v // = // v //.
It should be noted, that the monetary value vi of an outcome does not refer to the financial state of the decision maker after the outcome has materialised but to the difference between the financial states after the outcome and before the decision, i.e. to monetary gains and losses incurred. Thus, the monetary value of his state before the decision is always considered to be zero, v0 = 0, as well as its subjective utility, u(v0) = 0. That makes sense, because it is a general rule found in the psychology of happiness, perception and even learning that we respond more to changes than to states. Owning, say, 10000 EURO you may feel rich or poor depending on what you were used to before. According to the experimentally demonstrated Crespi Effect a high reward group of rats performs better than a low reward group in a learning experiment as expected, but after both goups are shifted to a common intermediate reward level the low reward group performs better that the high reward group before and the high reward group worse than the low reward group before. Even animal performance is not related to reward level but to changes of reward level.
The, say positive, subjective utility of any outcome oi of a choice for a person, e.g. the utility of a romantic meeting, can be expressed in terms of monetary value, by letting the person choose between that outcome and a gamble which offers the gain of a monetary amount, v, large enough to be clearly preferable to the outcome in question. For large probabilities of winning, the gamble is more valuable than the outcome and will be preferred. For small probabilities, i. e. large probabilities of obtaining nothing, the outcome oi will be chosen. For some probability p, the person will switch from preferring the gamble to preferring the outcome or vice versa. For that probability, gamble and outcome are equivalent, i.e.
v(oi)= u(// p, v, 0 //) or v(oi) = p . v (+ (1-p) . 0).
Negative utilities can be determined in an analogous manner. Consequently the utility of any outcome of a choice appeared to be measurable in terms of money on a ratio scale.
Humans, however, do not behave like economists, otherwise they would neither buy lotteries nor insurances, which are always gambles of negative expected value because part of their price or premium covers expenses for administration, taxes etc. and the profits of the seller.
That discrepancy between normative decision theory and human behaivour was explained at first by the assumption that the subjective utility of money is not a proportional but a concave, i.e. negatively accelerated, function of its amount for losses and a convex, i.e. positively accelerated, function for gains. If so, the comparatively large gains obtainable by winning in a lottery and the comparatively large losses incurred in the case of damage have a larger positive resp. negative subjective utility, in proportion to their monetary values, than the small losses incurred by paying even an unfair price of the lottery or premium of the insurance. That assumption had to be modified as will be shown presently.
Here, we restrict ourselves to discussing the Prospect Theory of Kahneman and Tversky, the most recent and most advanced theory of subjective utility and subjective probability assuming that humans choose among gambles not on the basis of their expected monetary values but by evaluating more or less accurately their expected subjective utilities, i. e. by selecting the gamble with maximum expected utility (instead of value). Let u(v) denote the subjective utility of some monetary value v and (p) the subjective probability or subjective estimate of some outcome probability p of a gamble. Kahneman and Tversky prefer to call the weight of a probability instead of subjective probability, because does not behave like a probability function, as we shall see later. With some exceptions to be discussed, they assume that
u(// p1, v1 / ... / pi, vi / ... / pn //) = (pi) . u(vi).
Before that rule is applied to predict choices between gambles a few Editing Rules simplifying the comparison of alternatives must be taken into account, which subjects apparently apply before comparing and evaluating the gambles of choice:
As reported above, the monetary values of the outcomes are coded as gains or losses, to be added to the present property.
Rounding of Numbers is a natural simplification, e.g. the outcome /.49, 101 / is treated as /.50, 100 /.
At first, Dominated Alternatives will be deleted from Choice, i.e. alternatives yielding an equal or inferior result for every outcome than another available alternative.
The Combination of Outcomes with Identical Results makes subjects simplify e.g. the gamble // .25, 200 / .25, 200 / .50, 0 // to the gamble // .50, 200 / .50, 0 // = // .50, 200 //
The Segregation of the Riskless Component transforms a gamble like // .80, 300 / .20, 200 // into the "gamble" 200 + // .80, 100 // or the gamble // .40, -400 / .60, -100 // into -100 + // .40, -300 //.
Because of the segregation of the riskless component, Strictly Positive or Strictly Negative Gambles must be discriminated from Regular Gambles. If in the gamble // p1, v1 / p2, v2 //, we have p1+p2 = 1 as well as v1 > v2 > 0 or v1 < v2 < 0, then
u(// p1, v1 / p2, v2 //) = u(v2)+u(// p1, v1-v2 //) = u(v2) + (p1).u(v1 - v2). >
It is only in Regular Gambles with either p1 + p2 < 1 or with v1 >= 0 >= v2 or v1 <= 0 <= v2 that u(// p1, v1 / p2, v2 //) = (p1).u(v1) + (p2).u(v2).
Another simplification is the Cancellation of Gamble Components shared by the Choice Alternatives. Comparing e. g. the gambles // .20, 200 / .50, 100 / .30,-100 // and // .20, 200 / .50, 150 / .30 ,-100 // in fact means ignoring the common first component of the gambles and looking for the larger of the values (50).u(100) + (.30).u(-100) and (50).u(150) + (.30).u(-100). A special case of that cancellation is the cancellation of a common bonus or a common first stage in a two-stage game. Examples of two-stage games will be given later.
Now we can discuss the form of the utility function u(v), i.e. the subjective utility u of the amount of money v. To that purpose make a choice between the following two gambles obtainable at the same price:
// .90, 3000 // or // .45, 6000 //,
and similarly between:
// .90, -3000 // or // .45, -6000 //
Most subjects prefer
// .90, 3000 // > //.45, 6000 // and // .90, -3000 // < // .45, -6000 //,
i.e. they prefer a smaller but probable gain to a larger but less probable gain of the same expected value and a larger but less probable loss to a smaller but more probable loss. These preferences are at variance with the assumptions of a negatively accelerated utility function in the negative region and positively accelerated utility function in the positive region of monetary value, which were used above to explain lottery and insurance buying. On the contrary, they point to a positively accelerated utility function in the negative and a negatively accelerated one in the positive region of monetary value as shown in Fig. 1 below. Otherwise we must assume (.90) > 2.(.45)
Now, make a choice between the following two gambles available at the same price:
// .25, 6000 / .75, 0 // or // .25, 4000 / .25, 2000 / .50, 0 //,
and similarly between:
// .25, -6000 / .75, 0 // or // .25, -4000 / .25, -2000 / .50, 0 //.
Most subjects prefer:
// .25, 6000 / .75, 0 // < // .25, 4000 / .25, 2000 / .50, 0 //,
// .25, -6000 / .75, 0 // > // .25, -4000 / .25, -2000 / .50, 0 //.
If we assume that subjects estimate the expected utilities of the gambles more or less accurately in the appropriate manner by multiplying the probability weight and utility of each outcome and summing the products over the outcomes, then the first preference implies the inequality
u(// .25, 6000 / .75, 0 //) < u(// .25, 4000 / .25, 2000 / .50, 0 //)
(.25).u(6000) + 0 < (.25).u(4000) + p(.25).u(2000) + 0
u(6000) < u(4000) + u(2000)
If that result is generalized to u(vi+vj) < u(vi) + u(vj), vi,vj > 0, it is compatible only with a negatively accelerated utility function of money gains.
Similarly, the second preference above, involving probable losses, implies the inequality
u(// .25, -6000 / .75, 0 //) > u(// .25, -4000 / .25, -2000 /.50, 0 //)
(.25).u(-6000) + 0 > (.25).u(-4000) + p(.25).u(-2000) + 0
u(-6000) > u(-4000) + u(-2000)
which, when generalized to u(vi+vj) > u(vi) + u(vj), vi,vj < 0, requires a positively accelerated utility function for decreasing losses, in other words a negatively accelerated grief function for increasing losses.
That result is in accordance with intuition, since a gaior loss of, say, 200 EURO may be felt almost twice as much as a gain resp. loss of 100 EURO, but a gain or loss of 2000000 EURO will hardly be felt twice as much as that of 1000000 EURO and for most of us there will not be much difference between gains or losses of 2000000000 EURO and 1000000000 EURO.
Now, make a choice between the following two alternatives for any values of v that are not small enough to be neglected:
// .50, v / .50, -v // or // 0 //.
Most subjects reject the gamble to the left, implying by the same argument and computation as above u(v) - u(-v) < u(0) = 0. Thus, losses of any amount weigh heavier than gains of the same amount, implying that the utility function is steeper for losses than for gains.
We thus arrive at the following approximate (subjective) utility function for the (objective) value of money (Fig 1)
Fig. 1. Utility function, u(v), of monetary value (gains and losses).
When determining the weighting function (p), i.e. the relationship between a sort of subjective estimate of objective outcome probabilities p, we have to consider some properties of that function:
Make a choice between the following two gambles:
// .33, 2500 / .66, 2400 / .01, 0 // or // 2400 // and
// .34,, 2400 // or // .33, 2500 //
Most subjects prefer // .33, 2500 / .66, 2400 / .01, 0 // < // 2400 //
and // .34, 2400 // < // .33, 2500 // implying together
(.33).u(2500) + (.66).u(2400) + 0 < u(2400) or
(.34).u(2400) < (.33).u(2500) < (1 - (.66)) . u(2400)
(.34) < 1 - (.66) or (.34) + (.66) < 1
This property, when generalized to (p) + (1-p) < 1 was called the Subcertainty of , i.e. the objective probabilities p are underestimated, of course with the exception of (0) = 0, (1) = 1 and with the exception of very small p as will be shown presently.
Choose one gamble from each of the following pairs of gambles:
// .80, 4000 // or // 3000 //
// .80, -4000 // or // -3000 //
// .20, 4000 // or // .25, 3000 //
// .20, -4000 // or // .25, -3000 //
Most subjects prefer
// .80, 4000 // < // 3000 //
// .80, -4000 // > // -3000 //
// .20, 4000 // > // .25, 3000 //
// .20, -4000 // < // .25, -3000 //
while the latter two preferences reflect the relative sizes of the expected utilities of the gambles to be chosen from, the first two do not. (Since expected utilities are reasonably assumed to be monotonously related to expected monetary values, we obtain for the latter two choices u(800) > u(750), and u(-800) < u(-750), but for the first two u(3200) > u(3000) and u(-3200) < u(-3000).) In fact the probabilities of winning resp. loosing in the two gambles above have merely been divided by four to yield the two gambles below. The resulting, at first sight counterintuitive, reversal of the preferences is called the Certainty Effect, referring to the fact that smaller but certain gains are preferred to larger but merely probable expected gains and that larger probable expected losses are preferred to smaller certain losses.
The certainty effect can be attributed to the above subcertainnty of estimated probabilities, or probability weights 0 < (p) < p < 1 as opposed to (1) =1. In the latter four gambles all (p) < p, probably to about the same extent in view of the small difference of the p-values .20 and .25. In the first two choices, however, (.80) < .80 but (1) = 1, and the right hand gamble is more attractive, if (.80) < .75, because in that case the expected utility of the left hand gamble is smaller than that of the right hand gamble, as can be seen from (.80).u(4000) + 0 < .75.u(4000) < u(3000), the second inequality being due to the negatively accelerated utility function. Thus, the right hand gamble can be preferred because of its larger expected utility, although its expected monetary value is smaller, as can be seen from .80x4000 + 0 = 3200 > 3000.
The certainty effect together with the above mentioned editing rule of cancellation of the common first stage in two-stage gambles can have apparently contradictory effects. In a two-stage game the choice for the second stage has to be made before the outcome of the first stage is known.
Choose between the two- stage games differing in the 2nd stage:
1. stage: // .25, second stage game .75, 0 //
2. stage: // .80, 4000 // or // 3000 //
1. stage: // 1000 //
2. stage: // .50, 1000 // or // 500 //
1.stage: // 2000 //
2. stage: // .50, -1000 // or // -500 //
The first choice between two stage gambles is in fact equivalent to a choice between two one-stage gambles discussed before, in which most subjects show a preference for the left alternative
// .20, 4000 // > // .25, 3000 //,
because the probabilities of receiving 4000 or 3000 instead of 0 are .25 x .80 = .20 and .25 x 1.00 = .25 respectively. Nevertheless most subjects show a reversed preference in the two-stage gamble, i.e.
1. stage: // .25, second stage game / .75, 0 //
2. stage: // .80, 4000 // < // 3000 //.
That inconsistency is due to the cancellation of the common first stage of the gamble and the certainty effect operating in its second stage.
In the second and third choice between two stage gambles, all outcomes are exactly are equivalent, since in both cases the expected monetary gain of the left alternative is equal to the certain monetary gain of the right alternative, namely 1500, as can be computed easily. Nevertheless most subjects show opposite preferences
1. stage: // 1.00, 1000 //
2. stage: // .50, 1000 // < // 500 //
1.stage: // 1.00, 2000 //
2. stage: // .50, -1000 // > // -500 //
That inconsistency is due to the cancellation of the first stage, which is common to both alternatives, and the already well known preference for smaller but certain gains and larger but merely probable losses in the second stage.
Next, choose one gamble from each of the subseq uent six pairs:
// .90, 3000 // or // .45, 6000 //
// .90, -3000 // or // .45, -6000 //
// .002, 3000 // or // .001, 6000 //
// .002, -3000 // or // .001, -6000 //
// .001, 5000 // or // 5 //
// .001, -5000 // or // -5 //
Most subjects prefer
// .90, 3000 // > // .45, 6000 //
// .90, -3000 // < // .45, -6000 //
// .002, 3000 // < // .001, 6000 //
// .002, -3000 // > // .001, -6000 //
// .001, 5000 // > // 5 //
// .001, -5000 // < // -5 //
The first two preferences would be expected, although the expected monetary values of the two gambles are the same - at least if the underestimation of the probabilities due to the subcertainty effect of is small, roughly equal or proportional to p - because from the form of the utility function, follows 2.u(3000) > u(6000) and 2.u(-3000) < u(6000). That a mere division of all probabilities by 450 brings about a reversal of the preferences is surprising and is called the Probability Effect. It refers to the fact that in the case of probable gains people prefer a larger probability of gain to a larger gain resulting in about the same expected utility of course. Similarly, in the case of probable losses a smaller probability of loss is preferred to a smaller loss. On the contrary, if gains or losses are very improbable, people prefer a larger gain to a larger probability of gain and a smaller loss to a smaller probability of loss. The probability effect points to another property of subjective estimates of probabilities or probability weights, the Overweighting of Small Probabilities p, i.e. (p) > p for small p as opposed to the underestimation of medium and large probabilities discussed above.
The probability effect even overturns the certainty effect discussed before, which is obtained with intermediate or large probabilities, as can be seen from the reversed preferences in the last two choices above.
Another effect which can be explained by the overweighting of small probabilities is the Probabilistic Insurance Effect referring to the fact that people prefer full insurance to probabilistic insurance. Probabilistic insurance means e.g. 50% coverage of damages or full coverage only every second day for 50% premium. The above form of the utility function should support probabilistic insurance, but the probability effect, i.e. the overestimation of small probabilities of damages, is overriding.
A further property of subjective probability estimates or probability weights is the Subadditivity of Small Probabilities, i.e. (r.p) < r. (p). The effect can be inferred from the intermediate pair of the six choices above. The observed preference implies that for the expected utilities e.g. in the third choice the inequalitiy
(.002).u(3000) < (.001).u(6000)
(.002)/ (.001) < u(6000)/u(3000) < 2
the second inequality in the preceding line being due to the negative acceleration of the utility function. By rearrangement we obtain (.002) < 2. (.001) or (.002) < (.001) + (.001).
The last property of subjective probability estimates or probability weights to be discussed is the Subproportionality of (p), which, too, can be derived from the probability effect. We can always choose two probabilities 0< p1 < p2 <1 and two positive monetary gains v1 > v2 > 0 such that subjects consider the following gambles equivalent:
// p1, v1 // = // p2, v2 //.
If we reduce both probabilities by multiplying them with some sufficiently small constant 0 < r << 1, we would obtain a preference for the right hand gamble because of the above probability effect, i.e.
// rp1, v1 // > // rp2, v2 //>.
Equating the utility of the games u(// p, v //) with their expected utilities (p).u(v) + 0 we obtain
(p1).u(v1) = (p2).u(v2)
(rp1).u(v1) > (rp2).u(v2).
Dividing the lower inequality by the upper one yields
(rp1)/ (p1) > (rp2)/ (p2)
demonstrating that the subjective estimate of a larger probability p1 suffers more from a proportional reduction than a smaller one p2.
We, thus, arrive at the following weighting function (p) or relationship between probabilities and their subjective estimates:
Weighting function (p) or relationship between probabilities and their subjective estimates:
Height of curve shows: Subcertainty (except for small probabilities).
Height of first point shows: Overestimation of small probabilities.
Slope of curve shows: Subadditivity of small probability estimates.
Curvature shows: Subproportionality of all probability estimates.
3.) D e c i s i o n s u n d e r U n c e r t a i n t y
A number of principles have been suggested to guide the choice between alternatives the outcome probabilities of which are unknown. Consider e.g. the choice among the following four alternatives with three or four outcomes of unknown probabilities, an outcome being represented by /, vi / instead of / ?, vi / as would be required in our former notation:
//, -3 /, 3 /, 0 // or //, -5 /, -5/, -5 // or
//, -4 /, -1 /, 6 /, 10 // or //, -2 /, 0 /, -8 /, 13 //.
A principle requiring no information about outcome probabilities is the Deletion of Dominated Alternatives which yield for every possible outcome a worse monetary value or subjective utility than another so called dominating alternative. (Since monetary value and subjective utility are monotonously related, we may refer to one of these measures only, say to monetary value, as long as we make use of ordinal information only.) In our example the second alternative is dominated by the first alternative and can be discarded right away, because it would never be selected or preferred to the dominating one.
Thereafter, an optimist might apply the Maximax Principle to the remaining alternatives and select the alternative possibly providing the highest monetary value. In our example that would be the fourth alternative yielding a monetary value of 13 at best, whereas the respective values for the other remaining alternatives are 3 and 10 respectively. A person applying that principle, however, must be very optimistic indeed, because in our case that alternative may just as well yield a monetary value of -8, i.e. the worst of all outcomes.
A more prudent decision maker may apply the Maximin Principle. That principle prescribes the registration of the value of the worst outcome of each choice alternative, i.e. -3, -4 and -8 in our example (after deletion of the dominated second alternative), and thereafter the selection of the alternative with the largest, i.e. most positive or least negative, of these (worst) values. In our example that would be the first alternative yielding a monetary value of -3 at worst. The maximin principle avoids all negative outcomes that can be avoided and guarantees the decision maker a minimum yield which may be regarded as his value of the choice. It is a pessimistic principle, of course, because its application may cause the miss of many most attractive outcomes, e.g. 6, 10 and 13 in our example or even much higher values that might be associated with these outcomes in other examples.
The maximin principle makes most sense if an adversary determines the outcome of the selected alternative that will occur. That would be the case in zero-sum games like chess or other competitions for which alternatives favouring both adversaries are not available. In such situations each player must be expected to select the outcome that is best for himself and consequently worst for his partner.
The application of the maximin or maximax principle requires only ordinal information about the outcome utilities, cum grano salis that applies to the subsequent Hurwicz- principle as well.
The Hurwicz- Principle is a compromise between the maximax and maximin principle: According to that principle, instead of comparing either the maximal or the minimal monetary values of the choice alternatives, one computes their weighted mean for each alternative, i.e. .u(vmax)+(1-).u(vmin), 0 £ £1, and selects the alternative with the largest of these weighted means. Obviously, - is a measure of optimism. For - = 0 the Hurwicz-- principle equals the maximin principle, for - = 1 it equals the maximax principle. The smaller -, the more unpleasant outcomes are avoided at the expense of missing pleasant ones, the larger -, the more pleasant ones are enabled to occur at the expense of enabling unpleasant ones too.
In our example, - =.5 would give equal weight to the best and worst outcome and yield the means (3 - 3)/2 = 0, (10 - 4)/2 = 3 and (13 - 8)/2 = 2.5 (apart from the mean -5 for the deleted alternative). That would lead to the selection of the second alternative and, thus, would make the best and worst results, 13 and -8, impossible. However, as opposed to the maximin principle, the Hurwicz principle with - = .5 would enable the two next to optimal outcomes to occur, which yield 10 and 6 respectively. On the other hand, it also enables the result of -4 to occur which, however, is not much worse than the result of -3 enabled by the maximin principle as well.
An effect somewhat similar to that of the Hurwicz- (=.5) principle has the Principle of Insufficient Reason which assigns equal probabilities to all outcomes of an alternative as the best guess available in view of the lack of better information. It permits the computation of expected utilities if outcome utilities are measured at least on an interval scale - however with doubtful probability estimates.
Situations in which the principle of insufficient reason is applicable are rare, because even if information about outcome probabilities is not provided, decision makers are likely to make at least crude and possibly erroneous estimates, based on previous experiences or world knowledge in general. Recently, the interest of decision research was focussed on how subjective probability estimates are made, which errors they are prone of, and how these errors come about. People appear to employ simplifying Heuristics which frequently produce acceptable estimates but sometimes fail completely, because they are not based on the laws of probability theory and sometimes are at variance with them.
Before mentioning a few of these heuristics it should be noted that humans are not too bad at estimating relative frequencies and distribution parameters when considering populations or series of events. The errors occur when probabilities of single events are estimated.
The Representativeness Heuristic makes people consider an event x more likely to be a member of an event class A, which is characterised by many of the properties of x, than of a class B characterised by few of them. It considers more typical events as more probable. That heuristic produces erroneous estimates if the former class is a subset of the latter, i.e. AB, and consequently according to probability theory p(xA) <= p(xB). In some survey, e.g., people produced on the average higher estimates when asked for the probability of an atomic war between USA and USSR than when asked for the probability of a war. The representative heuristic is also considered to be responsible for the so called "Law of Small Numbers", referring to the fact that people believe the law of large numbers to hold for small samples as well and therefore underestimate the probability of "untypical" large deviations of small sample statistics from their respective population parameters. An example of the "law of small numbers" is the Gambler´s Fallacy of underestimating e.g. the probability of long series of "red" in roulette and therefore expecting "blue" the more the longer a preceding series of "red" was, although its probability of occurrence is always the same. Another consequence of the representative heuristic is the Neglect of Base Rates. People do use base rates, i.e. relative class frequencies, when estimating the probability of an event belonging to an event class, but they do so only if no information about the event to be classified is given. Otherwise they tend to ignore base rates and estimate that probability on the basis of the similarity between the descriptions of the event and the event class. A man in a white coat, as opposed to a man, is more likely to be identified as a medical doctor than as a soldier although there are much more soldiers around than medical doctors. Finally, people ignore random regression towards mean effects in unreliable performance, i.e. they overestimate the probability of e.g. a person´s future (extremely) good or bad performance after having observed him perform (extremely) well or badly. They consider the previous performance as typical for the future performance and ignore that random fluctuation makes a worse performance more likely than a better one after a good performance and a better one more likely than a worse after weak performance.
The Availability Heuristic relates the size of event probabilities to the easiness or speed of the event coming to mind. That heuristic, too, may be successful in many cases but fail in others. It predicts the well known overestimation of the probability of events receiving extensive coverage by the media such as accidents, disasters, crimes etc. It also predicts that vividly imagined events are judged more likely to occur or to have occurred than unimagined events or events considered merely in abstract terms. That prediction was experimentally confirmed to some but not yet sufficient extent, except for difficult and unpleasant imaginations which can be expected not to come to mind very easily. Perhaps, the experimental finding that the probability of pleasant, desirable events tends to be overestimated and that of unpleasant, undesirable events to be underestimated can be attributed to the availability effect, because, as a rule, pleasant events tend to be remembered more easily.
The Anchoring Heuristic is a mechanism to which a well demonstrated experimental effect is attributed: In the first phase of an experiment some subjects are asked whether a certain measure (price, size, proportion, probability) is larger or smaller than a high value x, and other subjects are asked whether it is larger or smaller than a low value y (x>y). In a second phase, all subjects are asked to estimate the exact value of that measure. The subjects of the first group will give significantly larger estimates than those of the second group. The anchoring effect is robust and sometimes dramatic, if little experience with the measure is involved. Perhaps, it is responsible for the fact that the probabilities of independent conjunctive events, requiring multiplication of the probabilities of the components, tend to be overestimated, whereas those of disjunctive events, requiring addition of the probabilities of the components, tend to be underestimated. People still have the probabilities of the components in mind when estimating the products or sums asked for. The anchoring heuristic also explains the observed quite general conservatism in revising probabilities in view of new experiences as compared with the appropriate revision according to Bayes Rule.
Recently, the psychology of decision making has moved away from examining to what extent people are able to meet the requirements of normative decision theory. It has become obvious that the cognitive demand of classical decision methods is too severe for humans who cannot resort to computer simulation methods. The estimation of outcome probabilities may require information which is too costly to obtain. Interest of decision researchers, therefore, has moved from strategies which people employ to attain optimal decisions to strategies which they might employ to attain satisfactory decisions which require less information to be considered. Frequently, the expenses of the information needed to increase the probability popt of finding the optimal alternative of action is a positively accelerated function of that probability, i.e. an increase from, say, .90 to .95 is more expensive than an increase from, say, .20 to .25. The expected utility popt. uopt + (1- popt). pother, however, increases linearly with popt. Thus we have two thresholds at which the search for information may stop. One is the probability at which the negative utility of the expenses exceeds the additional positive utility of the optimal alternative and the other, smaller threshold is the probability beyond which the negative utility of the expenses begins to increase faster than the additional utility of the optimal alternative. A stop at the second threshold yields the best cost/efficiency ratio of the search for information.
Many decision processes, in fact, consist of sequences of choices, initial, intermediate and final ones, between the different action alternatives available at any one stage. Rarely all choices of such a sequence are made at the time of the first choice and, mostly, not all choice sequences can be considered, because their number increases exponentially with the number of stages. Therefore strategies for reducing the number of alternative sequences to be considered are involved in such decision processes and, consequently, are a topic in decision research.
Finally, the role of memory in decision making has become more and more apparent. Accordingly, models and findings of memory research became more and more prominent in decision research. Recognition Models assume that decision makers perform a memory search for similar decision situations experienced previously, evaluate possible alternatives of action on the basis of learned if-then rules and, if possible, select an alternative that proved (most) successful in the past. The more experience a person has with a type of decision situation, the more he can rely on that recognition mechanism. Similarly, Scenario Models assume that decision makers retrieve sets of relevant information (frames) from memory, construct a network from causal if-then rules retrieved or inferred, assign appropriate starting values to some if-parts of the rules and run the network in order to construct scenarios. Rules and starting values that lead to scenarios which are at variance with past experience are discarded and hitherto apparently plausible scenarios are extended into the future in order to predict the likely results of alternative choices of action. Story models are sorts of scenario models for past events that attempt to predict jury members verdicts from their fitting incoming information in the course of a trial into story-schemas, a process which has been studied extensively in the psychology of verbal memory. Argument Driven Action Models assume that alternative courses of action are evaluated by confronting them with arguments in favour and against. Mostly these models have been developed by studying how decisions are described, explained and defended afterwards. In an experiment on personnel hiring, candidates were matched against 16 job requirements. There was a threshold of about four requirements that a candidate could violate before being finally rejected. No such threshold was found for accepting a candidate or retaining him on the list of candidates.
R e f e r e n c e s
Beach, L. R.: The Psychology of Decision Making, Thousand Oaks, Sage Publications, 1997.
Plous, S.: The Psychology of Judgement and Decision Making, New York, McGraw-Hill Inc., 1993.
Rapoport A.: Decision Theory and Decision Behaviour, Dordrecht, Kluwer Academic Publishers, 1989.