2. Introduction to Individual Decision Making

2. Introduction to Individual Decision Making

2.1 Basic concepts: Preferences and utility 2.2 Choice under uncertainty: Lotteries and risk aversion 2.3 Value of information: Decision trees and backward induction Outline

2.1Basic concepts: Preferences and utility

2.1 Basic concepts: Preferences and utility 2.2 Choice under uncertainty: Lotteries and risk aversion 2.3 Value of information: Decision trees and backward induction Outline

Every choice we make is based on our preferences and our constraints: Among all the available alternatives we choose the most preferred one 2.1Basic concepts: Preferences and utility • We buy the car we like, and that we can afford • In a buffet, we choose the meals we like

Were do they come from? Do they make sense Why do I like chocolate and hate seafood? When do they form? Are they stable? Can I change myself? Preferences ...

What type of preferences can we work with ? Many disciplines have theories on preferences (psicology, sociology, economics, anthropology) We will take “preferences” asgiven

To assume that individuals are able to completely rank large sets of alternatives is, probably, very unrealistic • Although the final choice has to be made from within a set of many alternatives, it suffices to require individuals to be able to express their preferences in binary choices Given two different alternatives a and b, • Do I like a better than b ? • Do I like b better that a ? • Do I like both the same ?

Is this enough ? Example 1: Consider a situation in which there are only 3 alternatives: a, b, and c Suppose that the “declared” preferences are: • I like b better than a • I like c better than b • I like c better than a • What will be the final choice among the • three alternatives ?

Is this enough ? Example 2: Consider a situation in which there are only 3 alternatives: a, b, and c Suppose that the “declared” preferences are: • I like b better than a • I like c better than b • I like a better than c • What will be now the final choice among • the three alternatives ?

Restrictions on Preferences 1. Complete Given any two alternatives a and b, etiher a is preferred to b, b is preferred to a, or they are indifferent That is, no answers like: I do not know . . . It depends . . . Uf . . .

Restrictions on Preferences 2. Transitive Given any three alternatives a, b, and c, if a is preferred (or indifferent) to b and b is preferred (or indifferent) to c, then a should be preferred (or indifferent) to c That is, no “cycles” like: • I like b better than a • I like c better than b • I like a better than c

Question . . . Does it make sense to have “transitive indifferences” ?

2. Order the options from best to worse Cheese Ham Bread Onion Eggs Butter Test your preferences • 1. Which one do you prefer? • Cheese or ham • Butter or bread • Onion or bread • Butter or onion • Ham or eggs • Cheese or eggs

Regular Preferences Preferences that are both Complete and Transitive are called Regular Preferences If we knew the (regular) preferences of an individual, we could reproduce the ordered list (ranking) of alternatives according to her “tastes” (preferences) From such list we could predict what would be the best choice for the individual But . . . It might be a lot of work !!

Utility Function In complex scenarios (many alternatives), working with rankings of preferences might be useless ! Having a different representation of preferences might be useful.

Utility Function Decision Theory uses the so-called Utility Function as a workable instrument to represent individuals' preferences Utility is a term that should be understood as . . . Happiness Satisfaction Wellbeing . . .

Utility Function In this sense, the following sentences have the same meaning in Decision Theory: I like a better than b I am happier with a than with b I am more satisfied with a than with b The Utility of a is bigger than the Utility of b Or . . . in a more mathematical way . . . u(a) > u(b)

Utility Function The Utility Function is a mathematical formula that represents the preferences of an individual in the following sense: If a is preferred to b, then u(a) > u(b) If b is preferred to a, then u(b) > u(a) If a and b are indifferent, then u(a) = u(b) and vice versa, If u(a) > u(b), thena is preferred to b If u(b) > u(a), thenb is preferred to a If u(a) = u(b), thena and b are indifferent

Question: Are you sure that you can “capture” what is inside the human mind by means of a simple mathematical formula ?Answer:It depends . . . (on what these preferences are)

Example Think of an individual that loves traveling. She does not care where, the more days traveling the better. She has the opportunity to travel to Paris and/or London Given the following alternatives: a=(3,4), b=(1,7), and c=(5,2) (where (x,y) means “x days in London and y days in Paris”) What is her ranking ? Which one would be the preferred one ?

Example a=(3,4) 7 days traveling b=(1,7) 8 days traveling c=(5,2) 7 days traveling Thus, her ranking would be b aandc

Example Could you “capture” such preferences in a simple mathematical formula ??

Yes ! u(x,y) = x + y So, • u(a) = u(3,4) = 3 + 4 = 7 • u(b) = u(1,7) = 1 + 7 = 8 • u(c) = u(5,2) = 5 + 2 = 7 • b is the best alternative for this individual (highest utility). • Then, a and c are tied in second place

Question: What about other cases ?Answer:Once again, it depends. In general, yes !

Mathematical Fact from Decision Theory If the preferences of an individual are regular (complete and transitive), then they can be represented by a Utility Function

Caveats Decision Theory does not claim that individuals behave according to some utility function Decision Theory says that individuals behave “as if” they knew his or her utility function

Caveats Justthe same as . . . Billiard (pool) players do not behave according to the laws of trigonometry Billiard players behave “as if” they knew the laws of trigonometry

Utility can’t be measured: there is no such thing as an “utilitometer” • The scale of measure, the absolute value of the utility, is not important • What matters is the ordering induced by the utility scores: the higher the utility the higher the preference • Graphical representations are useful in simple cases

Example: Utility of Water Utility increase utility water water • When you are thirsty, more water means more utility • Each additional unit of water is less valuable that the previous one

Example: Utility of Money Similarly with other items ... Utility increase utility money money

Decision Theory: main message “When facing an individual decision problem, rational individuals will choose the feasible alternative that yields the highest utility”

Summary: • Regular Preferencesare simplified representation of individuals’ tastes • Utility functionsare convenient mathematical representations of individuals’ preferences • Individuals act as ifthey were choosing the feasible alternative that provides the highest utility

2. Introduction to Individual Decision Making