F4160-2 - 2 Notes on the Theory of Choice 2.1 Choice under...

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2. Notes on the Theory of Choice 2.1. Choice under Certainty In this section we review the basic aspects of decision making under certainty. The objective is to develop a logical coherent model for decision making based on a minimal set of axioms (“atoms” of theory formulation). The fundamental question asked is, given a set of alternative choices, how does an individual determine an optimal choice? 2.1.1. Elements of the Decision Problem Whereas the objective and question asked seem to be almost trivial, an abstract rigorous treatment is extremely simpli fi ed by the introduction of some mathematical elements. De fi nition 1 (Choice Problem). A choice problem is a triple ( X , R , O ) where the three elements are given as follows: Choice set X : Set of possible alternatives, which can be fi nite, countable or uncountable. It can be an abstract set or a topological space. Relation operator R : Device to sort alternatives. Choice criterion O : Rule to discriminate between ordered elements of choice set. A relation operator does de fi ne an ordering on the choice set. If for X, Y X we have that X R Y then the relation operator establishes a property that holds true between the choice elements X and Y . Such a property therefore does establish a binary relation between two elements. Typical examples of the relation operator R will be “weakly preferred” or “indi ff erent”. Other important examples in the theory of decision making will be “more risky”, etc. Those among the readers trained in mathematical analysis will clearly recognize the relation with equivalence relations established on sets of numbers. The most universal choice criterion in neo-classical economics and fi nance is optimality. We quite often will assume that an individual will choose from all elements of the choice set the optimal one, where optimality will be de fi ned by a relation operator that acts on the choice set. Example 1 (Project Evaluation / Capital Budgeting). This problem is almost trivial but let us describe the di ff erent elements: • X : Possible projects available characterized by their cash fl ows. • R : Net present value rule. For X, Y X , X R Y NPV ( X ) NPV ( Y ) . • O : Among projects with positive NPV do the one with highest NPV. This example shows that the de fi nition of the choice set and especially the choice criterion is straightforward once the relation operator is de fi ned. In the next section we discuss some properties of relation operators. 1
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2.1.2. Binary Relation Operators The following list gives the most common relation operators that we use in fi nancial economics. This relation operator are called binary since they act only on two alternatives at a time. 1. “Strictly preferred” Â : For X, Y X , X Â Y X strictly better than Y ”.
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