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Game Theory: Preferences and Expected Utility

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Game Theory:Preferences and Expected UtilityBranislav L. SlantchevDepartment of Political Science, University of California – San DiegoApril 4, 2012Contents.1 Preferences 22 Utility Representation 43 Choice Under Uncertainty 53.1 Lotteries.................................... 63.2 PreferencesOverLotteries .......................... 83.3 The Expected Utility Theorem ........................ 123.4 How to Think about Expected Utilities . ................... 203.4.1 Ride’n’MaimExample........................ 203.4.2 An Example with Multiple Outcomes . . .............. 223.5 Expected Utility as Useful Fiction . . . ................... 274 Risk Av ersion 301 PreferencesWe want to examine the behavior of an individual, called a player, who must choose fromamong a set of outcomes. Begin by formalizing the set of outcomes from which this choiceis to be made.Let X be the (finite) set of outcomes with common elements x;y;´. The elements of thisset are mutually exclusive (choice of one implies rejection of the others). For example, Xcan represent the set of candidates in an election and the player needs to chose for whom tov ote. Or it can represent a set of diplomatic and military actions—bombing, land invasion,sanctions—among which a player must choose one for implementation.The standard way to model the player is with his preference r elation , sometimes calleda binary r elation . The relation on X represents the relative merits of any two outcomes forthe player with respect to some criterion. For example, in mathematics the familiar weakinequality relation, ’’, defined on the set of integers, is interpreted as “integer x is at leastas big as integer y” whenever we write x  y. Similarly, a relation “is more liberal than,”denoted by ’P ’, can be defined on the set of candidates, and interpreted as “candidate x ismore liberal than candidate y” whenever we write xP y.More generally, we shall use the following notation to denote strict and weak preferences.We shall write x y whenever we mean that x is strictly preferred to y and x  y when-ever we mean that x is weakly preferred to y. We shall also write x  y whenever we meanthat the player is indifferent between x and y. Notice the following logical implications:x y , x  y ^:.y  x/x  y , x  y ^ y  xx  y,:.y x/:Suppose we present the player with two alternatives and ask him to rank them according tosome criterion. There are four possible answers we can get:1. x is better than y and y is not better than x2. y is better than x and x is not better than y3. x is not better than y and y is not better than x4. x is better than y and y is better than xAlthough logically possible, the fourth possibility will prove quite inconvenient, so weimmediately exclude it with a basic assumption.ASSUMPTIO N 1. Preferences are asymmetric: There is no pair x and y from X such thatx y and y x.We shall also require the player to be able to make judgments about every option that isof interest to us. In particular, he should be able to compare a third option, ´, to the originaltwo options. This assumption is quite strong for it implies that the player cannot refuse torank an alternative.2ASSU MPTION 2. Preferences are negatively transitive:Ifx y, then for any third ele-ment ´, either x ´,or´ y, or both.To understand what this assumption means, observe that it requires the player to rank ´with respect to both x and y. The easiest way to illustrate this is by placing the alternativesalong a line such that x y implies x is to the right of y. Then, we have the three possiblerankings of ´ from the assumption: (i) .x ´/ ^:.´ y/, (ii) .´ y/ ^:.x ´/;and(iii) .x ´/ ^ .´ y/. Each is shown in the picture in Figure 1.yx.´ y/ ^:.x ´/.x ´/ ^:.´ y/ .x ´/ ^ .´ y/Figure 1: Illustration of Negative Transitivity.As you can see in , this covers all possibilities of placing ´ somewhere along that line.For example, if (i) is true, then ´ is the left of y, if (ii) is true, then ´ is to the right of x,andif (iii) is true, ´ is between y and x.There are several properties that we now define and all of which are implied by the twoassumptions above. The three properties are:1. Irreflexivity: For no x is x x.2. Transitivity: If x y and y ´,thenx ´.3. Acyclicity: If, for a given finite integer n, x1 x2;x2 x3;:::;xn1 xn,thenxn¤ x1.Let’s prove that transitivity is implied by negative transitivity and asymmetry. Suppose thatthe two properties are satisfied, and assume (a) x y,and(b)y ´. Prove x ´.1. :Œ´ y from (b), asymmetry2. Œx ´ _ Œ´ y _ŒŒx ´ ^ Œ´ y from (a), negative transitivity3. Œx ´ _ ŒŒx ´ ^ Œ´ y from (1) and (2), disjunctiv e syllogism4. ŒŒx ´ _ Œx ´ ^ ŒŒx ´ _ Œ´ y from (3), distribution5. Œx ´ _ Œx ´ from (4), simplification6. x ´ from (5), tautologyBy the rule of conditional proof, we’re done. Hence, asymmetry and negative transitivityimply transitivity. The follo wing proposition states some other implications of these twoassumptions for the weak and indifference preference relations.PRO POSITION 1. If ‘ ’ is asymmetric and negatively transitive, then31.  is complete: For all x; y 2 X; x ¤ y, either x  y or y  x or both;2.  is transitive:Ifx  y and y  ´,thenx  ´;3.  is reflexive: For all x 2 X, x  x4.  is symmetric: For all x; y 2 X, x  y implies y  x;5.  is transitive:Ifx  y and y  ´,thenx  ´;6. If w  x;x y, and y  ´,thenw y and x ´.2An important concept we hear a lot about is that of rationality. Here is its precise definitionin terms of preference relations:DEFINITION 1. The preference relation  is rational if it is complete and transitive.This means that gi ven any two alternatives, the individual can determine whether helikes one at least as much as the other (completeness) and no sequence of pairwise choiceswill result in a cycle (transitivity). As the above proposition shows, these two propertiesare implied by the more basic asymmetry and negative transitivity of the strict preferencerelation. This is the precise definition of rationality we shall use in this course.If you are interested in social choice theory, the next step is to examine conditions thatpreference relations must meet in order for the set X to have maximal elements.