September 2 Exercises: Problem 2 (p. 21) Efficiency: p. 28-29: 1, 4, 5, 6 0.0.2 Pareto Efficiency (Sec. 4, Ch. 1 of text) We discuss here a notion of efficiency that is rooted in the individual preferences of group members over choices that may be made for the group. The word "efficiency" has other meanings in economics, such as "cost minimization" or "optimization." In the above example, for instance, it is efficient for the government to assign the task of reducing emissions to the firm that can do so at the lowest cost. Example 4 Suppose there are 3 people (1,2,3) and 4 possible choices of a restaurant as a choice for dinner for the group of three people: I (Italian), C (Chinese), S (seafood), and T (Thai). The three people rank the 4 restaurants as follows: 1 I C S T 2 I S=T C Here, " " indicates strictly prefers while "=" indicates that the individual is indifferent between the two alternatives. Q: What are the efficient choices of restaurants? The point of answering this question is to illustrate what economists typically mean by "efficient." Definition. Suppose a group of people can choose from a set of alternatives. Alternative is efficient if it is not possible to switch to some other alternative and in the process make some people in the group strictly better off and without making anyone strictly worse off. An alternative with this property is often referred to as Pareto efficient or Pareto optimal, after the 19th century Italian economist Vilfredo Pareto, who came up with the idea. I personally prefer this terminology, which is more precise than simply "efficient". As noted above, "efficiency" has other meanings in economics. If an alternative exists such that is ranked at least as good as by every person, and at least one person strictly prefers to,then is said to Pareto dominate. 1. Is I efficient? Yes, because switching to either of C, S, or T would make person 1 (and person 2) strictly worse off. 2. Is C efficient? Looking at person 1 s preferences, we can see that the only possible improvement from his perspective is switching from C to I. Switching to I from C, however, makes person 3 strictly worse off. It is therefore not possible to switch from C to some other alternative and make some people strictly better off without hurting some other person. Yes, C is efficient. 3. Is S efficient? Looking at person 1 s preferences, we can see that the only possible moves that would not hurt him are I and C. Moving from S to I would make persons 1 8
and 2 strictly better off and it would not hurt person 3. No, S is not efficient. 4. Is T efficient? No, because every person is made strictly better off by switching from T to I. Incidentally, it is also also true that switching from T to S shows that T is not efficient. We conclude that I and C are efficient but that S and T are not. What does this mean? First, it would be really dumb for the group to go out together to either the seafood (S) or the Thai (T) restaurants. Second, efficiency does not help the group to choose between I and C. You might be tempted to say, "But 1 and 2 rate I as their best choice, and 3 rates I as second best; 2 in fact ranks C as his worst choice!" But this argument presumes either (i) a method by which the choice is made such as majority rule, or (ii) the assumption that each individual s well-being counts equally. As to (ii), it bears emphasizing that this notion of efficiency respects the preferences of each individual. There is no weighing of one person s interests against anothers. Efficiency rules out the stupid options but leaves the problem of weighing one person s interests against another s in selecting among the efficient choices. Notice also that I and C are each rated as strictly better than every other choice by at least one individual. This is a sufficient condition for an alternative to be efficient in the above sense: if some individual ranks some alternative as strictly better than every other alternative, then it is clearly impossible to switch to some other alternative without hurting that individual. Consequently, it must be efficient. Example 5 Can an alternative be efficient even if every individual has some other alternative that he thinks is strictly better? Yes, though not in the above example. Consider 1 C I S T 2 S=T I C This changes the above table by lowering I in the rankings of persons 1 and 2, switching it with the next best alternative(s) for those individuals. Notice that I remains efficient: looking at person 1 s ranking, the only possible improvement would be to switch from I to C, but that would make person 2 worse off. Since 3 ranks C as strictly best, it is also efficient. Consider S. Looking at 2 s ranking, we see that a switch to T is the only possible way to not hurt him. But this would strictly hurt both persons 1 and 2, and so S is also efficient. Finally, what about T? A switch from T to S doesn t hurt person 2, but it makes persons 1 and 3 strictly better off. Therefore, T is not efficient in this second table. This makes an important point: if some individual ranks an alternative as best but also equivalent to some other alternative,then is not necessarily efficient. 9
0.0.2.1 Efficiency with Numerical Representation of Preferences (Utility) Example 6 We will typically assume that each individual assigns a number to each alternative so that his ranking of two alternatives simply reflects the relative size of the two numbers. This assumes that each agent has a utility function over the set of alternatives. Recall the first example from above: 1 I C S T 2 I S=T C There are many utility functions that represent each of these person s preferences. Here s one set: person utility values 1 1 ( ) =5 1 ( ) =4 1 ( ) =3 1 ( )=2 2 2 ( ) = 200 2 ( ) =0 2 ( ) = 100 2 ( ) = 100 3 3 ( ) = 10 3 ( ) = 5 3 ( ) = 10 3 ( )= 50 Q: Where did these numbers come from? A: I just made them up. The ordering of the utility numbers for each individual is the same as the ordering of restaurants in the above table. This is the point of representing preferences with a utility function. Q: Are there other possible representations of these same preferences? A: Of course. Q: What do the numbers mean? A: In general, nothing. We don t necessarily interpret utility as measuring something. No units are specified. In particular, the fact that 2 ( ) = 200 and 1 ( ) =5does not mean that person 2 likes Italian food 40 times as much as person 1! Utility numbers do not necessarily measure anything, though in some examples they do. Q: What s the point of numerically representing preferences with utility functions? A: There isn t much point to it in this example. More generally, however, utility representations can allow you to use the mathematics of functions to analyze problems. Utility therefore facilitates the use of mathematics to analyze economic problems. Assume we have some utility representation of each individual s preferences. An alternative is Pareto efficient if it solves the maximization problem ( ) max Here, denotes the number of individuals in the group and ( ) denotes the utility function of individual. An alternative can, however, be efficient even it fails to maximize this sum. This is a second sufficient condition for Pareto efficiency (the first, discussed above, is that some person rank the alternative as strictly better than any other alternative). 10
Proof. Suppose ( ) ( ) for all (1) We prove the result by contradiction. Assume that is not Pareto efficient. This means that there exist some such that Pareto dominates,i.e., ( ) ( ) for each person, with the inequality strict for at least one. Therefore, ( ) ( ) which contradicts (1). Exercise. There are 4 people (1, 2, 3, and4) and 3 alternatives (,, and ). The people rank the alternatives as follows: 1 a b c 2 b a c 3 a=c b 4 c=b a What are the Pareto efficient alternatives? 0.0.3 Chapter 1, Section 5 Exercises: Nash equilibrium and dominant strategy equilibrium: p. 44: 1, 3, 4 Chapter 1, Section 5 of Campbell s book is an introduction to noncooperative game theory. "Noncooperative" here means that each individual is assumed to care only about his own well-being, i.e., everyone is selfish. A game requires the specification of the players, the actions available to each player, and the order in which actions are taken. We might consider this to be the rules of the game. Games are interesting when some outcome affects the well-being of the different agents (i.e., what any player receives depends upon the actions of all players). solution concept: A story or theory of what happens when the game is played a notion of equilibrium. Solution concepts typically depend upon what the players know and different senses in which an action can be interpreted as in the best interest of a player (e.g., does the optimality of an agent s strategy depend upon some specification of strategies for the other agents?) game theory: precise language of incentives. Today we ll discuss two solution concepts: dominant strategy equilibrium and Nash equilibrium. 0.0.4 Dominant and Dominated Strategies Example 7 Prisoner s Dilemma 11
1 2 2 2 10 1 1 10 5 5 The choice C is a dominant strategy for each of the two players. The prisoner s dilemma is a classic example because it illustrates the conflict between the best outcome for a group (here, 2 2) as compared to the outcome that results when every person pursues his own self-interest ( 5 5). It has been used to model arms races between nations. Definition of a dominant strategy: 0 is a dominant strategy iff for and, ( 0 ) ( ) A player can have more than one dominant strategy with this definition, though any two of his dominant strategies must be payoff-equivalent in the sense of providing him with exactly the same utility for all choices of his opponents strategies: if 0 and 00 are both dominant strategies, then ( 0 )= ( 00 ) for all. strictly dominate, (weakly) dominate, (weakly) dominated, strictly dominant, strictly dominated. Recall the VCG mechanism that was discussed last week. Honest reporting is a dominant strategy for each firm, i.e., the honest report maximizes a firm s profit or payoff given the reports of the other firms, regardless of the specification of those reports by the other firms. 0.0.5 Nash Equilibrium An -tuple ( 1 ) of pure strategies is a pure strategy Nash equilibrium if, for each player, ( ) ( 0 ) for all other pure strategies 0 of player. This idea is due to John Nash (1951). Notice that a dominant strategy equilibrium is necessarily also a Nash equilibrium. Example 8 MeetinginNY: 1 2 100 100 0 0 0 0 100 100 Example 9 Battle of the Sexes 2 1 0 0 0 0 1 2 12