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Unformatted text preview: Economics 101, UCLA Fall 2010 Jernej Copic Lecture 6, 10/16/10. Notes on the equilibrium principle and Pareto efficiency; For a moment our story changed to a direct commentary. 1. The equilibrium principle. We have seen examples of equilibria of various games. Before moving on, let's formulate a general way to state that a "social situation" is in equilibrium, call this the equilibrium principle. An equilibrium situation is one in which no individual can benefit by changing his/her own behavior given available information. This principle applies entirely generally, not only in games but also in other models of social interaction (later we will see that in the context of general equilibrium), as long as economic individuals are able to decide what is the best thing to do given their objective (aka utility function). In other words, none of the economic individuals could benefit by unilaterally acting differently than they do. Important! Of course it could still be that everyone together could benefit if they could somehow coordinate their behavior. Unfortunately, there is no guarantee that such joint coordination would be in equilibrium - for example, in the game of a study group (aka prisoners' dilemma) both players would benefit if they could somehow agree to play the strategy where each studies. But such agreement would not be in equilibrium, as each would have an incentive to change his own behavior to not studying. For a second, let's just focus on the last part of this principle: "given available information." Remember for instance the sequential game between Jeremiah and No1. There, when No1 decides his action, he has no particular information. Jeremiah, however, when it is his turn to take his action, knows what No1 did. Thus, in that case "available information" for Jeremiah would be the history, or in what situation he finds himself is what No1 did (remember that Jeremiah's strategy specifies what he does for every possible situation or history - in our narration, that was embodied in the instruction set that Jeremiah gives to Clodoveo). In a model of a simultaneous-move game, none of the players has any particular information on which to base his decision, so we can then simply disregard the last part of the principle. If we take a mixed equilibrium of a game, as we have seen in the example of the coordination game, then none of the players could benefit by changing their behavior. That is because in the mixed equilibrium, each player is indifferent between both of his actions 1 so it would make no difference for him if instead of flipping a coin between the actions, he took either one for certain (and it would for the same reason make no difference to him if he played them with different probabilities - the only problem is that then the other player would no longer be indifferent so that would not be an equilibrium situation). In any case, none of the players can benefit by changing his/her own behavior. Of course one could ask: "but why this whole fuss around the equilibrium in Economics?" Put yourself in the shoes of a LA city commissioner for transportation, and suppose you were given the assignment to fix the traffic jam problem (In fact, in a moment, we will put ourselves in the shoes of LA city police supervisor). Now, if the final policy prescription were made under the assumption of behavior by the drivers which was not equilibrium behavior, then the "fix" would be a mere wishful thinking since the drivers would do whatever they wanted, not whatever the commissioner assumed they would. For sooner or later the drivers, and humans in general, will figure out whatever is best for them and do that. The equilibrium behavior is therefore an important concept to keep in mind for any situation involving human behavior because in an equilibrium each person really does what is best for them given what the others are doing. 2. Pareto efficiency. Another important concept of Economics is the concept of how well off everyone is in a particular arrangement. For if we could all agree to arrange things differently, then why not? The concept which formally describes this is called Pareto efficiency (after the Italian economist Vilfredo Pareto 1848-1923) and is also very simple to state completely generally: A situation is Pareto optimal if in order to improve the well being of one individual, another one must necessarily be made worse off. To understand why this is a sensible principle, think about the opposite, and forget for a moment about the equilibrium question. Meaning, forget about the fact that each person is do what was best for them and that and take a situation in which everyone could be made better off by coordinating their action and achieving a different outcome. Well then: if everyone could be made better off, then everyone would agree to such a change (of course, keeping in mind that they would all be somewhat naive in not worrying about the fact that people may choose to behave differently since in their meeting they do not consider the equilibrium problem). So we can paraphrase our Pareto efficiency. If a situation is a Pareto-optimal one, then for any proposed change at least one of the individuals would object to it (the one who would be hurt by such change) so that the individuals could never (unanimously) all agree to a change from a Pareto-optimal situation. 2 To make things a bit clearer, suppose that No1 told two members of his team, Alan and Bob, that he would give them some cash on hand, and they were to choose between four different possibilities (the first number represents cash payment to Alan, and the second one the payment to Bob): ($1, $1), ($25, $0), ($0, $25), ($20, $20), ($23, $0). The only condition that No1 requires is that they both agree to the arrangement. Alan and Bob would clearly be silly if they jointly decided to take $1 each - they could instead just take $20 each. In other words, the arrangement ($20,$20) Pareto dominates the ($1,$1) since both Bob and Alan are better off in the former. Similarly, Alan and Bob would be silly if they took the payments ($23,$0) since they could take ($25,$0) which would improve Alan's well-being without any cost to Bob (assuming that Bob cares only about the money he gets, as an economist would, and holds no spite or grudge regarding things which are to him irrelevant - thus since he gets $0 in either case, he would not object). Any of the other three arrangements is Pareto optimal: for example, if we take the arrangement ($0,$25), Bob would object to changing from that regardless of what change (among those available) were proposed. One could also tell a story as to how such an arrangement would come about: for instance if Bob were the one in control, by holding a gun to Alan's head, or by having some magical power over Alan, then there would simply be no discussion and Bob's consent would be the only important issue. On the other hand, if Alan and Bob both agreed to maximize the total sum of their utilities, then ($20,$20) would be the arrangement they would choose. For a chicken economist Jeremiah who is just a careful philosopher and is afraid to take any fire from either side, the most prudent thing to do is to simply enumerate which arrangements are Pareto optimal without making any particular recommendation of which one of them should be chosen. If Jeremiah were to choose one of them, he would be taking a stand on welfare redistribution (and thus become more of a social activist rather than a non-paternalistic economist), and either Alan and Bob may see a reason to object. Indeed, if Bob held a gun, he may want to shoot Jeremiah first for voicing heretic thoughts. Let's state this as a paraphrase of Pareto efficiency: All Pareto efficient allocations are precisely those which a non-paternalistic economist could put on the table as all reasonable arrangements if he didn't want to take a stand on welfare redistribution. But do not forget that in this example, we did not think of any games or equilibria thereof and just thought purely of the distribution of resources (or welfare). Of course, for a more sophisticated economist, the right question to ask is the following: 3 If there is an interaction (a game) between the individuals in this economy, which outcomes are the equilibrium outcomes of this interaction and which of these are Pareto efficient. In any allocation satisfying both of these two conditions, one can rest assured that the individuals would not want to change their behavior, and that it would be impossible to simultaneously improve everyone's welfare. Going back to the previous example, No1 may be a clever cat. He could take the first four arrangements and instead of just having Alan and Bob simply agree to one of them, he could have them play a game which he cleverly constructs. Alan and Bob each have two actions available, and each has to secretly write his action in the game on a piece of paper and hand it over to No1 - thus this is a simultaneous-move game between Alan and Bob. The payoffs they obtain are as follows (you will recognize this as the prisoners' dilemma, Alan is player 1): if they play (A1 , A1 ), they get $1 each; if the actions turn out to be (A1 , A2 ) 1 2 1 2 they get payoffs ($25,$0); if they play (A2 , A1 ) they get payoffs ($0,$25); and if they play 1 2 1 1 (A1 , A2 ), they get $25 each. Then we already know what would happen! Namely, it is a dominant strategy for each of them to play their first action so that (A1 , A1 ) is the only 1 2 Nash equilibrium of this game, and as much as they may wish to get something better, in equilibrium they will get $1 each. The moral of this story is that when we think of how well off our economic individuals are in different arrangements, Pareto efficiency is a good way to compare these. But when these are actually outcomes of some game or some other interaction we may first want to ask the question of what are the equilibria of this interaction, and then only consider the equilibrium arrangements as relevant to our consideration in the first place. Thus, for example in a prisoners' dilemma game, there wouldn't really be much of a choice, and unfortunately, the only equilibrium outcome is the only outcome which is not Pareto optimal. Comparing the outcome of a mixed-strategy equilibrium. What about when an outcome of a game is a result of some randomness, for instance when it is an outcome of a mixed-strategy equilibrium? Then we can simply consider the expected utility of each player in such mixed-strategy equilibrium, when making the Pareto comparison. For example, take the following game, which you will recognize as a version of the battle of the sexes (only that we have substituted y as a parameter for one of the payoffs; assume that y > 0): (A1 , A1 )...u1 = y, u2 = 1, 1 2 (A1 , A2 )...u1 = 0, u2 = 0, 1 2 2 (A1 , A1 )...u1 = 0, u2 = 0, 2 (A2 , A2 )...u1 = 1, u2 = y. 1 2 It has two pure-strategy equilibria, (A1 , A1 ), and (A2 , A2 ). Clearly, these two do not Pareto1 2 1 2 4 dominate each other, regardless of y - for instance, if y > 1 then player 1 would object to changing from the first one to the second one and player 2 would object to the opposite change (note that if y = 1, then players would obtain the same payoff in both equilibria so that these two equilibria would again not Pareto-dominate one other). Now let's compute the mixed-strategy equilibrium. Suppose that player 1 puts probability p on A1 , and (1 - p) on A2 , and player 2 plays action A1 with probability q and A2 1 1 2 2 with (1 - q). Then if player 2 were to play A1 his payoff would be 2 u2 (pA1 + (1 - p)A2 , A1 ) = 1p + 0(1 - p) = p, 1 1 2 and if he were to play A2 , he would obtain 2 u2 (pA1 + (1 - p)A2 , A2 ) = 0p + y(1 - p) = y(1 - p). 1 1 2 Since we assumed in the first place that he is choosing randomly between these two, it must be that he is indifferent (otherwise he would simply choose the one that he strictly prefers). Hence, in a mixed-strategy equilibrium, u2 (pA1 + (1 - p)A2 , A1 ) = u2 (pA1 + (1 - p)A2 , A2 ), 1 1 2 1 1 2
y so that p = y(1 - p), and p = 1+y . Similarly, player 1 must be indifferent between his two 1 actions in the mixed strategy equilibrium, which gives us yq = (1 - q), so that q = 1+y . Now y 1 notice also that since 1+y + 1+y = 1, we have that q = 1 - p. Now let's compute the expected payoff to each of the two players in this mixedstrategy equilibrium. To do this, interpret their mixed strategy as a result of a flip of a coin. y Player 1 has a coin which gives heads with probability p = y+1 , and he chooses his action according to the coin flip: if the result is heads then he plays A1 , else he plays A2 . Player 2 1 1 1 has his own coin which gives heads with probability q = 1+y , and implements his strategy in a similar way. Now take for instance the outcome (A1 , A1 ). In the equilibrium we constructed, 1 2 2 y 1 1 this outcome occurs with probability p q = 1+y = 1+y y (since it happens only 1+y when both private coins simultaneously happen to show heads). Similarly, outcome (A1 , A2 ) 2 21 2 1 1 2 2 1 happens with probability p (1 - q) = 1+y y , (A1 , A2 ) with (1 - p) q = 1+y , and 2 1 2 2 (A1 , A2 ) with (1 - p) (1 - q) = 1+y y. Finally, we can compute the expected utility of player 1, EU1 (s , s ) = pqu1 (A1 , A1 )+p(1-q)u1 (A1 , A2 )+(1-p)qu1 (A2 , A1 )+(1-p)(1-q)u1 (A2 , A2 ) 1 2 1 2 1 2 1 2 1 2 5 = 1 1+y 2 yy+ 1 1+y 2 y1= 1 1+y 2 y (y + 1) = y . y+1 For player 2, such computation would give us EU2 (s , s ) = 1 2 y , y+1 y so that before the players flip their coins this equilibrium gives them a utility of y+1 each. But notice that if y > 1 then in this mixed-strategy equilibrium, each player obtains less than in either of the pure strategy equilibria, so this equilibrium is not Pareto optimal. If y y y 1, then y + 1 > 1, so that y+1 < y and y+1 < 1, so that again this mixed-strategy equilibrium makes both players worse of than either of the pure-strategy equilibria. In other words, in this particular game, the mixed-strategy equilibrium is not Pareto optimal among the equilibria of our game, regardless of the value of y. Important! It may be that a mixed-strategy equilibrium is a Pareto optimal one. For example, when a game only has a mixed strategy equilibrium, then this clearly will be a Pareto-optimal one among all the equilibria, since it is then the only one. 6 ...
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This note was uploaded on 02/06/2012 for the course ECON 101 taught by Professor Buddin during the Spring '08 term at UCLA.
- Spring '08