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ÆF¤s¦B¬M 2007-12-01 21:16

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ÆF¤s¦B¬M 2007-12-01 21:53
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Theorem (Intermediate-value theorem, Rudin) oNK-^N?-T  
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Let f be a continuous real function on the interval [a, b]. If f(a) < f(b) and if c is a number such that f(a)<c<f(b), then there exists a point x belongs to (a, b) such that f(x)=c. zy!mP  
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What is neuroeconomics? 1Dt"Rcn"4  
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Neuroeconomics is the use of data on brain processes to suggest new underpinnings for economic theories, which explain how much people save, why there are strikes, why the stock market fluctuates, the nature of consumer confidence and its effect on the economy, and so forth. $nn~K  
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Until recently, economists have always been content to treat the human brain as a "black box" and suggest mathematical equations which simplify what the brain is doing. Most empirical studies of economic behavior have therefore relied on measuring inputs, like prices, and predicting outputs, like how much people will buy, from a simplified theory of brain processes. This approach reflects a bias traceable to the 1880¡¦s, when Jevons wrote ¡§I hesitate it is impossible to measure the feelings of the human heart¡¨. n!eg"pL  
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This ¡§rational choice¡¨ approach has been enormously successful. But now advances in genetics and brain imaging (and other techniques) have made it possible to observe detailed processes in the brain better than ever before. Brain scanning (ongoing at the new Broad Imaging Center at Caltech) shows which parts of the brain are active when people make economic decisions. This means that we will eventually be able to replace the simple mathematical ideas that have been used in economics with more neurally-detailed descriptions. For example, when economists think about gambling they assume that people combine the chance of winning (probability) with an expectation of how they will value winning and losing (¡§utilities¡¨). If this theory is correct, neuroeconomics will find two processes in the brain¡Xone for guessing how likely one is to win and lose, and another for evaluating the hedonic pleasure and pain of winning and losing¡Xand another brain region which combines probability and hedonic sensations. More likely, neuroeconomics will show that the desire or aversion to gamble is more complicated than that simple model. Research already shows that pathological gamblers tend to lack a certain gene which limits how much pleasure (in the form of the amount of ¡§dopamine¡¨ neurotransmitter that is released when they win) they get from winning. Not getting enough dopamine from everyday pleasures means gamblers need bigger and bigger ¡§fixes¡¨ to feel stimulated. In our lab at Caltech, we are also investigating the ¡§fear of the unknown¡¨ or ¡§tolerance for ambiguity¡¨¡Xhow willing are people to gamble, invest, or take a social risk (like going to a party where they don¡¦t know anybody)? Our hunch is that fear of the unknown is triggered by activity in the ¡§amygdala¡¨, an almond-shaped region (common to most mammals) which is active in registering very rapid sensations of fear, and in both learning and unlearning what to be afraid of. Understanding the neural basis of investing in the face of unknown odds is important for understanding economic phenomena like entrepreneurship, since entrepreneurs start businesses knowing little about their odds¡Xthey are economically fearless in a way that most people are not. =7e8N&-nv  
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Another example is discounting future rewards. The standard theory, which was invented in the 1950s, is that people apply a single declining ¡§discount factor¡¨ to future rewards when weighing present rewards against future ones. New theories suggest that there are *two* components to ¡§time discounting¡¨, not one: One component is the steady discounting of future rewards, and the second is a preference for immediate rewards. The second factor can explain why people struggle with temptations and procrastinate. Using fMRI imaging of brain activity when people choose between immediate and future rewards, we will be able to see whether there are two components of time discounting (as the new theories predict) or just one (as the old theories predict). If we find two components, we will know more about the nature of the preference for immediate reward. It may be an emotional desire, or even a physical instinct like grasping for food that is within your reach. The research could yield dramatic insights in helping people resist temptation, saving more and spending less. J3:P/n&  
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A third area we are actively exploring is trust. In a typical ¡§trust game¡¨ one person has some money, say $10, which they can invest part of, keeping the rest. The amount they invest is tripled (representing the return to a productive investment, like investing in a factory in a rapidly-growing foreign country). But that tripled amount rests in the hands of a second person, or ¡§trustee¡¨, who is free to repay as much as she likes and keep the rest. To make the game challenging and scientifically interesting, if the second trustee keeps all the money there is nothing the first person can do about it. (Economists call this ¡§weak enforcement of property rights¡¨ which is often characteristic of less-developed economies with weaker legal systems.) This game enables us to measure loose concepts like trust and trustworthiness in a crisp numerical way. The amount the first person invests is a measure of how much she expects the trustee to repay her. The amount the trustee repays is a measure of trustworthiness, moral obligation, or reciprocity. _'I9rGlx3  
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Trust is important in the economy because even large, complex economies rely largely on trust every day. When you deposit a large sum in a bank ATM, or donate money to a charity assuming it will be spent on good causes, or invest in an overseas business partner you barely know, you are trusting that your money will be safe. Studies show that simple measures of trust, like asking people ¡§In general, do you trust people?¡¨ are highly correlated with how wealthy a society is. (The poorest countries, like many in Asia and Africa, have very low levels of trust, which reflects and tolerates corruption, prevents investment, and causes the most productive workers to emigrate.) So understanding trust in the brain may enable us to understand economic behavior from synapse to society. GC4$9q}C4Z  
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In early fMRI brain scanning, with collaborators at Baylor Medical Center, we have studied what goes on in peoples¡¦ brains when they trust and decide how much to repay. We found a surprising effect of gender. When men decide how much to trust or repay, an area called the ¡§medial cingulate sulcus¡¨ is active. This is an area used to process potential reward, and calculate numbers. The male brains are just ¡§doing the math¡¨ and turn off after they have made a decision. The female brains are quite different. After women have decided how much to repay, but before they know how their partner reacted to their decision, areas of the brain active in processing potential reward (ventromedial prefrontal cortex and ventral striatum) and in regulating worry and error-detection (caudate nucleus) are active. The women are worrying, and thinking about the reward consequences, after they have decided how much to repay. ]G1j\wnF  
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The difference in brain activity in the two genders is like the kind of behavior you might see after a couple gets home from a potluck dinner and rehashes the event. The man wants to turn on the TV and catch some sports scores (his medial cingulate is turned off). The woman is more likely to rehash the evening¡¦s events, and worry about whether she said the right thing and whether the hostess was happy with the dish she brought, and whether plans for having lunch later in the week are genuine. 7PQedZ<\  
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