Talk:Expected utility hypothesis/Archive 1

Merger

Hi,

Would a redirect to Utility#Expected_utility be appropriate? Tony Bruguier

Also, there's this guy who wrote this inside the article. I deleted it and copied it here:

EDIT -- I don't have the time to fix this but the author who typed the above is confusing Von Neumann and Morgenstern's expected utility theorem (which derives an expected utility function from preferences over lotteries) and Savage's expected utility theorem (which derives an expected utility function from preferences over consumption streams)

Yes, we should merge both, but best under the name "Expected Utility Theory", I think that's the most common name. I also plan to improve the article, since it is a central topic and is in a poor state.Rieger 09:18, 23 March 2007 (UTC)

Move to "Expected Utility"

I propose moving this article to "Expected utility." That's how people usually reference the topic in conversation and text, and that's also what about half of the incoming links direct to now. Comments or reactions? Jeremy Tobacman 20:41, 5 July 2007 (UTC)

Totally agree. Even "Expected utility theory" is more common than "hypothesis". 83.77.96.210 (talk) 11:32, 6 February 2008 (UTC)

Fair use rationale for Image:Pyat rublei 1997.jpg

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BetacommandBot 11:26, 6 July 2007 (UTC)

Merged text needs cleanup

Following Talk:Utility#Merger (Expected utility), I have merged the content from Neumann-Morgenstern utility into here. It's only a very rough copy-paste merger, and the text definitely needs cleanup by an expert. Also, it might be appropriate to merge some of the mathematical details from Utility#Expected utility into here. --B. Wolterding 11:05, 14 July 2007 (UTC)

Poker - better removing?

The added text about poker doesn't seem to fit well to the usual use of expected utility theory (in the sense of utility for a given wealth), since the utility in a poker tournament is 1 for winning and 0 for losing, as far as I see... If you play on money, however, risk-neutral play may never be the right thing, at least if it's about large amounts and you are risk-averse as most people... So in other words, it might not be completely wrong what is written, but is definitely not helping to understand the main ideas. A simple example (where one cannot discuss about the validity) would be much better there! (If I only had more time!) In conclusion: I would suggest to remove this paragraph. What do you think? Rieger 12:14, 20 October 2007 (UTC)

I'm the one who put it in, and I think it's a good example. I hope it's not hard for non-poker players to understand what I meant. I'll take a look at it and possibly clean it up. The utility in poker isn't 1 for winning and 0 for losing, it's much more complicated. There are generally multiple places that pay out in a tournament. For a large tournament there may be as many as 1000 players, and the top 100 might win money, with the highest finishers getting the most money. So, when making decisions regarding whether or not to put your chips at risk in a hand, you have to take into account many factors: first is obviously the odds that you will win that particular hand, but you also must consider the amount of money you stand to win in the tournament if you win the hand, and weigh this against the amount of money that you can win if you lose the hand (or if you decide to just fold, and not play the hand). You even have to look at the amount of chips that your opponents have left, in order to determine how risk-averse THEY are likely to be, and adjust your strategy accordingly. Dan Harrington has discussed these ideas in his poker books, although I'm not sure if he actually used the words "expected utility".

For non-tournament, "cash games" of poker, a risk-neutral strategy really is the best, assuming you are properly bankrolled. A properly bankrolled poker player should be risk-neutral, meaning they NEVER want to turn down a bet with a positive expected value. If they can't afford to play that way, they would be best served by moving down to a lower-stakes table. Never sit down at the table with more money than you can afford to lose. Deepfryer99 16:15, 29 October 2007 (UTC)

I see why you think the example is good, but I still see a a problem with the poker paragraph - and also one with your above comment: (1) The poker example is rather involved, as becomes clear from your explanations above. Lots of possible payoffs, that are all not really linear in the money amounts, thus it is very difficult to say what "risk-neutral" really means here. It is interesting, sure, but maybe too involved for an example for people who want to understand the concept of expected utility. - I think it would be better to have a simple example that is clear and easily accessible for the non-expert. (2) You wrote: "For non-tournament, "cash games" of poker, a risk-neutral strategy really is the best, assuming you are properly bankrolled." - This is wrong: actually, expected utility is all about why this is not true. It is the expected utility that you want to maximize, not the expected amount of cash you make. (For small amounts the utility function should be nearly linear, so it shouldn't make a difference. For larger amounts it will make a difference, long before you will be broke.) Sorry for the late answer. Rieger 28 January 2008 —Preceding unsigned comment added by 83.76.74.157 (talk) 20:50, 28 January 2008 (UTC)

New material

The following text on expected utility could be used to improve the article. (It's based on something a student of mine wrote in the appendix of her thesis and is explicitly copyright-free, thus suitable for Wikipedia.) Maybe somebody else could do the cut- and paste, since I don't want to change other people's work just by myself... Moreover this piece needs clean-up and extensions, and I may not have time to do this by myself (sorry, I wished I had!). - But I hope it can contribute to the improvement of this important article!

Here it goes:

The idea of the expected utility theory began in the 17th century when Daniel Bernoulli, Pierre de Fermat, Blaise Pascal and Christian Huyges began working on the concept of probabilities. John von Neumann and Oskar Morgenstern furthered the formal economic theory of choice between uncertain alternatives. These uncertain alternatives were mathematically explained as the expected value and may be calculated as follows: $U=\sum _{i}p_{i}u(x_{i}),$ where $p$ is the probability of a simple gamble with outcomes $x_{i}$ and corresponding probabilities $p_{i}$ and $U$, represents the expected utility. For example if a dice is rolled, each probability $p_{i}$ is $1/6$ and the $x_{i}$ are the (monetary) outcomes when the dice shows the number $i$.

Individuals that decide rationally under risk are said to abide by the expected utility theory. There are four axioms of the expected utility theory that define such a rational decision maker. They are completeness, transitivity, independence and continuity. Completeness assumes that an individual has well defined preferences and can decide between two alternatives.

Axiom(Completeness): For every A and B either $A<B,A>BorA=B$ (this means: A is worse than B, or better ,or equally good) holds.

Transitivity assumes that, as an individual decides according to the completeness axiom, the individual also decides consistently.

Axiom(Transitivity): For every A, B and C with $A>B$ and $B>C$ we must have $A>C$.

Independence also pertains to well-defined preferences and assumes that the preference order of two gambles mixed with a third one maintains the same preference order as when the two are mixed independently.

Axiom(Independence): Let A and B be two lotteries with $A>B$, and let $t\in [0,1]$ then $tA+(1-t)C>tB+(1-t)C$ .

Continuity assumes that when there are three lotteries (A, B and C) and the individual prefers A to B and B to C, then there should be a possible combination of A and C in which the individual is then indifferent between this mix and the lottery B.

Axiom(Continuity): Let A, B and C be lotteries with $A>B>C$ then there exists a probability p such that B is equally good as $pA+(1-p)C$.

If all these axioms are satisfied, then the individual is said to be rational and the preferences can be represented by a utility function. In other words: if an individual always chooses his/her most preferred alternative available, then the individual will choose one gamble over another if and only if the expected utility of one exceeds the other; thereby maximizing his/her utility. The utility of any gamble may be expressed as a linear combination involving only the utility of the outcomes and their respective probabilites. Utility functions are also normally continuous functions. Such utility functions are also referred to as von Neumann-Morgenstern (VNM) utility functions. This is a central theme of the expected utility in which an individual chooses not the highest expected value, but rather the highest expected utility. The expected utility individual makes decisions rationally based on the axioms of the theory. The expected utility theory generally accepts the assumption that individuals are risk averse, meaning that the individual would refuse a fair gamble (a fair gamble has an expected value of zero), and also implying that their utility functions are concave and show diminishing marginal wealth utility. The risk attitude is directly related to the curvature of the utility function: risk neutral individuals have linear utility functions, while risk seeking individuals have convex and risk averse have concave utility functions. The degree of risk aversion can be measured by the curvature of the utility function. Since the risk attitudes are unchanged under affine transformations of u, this has to be normalized by u'. This leads to the definition of the Arrow-Pratt measure of absolute risk aversion:

$ARA(x)=-{\frac {u''(x)}{u'(x)}}$

Special classes of utility functions are the CRRA (constant relative risk aversion) functions, where ARA(x)/x is constant, and the CARA (constant absolute risk aversion) functions, where ARA is constant. They are often used in economics for simplification purposes.

While expected utility theory is the rational benchmark for decisions under risk, there are nowadays modifications that aim to model decisions of real individuals (taking into account behavioral, non-rational factors). Examples include rank dependent utility, prospect theory, cumulative prospect theory and SP/A theory.

Rieger (talk) 19:08, 5 May 2008 (UTC)