Probability and expected utility, Game Theory


Most students know the elementary combinatorial rules for probability algebra and need only a refresher with some exam- ples. We have used card examples; you can easily construct similar ones with coins or dice.

The concept of risk aversion is simple at an intuitive level, but its treatment using expected utility can be difficult to get across. We have found it useful to involve the students. Take a particular utility function, say the logarithmic, and calculate the sure prospect that gives the same utility as the expected utility of a particular lottery. The logarithmic utility function is shown in the diagram below, with payoffs on the vertical axis representing the log of the dollar amount on the horizontal axis:

1882_probability and expected utility.png

In this case, U(10) = 1 and U(100) = 2. One possible lottery to consider might be that in which there is a 50-50 chance of getting 10 or 100 (55 on average). With risk aversion, U(0.5 ´ 10 + 0.5 ´ 100) = 1.74 > 0.5 ´ U(10) +  0.5 ´ U(100). Rather, 0.5 ´ U(10) + 0.5 ´ U(100) = 1.5 = U(31.6). Thus, $31.60 gives the same amount of utility as the 50-50 lottery between $10 and $100 under  this utility function. Now ask for a vote on how many students would accept the sure  prospect  ($31.60)  and  how  many  the  lottery  (50% chance of $10 and 50% chance of $100). If a majority would accept the sure prospect, say, "Most of you seem more risk- averse than  this. Let us try a more concave function, say U(x) = -1/x" and repeat the experiment. You can use this process to try to find the risk aversion of the median student.

A few students get sufficiently intrigued by this to want more. If your class gets interested, and if you have time, you can talk about the history of the subject (St. Petersburg para- dox and all that) or about the recent work in psychology and economic theory on non-expected-utility approaches. For a discussion of the St. Petersburg paradox, or consider using the following simple example of the Allais paradox that can help students see that they do not always make choices consistent with maximizing their expected utility.

Describe first a choice between two lotteries: Lottery A pays $3,000 with probability 1 and Lottery B pays $0 with probability 0.2 and $4,000 with probability 0.8. Ask stu- dents to choose which lottery they would prefer to enter at a price of zero (and ask them to make note of their choices). Most choose A over B. Then describe a choice between two different lotteries: Lottery C pays $0 with probability 0.8 and

$4,000 with probability 0.2; Lottery D pays $0 with proba- bility 0.75 and $3,000 with probability 0.25. Again ask students to pick. Most choose C over D.

Now consider how the paired choices fit with the idea that people maximize expected utility. Set U(0) = 0. For those who chose A and C, this implies that EU(A) > EU(B) or that 1U(3,000) > 0.8U(4,000); but choosing C implies that EU(C) > EU(D) or that 0.2U(4,000) > 0.25U(3,000). The latter is equivalent to 0.8U(4,000) > 1U(3,000). This is in direct contradiction to the implication made when choos- ing A over B. Similar calculations can be used to show that those who choose B and D also violate the expected utility hypothesis. The choices of both A and D, or both B and C are consistent with maximization of expected utility.

Posted Date: 9/27/2012 4:19:18 AM | Location : United States

Related Discussions:- Probability and expected utility, Assignment Help, Ask Question on Probability and expected utility, Get Answer, Expert's Help, Probability and expected utility Discussions

Write discussion on Probability and expected utility
Your posts are moderated
Related Questions
A type of initial worth auction during which a "clock" initially indicates a worth for the item for sale substantially beyond any bidder is probably going to pay. Then, the clock g

An auction during which bidders simultaneously submit bids to the auctioneer while not information of the number bid by different participants. Usually, the very best bidder (or lo

A non-credible threat may be a threat created by a player in a very Sequential Game which might not be within the best interest for the player to hold out. The hope is that the thr

the first three words are ''''the boys'' down''''. what are the last three words?

Consider two quantity-setting firms that produce a homogeneous good. The inverse demand function for the good is p = A - (q 1 +q 2 ). Both firms have a cost function C = q 2 (a

Living from 1845 to 1926, Edgeworth's contributions to Economics still influence trendy game theorists. His Mathematical Psychics printed in 1881, demonstrated the notion of compet

Explain about the term Game Theory. Game Theory: While the decisions of two or more firms considerably influence each others’ profits, in that case they are into a situation

In a positive add game, the combined payoffs of all players aren't identical in each outcome of the sport. This differs from constant add (or zero add) games during which all outco

a) Define the term Nash equilibrium b) You are given the following pay-off matrix:   Strategies for player 1   Strategies for player 2