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# GAMETHEORY ECON1200

Pitt

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This 23 page Class Notes was uploaded by Quentin Huel Jr. on Monday October 26, 2015. The Class Notes belongs to ECON1200 at University of Pittsburgh taught by Staff in Fall. Since its upload, it has received 22 views. For similar materials see /class/229427/econ1200-university-of-pittsburgh in Economcs at University of Pittsburgh.

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Date Created: 10/26/15

Probability Expected Payoffs and Expected Utility In thinking about mixed strategies we will need to make use of probabilities We will therefore review the basic rules or probability and then derive the notion of expected value We will also develop the notion of expected utility as an alternative to expected payoffs Probabilistic analysis arises when we face uncertainty In situations where events are uncertain a probability measures the likelihood that a particular event or set of events occurs e g The probability that a roll of a die comes up 6 The probability that two randomly chosen cards add up to 21 Blackjack Sample Space or Universe Let S denote a set collection or listinw of all possible states of the environment known as the sample space or universe a typical state is denoted as s For example Ss1 s2 successfailure or lowhigh price Ss1 s2sn1sn number ofn units sold or 11 offers rece1ved SO stock price or salary offer continuous positive set space Events An event is a collection of those states s that result in the occurrence of the event An event can be that state s occurs or that multiple states occur or that one of several states occurs there are other possibilities Event A is a subset of S denoted as A C S Event A occurs 11 the true state s is an element of the set A written as seA Venn Diagrams Illustrates the sample space and events S is the sample space and A1 and A2 are events Within S Event A1 does not occur Denoted A1c Complement of A1 Event A1 or A2 occurs Uenoted A1 3 A2 For probability use Addition Rules Event A1 and A2 both occur denoted A1 5 7 A2 For probability use Multiplication Rules Probability To each uncertain event A or set of events eg A1 or A2 we would like to assign weights which measure the likelihood or importance of the events in a proportionate manner Let PAi be the probability of A1 We further assume that UAiS all i PU Al1 all i PAl20 Addition Rules The A robabilit of event A or event B PAIIXB If the events do not overlap ie the events are disjoint subsets of S so that A9Blo then the probability of A or B is simply the sum of the two probabilities PALIXB PA PB If the events overlap are not disjoint AyB u o use the modi ed addition rule PAAB PA PB PAyB Example Using the Addition Rule Suppose you throw two dice There are 6X636 possible ways in which both can land Event A What is the probability that both dice show the same number A11 22 33 44 55 66 so PA636 Event B What is the probability that the two die add up to eight B26 35 44 53 62 so PB536 Event C What is the probability that A or B happens 1e t tA B First note that AyB 44 so PA B 136 PA AB PAPBPAy B 6365361361036 518 Multiplication Rules The probability of event A and event B PAf B Multiplicauon rule appnes If A and B are independent events A and B are independent CVCllLb u I U X uUCb uut depend on Whether B occurs or not and PB does not depend on Whether A occurs or not PAyB PA PBPAB Conditional plowwnv for non independent events The probability of A given that B has occurred is PAB PABPB Examples Using Multiplication Rules An unbiased coin is ipped 5 times What is the probability of the sequence TTTTT PT5 5 independent ips so 5X5X5X5X503125 Suppose a card is drawn from a standard 52 card deck Let B be the event the card is a queen PB4 52 Event A Conditional an Event B What is the probability that the card is the Queen of Hearts First note that PABPA 39 B 152 Probability the Card is the Queen of Hearts P AB P AB P B 152 452 14 B aye 5 Rule Used for making inferences given a particular outcome event A can we infer the unobserved cause of that outcome some event B1 B2Bn buppose we Know the prior probabilities PBi and the conditional probabilities PABi Suppose that B1 B2Bn form a complete partition of the sample space V W um Ai Bi V we uinjp for any i 131 j In this case we have that PAZPAiBiPBi i1 Bayes rule is a formula for computing the posterior probabilities e g the probability that event Bk was the cause of outcome A denoted PBkA PBk A PBk m APA Usingtheconditional P B k k P probability rule PltABkgtPltBkgt Thlsls Usmg expresswn l n B R 1 above 2 ayes ue i1 Bayes KUlCprClal base Suppose S consists ofjust B and not B ie Bc Then Bayes rule can be stated as PM BPB PA BPB PA BCPBquot Example Suppose a drug test is 95 effective the test will be positive on a drug user 95 of the time and will be negative on a nondrug user 95 of the time Assu 5 of the popul mug users Suppose an individual tests positive What is the probability he is a drug user PB A Jjayes K1116 Example Let A be the event that the individual tests positive Let B be the event individual is a drug user Let B0 be the com lementar event that the individual is not a drug user Find PBA PAB 95 PABC 05 PB05 P 0 PM BP B PA BPB PA BCPBC 9505 9505 0595 39 PBA Monty Hall S 3 Door Problem There are three closed doors Behind one of the doors is a brand new sports car Behind each of the other two doors is a smelly goat You can t see the car or smell the goats You win the prize behind the door you choose The sequence of play of the game is as follows You choose a door and announce your choice The host Monty Hall who knows where the car is always selects one of the two doors that you did not choose which he knows has a goat behind it Monty then asks if you want to switch your choice to the unopened door that you did not choose Should you switch You Should Always Switch Let Ci be the event car is behind door i and let G be the event Monty chooses a door with a goat behind it Suppose without loss of generality the contestant chooses door 1 Then Monty shows a goat behind door number 3 According to the rules PGl and so PGC1l Initially PC1PC2PC3l3 By the addition rule we also know that PC2 Bl C323 After Monty s move PC3O PC1 remains 13 but PC2 now becomes 23 Accordin39 to Ba es Rule PC1G PGC1PC11gtltl3 PG 1 1t IOllows that PC2G23 so the contestant always does better by switching the probability is 23 he wins the car l3 Here is Another Proof Let WXyz describe the game wwour initial door choice xthe door Monty opens ythe door ou nall decide u on and zWL whether ou win or lose Without loss of generality assume the car is behind door number 1 and that there are goats behind door numbers 2 and 3 Suppose you adopt the never switch strategy The sample space under this strategy is S121W131W232L323L If you choose door 2 or 3 you always lose with this strategy But if you initially choose one of the three doors randomly it must be that the outcome 232L and 323L each occur with probability 13 That means the two outcomes 121W and 131W have the remaining 13 probability 2 you win with probability 13 Suppose you adopt the always switch strategy The sample space under this strategy is S123L132L231W321W Since 39 ou initially choose door 2 with A robabilit 13 and door 3 with probability 13 the probability you win with the switching strategy is l31323 2 you should always switch Expected Value or Payoff One use of A robabilities to calculate ex ected values or payoffs for uncertain outcomes Suppose that an outcome e g a money payoff is uncertain There are 11 possible values X1 X2XN Moreover we know the probability of obtaining each value The expected value or expected payoff of the uncertain outcome is then given by PltX1gtX1PltX2gtX2PltXNgtXN An Example You are made the following proposal You pay 3 for the right to roll a die once You then roll the die and are paid the number of dollars shown on the die Should you accept the proposal The expected payo of the uncertain die throw is lgtlt1lgtlt2lgtlt3lgtlt4lgtlt5lgtlt6350 6 6 6 6 6 6 The expected payoff from the die throw is greater than the 3 price so a risk neutral player accepts the proposal Extensive Form Illustration Nature as a Player Payoffs are in net terms 3 7 winnings R2 21 Pmusal Accounting for Risk Aversion The assumption that individuals treat expected payoffs the same as certain payoffs ie that they are risk neutral may n0t 1011 1n pracuce Recall our earlier examples A risk neutral person is indifferent between 25 for certain or a 25 chance of earning 100 and a 75 chance of earning 0 A risk neutral person agrees to pay 3 to roll a die once and receive as payment the number of dollars shown on the die Many people are risk averse and prefer 25 with certainty to the uncertain gamble or might be unwilling to pay 3 for the right to roll the die once so imagining that people base their decisions on expected payoffs alone may yield misleading results What can we do to account for the fact that man 1 e0 1e are risk averse We can use the concept of expected utility Utility Function Transformation Let x be the payoff amount in dollars and let Ux be a continuous increasing function ofX The function UX gives an individual s level of satisfaction in fictional utils from receiving payoff amount X and is known as a utility function If the certain payoff of 25 is preferred to the gamble due to risk aversion then we want a utility function that satis es U25 gt 25 U100 75 UO The left hand side is the utility of the certain payoff and the right hand side is the expected utility from the gamble In this case any concave function U x will work eg U X J J25 gt 25 xIOO 75 Jim 5 gt 25 Graphical Illustration 0 The blue line shows the utility of any certain monetary payo between 0 and 100 assuming U X J Utility Function Transformation Illustrated Utility Level Dollars Utility diminishes with increases in monetary payoff this is just the principle of diminishing marginal utility requires risk aversion Black dashed line shows the expected utility of risky payoff At 25 the certain payoff yields higher utility than the risky payoff Another Example If keeping 3 were preferred to rolling a die and getting paid the number of dollars that turns up expected payoff 35 we need a utility function that satis ed HongtUnUmUmUoUomUom In this case where the expected payoff 35 is strictly higher than the certain amount the 3 price the utility function must be sufficiently concave for the above relation to hold If we used U x J x we would nd that the lefthandside of the expression above was J3 1732 while the righthand side equals 1805 so we need a more concave function We would need a utility function transformation of U x x 1100 for the inequality above to hold 50 times more risk aversion Summing up The notions of probability and expected payoff are frequently encountered in game theory We mainly assume that players are risk neutral so they seek to maximized expected payoff We are aware that expected monetary payoff might not be the relevant consideration that aversion to risk may play a role We have seen how to transform the objective from payoff to utility maximization so as to capture the possibility of risk aversion the trick is to assume some concave utility function transformation Now that we know how to deal with risk aversion we are going to largely ignoru 1L auu aboulue riSk uuutral uuuuViOf

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