ch18 不确定性和风险转移

– probability of obtaining a head on the fair-flip of a coin is 0.5
• If a lottery offers n distinct prizes and the probabilities of winning the prizes are i (i=1,n) then
• Because no player would pay a lot to play this game, it is not worth its infinite expected value
8
Expected Utility
• Individuals do not care directly about the dollar values of the prizes
U(xi) = i ·U(xn) + (1 - i) ·U(x1)
13
The von NeumannMorgenstern Theorem
• Since U(xn) = 1 and U(x1) = 0
U(xi) = i ·1 + (1 - i) ·0 = i
• The utility number attached to any other prize is simply the probability of winning it • Note that this choice of utility numbers is arbitrary
16
Expected Utility Maximization
• Substituting the utility index numbers gives
expected utility (1) = q ·2 + (1-q) ·3 expected utility (2) = t ·5 + nMorgenstern Theorem
• Suppose that there are n possible prizes that an individual might win (x1,…xn) arranged in ascending order of desirability
• When faced with two gambles with the same expected value, individuals will usually choose the one with lower risk
19
Risk Aversion
• In general, we assume that the marginal utility of wealth falls as wealth gets larger
• we will assume that this is not the case
6
St. Petersburg Paradox
• A coin is flipped until a head appears • If a head appears on the nth flip, the player is paid $2n
1 1 2 2 n n
E ( X ) i xi
i 1
n
• The expected value is a weighted sum of the outcomes
– the weights are the respective probabilities
3
Expected Value
• Suppose that Smith and Jones decide to flip a coin
4
Expected Value
• Games which have an expected value of zero (or cost their expected values) are called actuarially fair games
– a common observation is that people often refuse to participate in actuarially fair games
– this would measure how much the game is worth to the individual
9
Expected Utility
• Expected utility can be calculated in the same manner as expected value
x1 = $2, x2 = $4, x3 = $8,…,xn = $2n
• The probability of getting of getting a head on the ith trial is (½)i
1=½, 2= ¼,…, n= 1/2n
7
St. Petersburg Paradox

i 1
n
i
1
2
Expected Value
• For a lottery (X) with prizes x1,x2,…,xn and the probabilities of winning 1,2,…n, the expected value of the lottery E( X ) x x ... x is
12
The von NeumannMorgenstern Theorem
• The von Neumann-Morgenstern method is to define the utility of xi as the expected utility of the gamble that the individual considers equally desirable to xi
5
Fair Games
• People are generally unwilling to play fair games • There may be a few exceptions
– when very small amounts of money are at stake – when there is utility derived from the actual play of the game
E ( X ) i U ( xi )
i 1 n
• Because utility may rise less rapidly than the dollar value of the prizes, it is possible that expected utility will be less than the monetary expected value
15
Expected Utility Maximization
• Consider two gambles:
– first gamble offers x2 with probability q and x3 with probability (1-q) expected utility (1) = q ·U(x2) + (1-q) ·U(x3) – second gamble offers x5 with probability t and x6 with probability (1-t) expected utility (2) = t ·U(x5) + (1-t) ·U(x6)
– they care about the utility that the dollars provide
• If we assume diminishing marginal utility of wealth, the St. Petersburg game may converge to a finite expected utility value
18
Risk Aversion
• Two lotteries may have the same expected value but differ in their riskiness
– flip a coin for $1 versus $1,000
• Risk refers to the variability of the outcomes of some uncertain activity
• The expected value of the St. Petersburg paradox game is infinite
1 E ( X ) i x i 2 2 i 1 i 1
i i
E ( X ) 1 1 1 ... 1
Chapter 18
UNCERTAINTY AND RISK AVERSION
1
Probability
• The probability of a repetitive event happening is the relative frequency with which it will occur
– x1 = least preferred prize U(x1) = 0 – xn = most preferred prize U(xn) = 1
11
The von NeumannMorgenstern Theorem
• The point of the von NeumannMorgenstern theorem is to show that there is a reasonable way to assign specific utility numbers to the other prizes available
– heads (x1) Jones will pay Smith $1 – tails (x2) Smith will pay Jones $1
• From Smith’s point of view,
E( X ) 1x1 2 x2
1 1 E ( X ) ($1) ( $1) 0 2 2
• The individual will prefer gamble 1 to gamble 2 if and only if
q ·2 + (1-q) ·3 > t ·5 + (1-t) ·6
17
Expected Utility Maximization
• If individuals obey the von NeumannMorgenstern axioms of behavior in uncertain situations, they will act as if they choose the option that maximizes the expected value of their von Neumann-Morgenstern utility index
14
Expected Utility Maximization
• A rational individual will choose among gambles based on their expected utilities (the expected values of the von Neumann-Morgenstern utility index)
– a flip of a coin for $1,000 promises a small gain in utility if you win, but a large loss in utility if you lose – a flip of a coin for $1 is inconsequential as the gain in utility from a win is not much different as the drop in utility from a loss