Decision Making under Risk

1
Decision Making under Risk

• Lecture 8

2
Todays Topics

• Difference between risky and uncertain situations
• Alternative theories of decision making (using
  survey results)
• Maximizing Expected Wealth
• Risk to Return tradeoff
• Maximizing Expected Utility
• Algebra of Expectations (and Geometry)

3
Risk versus Uncertainty

• Let us assume that the individual may experience
  two or more possible events (state of worlds)
  over which the individual has no direct control.
• If the individual knows or can assign
  probabilities to each of these possible events
  then the individual is said to face a risky
  situation.
• If the individual cant assign probabilities to
  these events then the individual is said to be
  making decision under uncertainty.

4
Decision to Gamble
WIN WealthWoAW
p
Accept Gamble
(1-p)
LOSE WealthWo-AL
Reject Gamble
WealthWo
5
Gambles

• Risky situations are often referred to as a
  gamble -- while you know what you win or lose
  as well as the probabilities of winning or
  losing, you dont know if you are going to win or
  lose! In general, a gamble can be characterized
  as
• Event Probability Amount
• Winning p AW
• Losing (1-p) - AL
• Expected Winnings
• EG p AW (1-p)(-AL)
• Variance
• VG p(1-p) (AW-AL)2

6
Alternative Forms of Gambles

• Often we think of gambling in slightly a
  different manner. In order to gamble, you have
  put up L dollars. If you win then you walk away
  with W dollars. Your net winnings (Aw) are W-L
  dollars. If you lose then you walk away with
  nothing but you have lost the L dollars you
  wagered to play the game. Consequently, your
  expected gain from playing the game
• p (W-L) (1-p) (-L) p W - L
• Should how we state the game affect the decision
  of an individual to either accept or reject the
  gamble?

7
Should how the gamble is stated affect behavior?

• Gamble 1 I flip a fair coin and I will pay
  you 1,000 if it comes up heads and you will pay
  me 500 if it comes up tails.
• Gamble 2 I will put 1,000 on the table and you
  will put 500. We will flip a fair coin and if
  it comes up heads you take what is on the table
  otherwise (tails) I will take what is on the
  table.

8
Hypothesis of Maximizing Expected Income or
Wealth

• How will individuals choose between accepting the
  gamble or rejecting it?
• Hypothesis 1 Individuals will only accept the
  gamble if it is expected to increase their
  expected wealth or income -- in other words if
  the gamble has a positive expected winnings
• Accept if EG PAW -(1-P)AL gt 0
• Reject if EG PAW -(1-P)AL lt 0
• Note
• EIncome P(MAW) (1-P)(M- AL) M (PAW
  -(1-P) AL) M EG

9
2nd Hypothesis -- Tradeoff between risk and reward
Return
Risk
10
Third Hypothesis

• The individual will accept the gamble only if the
  expected utility of the gamble exceeds the
  utility they would enjoy if they rejected it.
  Let us assume that the individual has M dollars
  of income. Instead of focusing upon the dollar
  values of winnings and losses, let us formulate
  the utility of the gamble where U(X) is the
  utility of having X with certainty
• Event Probability Net Income
• Winning p MAW
• Losing (1-p) M-AL
• Expected Utility
• EUGamble p U(MAW) (1-p) U(M-AL)

11
Maintained Hypothesis

• The hypothesis we will pursue when considering
  decision making under risk is often called the
  von Neumann-Morgenstern hypothesis that
  individual facing risk will choose the option
  that maximizes their expected utility.

12
Graph of Expected Utility
U
UW U(MAW)
E(Ugamble)
U(M)
UL U(M-AL)
Income
13
Risk Aversion

• Notice in the last graph, if the E(G) 0 then
• U(M) gt E(Ugamble)
• Reject a fair bet.
• Are there individuals would like to gamble? Or
  would be indifferent?

14
Risk Averse (E(G)0)
U
UW U(MAW)
U(M)
E(Ugamble)
UL U(M-AL)
Income
15
Risk Lovers (E(G)0)
U
UW U(MAW)
E(Ugamble)
U(M)
UL U(M-AL)
Income
16
Risk Neutral (E(G)0)
U
UW U(MAW)
E(Ugamble)
U(M)
UL U(M-AL)
Income
17
For Next Lecture

• Chapter 6 pages 187 to 193 and Appendix
• Topic Insurance and Search Behavior (if time
  permits)