Jonathan

Schiller

Decision under
Risk
Maximizing what?
Interactive Case Study Seminar: Introduction
Why maximize
Utility?
Some Paradoxes
to Decision and Game Theory
Exercises IV

Jonathan Schiller

University of Bayreuth
jonathan.schiller@uni-bayreuth.de
 Outline

Jonathan
Schiller

Decision under
Risk
Maximizing what?
Why maximize
Utility?
Some Paradoxes
Exercises IV
1 Maximizing what?
2 Why maximize Utility?
3 Some Paradoxes
 Learning Goals

Jonathan
Schiller

Decision under
Risk
Maximizing what?
Why maximize
Utility?
Some Paradoxes
Exercises IV Get used to decisions under risk
Find justification for the idea of maximizing utility
 Basics

Jonathan
Schiller

Decision under
Risk Decisions under risk imply that the decision maker knows
Maximizing what?
Why maximize the probabilities of outcomes
Utility?
Some Paradoxes
Exercises IV
This knowledge can be tacit
Example: Roulette
Is it rational to play a game where you lose money on
average?
Why? Why not?
 Basics

Jonathan
Schiller

Decision under
Risk Decisions under risk imply that the decision maker knows
Maximizing what?
Why maximize the probabilities of outcomes
Utility?
Some Paradoxes
Exercises IV
This knowledge can be tacit
Example: Roulette
Is it rational to play a game where you lose money on
average?
Why? Why not?
Is money everything that counts? Why should we
maximize the expected value of anything?
 Maximize what?

Jonathan
Schiller
1 Maximize Money?
Decision under
Risk
Maximizing what? EMV = p1 ∗ MV1 + p2 ∗ MV2 + ... + pn ∗ MVn
Why maximize
Utility?
Some Paradoxes
Exercises IV 2 Maximize Value?

EV = p1 ∗ V1 + p2 ∗ V2 + ... + pn ∗ Vn
 Maximize what?

Jonathan
Schiller

Decision under 1 Maximize Money?

Risk
Maximizing what?
Why maximize
Utility?
EMV = p1 ∗ MV1 + p2 ∗ MV2 + ... + pn ∗ MVn
Some Paradoxes
Exercises IV
2 Maximize Value?

EV = p1 ∗ V1 + p2 ∗ V2 + ... + pn ∗ Vn

3 Maximize Utility!

EU = p1 ∗ U1 + p2 ∗ U2 + ... + pn ∗ Un
 Utility

Jonathan
Schiller

Decision under
Risk
Maximizing what?
Why maximize
Utility?
Some Paradoxes
Utility is an abstract entity and unobservable
Exercises IV
Utility is the value of an action from the decision maker’s
point of view
Good concept to describe human behavior?
 Why maximize Utility?

Jonathan
Schiller

Decision under
Risk
Maximizing what? There are different approaches to justify the principle of
Why maximize
Utility? maximizing expected utility
Some Paradoxes
Exercises IV Law of Large numbers
Axiomatic approach
Basic question: Why is it rational to maximize expected
utility?
Discussion mostly philosophical (analytical philosophy)
 Law of Large Numbers

Jonathan
Schiller

Decision under
Risk
Maximizing what?
Idea: Everyone who maximizes his expected utility is
Why maximize
Utility?
better off in the long run
Some Paradoxes
Exercises IV Example: toss a coin infinitely many times What do you
expect the ratio Head/Tail to be?
Criticism:
Keynes: ”In the long run we are all dead”
Gambler’s ruin
Many decisions under risk are unique!
 Axiomatic Approach

Jonathan Is there a good reason to maximize expected utility even if we

Schiller
face a decision just once?
Decision under
Risk
EU1: If all outcomes of an act have utility u, then the
Maximizing what? utility of the act is u
Why maximize
Utility?
Some Paradoxes EU2: If one act is certain to lead to better outcomes
Exercises IV
under all states compared to another act, then the utility
of the first act exceeds that of the latter
EU3: Every decision problem can be transformed into a
decision problem with equally probably states in which the
utility of all states is preserved
EU4: If two outcomes are equally probable and if the
better outcome is made slightly worse, then this can be
compensated for by adding some amount of utility to the
other outcome, such that the overall utility is preserved
 Axiomatic Approach

Jonathan
Schiller

Decision under Proof (Step 1): Show that EU4 implies that to preserve the
Risk
Maximizing what? utility of an act any change of utility of an act needs to be
Why maximize
Utility? compensated by adding the exact same amount ∗(−1) of utility
Some Paradoxes
Exercises IV to any other act

Let ε1 >(U1 − U2)>0, this implies ε1 = ε2 as anything else

would violate EU2!
 Axiomatic Approach

Jonathan
Schiller

Decision under
Proof (Step 2): Transform any decision problem into a decision
Risk problem with equally probable states and equal utility values for
Maximizing what?
Why maximize
Utility?
all states using EU4, Proof (Step 1) and EU1
Some Paradoxes
Exercises IV

1 Apply EU3 to make all states equally probable

2 Apply EU4 and ε1 = ε2 to equalize utility values
3 Apply EU1 to show that E (U(a)) = U(a)
 Allais’ Paradox

Jonathan
Schiller

Decision under
Risk
Maximizing what?
Why maximize
Utility?
Some Paradoxes
Exercises IV
 Allais’ Paradox

Jonathan
Schiller

Decision under
Risk
Maximizing what?
Why maximize
Utility?
Some Paradoxes
Exercises IV

Empirical studies show that people prefer G1 over G2 and G4

over G3. However this is irrational as the difference between
G1 and G2 is equal to the difference between G3 and G4

u(G 1) − u(G 2) = 0.11u(1M) − [0.01u(0) + 0.1u(5M)]

 St Petersburg Paradox

Jonathan
Schiller

Decision under 1 Toss a fair coin until it lands heads up. The player receives
Risk
Maximizing what?
2n units of utility, with n =number of the throw the coin
Why maximize
Utility? first lands heads up
Some Paradoxes
Exercises IV 2 How much utility would you be willing to pay for the
game?
Calculate expected utility!
∞
1 1 X 1
∗ 2 + ∗ 4 + ... = 1 + 1 + 1 + ... = ( )n ∗ 2n = ∞
2 4 n=1
2

Is that something you should do?

 Exercise IV

Jonathan
Schiller
You are in Las Vegas. The probability of winning a
Decision under jackpot of 350.000 $ is one in a million. As someone who
Risk
Maximizing what? maximizes expected utility, how much would you be willing
Why maximize
Utility? to pay to enter assuming that your utility function over
Some Paradoxes
Exercises IV money is u(x) = ln(x + 1)?
Thanks to some changes in the game above, the
probability of winning is now 1 in 1000. How much would
you be willing to pay now?
Compare the results from (1) and (2). Why do we see
such surprising results?
Why did we assume a utility function of ln(x + 1) rather
than just ln(x)?
 Exercise IV

Jonathan
Schiller

Decision under
Risk
Maximizing what?
Why maximize
Utility?
Some Paradoxes
Explain why the St Petersburg Paradox poses a difficulty
Exercises IV
for the principle of maximizing expected utility!
Construct a new version of the St Petersburg Paradox
using a normal six-sided dice instead of a coin!
 Exercise IV

Jonathan
Schiller
As a decision theorist you are approached by a pharmaceutical
company that is about to expand its supply. There are three
Decision under
Risk options: (1) Create a new drug (50M, 20% chance of success,
Maximizing what?
Why maximize
(2) buy a competing company that is about to release a new
Utility?
Some Paradoxes drug (120M, 80% chance of success) or (3) buy an already
Exercises IV
existing patent (170M, 100% chance of success). After their
decision there are two scenarios: (1) there is high demand for
the new drug and the company earns 200M (30% chance) or
there is low demand (100M profit). The company has a linear
and directly proportional utility gain from positive profits,
whereas (dis-)utility from negative profits is determined as
follows: u(x) = 2 ∗ x.
The company can also choose to do nothing. What would your
advise be? (Hint: Visualize the decision in a decision tree!)