Statistics and Data Analysis

1
Statistics and Data Analysis

• Professor William Greene
• Stern School of Business
• IOMS Department
• Department of Economics

2
Statistics and Data Analysis
Part 4 Expected Value
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Expected Value Expected Agenda
1/18

• Discrete distributions of payoffs
• Mathematical expectation
• Expected return on a bet
• Fair and unfair games
• Applications Warranties and insurance
• Litigation risk and probability trees
• Expected value and expected utility

4
The Expected Value
2/18

• A random experiment has outcomes (payoffs) that
  are quantitative (e.g., not boy/girl), monetary
  for simplicity.
• Probability distribution over outcomes is
• P1 P2 P3 PN
• 1 2 3 N
• Expected outcome (payoff) is
• P1 1 P2 2 P3 3 PN N.
• Note The average outcome a weighted average
  of the payoffs

5
An Expected Value
3/18

• I bet 1 on a (fair) coin toss.
• Heads, I get my 1 back 1.
• Tails, I lose the 1.
• Expected value Sipayoffs Pi i
• Epayoff(1)(1/2) (-1)(1/2) 0.
• (This is a fair game.)

6
A Risky Business Venture
4/18

• 4 Alternative Projects Success depends on
  economic conditions, which cannot be forecasted
  perfectly.
• Boom Recession Expected
• (Probability) (90) (10) Value
• Beer -10,000 12,000 -7,800
• Fine Wine 20,000 -8,000 17,200
• Both 10,000 4,000 9,400
• T-bill 3,000 3,000 3,000

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Which Venture to Undertake?
5/18

• Assume the manager is indifferent to variance
  she cares only about expected values
• BOTH surely dominates T-bill.
• T-bill produces a certain 3,000
• BEER has a negative expected value
• Choice is between FINE WINE and BOTH. Based only
  on expectation, choose FINE WINE
• Why might she not choose FINE WINE? Its more
  risky. It might lose a whole lot of money. The
  initial assumption is unrealistic.

8
American Roulette
6/18

• Bet 1 on a number (not 0 or 00)
• If it comes up, win 35. If not, lose the 1
• EWin (-1)(37/38) (35)(1/38)
• -5.3 cents.
• Different combinations (all red, all odd, etc.)
  all return -.053 per 1 bet.
• Stay long enough and the wheel will always take
  it all. (It will grind you down.)
• (A twist. Why not bet 1,000,000. Why do
  casinos have table limits?)

18 Red numbers 18 Black numbers
2 Green numbers (0,00)
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Caribbean Stud Poker
7/18
10
The House Edge is 5.22
8/18
http//wizardofodds.com/caribbeanstud
These are the returns to the player.
Its not that bad. Its closer to 2.5 based on a
simple betting strategy.
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The Business of Gambling
10/18

• Casinos run millions of experiments every day.
  
• Payoffs and probabilities are unknown (except on
  slot machines and roulette wheels) because
  players bet strategically and there are many
  types of games to choose from.
• The aggregation of the millions of bets of all
  these types is almost perfectly predictable. The
  expected payoff to an entire casino is known with
  virtual certainty.
• The uncertainty in the casino business relates to
  how many people come to the site.

12
Triple Damages in Antitrust Cases
11/18

• Benefit to collusion or other antisocial activity
  is B
• Probability of being caught is P
• Net benefit
• If they just have to give back the profits
• B-PB (1-P)B which is always
  positive!
• Under the treble damages rule
• B3PB (1-3P)B might still be gt 0 if P
  lt 1/3.
• How to make sure the net benefit is negative
  Prison!

13
Fair Games
12/18

• Define A game is defined to be a situation of
  uncertain outcome with monetary payoffs. Betting
  the entire company fortune on a new product is a
  game
• A fair game has Epayoff 0
• Fair has no moral (equity) connotation. It is
  a mathematical construction.

14
Actuarially Fair Insurance
13/18

• Insurance policy
• You pay premium F
• If you collect on the policy, the payout W
• Probability they pay you P
• Expected profit to them is
• EProfit F - P x W gt 0 if F/W
  gt P
• When is insurance fair? EProfit 0?
• Applications
• Automobile deductible
• Consumer product warranties

15
14/18
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Rational Use of a Probability?
For all the criticism BP executives may deserve,
they are far from the only people to struggle
with such low-probability, high-cost events.
Nearly everyone does. These are precisely the
kinds of events that are hard for us as humans to
get our hands around and react to rationally,
Quotes from Spillonomics Underestimating Risk By
DAVID LEONHARDT, New York Times Magazine, Sunday,
June 6, 2010, pp. 13-14.
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Litigation Risk Analysis
15/18

• Form probability tree for decisions and outcomes
• Determine conditional expected payoffs (gains or
  losses)
• Choose strategy to optimize expected value of
  payoff function (minimize loss or maximize (net)
  gain.

18
16/18
Litigation Risk Analysis Using Probabilities to
Determine a Strategy
Two paths to a favorable outcome. Probability
(upper) .7(.6)(.4) (lower) .5(.3)(.6) .168
.09 .258. How can I use this to decide
whether to litigate or not?
Suppose the cost to litigate 1,000,000 and a
favorable outcome pays 3,000,000. What should
you do?
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Summary
18/18

• Expected value average outcome (weighted by
  probabilities)
• Expected value is an input to business decisions
• Games can be fair or unfair (have negative
  expected value).
• Some agents worry about unfair games
• All casino games are unfair but people play them
  anyway.
• Product warranties are a hugely profitable unfair
  game. Consumers do not know much about
  probabilities. (Or about manufacturer
  warranties.)
• Many decision situations involve certain costs
  and random payoffs. The cost benefit test
  requires an evaluation of expected values.
• Decision makers also worry about risk (variance)
  and also about the utility of payoffs rather than
  the payoffs themselves.