Part 4: Expected Value 4-1/25 Statistics and Data Analysis Professor William Greene Stern School of Business IOMS Department Department of Economics

Part 4: Expected Value4-1/25

Statistics and Data Analysis

Professor William Greene

Stern School of Business

IOMS Department

Department of Economics


Statistics and Data Analysis

Part 4 – Expected Value


http://www.usatoday.com/story/money/cars/2014/09/15/rudest-drivers-insurecom-idaho/15671265/http://www.insure.com/car-insurance/rudest-drivers-by-state.html

Hence, the survey asked (2000) drivers to name those they think are rudest in other states. California drivers were the No. 1 haters of drivers from surrounding states. They also thought New York drivers are more rude than those who may have some actual knowledge of their driving behavior, those in New Jersey. When it comes to states with rude drivers, however, here's the top 10:1. Idaho2. Washington, D.C.3. New York4. Wyoming5. Massachusetts6.Delaware (tie)7. Vermont (tie)8. New Jersey9. Nevada10. Utah


Expected Value Expected Agenda

Discrete distributions of payoffs Mathematical expectation

Expected return on a bet Fair and unfair games

Applications: Warranties and insurance Litigation risk and probability trees


The Expected ValueA random “experiment” has outcomes (payoffs) that

are quantitative, monetary for simplicity.Probability distribution over M possible outcomes is

Probability P1 P2 P3 … PM

Payoff $1 $2 $3 … $M

Expected outcome (payoff) is

P1 $1 + P2 $2 + P3 $3 + … + PM $M.

Note: The “average” outcome = a weighted average of the payoffs


Fair Games

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 Expected payoff = 0 “Fair” has no moral (equity) connotation. It

is a mathematical construction.


A Fair ‘Game’ Has Zero Expected Value

I bet $1 on a (fair) coin toss.

Heads, I get my $1 back + $1.

Tails, I lose the $1.

Expected value = Σi=payoffs Pi $i

E[payoff]=(+$1)(1/2) + (-$1)(1/2) = 0.

This is a “fair” game.


A Risky Business Venture

4 Alternative “Projects:” Success depends on economic conditions, which cannot be forecasted perfectly.

For each project,

Expected Value = .9*(Result|Boom) + .1*(Result|Recession)

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


Which Venture to Undertake?

Assume the manager cares only about expected values (i.e., is indifferent to variance)

BOTH surely dominates T-bill. T-bill produces a certain

+3,000 BOTH > T-bill in either state.

BEER has a negative expected value. Dominated by the other 3.

Choice is between FINE WINE and BOTH. Based only on expectation, choose FINE WINE

Why might she not choose FINE WINE? It’s more risky. It might lose a whole lot of money. The initial assumption is unrealistic. People do care about ‘risk,’ i.e., variation in outcomes.


American Roulette

Bet $1 on any of the 38 numbers. If it comes up, win $35. If not, lose the $1 E[Win] = (-$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)


The Gambler’s Odds in RouletteEvery bet loses 5.3 cents/$


The Gambler’s Ruin

http://en.wikipedia.org/wiki/Roulette


Caribbean Stud Poker


The House Edge is 5.22%

It’s not that bad. It’s closer to 2.5% based on a simple betting strategy.

These are the returns to the player.

http://wizardofodds.com/caribbeanstud


The Business of Gambling

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.

http://www.foxnews.com/us/2014/09/09/decades-standing-pat-made-atlantic-city-loser-say-veteran-workers/


Triple Damages in Antitrust Cases 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-P*B = (1-P)*B which is always positive! Under the treble damages rule:

B–3*P*B = (1-3P)*B might still be > 0 if P < 1/3. How to make sure the net benefit is negative: Prison!


Actuarially Fair Insurance 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 E[Profit] = F - P x W > 0 if F/W > P When is insurance “fair?” E[Profit] = 0? Applications

Automobile deductible Consumer product warranties Health insurance


A proposal to replace Fannie Mae and Freddie Mac with government created insurance for mortgage backed securities funded by actuarially fair insurance.

(Think FDIC.)



Since Hamman founded the Dallas-based company in 1986, SCA has grown into the world's go-to insurer of stunts that give fans the opportunity to win piles of dough if they make a hole-in-one, kick a field goal or sink a half-court shot. SCA determines the odds for each contest and charges event sponsors a fee based on the probability of someone succeeding, prize value and number of contestants. If the contestants win, SCA pays them the full prize amount. If they don't, SCA pockets the premium. "The primary distinction between us and [traditional] insurance businesses is that they restore something economically when something bad happens," Hamman says. "When you put on a promotion, you're simply taking a position on the likelihood of an event occurring."

How do they ‘determine the odds?’


Fitzgerald said he estimates that Hamilton's blast probably saved customers a little more than $500,000. His company bought insurance for the promotion. The company issued an explanation of the Grand Slam Payout, which said anyone who purchased flooring or countertops starting Aug. 29 would get a refund if Hamilton hit a bases-loaded home run during the promotion period, which was to run until Sept. 28 or as soon as Hamilton hit a grand slam.


Health Care Insurance Vocabulary

Insurance companies provide insurance against adverse health outcomes. Under new methodology, they also insure against future adverse outcomes by insuring against costs of prevention.

Government guarantees that there will be a large pool of customers and in return insurance companies agree to a rate structure.

Rates are ‘fair’ if the pool contains a good mix of ‘heavy’ users (old, sick) and ‘light’ users (young, healthy) who will not use the system (on average)

Adverse selection: Not enough young people join. The rates become ‘unfair’ against the insurance companies. Insurers must raise rates – then the situation becomes worse. Death spiral.

Moral hazard: People change their behavior because they have insurance – and rates do not reflect the changed behavior. Again, rates become unfair.


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 RiskBy DAVID LEONHARDT, New York Times Magazine,

Sunday, June 6, 2010, pp. 13-14.

E[benefit of building and operating rig]= very low probability * very high cost +very high probability * huge profits.The expected benefit is positive under any realistic calculations of costs and profits. This insurance ‘market’ fails. Food for thought: Why does it fail?


Litigation Risk Analysis

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.


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?

P(Upper path) = P(Causation|Liability,Document)P(Liability|Document)P(Document) = P(Causation,Liability|Document)P(Document) = P(Causation,Liability,Document) = .7(.6)(.4)=.168. (Similarly for lower path, probability = .5(.3)(.6) = .09.)


Summary 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.

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Part 4: Expected Value 4-1/25 Statistics and Data Analysis Professor William Greene Stern School of Business IOMS Department Department of Economics