Expected Value Choice Principle

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Expected Value Choice Principle

• It applies when a decision maker must choose
  between two or more prospects. It says a
  decision maker should select the prospect with
  the highest expected value.

2
Expected Value

• The expected (monetary) value (EMV or EV) of a
  gamble (G) with several possible outcomes is
  obtained by multiplying each possible (cash)
  outcome (ci) by its probability (pi) and summing
  these products over all the possible outcomes.
• EV(G) ? ci pi

3
Decision Tree

• Simple Bet - I will bet you 1.00. I will flip a
  coin once. If it comes up heads, you win my
  1.00. If it comes up tails, I win your 1.00.
• Should you take this bet?

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A 1.00 bet on the flip of a coin.
DONT TAKE BET
1.00
COIN FLIP
TAKE BET
HEADS
TAILS
2.00
0.00
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Decision Trees
A 1.00 bet on the flip of a coin.
CHOICE NODE - points where the decision maker
must choose.
CHANCE NODE - points where uncertainty or
chance is introduced.
COIN FLIP
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Decision Trees

• A decision tree should lay out the logical
  sequence of events that will occur.
• At the end of a tree branch is the payoff
  consequence. What is the final state of wealth
  if this sequence of events occurs?

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Decision Trees

• EV(G) ? ci pi
• EV(Take Bet)
• (2.00 0.50) (0.00 0.50)
• 1.00
• EV (Dont Take Bet)
• (1.00 1.0) 1.00

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EV Choice Principle

• ... a decision maker should select the prospect
  with the highest expected value.

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EV of a Fair Bet

• EV(Take Bet) 1.00
• EV(Dont Take Bet) 1.00
• Conclusion - If offered this bet, a decision
  maker is indifferent between take and dont take
  since the expected value of each is equal.

10
Roulette Gamble

• What is the EV of betting 1 on black 13 in
  Roulette?

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Roulette Gamble
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Roulette Gamble

• What is the EV of betting 1 on black 13 in
  Roulette?
• (35 1/38) (-1 37/38)
• -0.0526
• Each time you place a bet on the roulette table
  you should expect to lose 5.26.

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Lets Make a Deal!
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Lets Make a Deal!

• Assume your are a contestant on this game show.
• There are three doors

3
1
2
15
Lets Make a Deal!

• You have been asked to select one of these three
  doors.

3
1
2
16
Lets Make a Deal!

• Behind one of the doors is the grand prize!

1
17
Lets Make a Deal!

• Behind one of the doors is the booby prize!

2
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Lets Make a Deal!

• Now assume you have selected Door 1.
• Monty Hall (the host) will now open one of the
  other doors (door 2 or 3) and show you the
  prize behind that door (it wont be the grand
  prize).

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Lets Make a Deal!

• He will now ask you whether you would like to
  keep your original selection (Door 1) or switch
  to the other unopened door (Door 2 or 3).
• Should you switch?

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Lets Make a Deal!

• Marilyn vos Savant (9/90) - When you first choose
  door No. 1 from among the three, theres a 1/3
  chance that the prize is behind that one and a
  2/3 chance that its behind one of the others.
  But then the host steps in and gives you a clue.
  If the prize is behind No. 2, the host shows you
  No. 3 and if the prize is behind is behind No.
  3, the host shows you No. 2. So when you switch,
  you win if the prize is behind No. 2 or No. 3.
  YOU WIN EITHER WAY! But if you dont switch, you
  win only if the prize is behind door No. 1.

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Lets Make a Deal!

• Grand Prize You choose Host Opens You Switch
  Result
• Is In To
• 1 1 2 or 3 2 or 3 Lose
• 1 2 3 1 Win
• 1 3 2 1 Win
• 2 1 3 2 Win
• 2 2 1 or 3 1 or 3 Lose
• 2 3 1 2 Win
• 3 1 2 3 Win
• 3 2 1 3 Win
• 3 3 1 or 2 1 or 2 Lose

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Lets Make a Deal!

• You lose in 3/9 cases - p(lose) 1/3
• You win in 6/9 cases - p(win) 2/3.
• Still not convinced? Lets try and simulate the
  result using expected value concepts.
• Simulation