Games Against Nature

1
Games Against Nature

• Lecture Notes
• Game Theory
• February 7, 2008

2
Strategy Choices

• How should a person select a strategy to play
  against nature if the ratios of natures strategy
  choices are UNKNOWN?
• Four Ideas
• LaPlace Optimist
• Wald Pessimist
• Hurwicz Gamblers Coefficient
• Savage No Regrets

3
Example What to do?
4
LaPlaces Solution

• Choose row with maximum possible payoffs.
• Choose row with highest sum
• Equivalently, this is the row with the highest
  average payoffs
• We can think of LaPlaces solution as one for the
  optimist
• High reward, but may also be highly risky

5
Example LaPlace Solution
6
Walds Solution

• Find minimum payoffs for each row
• Choose maximum
• Similar to minimax strategy choice in zero-sum
  games
• We can think of Walds solution as one for the
  pessimist
• Low reward, but very little risk

7
Example Wald Solution
8
Hurwiczs Solution

• Determine your coefficient of optimism
• Call it p
• For each row calculate
• (row max) p (row min) (1 - p)
• Choose row with highest adjusted payoff
• We can think of Hurwiczs solution as one that
  allows for any level of gambling
• Strategy choices can be infinitely individualized

9
Example Hurwicz Solution
Assume p 3/4
10
Savages Solution

• Determine regret matrix
• For each new entry, subtract the original entry
  (in that position) from the row maximum
• Represents loss player faces due to lack of
  foreknowledge
• Minimize regret
• Find all row maximums for regret matrix
• Choose row with minimum regret
• We can think of Savages solution as one for the
  second-guesser type
• Monday morning quarterback

11
Example Savage Solution
12
So what should we do?

• Notice Straffins exquisite writing
• Game in Chapter 10 has four different solutions,
  depending on which method you use to determine
  your strategy
• Each of the four may be appropriate in different
  settings
• Try to think of at least of a game application
  where each one of these strategy selection tools
  might make sense
• Read about Milnors axiomatic approach to
  evaluating strategy choices in games against
  nature