L3 - Game Theory: Expected Utility

Question 1

What is the formula for Expected Value?

Answer 1

Question 2

What the formula for Expected Utility?

Answer 2

• where utility is a function of the income of during that state
  
  • weighting utilities (whereas Expected Value is weighting incomes)

Question 3

What does the utility function of a risk neutral person look like?

Answer 3

• For a risk neutral individual the utility of the expectedvalue (EV(R)) between two incomes is equal to the expected utility of the expected value EU(R)
• In general, other risk neutral utility functions can include u(c)=bc where b > 0
• Risk neutraility is often assumed in game theory for simplicity

Question 4

What does the utility function of a Risk-Averse individual look like?

Answer 4

• Other risk-averse functions can include u(c) = c^a where 0 < a < 1
• For smaller values of an in this range, the utility function is more curved and the individual is relatively more risk-averse.
  
  • Steeper at the start too

Question 5

What does the utility function of a risk-loving individual look like?

Answer 5

Question 6

How do we choose between actions with expected payoffs and risks?

Answer 6

The last line means that they will stop gambling and accept the banker’s safe option when x is over that amount

A risk-averse individual would have a lower value of x that they would be willing to accept so they can stop gambling

Question 7

What are the limitations of expected utility theory?

Answer 7

All from Carmichael (2004)

• Theoretical Limitations –< arise from when E(U) theory is unable to capture important elements of an individual choice problem
  
  • Portfolio Effect –> probability distributions of gambles under consideration are in isolation and do not consider ‘other gambles’ that are already faced by the individual but may be relevant (invest in M&S or UNited football) –> already hold stocks in Tesco so may be more weighted to M&S
  • Temporal considerations –> some people don’t like the uncertainty of waiting to find out a result or when they are going to receive a payment –> people may not act rationally because of this (E(U) theory doesn’t consider this)
    
    • Finding out the result of the lottery today and getting paid in a year, finding out the result of the lottery in one years time and getting paid then or a certainity
      
      • Should be indifferent between the first two
• Descriptive Limitations–> follow from experimental evidence of a violation of the underlying assumptions of E(U) theory
  
  • independence axiom –> Common elements involved in all payoffs (may research theses)
    
    • common consequence effects
    • common ratio effect
  • Transitivity
    
    • preference reversal when choices are over risky prospected leading to violation of transitivity
    • Lichtenstein & Slovic (1971) –> some people prefer lotteries with high certainty of a small price less than a small chance of winning a large prize
      
      • Yet when asked directly they frequently preferred the opposite

Certainty effect (Kahneman, 2011)

• Changes in the probability of gains or losses do not affect people’s subjective evaluations in linear terms
• under behavioural economics, a move from 50-60% has a smaller emotional impact than a move from 95-100% (certainty)
• Equally a move from 0-5% is valued more than 5-10% –> people over-weight small probabilities