Chapter 2 Games of Chance A short questionnaire part Question Rank the following gambles: A: win $5 million with probability win $ million with probability win $ with probability B: win $5 million with probability. win $ million with probability.89 win $ with probability. 2
A short questionnaire part 2 Question 2 Rank the following gambles: C: win $5 million with probability win $ million with probability. win $ with probability.89 D: win $5 million with probability. win $ million with probability win $ with probability.9 3 Chance in Games Chance poses additional challenges to players Uncertainty of outcomes Expected values instead of known payoffs fair games unfair games 4
Fair Game of Chance between gambler and casino.5, -.5 -, 5 Unfair Game of Chance between gambler and casino 8/37, - 9/37 -, 6
Utility Ranking gambles (or lotteries) Gamble A Gamble B Which is better? $45 vs..5 $.5 $ Definition of expected utility Desirable properties of expected utility rationality (consistency) monotonicity (more is better) measurability (any two gambles can be ranked) 7 Equilateral triangle All possible gambles with three outcomes (,,) (/3,/3,/3) (/4,3/2,3/5) (,,) (,,) 8
Expected utility, indifference lines (,,) (.75,,.25) Eu (p) = 2 (.5,,.5) (.5,.5,) Eu (p) =.5 (,,) (,,) Eu (p) = Eu (p) = 9 Attitudes towards Risk Utility of monetary payoffs Uncertainty of outcomes Expected values instead of known payoffs fair games unfair games Different types of players: risk-neutral risk-seeking risk-averse
A risk seeker seeks out risk 8/37.5, - enter a 9/37 -, stay out, Small Business, imperfect information 5, g a p(good) Not, p(bad) b Not 2
Small Business, risk-neutral player u(x) = x g 5, a.2 Not,.8 b Not, 3 Small Business, risk-averse player u(x) = x ½ g 447.2 a.2 Not.8 b Not 4
Small Business, risk-seeking player u(x) = ½ x 2 g.25 9 a.2 Not 5 7.8 b Not 5 Subsidized Small Business g u(5,+subsidy) a p(good) Not u(,) u(+subsidy) p(bad) b Not u(,) 6
Indifference Curves for Expected Utility Theory Risky situations can be seen as probability distributions Expected utility to evaluate probability distribution 7 Indifference curves for risk-neutral player Probability, win $ Eu = B Eu =.5.5 Eu =.5 A Eu = -.5 Eu = - C Probability, 8
Probability, win $ Indifference curves for risk-seeking player.5.4 Eu =.5 Eu =.5 Eu = - Eu = -.5 Eu = - Probability, lose $ 9 Indifference curves for risk-averse player Probability, win $ Eu = Eu =.5.4 Eu = - Eu = - 2.5 Probability, 2
Alternatives to Expected Utility Evidence against expected utility as predictive theory Probability bias Systematic mistake in assessing probabilities Extreme probs ( & ) tend to move toward /2 Framing effect Framing of gamble matters Nonlinearity Payoffs not linear in probabilities The Allais Paradox 2 A short questionnaire part Question Rank the following gambles: A: win $5 million with probability win $ million with probability win $ with probability B: win $5 million with probability. win $ million with probability.89 win $ with probability. 22
A short questionnaire part 2 Question 2 Rank the following gambles: C: win $5 million with probability win $ million with probability. win $ with probability.89 D: win $5 million with probability. win $ million with probability win $ with probability.9 23 Allais Paradox Highest High Liability B D A D preferred to C C Lowest 24
Appendix. Beat the Dealer The game Ten, a model of casino Blackjack Constructing a strategy that beats the dealer in Ten The principles of Ten apply to Blackjack 25 Ten (Simplified Blackjack) Player : You; Player 2: Me (the dealer) Deck of cards: 4 Tens 7 Fives Two cards are burned (removed and shown) You bet min of $; max of $5 One card dealt to You (face down) One card dealt to me (face up) You win your bet if you are closer to Dealer must draw a card with 5 26
The burn in Ten (4, 5) = (#of s, #of 5s) 42/ 56/ (3, 6) 2/ =4/ 3/ (2, 7) 27 Ten after the pack (4, 5) (, ) (-, ) 52/72 4/7 2/3 Draw Hit 3/7 2/72 2 /3 Both dealt 5s 4/7 (, -) Stand Hit Pat 2 3/7 (-, ) EV = 52/72 + 5/9 4/8 (4/7 3/7) = 2/54 (, -) (, ) 28
Ten after the pack (3, 6) (, ) (-, ) 42/72 3/7 Draw /2 4/7 Hit 3/72 2 /2 Both 3/7 dealt 5s (, -) Stand Hit Pat 2 4/7 (-, ) EV = 42/72 + 6/9 5/8 (3/7 4/7) = -3/54 (, -) (, ) 29 Ten after the pack (2, 7) 3/72 (, ) 2/7 (-, ) Draw Hit 5/7 42/72 2 Both dealt 5s 2/7 Stand Hit Pat 2 5/7 EV = 3/72 + 7/9 6/8 ( /2) = -4/54 2/6 4/6 (, -) (-, ) (, -) (, ) 3