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# Why Do People Play the Lottery?

Why Do People Play the Lottery?. The economics of gambling and insurance. The Facts. Overall odds of winning a prize: 1 in 54 Individual prizes Match 6: 13,983,816 Match 5+bonus 2,330,636 Match 5: 55,492 Match 4: 1,033 Match 3: 57 Average payout (over 17 draws Nov, Dec 2002, Jan2003)

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## Why Do People Play the Lottery?

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1. Why Do People Play the Lottery? The economics of gambling and insurance

2. The Facts • Overall odds of winning a prize: • 1 in 54 • Individual prizes • Match 6: 13,983,816 • Match 5+bonus 2,330,636 • Match 5: 55,492 • Match 4: 1,033 • Match 3: 57 • Average payout (over 17 draws Nov, Dec 2002, Jan2003) • £26.18

3. The Choice (simplified) • Individual starts with wealth £1000. • Cost of participating is £1 • Prize is £100 • The choice: • Don’t participate: certainty of £1000. • Participate: uncertainty • Lose  £999 • Win  £1100

4. How is the choice made Assume Maximisation of Utility Utility U(1100) U(1000) U(999) 999 1000 1100 Wealth

5. Expected Utility 1 • Suppose: • U(999)=10 • U(1100)=100 • Play the lottery 1000 times, given the odds (p=0.01) you expect: • Win 10 times • Lose 990 times • You expect to have utility • 10*100+990*10=10900 • Average utility per play (Expected Utility) • 10.9

6. Expected Utility 2 Formalising the calculation of expected utility Let: Then: Expected utility is a weighted average of the utilities obtained when winning and losing with the weights given by the probability of winning and losing.

7. Expected Wealth Defined similarly to Expected Utility Average wealth per play if the lottery is played a large number of times.

8. Graphing E(U) and E(W) 110 Pw = 0.5 PL = 0.5 Ww= 110 WL = 10 Uw= 110 UL = 10 60 Pw = 0.2, PL = 0.8 30 10 10 30 60 110

9. Utility The ‘Fair’ Bet U(1100) Fix the odds in the lottery so that: E(W)=1000 U(1000) E(U) U(999) The bet is irrational because: E(U)<U(1000) 999 1000 1100 E(W) Wealth Interpretation: Whilst we know there’s a chance of getting a big win and over time we expect to break even, the value we place on the money we might lose is higher than that placed on what we might win

10. Increasing Marginal Utility Utility The high and increasing value we place on increased income persuades us its worth taking the risk. E(U) U(1000) 999 1000 1100 Wealth E(W)

11. Insurance • The reverse of a lottery: • People pay to avoid uncertainty. • Car: • Worth £1100 intact • Worth £100 after an accident • Probability of an accident 0.1 • E(W)=1000

12. Economics of Insurance Utility U(1100) As long as the premiums are not excessive it is rational to insure E(U) Max Premium U(100) 100 1000 1100 Wealth

13. Why play the lottery and insure • A person with the type of utility function for which insurance is rational finds playing the lottery irrational. • Why play the lottery? • Utility is gained from the act of participating. • Marginal utility is locally increasing. • Where losses are expected to be small we are willing to take risks.

14. Locally increasing MU Utility E(U2) U(WI) Premium Wealth with insurance Wealth

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