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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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**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) • £26.18**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**How is the choice made**Assume Maximisation of Utility Utility U(1100) U(1000) U(999) 999 1000 1100 Wealth**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**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.**Expected Wealth**Defined similarly to Expected Utility Average wealth per play if the lottery is played a large number of times.**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**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**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)**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**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**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.**Locally increasing MU**Utility E(U2) U(WI) Premium Wealth with insurance Wealth

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