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 madeAssume Maximisation of Utility

Wealth

Utility

1000

U(1000)

1100

U(1100)

999

U(999)

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 2Formalising the calculation of expected utility

1000

10990100109.10

1099.010001.0

101000

990100

1000

10

Let:

wP1000

10LP

1000

990

WU100 LU10

Then: LLWw UPUPUE

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 WealthDefined similarly to Expected Utility

Average wealth per play if the lottery is played a large number of times.

01.10001000

98901011000

1000

999990110010

LLWw WPWPWE

Graphing E(U) and E(W)

Pw = 0.5

PL = 0.5

Ww = 110

WL = 10

Uw = 110

UL = 10

10

10

60

110

60 110

Pw = 0.2, PL = 0.8

30

30

The ‘Fair’ Bet

Wealth

Utility

1000

U(1000)

1100

U(1100)

999

U(999)

Fix the odds in the lottery so that:

E(W)=1000

E(U)

E(W)

The bet is irrational because:E(U)<U(1000)

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

Wealth

Utility

1000 1100999

U(1000)E(U)

E(W)

The high and increasing value we place on increased income persuades us its worth taking the risk.

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 InsuranceUtility

Wealth100 11001000

Max Premium

As long as the premiums are not excessive it is rational to insure

U(1100)E(U)

U(100)

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 MUUtility

WealthWealth with insurance

E(U2)U(WI)

Premium