Why Do People Play the Lottery?

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 5bonus 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
• Dont 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)
1000
1100
999
Wealth
5
Expected Utility 1

• Suppose
• U(999)10
• U(1100)100
• Play the lottery 1000 times, given the odds
  (p0.01) you expect
• Win 10 times
• Lose 990 times
• You expect to have utility
• 101009901010900
• 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
60
110
30
9
The Fair Bet
Utility
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)ltU(1000)
1000
1100
999
E(W)
Wealth
Interpretation Whilst we know theres 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)
1000
1100
999
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
1100
1000
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