Question

1. Suppose that the causal relationship between smoking and lung cancer is as follows. Whether or not you get lung cancer depends on whether you have a particular genetic makeup: if you do, then smoking will cause you to get lung cancer and if you don't then smoking will not give you lung cancer. Furthermore, smoking is a necessary condition for lung cancer. Under these hypotheses, the decision whether or not to smoke looks as follows: state -genetic predisposition s: no genetic predisposition act smoke cancer+pleasure no cancer+ pleasure not smokez3: no cancer, no pleasure z: : no cancer, no pleasure Suppose that 8% of the population has the genetic predisposition to get lung cancer as a consequence of smoking. Emily knows this and satisfies the axioms of expected utility. She ranks the outcomes as follows: z (where means 'better than'). Emily says that she is indifferent between smoking and not smoking. Construct her von Neumann- Morgenstern utility function.

Answer 1

From the question we understand that,

• Probability of Emily having genetic predisposition = s₁ = 0.08
• Probability of Emily not having genetic predisposition = s₂ = 1 - s₁ = 1 - 0.08 = 0.92

Let the event space for the population be Z, for solving the problem we would be considering the below 3 events for Emily,

• z₁ : smokes + genetic predisposition
• z₂ : smokes + no genetic predisposition
• z₃ : not smokes + genetic predisposition
• z₄ : not smokes + no genetic predisposition
• z₃ and z₄ are equivalent events
• also, the events are ordered in terms of preference as z₂ > z₃ > z₁
• Emily is indifferent between smoking and not smoking

The expected utility function against each event is a function of smoking (denote by smoke) and pleasure (denote by ples), such that,

U(z_i) = U(smoke_i, ples_i) where i is an event belonging to event space Z

By definition, the von Neumann-Morgenstern utility theorem tells that, under certain axioms of rational behavior, a decision-maker will act in order to maximise the expected utility defined over the potential outcomes. Main axioms to note are,

• Completeness, which tells that exactly one of z₁ > z₂, z₁ < z₂ and z₁ ~ z₂ holds
• Transitivity, tells us that if z₂ > z₃ > z₁ holds, then we can say that z₂ > z₁
• Continuity, tells that if z₂ > z₃ > z₁ holds, then there exists a probablity p such that pz₂ + (1-p)z₁ ~ z₃
• Independence states that occurance of any event is not dependent on occurance of other event in the same event space

We also know that either Emily can have cancer or not have cancer meaning that p_cancer = p_nocancer = 0.5  

Now computing the probabilities corresponding to each event z_i,

• p₁ = p_cancer X s₁ = 0.5 X 0.08 = 0.04
• p₂ = p_cancer X s₂ = 0.5 X 0.92 = 0.46
• p₃ = p_nocancer X s₁ = 0.5 X 0.08 = 0.04
• p₄ = p_nocancer X s₂ = 0.5 X 0.92 = 0.46

Also, if z₁ > z₂ then U(z₁) > U(z₂) as the theorem assumes monotonic increasing utility function

Hence the utility function is given by,

U = p₁U(z₁) + p₂U(z₂) + p₃U(z₃) + p₄U(z₄)

As z₃ and z₄ are equivalent events,

U = p₁U(z₁) + p₂U(z₂) + p₃U(z₃) + p₄U(z₃)

= p₁U(z₁) + p₂U(z₂) + (p₃ + p₄)U(z₃)

= 0.04U(z₁) + 0.46U(z₂) + (0.04+ 0.46)U(z₃)

Hence, the von Neumann-Morgenstern utility function for Emily is,

0.04U(z₁) + 0.46U(z₂) + 0.50U(z₃) where U(z_i) = U(smoke_i, ples_i)