📎

Example

Game show host gives you a coin

  • tail 2.000.000$
  • heads 0$
  • not playing 1.000.000$

EMV

EMV=12($0)+12($2,500,000)=$1,250,000 \text{EMV} = \frac{1}{2}(\$ 0)+ \frac{1}{2}(\$ 2,500,000)=\$ 1,250,000 

Expected utilities

Lets say SnS_n is the state of owning nn dollars.

Our current financial status is SkS_k .

EU( Accept )=12U(Sk)+12U(Sk+2,500,000)EU( Decline )=U(Sk+1,000,000) \begin{aligned}E U(\text { Accept }) &=\frac{1}{2} U(S_{k})+\frac{1}{2} U(S_{k+2,500,000}) \\E U(\text { Decline }) &=U(S_{k+1,000,000})\end{aligned} 

The Utility for getting your first million dollars is very high, but the utility for the additional million is smaller.

exactly proportional to the logarithm

Lets say

U(Sk)=5U(S_k) = 5

U(Sk+1,000,000)=8U(S_{k+1,000,000}) = 8

U(Sk+2,500,000)=9U(S_{k+2,500,000}) = 9

Then the rational agent (that is not a billionaire) would decline although the EMV is higher.

EU( Accept )=12⋅5+12⋅9=7EU( Decline )=8 \begin{aligned}E U(\text { Accept }) &=\frac{1}{2} \cdot 5+\frac{1}{2} \cdot 9 = 7\\E U(\text { Decline }) &=8\end{aligned} 

A billionare would have a locally linear utility function and accept.

Conclusion:

In general we can say that in the positive part of the curve where the slope is decreasing, for any lottery LL :

U(L)<U(SEMV(L))U(L) < U(\textcolor{pink}{S_{EMV(L)}})

the utility of being faced with that lottery << than the utility of being handed the expected monetary value of the lottery with absolute certainty