I got somewhat puzzled after watching this video on Kelly Criterion in economics and the associated "just one more" paradox. This question should be self-contained, so watching the video itself is not necessary.
Setup: We have $V_0 = 100\$$ to start with, and throw an unbiased coin. In case of heads, we gain 80%, resulting in $V_1 = 180\$$, in case of tails we lose 50%, resulting in $V_1 = 50\$$. While the video considers playing multiple consecutive games, this question relates to playing only a single game.
We can find the expected value of this game, namely $$\langle V_1 \rangle = 0.5\cdot 180 + 0.5\cdot 50 = 115$$ And thus the expected gain is positive $$\langle V_1 - V_0 \rangle = 115 - 100 = 15$$
Problem: Now, I propose to convert this game to a logarithmic scale. The original equation for this game is $$V_1 = V_0 X$$ where $X$ is the multiplier of the coin toss, namely $1.8$ or $0.5$. We can take the base 10 logarithm of both sides $$\log_{10} V_1 = \log_{10} V_0 + \log_{10} X$$ We can rewrite the above equation, defining new variables for the corresponding terms $$W_1 = W_0 + Y$$ Finally, we can calculate the expected gain in this log-view of our game $$\langle W_1 - W_0 \rangle = \langle Y \rangle = 0.5 \cdot 0.255 + 0.5 \cdot (-0.301) = -0.023$$
Question: So, in the original view this is a winning game, in the logarithmic view it is losing game. Why is that? Do I have a mistake in my calculation, or maybe the assumption that the two games are equivalent is wrong?
Original Case :
Proportional Change $ W = + 0.5 \times (0.8) - 0.5 \times (0.5) = 0.40 \times 0.25 = 0.15 $
Actual Change $ 0.15 \times 100 = 15 $ : POSITIVE !
New Case :
Proportional Change $ W = + 0.5 \times \log [ (0.80) ] - 0.5 \times \log [ (0.50) ] = 0.5 \times 0.204 = 0.102 $
Actual Change $ 0.102 \times 100 = 10.2 $ : POSITIVE !
We get Positive Change in linear & logarithmic Calculations.
No Change Case :
Let us say winning gives $+80$ , while losing give $-80$ :
Proportional Change $ W = + 0.5 \times \log [ (0.80) ] - 0.5 \times \log [ (0.80) ] = 0.5 \times 0 = 0 $
We get $0$ Change in linear & logarithmic Calculations.
Negative Case :
Let us say winning gives $+80$ , while losing give $-90$ :
Proportional Change $ W = + 0.5 \times \log [ (0.80) ] - 0.5 \times \log [ (0.90) ] = 0.5 \times [8/9] = -0.051 $ : Negative !
Actual Change $ -0.051 \times 100 = -5.1 $ : Negative !
We get Negative Change in linear & logarithmic Calculations.
Summary :
That Change is logarithmic , hence the value will not match the linear non-logarithmic Change , in general.
It will always match the SIGN.
ADDENDUM :
Why did I calculate Proportional Change & then Multiplied that by 100 to get Actual Change ?
(A) When Subtracting $\log$ values , the $+ \log 100$ will cancel with the $- \log 100$ ( It will cancel during Division ) & hence it will not matter.
(B) When Adding the $\log$ values , it will generate $+ \log 100 + \log 100$ which is $+ \log 100^2$ ( Involving the Square of the Initial Amount ) which is totally unrelated to the Calculations.
Hence , I ignored that Constant , introducing it when necessary.