I found a question here that I am skeptical about the accepted answer. And I am unsure about finding the most optimal strategy.
Here is the question essentially:
You are given the opportunity to make money by betting a total of 100 bucks on the outcome of two simultaneous matches:
- Match A is between the Pink team and the Maroon team
- Match B is between the Brown team and the Cyan team
The Pink team's probability of victory is 40%. The Brown team's probability of victory is 70%. The betting odds are
- Pink: 7:4
- Maroon: 2:3
- Brown: 1:4
- Cyan: 3:1
How much money do you bet on each team? You do not have to bet all 100 bucks, but your bets must be whole numbers and the total of all five blanks (bets on the four teams and the unbet amount) must sum to 100. There is no single "correct" answer, but there are many "wrong" answers. As a reminder, a hypothetical team having 2:7 odds means that if you bet 7 on that team and they win, you get your 7 bucks bet back and win an additional 2 bucks.
What is the expected value or expected payoff or expected profit? Do these mean the same thing?
The author said that the equation of expected payoff is below and the answers to this post accepted that this equation is true. The author is betting size $B$ on team B, size $P$ on team P etc. $\#1$
$(P+ \frac{7}{4})0.4 + (M + \frac{2}{3}M)0.6+(B+\frac{1}{4}B)0.7+(C+3C)0.3 + R(unbet)$
But why is this correct? I am skeptical about this accepted answer. Don't you have to consider the outcome of losing bet size $P$, $M$, $B$ and $C$?
For example should the expected payoff be $\#2$*:
$(P+ \frac{7}{4}P)0.4 - (P \times 0.6)+ (M + \frac{2}{3}M)0.6 - (M \times 0.4)+(B+\frac{1}{4}B)0.7- (B \times 0.3)+(C+3C)0.3 - (C \times 0.7)+ R(unbet)$
or should I even consider adding the sizes of my bet or my stake back?
So should the expected payoff be this instead by not accounting my original stake since I own the stake to begin with?: $\#3$
$(\frac{7}{4}P)0.4 - (P \times 0.6)+ (\frac{2}{3}M)0.6 - (M \times 0.4)+(\frac{1}{4}B)0.7- (B \times 0.3)+(3C)0.3 - (C \times 0.7)+ R(unbet)$
This makes the most sense to me since these are my actual payoff, my actual profits, not including the stake the I own. Should this be called the expected profits instead?
Doing the multiplication, this reduces to $0.1P + 0M - 0.125B + 0.2C$
How do I maximize this equation (with or without computer) and hence find the most optimal strategy? Am I even on the right path?
EDIT: Also Is there a way to mathematically reason through the problem and find a balance between the expected payoffs and the expected variance? I could just compute the expected value for each of the team and bet all my money on the team that has the highest expected value, but this will also maximizes my potential loses. How do I balance between expected value and expected variance. Thanks.
I believe the source of the confusion is indeed simply defining what you mean by "payoff."
Let $X$ denote "how much money you have at the end" and $Y$ denote "how much did your wealth change." Since you started with 100 bucks, $X=Y+100$, or equivalently $Y=X-P-M-B-C-R$.
Either of these is a reasonable way to think about the payoff, and maximizing one of them is the same as maximizing the other.
The quoted answer is correctly finding the expected value of $X$, but you seem to want to find the expected value of $Y$, the change in wealth.
So I believe your formula #3 is correct for $E[Y]$ except you should remove the $R(unbet)$ term at the end, because the unbet portion does not cause a change in wealth.