In a given lottery, let's say my odds of winning a grand prize are $p_0$, and the grand prize has value $V$. I can either buy more tickets, at a cost $C_1$, or for each ticket, I can buy a multiplier for each ticket, at a cost $C_2$ per ticket. The multipliers are structured as follows:
5X
1 in 10
4X
1 in 10
3X
1 in 3.33
2X
1 in 2
That is, if I pay $C_2$ extra for a ticket for a multiplier, then I have a $1/2$ chance of doubling my prize, a $1/3$ chance of tripling, etc.
For simplicity let's assume there is only the grand prize of value $V$. Should I buy more tickets, or buy multipliers on all my tickets?
My Thoughts
My average payout per ticket is $Vp_0$, which costs $C_1N$, if I buy $N$ tickets. So if I buy multipliers on $M \leq N$ tickets, my profit equation takes the form:
$$
\text{Profit} = NVp_0 – C_1N – C_2M
$$
I suppose the question is, at what point does the incremental payout from an additional ticket become less than buying a multiplier on an existing ticket. If $m$ is the multiplier, and $P_m$ is the probability of that multiplier, then the question is at what point does $Vp_0 – C_1 \lessgtr mP_mVp_0 – C_2$
Best Answer
On the assumption that there are lots of tickets sold (so your buying one more does not change the value of the existing tickets), each ticket has some expected value $v$. This includes all the prizes. If you buy another ticket, you get $v$. If you buy a multiplier, you get $v(\frac 12 \cdot 1 + \frac 3{10}\cdot 2 + \frac 1{10}\cdot 3 + \frac 1{10}\cdot 4)=\frac {18}{10}v$ added to your value. Your loss on buying another ticket is $C_1-v.$ Your loss on buying a multiplier is $C_2-\frac {18}{10}v.$ Compare these and you will know which way you are less badly off.