One of the toughest questions facing an incentive website or any e-commerce website for that matter, is the question of price. In the case of the former, price refers to the size of the cash reward (rebate) offered to users. The goal is to size the rebate such that it maximizes net income. If the goal is to maximize ROI, this sweet spot is critical. And if the goal isn’t to maximize ROI, there are probably greater things to worry about. :) Most incentive marketers will size their cash rebates based on trial and error, but there’s a far more precise way to determine the optimal rebate. The following is a walk through of how to solve for the optimal incentive rebate. Using some high school algebra and calculus. I applied this technique to determine optimal rebates for the network of incentive websites I created in 2003. This technique allowed Topaz Financial to drive more than 100,000 completed advertiser actions at margins that would not have otherwise been possible. If the math looks daunting, there are many tutorials for solving these equations. A search for ’solving systems of equations’ and ‘differential equations’ should turn up helpful resources.
To illustrate how it works, let’s create a hypothetical situation. We’ll assume that your incentive website offers a cash rebate for each completed action, in this case, the purchase of a pair of shoes. Further, let’s assume the merchant pays you a $25 commission for each completed sale. We want to determine the size of the cash rebate that maximizes total profit. Your first inclination might be to offer visitors a large share of your commission to entice a greater number of users to complete the purchase (action). Yet paying out a large share of the commission could cause the reduction in net income that outweighs the increase in volume of actions. On the one hand you want to offer a rebate that entices a large number of visitors to complete the offer. On the other hand you want to offer a rebate small enough to keep your net commission high. Likewise, there is a positive relationship between the size of the rebate and volume of actions completed. To make the math simpler we will assume that this positive relationship is linear. In other words, we will assume that a given change in the rebate will always induce the same increase in purchases. Admittedly at extremely high or low rebates this assumption may not hold, but for our purposes it is a fair assumption.
To start, we need to collect a few data points. You’ll need to experiment to see how users react to a few different rebates. The benefit of the linear assumption is that we only test 2 prices in order to calculate the slope of our line.
Assuming the offer is currently running, you already have one set of coordinates. Let’s assume that when an offer has a rebate of $5 there are 15 completed actions. To find the second set of coordinates you’ll want to set a new rebate and measure the number of completed actions. Let’s say that when we increase the rebate to $10, there are 40 completed actions. To put it in math terms:
Let us denote pairs of shoes sold as Y and rebate as X.
The equation of Y in terms of X is a linear function of the form Y = A + B X, where A and B are constants.
This equation passes through (5,15) and (10,40)
Thus,
15 = A + 5 B Equation 1
40 = A + 10 B Equation 2
Subtracting Equation 1 from Equation 2,
40 = A + 10 B Equation 2
15 = A + 5 B Equation 1
25 = 0 + 5 B
Or B= 5 =25/5
Substituting the value of B in Equation 1,
15 = A + 25
or A = -10 =15-25
Thus, the equation is of the form
Y = 5 X – 10
where X is the reward and Y is the number of shoes sold
We have to maximize profits Z= (Commission-Rebate) x Number of shoes sold= (25-X) Y
but Y= 5X -10
Therefore,
Z=(25-X) (5X -10) = 125 X -250 – 5 X^2 + 10 X = O r Z= -5 X^2 + 135 X -250
Our task is to maximize profits or maximize Z= -5 X^2 + 135 X -250
To find the maximum value of Z we differentiate Z with respect to X and equate it to zero:
dZ / dX = – 10X +135=0
or X = 135/10= 13.5
Thus to maximize profits, the rebate should be 13.5
Profit = Z= – 5 X^2 +135 X -250 = 661.25
Since a rebate of 13.5 would mean selling a fraction of pairs, we can offer a rebate of 13 or 14 which would give an identical profit of $660 (see table).The most challenging part of the process is holding the traffic sources and number of clicks to the offer constant while you are testing. If there are huge swings in the number of users exposed to the offer, or if the composition of traffic changes drastically, your results will be less trustworthy.
Posted by keyenfarrell 
