# Where lock-in feedback finds and tracks optimal selling price After my bare bones Python implementation of Davide Iannuzzi’s and Maurits Kaptein’s “Lock in Feedback” (LiF) algorithm, I thought it interesting to get a better feel for the algorithm’s capabilities and constraints by putting it to the test in finding, and subsequently “locking in” to the optimal price of a product in a R simulation (inspired by one of Francis Smart’s blogposts) of a noisy, yet monopolistic market.

Before we start, let’s first define two helper functions. MatrixShift is used in LiF. It enqueues a row at the end of a matrix M while dequeuing a row from the start of the matrix. It then returns the new matrix. PlotAnimation returns the animated GIF at the end of this page (if doPlot is set to TRUE). This function makes use of R libraries “animation” and “cairo”.

``````doPlot <- FALSE

matrixShift <- function(M, row){
if(any(is.na(A))){
i <- sum(!is.na(A))
A[i+1] <- row
} else {
A <- c(A,row)
A <- A[-1]
}
return(A)
}

plotAnimation <- function() {

Cairo(file = paste0("S", i/200, ".png"), bg = "white", width = 1300,height = 900)
par(mfrow=c(3,2))
plot(qd, op, type="l", xlab="Quantity", ylab="Dollars", main="Demand", lwd=2)
abline(h=0, lwd=2)
lines(qd, mc, col="red", lwd=2)
plot(minmax(op),minmax(tr,tp), type="n", ylab="", xlab="Price", main="Optimal pricing")
grid()
abline(h=0, lwd=2)
abline(v=optprice, col="red", lwd=2)
lines(op,tr, col="blue", lwd=2)
lines(op,tp, col="red", lwd=2)
plot(arr.wtp, ylim=c(35,60))
plot(arr.P, type="l",ylim=c(35,60))
plot(arr.p0, type="l", xlab="Time in stream", ylab="selected price",ylim=c(35,60))
lines(arr.p0, col="red",ylim=c(35,60))
dev.off()
}
minmax <- function(...) c(min(...),max(...))``````

Now on to the meat of our simulation! To keep our model relatively simple, we will simulate a single price monopolist model of pricing, marginal cost, demand and revenue.

The general assumptions of a monopoly market simulation are:

• Only one firm is in the market, with a market share of 100%.
• A unique product with no close substitutes
• Entry of firms is not permitted (completely blocked).

In other words, a single price monopolist is the only supplier of a particular good. The monopolist therefore has the power to set a price to sell the product at. Those who have a willingness to pay which is greater than the price will buy the good. Those who have a willingness to pay for the good which is less than the chosen price will not buy it.

Let’s first generate our consumers:

``````npeep <- 2000                           # Number of potential consumers
wtp <- 45 + rnorm(npeep,0)*4            # Each person has a different willingness to pay
wtp[wtp<0] <- 0                         # if less than zero, they really don't want to buy, set to zero
wtp_sliced <- c()``````

To figure out the demand curve we count the number of people willing to pay at least as much as the offering price.

``````maxop <- 120                            # Maximum offering price
op <- 0:maxop                           # Offering Prices (ranges from 0 to maxop)
qd <- rep(NA,maxop+1)                   # Quantity Demanded``````

Set LiF parameters:

``````TT <- npeep                             # Length of stream
T <- 10                                 # Integration time
w <- (2*pi)/T                           # Omega
A <- 3                                  # Amlitude LIF
gamma <- 0.05                           # Learnrate
p0 <- 55                                # Startvalue of price``````

Some variables to help us log our sales:

``````arr.P <- rep(NA, TT)                    # Price log
arr.p0 <- rep(NA,TT)                    # Startprice log
arr.wtp <- rep(NA,TT)                   # Willingness To Pay log

yws <- rep(NA, T)

We will also need Total Revenue, Total Profit, and Total Cost vectors:

``tr <- tp <- tc <- rep(NA,maxop+1) ``

Now iterate through all of our customers, where each iteration selling prices is calculated by LiF while logging our sales:

``````for(i in 1:npeep){

# Log willingness to pay
wtp_sliced <- c(wtp_sliced, wtp[i])
arr.wtp[i] <- wtp[i]

for (j in 1:maxop+1) qd[j] <- sum(wtp_sliced>=op[j])

# Marginal cost is increasing in our model - let's see if LiF can keep up!
mc <- qd*0.005

# Calculate Quantity Demanded
qd.gain <- qd[-length(qd)]-qd[-1]
qd.gain[length(qd.gain)+1] <- qd.gain[length(qd.gain)]

# Calculate Total Cost
for (j in 1:length(op)) tc[j] <- sum((mc*qd.gain)[length(qd):j])

# Calculate Total Revenue
tr <- qd*op

# Calculate Total Price
tp <- tr-tc

if (doPlot) {
indexmax <- which.max(tp)
optprice <- op[indexmax]
}

# LIF: Oscillate around P
P <- p0 + A*cos(w*i)
arr.P[i] <- P

# If willingness to pay is smaller than the price, don't buy
# If willingness to pay is equal to, or greater than the price, buy
if (wtp[i]<P) {
} else {
}

# Compute Profit:
yws <- matrixShift(yws, yt * cos(w*i))

# LIF: Reset p0
if(i > T){
yw <- sum(yws) / T
p0 <- p0 + (gamma / T) * yw
}

# Log p0
arr.p0[i] <- p0

# plot a snapshot of the logs every 200 iterations
if (i %% 200 == 0 && doPlot) {
plotAnimation
}
}``````

Now let’s see if LiF has indeed found, and kept track of, our selling price:

``print(p0)``
``##  54.6034``

Seems a reasonable enough a price, but to really know how the price as determined by LiF corresponds to the optimal value we can infer from our monopolist assumptions, we need to take a closer look at our logs.

Or maybe we can do better – let’s plot the log files as animated GIFs, to get an even better feel for how LiF, and our simulated model, behaved:

``````if (doPlot) {
ani.options(convert = shQuote("convert.exe"),ani.width = 1300, ani.height = 900, interval = 0.05, ani.dev = "png",ani.type = "png")
bm.files = sprintf("S%i.png", 1:(npeep/200) )
im.convert(bm.files, output = paste0("ani",format(Sys.time(), "%d%b%H%M%S"),".gif"), extra.opts = "-dispose Background", clean = T)
}`````` Let’s assume my plots are numbered from 1 to 6, left to right, then top to bottom, like so:

Plot 1Plot 2
Plot 3Plot 4
Plot 5Plot 6

… then the numbered plots would describe the following processes:

#### Plot 3

The randomized willingness to pay as per customer (per iteration, as we serve one customer at a time). This willingness to pay is randomly determined by rnorm – that is, normally distributed around a preset value of \$45. You can clearly see how the willingness to pay is scattered around this value.

#### Plot 5

Does the customer buy the product, or not? As you can see, LiF seems to find an optimum value where most customers seem willing to pay, while still making a profit for the monopolist. The amount of customers that are willing to pay becomes less when the price is forced up because of the slowly rising marginal cost, which has to be discounted into the prices, if the monopolist is to stay profitable.

#### Plot 1

Here you can observe how the marginal cost (red line) slowly rises as more and more products leave the factory (maybe, for example, as the result of natural wear and tear on the machine that produces the product) and how this changes the demand for the product at certain price points (black curve). The optimal price point for the monopolist is at the point where marginal revenue curve intersects the marginal cost curve.

#### Plot 2

Two graphs that allow you to determine the optimal price as per our monopolist model. Total Revenue at all relevant price points in in blue. And Total Profit in red. As you can see, as a result of rising Marginal Costs, the optimal price moves from somewhere below 40 towards 50.

#### Plot 5

LiF, which is, of course, unaware of our model and the optimal price we found in plot 2, can however search for the optimal price by oscillating around an internal price parameter, and use LiF’s magic to seesaw towards an accurate estimate of the optimal price as based on the willingness of customers to buy our product at that particular set price.

#### Plot 6

Which seems to work out quite well – LiF quickly moves towards an optimal price, and follows nicely follows the upward trajectory of the optimal price point. Yay for LiF!