After reading through quite a bit of literature -- or at least a weekend's worth -- on optimized pricing, it seems as though the same ideas are being circulated, again and again and again. I know there are good resources out there in terms of pricing in high-volume environments, such as online advertising, but for the most part, in retail and macro "bidding", such as winning large contracts every few years or applying for an RFP, the thought process has remained the same: what's the probability of "winning" -- i.e. getting the bid -- at price point $x$, and what's the price at which we optimize the expected return. This can basically be described as:
\begin{eqnarray}
P(y \vert \vec{x} ) &=& \frac{1}{1+e^{-\beta \cdot \vec{x}}}\\
\mathrm{E}\left(x \right) &=& x P(y \vert x )
\end{eqnarray}
Where, I"ve already modeled the "winning" probability as a logistic regression -- standard practice based on former papers. But, it's interesting to note that supply and demand curves have a very close connection here, and most often this function $p(y \vert x)$ needs to have some specific properties, such as:
A nice way to formulate this is of course with a right-sided CDF. Integrating $p(y \vert x)$, what some people call a "willingness to pay'' function $w(x)$ we have:
\begin{eqnarray}
d(x) &=& D \int_x^\infty dx \ \ w(x) \\
d(x) &=& D p(y \vert X \ge x)
\end{eqnarray}
So the ``demand'' at price point $x$ will now have some nice properties, such as being monotonically decreasing and most likely ``smooth'' due to integration. It's important to point out that when someone associates an ``elasticity'' with a supply and demand curve, such as:
\begin{eqnarray}
d(x) & \approx & \alpha + \beta x
\end{eqnarray}
With $\beta < 0$, what you're actually doing is imposing a constant ``willingness'' to pay function, which is interesting because my ``risk'' of saying no to any deal -- much like any consumer -- is certainly not constant over all price points.
Typical strategies for pricing a single customer i've read have :
Both methods work really well, since you can stratify by customer / bid-type, throw in extra variables for controls, etc., but why not expand the functional form? If the ``risk'' of drop out at all price points is non-uniform, why not use something that has ordered risk built into model, such as a weibull regression, or even an empirical kaplan meier estimator on price point $x$. (Particularly when you are worried about bumping price points on contracts to failure.)
\begin{eqnarray}
P(y \vert \vec{x} ) &=& \frac{1}{1+e^{-\beta \cdot \vec{x}}}\\
\mathrm{E}\left(x \right) &=& x P(y \vert x )
\end{eqnarray}
Where, I"ve already modeled the "winning" probability as a logistic regression -- standard practice based on former papers. But, it's interesting to note that supply and demand curves have a very close connection here, and most often this function $p(y \vert x)$ needs to have some specific properties, such as:
- Be monotically decreasing in $x$ -- for non status-associated or "Giffen" goods.
- Approach zero as $x \to \infty$.
- Approach the total supply, call it $D$, as $x \to 0$.
A nice way to formulate this is of course with a right-sided CDF. Integrating $p(y \vert x)$, what some people call a "willingness to pay'' function $w(x)$ we have:
\begin{eqnarray}
d(x) &=& D \int_x^\infty dx \ \ w(x) \\
d(x) &=& D p(y \vert X \ge x)
\end{eqnarray}
So the ``demand'' at price point $x$ will now have some nice properties, such as being monotonically decreasing and most likely ``smooth'' due to integration. It's important to point out that when someone associates an ``elasticity'' with a supply and demand curve, such as:
\begin{eqnarray}
d(x) & \approx & \alpha + \beta x
\end{eqnarray}
With $\beta < 0$, what you're actually doing is imposing a constant ``willingness'' to pay function, which is interesting because my ``risk'' of saying no to any deal -- much like any consumer -- is certainly not constant over all price points.
Typical strategies for pricing a single customer i've read have :
- Fit a logistic function / regression to the right-sided CDF, $p(y \vert X \ge x)$. This results in a logistic, or more or less bell-shaped willingness-to-pay function $w(x)$, where there is a ``sweet spot'' for price with some variance.
- Fit a linear regression to the demand function, $d(x)$. Constant $w(x)$.
Both methods work really well, since you can stratify by customer / bid-type, throw in extra variables for controls, etc., but why not expand the functional form? If the ``risk'' of drop out at all price points is non-uniform, why not use something that has ordered risk built into model, such as a weibull regression, or even an empirical kaplan meier estimator on price point $x$. (Particularly when you are worried about bumping price points on contracts to failure.)
