<p>Let's start with an environment in which there is only ever a single pledge level. The question at hand is: how do we choose the dollar amount of that level to maximise revenue (income)?</p>

<p>Observation 1: your probability distribution implies a demand curve.</p>

<p>We can plot the cumulative distribution of willingness to pledge and it will probably look roughly like this (with most of the mass at the bottom, as you speculate):</p>

<p><a href="https://i.sstatic.net/Hkhsi.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Hkhsi.png" alt="enter image description here"></a></p>

<p>The $y$ value in this figure tells us the probability that a given individual's willingness to pledge (WTP) is no greater than the dollar value on the $x$ axis.</p>

<p>By turning this upside down, we get the probability that somone is willing to pay <em>at least</em> this much:</p>

<p><a href="https://i.sstatic.net/UOgGS.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/UOgGS.png" alt="enter image description here"></a></p>

<p>This tells us, for example, that there is a rouly 55% chance of a randomly selected individual being willing to pledge <em>at least</em> \$10. If we take 100 people then, on average, 55 of them will pledge. Thus, we can think of the vertical axis as being an indirect way of measuring pledge <em>quantities</em>, corresponding with the pledge level (or <em>price</em>) on the horizontal axis. If we swap the two axes, we therefore get something that looks very much like a textbook demand curve:</p>

<p><a href="https://i.sstatic.net/YPtop.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/YPtop.png" alt="enter image description here"></a></p>

<p>Given this demand curve, we can use all of the usual tools from the kind of standard monopoly optimisation problem that you can find in and introductory economics textbook. For example, if the CDF in the first figure is given by function $F$ then$$\Pr(\text{WTP}&gt;x)=1-F(x)$$ so the revenue from setting the pledge level at $x$ when the audience consists of $n$ potential pledgers is $R(x)=[1-F(x)]xn$. Just like a standard monopolist's problem, we can compute the revenue-maximizing pledge level ("price") by setting marginal revenue equal to zero:</p>

<p>$$R'(x)=n[1-F(x)]-xnF'(x)=0.$$</p>

<p>Substituting your empirical distribution for $F$ in this equation and solving for $x$ gives your optimal pledge level.</p>

<hr>

<p>Now, if we allow for multiple pledge levels then things become much more complicated. Firstly, we need a model of the behaviour of people and why it is that some people might pledge more than others.</p>

<p>If, for example, we are working in a Kickstarter-like environment then the main motive would be to get the goodies that come with the higher pledge tier. But that means we need a model not only of the pledge levels, but what you get in return. A complete treatment of this is beyond the scope of this answer, but a fairly readable summary can be found in "<a href="http://people.ischool.berkeley.edu/~hal/Papers/version.pdf" rel="nofollow noreferrer">Versioning Information Goods</a>" by Hal Varian. If you want a more detailed, technical treatment then the best place to look is "<a href="https://mitpress.mit.edu/books/economics-contracts" rel="nofollow noreferrer">The Economics of Contracts: A Primer</a>" by Salanie.</p>

CC BY-SA 3.0
2016-07-18T10:50:29.883Z
There is some cheating when you cross from empirics to theory. The CDF you describe was estimated when multiple pledge levels were allowed and then you use this to estimate a one pledge level only demand function.
Giskard
CC BY-SA 3.0
2016-07-18T10:58:46.207Z
@denesp The first sentence of my answer states that I am assuming there is a single pledge level. I have edited the answer to make this a bit more clear.
Ubiquitous
CC BY-SA 3.0
2016-07-18T11:53:04.700Z
Sorry I did not notice that. But then how does one derive the CDF? Is it given or does one perform experiments with countless individual pledge levels?
Giskard
CC BY-SA 3.0
2016-07-18T12:45:06.980Z
@denesp I don't know! I took the luxuary of sidestepping this (difficult) question since the OP said "As to how to obtain this type of information, let's say I have found a model, or I can just draw out the histogram myself based on some reasonable guess about the relative potential of a person choosing a pledge level."
Ubiquitous
CC BY-SA 3.0
2016-07-18T10:31:14.570Z
2016-07-18T11:00:14.003Z
2016-07-18T11:00:14.003Z
Ubiquitous
<p>I'm going to offer an ambitious answer and say you can "generalize" your Kickstarter-esque problem as an (extremely special) <strong>auction</strong>. I say auction because you can presumably pledge on an essentially continuous scale, and although there are tiered rewards, you are also bidding towards the completion of a good along with other people, and presumably people know about the completion of this good.</p>

<p>Besides being a multi-good auction where you are looking to set optimal reservation prices for each tiered reward, and where I assume each higher tier gets you everything from the lower tiers, although the final good you are producing may not be a public good, let us say there is a tier where if you pledge that much, you get access/ownership to the final good when it is completed. There will likely be some sort of free-riding effect for how much you want to bid to ensure the project's completion, since there will usually be a lot of bidders for serious projects like this.</p>

<p>Bidding beyond that tier is, minus the value of the other tiered rewards, just bidding further on the completion of the project. Can we consider bids beyond this level to be contributing to an excludable public good at that point? I'd hesitate to designate those higher bids as venture capitalism.  If this is the case, you could find a mathematical framework for pricing such goods here, <a href="http://ac.els-cdn.com/S0022053199926030/1-s2.0-S0022053199926030-main.pdf?_tid=0721e754-7c32-11e6-b976-00000aab0f6b&amp;acdnat=1474046679_13604bb7a2c51e90faac657f86f9b795" rel="nofollow"><strong>Auction-Like Mechanisms for Pricing Excludable Public Goods</strong></a>. It talks about how you can price relative to people's reservation prices for a good that I think yours is like.</p>

<blockquote>
  <p>"We study the problem of strategy-proof mechanism design for indivisible and excludable public goods."</p>
</blockquote>

<p>(The references for this paper are also very good, e.g. Clarke's article on multi-part pricing, which sounds quite relevant for a tiered incentive structure like yours.)</p>

<hr>

<p>Your major concern still is in finding the distribution which bidders lie on. I think you are right that you will end up having to guess what the distribution looks like. You also talk about whether the distribution will change based on how many people see the project. My suspicion is that the distribution will be more heavily towards the low end of willingness to pay with fewer viewers, and as more people see the project (and its bidders), the more people will be willing to put in larger bids for the project, so the distribution will slowly "grow a tail" towards the higher end.</p>

CC BY-SA 3.0
2016-09-16T17:28:08.103Z
2016-09-16T17:28:08.103Z
Kitsune Cavalry
<p>Let's say that I have a project and I need to earn some target to be successful, let's say $50,000</p>

<p>The way I earn is by one time donations at various levels.  1, 5, 7.5 (some discounted item), 10, 15, 20, and 25.</p>

<p>If i say that i want the least number of pledges, then it seems obvious that I would like to maximally sell as much $25 items as possible.  </p>

<p>But in reality, there is some sort of distribution that tells me how likely it is that I'll be able to sell each type of items.  For example, if this were a kickstarter project, typically there are fewer higher tier bids and a lot more lower tier ones.  As to how to obtain this type of information, let's say I have found a model, or I can just draw out the histogram myself based on some reasonable guess about the relative potential of a person choosing a pledge level.  </p>

<p>My question is, how do I figure out, given some likelihood about how people tend to pledge, how to reach this particular target?  Perhaps let's say I assume that the distribution is the same no matter how many people end up bidding.  So I guess I really am trying to find the minimum number of bids necessary, given the liklihood of certain bids actually being sold.  </p>

<p>The reason I'm asking this is because I want to try to quantify in some way the importance of exposure.  The more people see something, there's some likelihood they'll be interested, and even if they are, they're going to pick the tier item that suits them, which is typically low.  I know it might be a bit silly to think about this without regard to the quality of the project itself, but I was wondering if thinking about this might provide some insight on how important it is to do proper marketing.  </p>

<p>I'm also interested in a reverse question.  Let's say I can only reach about 5000 people, based on the fact maybe that similar projects to mine only got an average of that maybe people to back them.  Is there a way to figure out how to shape the distribution to reach the goal in a way that sticks to the goal? Or perhaps to start with my initial distribution, and see how to morph it as little as possible to reach the target?</p>

<p>Thanks</p>

CC BY-SA 3.0
2016-07-18T07:48:21.663Z
This seems unclear. The exact details of 'morphing' as little as possible are not given. Obviously you would like to 'morph' the smallest donations into the biggest ones. It is not clear how much more difficult this is than morphing big ones into biggest ones. With such murkiness it is not possible to give an answer. (Perhaps you need a marketing solution.)
Giskard
<optimization>
Finding the minimum # of items to be sold to meet a goal, based on likelihood
notaneconomist
# Finding the minimum # of items to be sold to meet a goal, based on likelihood

(Asked by: username_0 on 2016-07-18)

Let's say that I have a project and I need to earn some target to be successful, let's say $50,000

The way I earn is by one time donations at various levels. 1, 5, 7.5 (some discounted item), 10, 15, 20, and 25.

If i say that i want the least number of pledges, then it seems obvious that I would like to maximally sell as much $25 items as possible.

But in reality, there is some sort of distribution that tells me how likely it is that I'll be able to sell each type of items. For example, if this were a kickstarter project, typically there are fewer higher tier bids and a lot more lower tier ones. As to how to obtain this type of information, let's say I have found a model, or I can just draw out the histogram myself based on some reasonable guess about the relative potential of a person choosing a pledge level.

My question is, how do I figure out, given some likelihood about how people tend to pledge, how to reach this particular target? Perhaps let's say I assume that the distribution is the same no matter how many people end up bidding. So I guess I really am trying to find the minimum number of bids necessary, given the liklihood of certain bids actually being sold.

The reason I'm asking this is because I want to try to quantify in some way the importance of exposure. The more people see something, there's some likelihood they'll be interested, and even if they are, they're going to pick the tier item that suits them, which is typically low. I know it might be a bit silly to think about this without regard to the quality of the project itself, but I was wondering if thinking about this might provide some insight on how important it is to do proper marketing.

I'm also interested in a reverse question. Let's say I can only reach about 5000 people, based on the fact maybe that similar projects to mine only got an average of that maybe people to back them. Is there a way to figure out how to shape the distribution to reach the goal in a way that sticks to the goal? Or perhaps to start with my initial distribution, and see how to morph it as little as possible to reach the target?

Thanks

**Tags:** optimization
> **Comments:**
> **username_1**: This seems unclear. The exact details of 'morphing' as little as possible are not given. Obviously you would like to 'morph' the smallest donations into the biggest ones. It is not clear how much more difficult this is than morphing big ones into biggest ones. With such murkiness it is not possible to give an answer. (Perhaps you need a marketing solution.)  

---

## 2 answers

### Answer by username_2 on 2016-07-18. Score: 1

Let's start with an environment in which there is only ever a single pledge level. The question at hand is: how do we choose the dollar amount of that level to maximise revenue (income)?

Observation 1: your probability distribution implies a demand curve.

We can plot the cumulative distribution of willingness to pledge and it will probably look roughly like this (with most of the mass at the bottom, as you speculate):

[![enter image description here](https://i.sstatic.net/Hkhsi.png)](https://i.sstatic.net/Hkhsi.png)

The $y$ value in this figure tells us the probability that a given individual's willingness to pledge (WTP) is no greater than the dollar value on the $x$ axis.

By turning this upside down, we get the probability that somone is willing to pay *at least* this much:

[![enter image description here](https://i.sstatic.net/UOgGS.png)](https://i.sstatic.net/UOgGS.png)

This tells us, for example, that there is a rouly 55% chance of a randomly selected individual being willing to pledge *at least* \$10. If we take 100 people then, on average, 55 of them will pledge. Thus, we can think of the vertical axis as being an indirect way of measuring pledge *quantities*, corresponding with the pledge level (or *price*) on the horizontal axis. If we swap the two axes, we therefore get something that looks very much like a textbook demand curve:

[![enter image description here](https://i.sstatic.net/YPtop.png)](https://i.sstatic.net/YPtop.png)

Given this demand curve, we can use all of the usual tools from the kind of standard monopoly optimisation problem that you can find in and introductory economics textbook. For example, if the CDF in the first figure is given by function $F$ then$$\Pr(\text{WTP}>x)=1-F(x)$$ so the revenue from setting the pledge level at $x$ when the audience consists of $n$ potential pledgers is $R(x)=[1-F(x)]xn$. Just like a standard monopolist's problem, we can compute the revenue-maximizing pledge level ("price") by setting marginal revenue equal to zero:

$$R'(x)=n[1-F(x)]-xnF'(x)=0.$$

Substituting your empirical distribution for $F$ in this equation and solving for $x$ gives your optimal pledge level.

---

Now, if we allow for multiple pledge levels then things become much more complicated. Firstly, we need a model of the behaviour of people and why it is that some people might pledge more than others.

If, for example, we are working in a Kickstarter-like environment then the main motive would be to get the goodies that come with the higher pledge tier. But that means we need a model not only of the pledge levels, but what you get in return. A complete treatment of this is beyond the scope of this answer, but a fairly readable summary can be found in "[Versioning Information Goods](http://people.ischool.berkeley.edu/~hal/Papers/version.pdf)" by Hal Varian. If you want a more detailed, technical treatment then the best place to look is "[The Economics of Contracts: A Primer](https://mitpress.mit.edu/books/economics-contracts)" by Salanie.

> **Comments:**
> **username_1**: There is some cheating when you cross from empirics to theory. The CDF you describe was estimated when multiple pledge levels were allowed and then you use this to estimate a one pledge level only demand function.  
> **username_2**: @denesp The first sentence of my answer states that I am assuming there is a single pledge level. I have edited the answer to make this a bit more clear.  
> **username_1**: Sorry I did not notice that. But then how does one derive the CDF? Is it given or does one perform experiments with countless individual pledge levels?  
> **username_2**: @denesp I don't know! I took the luxuary of sidestepping this (difficult) question since the OP said "As to how to obtain this type of information, let's say I have found a model, or I can just draw out the histogram myself based on some reasonable guess about the relative potential of a person choosing a pledge level."  

---

### Answer by username_3 on 2016-09-16. Score: 0

I'm going to offer an ambitious answer and say you can "generalize" your Kickstarter-esque problem as an (extremely special) **auction**. I say auction because you can presumably pledge on an essentially continuous scale, and although there are tiered rewards, you are also bidding towards the completion of a good along with other people, and presumably people know about the completion of this good.

Besides being a multi-good auction where you are looking to set optimal reservation prices for each tiered reward, and where I assume each higher tier gets you everything from the lower tiers, although the final good you are producing may not be a public good, let us say there is a tier where if you pledge that much, you get access/ownership to the final good when it is completed. There will likely be some sort of free-riding effect for how much you want to bid to ensure the project's completion, since there will usually be a lot of bidders for serious projects like this.

Bidding beyond that tier is, minus the value of the other tiered rewards, just bidding further on the completion of the project. Can we consider bids beyond this level to be contributing to an excludable public good at that point? I'd hesitate to designate those higher bids as venture capitalism. If this is the case, you could find a mathematical framework for pricing such goods here, [**Auction-Like Mechanisms for Pricing Excludable Public Goods**](http://ac.els-cdn.com/S0022053199926030/1-s2.0-S0022053199926030-main.pdf?_tid=0721e754-7c32-11e6-b976-00000aab0f6b&acdnat=1474046679_13604bb7a2c51e90faac657f86f9b795). It talks about how you can price relative to people's reservation prices for a good that I think yours is like.

> "We study the problem of strategy-proof mechanism design for indivisible and excludable public goods."

(The references for this paper are also very good, e.g. Clarke's article on multi-part pricing, which sounds quite relevant for a tiered incentive structure like yours.)

---

Your major concern still is in finding the distribution which bidders lie on. I think you are right that you will end up having to guess what the distribution looks like. You also talk about whether the distribution will change based on how many people see the project. My suspicion is that the distribution will be more heavily towards the low end of willingness to pay with fewer viewers, and as more people see the project (and its bidders), the more people will be willing to put in larger bids for the project, so the distribution will slowly "grow a tail" towards the higher end.


---

