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Everyone has their own reaction when discovering wrong data. It might start with a double take or maybe an itching feeling that the number should be a higher. However it starts, it usually leads to an investigation to discover what went wrong. While this is a very normal reaction, I offer an alternative. Before turning over every stone in your ETL, ask a few questions to discover if your “wrong” data really is wrong.

In this post I explore what wrong means when it comes to data (spoiler alert: it is not black and white). I also offer a few tricks to diagnose which of the buckets of wrong your problem falls into. Yes, this approach may add an extra step or two in your process, but it can also save a day of work trying to fix something that isn’t even broken.

## Begin with questions

Whoever came up with the saying “measure twice, cut once” probably was not thinking about troubleshooting the outputs of a reporting system. Origin aside, the saying is applicable when diagnosing the cause of wrong data – make sure you know what problem you are investigating before you start investigating.

The two questions I find most helpful to answer before beginning my investigation are:

• What results would I / you expect to see?
• Where would I / you check this number to know it is correct? (i.e., source of truth)

These questions are incredibly helpful because they help to identify the reference point that the consumer of the data uses to determine if the data is correct. This leads me to the thesis of my post – wrong does not actually mean wrong, it means a mismatch between expectation and reality. This means that for data to be viewed as wrong there is an issue with expected numbers, presented numbers or potentially both at the same time.

In additional to diagnosing where the issue lies (i.e., expectation or reality) these questions will also help you diagnose the degree to which expectation and reality are out of sync. If your stakeholder expects to see the number 23 and -23 is displayed it is potentially a simpler fix (find the negative you dropped) than seeing the number 1.35.

After evaluating the answers to the questions listed above, you will most likely find you land in one of the following three buckets of “wrong”:

• The data is confusing (but not wrong)
• The stakeholder is using a faulty source to “check” the data (which is not wrong)
• The data really is wrong

## The data is confusing (but not wrong)

This is the most common use case I have come across. Confusing data often occurs when the business logic is not surfaced to the stakeholder as it is hidden behind a reporting layer or occurs deep in the transformation layer.

Let’s take the KPI “sales”, for example. You can calculate sales as gross or net, at time of sale or at time recognized following GAAP, including returns or not – the variants are endless and often unique to the team and / or company. Ultimately, it is best to get the entire organization speaking the same language when it comes to KPIs, but that is a large enough topic for a standalone blog post (or book!). At minimum, I suggest exposing the definitions of your most important KPIs in some sort of data dictionary (perhaps as part of your data warehouse SLA).

Another potential cause of confusion could be time delays. Most reporting tools and data warehouses do not display real-time data. This can lead to a situation where there is no confusion about the metric definition, just confusion about how current the data is. For example, data in most website reporting tools (e.g., Google Analytics) is delayed by a few hours (I have seen websites with heavy traffic see delays as great as 12 hours). Stakeholders may want to look at the traffic volume of a big press hit or a new website feature and be confused when the data does not appear to be there.

Notice cause of confusion in these cases is in how the data is presented or taught to the stakeholder. Sending the data in an email, or building a report without providing context (metric definition, data freshness) can sometimes do more harm than good.

## The stakeholder is using a faulty source to “check” the data

Remember, wrong is relative and in this case, it is relative to data that is itself incorrect. One place you might see this is in web traffic statistics. Suppose you have an internal, server-side tracking system capturing site visits as well as an external tracking tool. On your way to the office an email hits you: “Google Analytics traffic looks higher than our internal tracking.” There are a number of regularly occurring reasons this could happen.

The most common is normal variance between data sources. In the example above you are comparing a near lossless data source (server-side tracking) with a fallible data source (client-side javascript tracking). The client-side session count could be lower because of ad blockers or javascript not loading due to a heavy website / slow internet connection. Alternatively, the server-side session count could be lower because your 3rd party client-side tracking tool automatically filters out bots and other suspect traffic.

In the case of normal variance, the different between the data sources typically falls in a predictable range (if it doesn’t you may want to run through the web analytics checklist). Once you have determined acceptable variance use it as a lens through which you evaluate whether data is actually wrong. Demonstrating the historical variation (through data) can help to defuse wrong data conversations. Setting the expectation up front that the systems will have different numbers can help prevent wrong data conversations before they start.

You may, however, come across cases when the internal traffic data is significantly lower than your external web tracking tool. There is the very real possibility to consider that the internal tracking tool could have issue. Perhaps the server crashed last night or was overwhelmed by the amount of traffic that was passing through the system during a traffic spike. While your internal services may be robust, it is always important to consider that all systems can break – internal and external.

## The data really is wrong

Sometimes data really is wrong. It is a result no one likes, but considering the complexity of translating your transactional database to a star schema and then calculating business KPIs, it is expected. I like to think my goal is not to prevent all errors, but to make sure the errors that do occur are small and caught / resolved quickly. To improve your odds of quickly finding errors, build your data quality flywheel – a culture where everyone feels responsible for data quality.

## Conclusion

Sometime wrong data is wrong and sometime it is actually right. Before embarking on a journey to the center of your data stack do your due diligence to understand why your stakeholder (or you) thinks the data is wrong. Armed with this information, apply the appropriate solution – teaching your stakeholder or fixing the data product.

Remember “wrong” data, regardless of whether the data is wrong or not, is your responsibility. If stakeholders look at a report and get the wrong idea of the business logic under the hood, that is as much of a problem as the math being wrong.

Reporting tools are great for a business (see: Reporting is a Gateway Drug), but with sharing information comes the responsibility to help people make sense of it. If you are digging into the data every day, you are all too familiar with the imperfections of the data stores you are touching. Just remember not everyone is as familiar with the data as you!


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# American Institute of Mathematical Sciences

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2015, 12(5): 1007-1016. doi: 10.3934/mbe.2015.12.1007

## Order reduction for an RNA virus evolution model

 1 Centre de Recerca Matemática, Campus de Bellaterra, Edifici C, 08193 Barcelona, Spain 2 Department of Applied Mathematics, Samara State Aerospace University (SSAU), 443086 Samara, 34, Moskovskoye shosse, Russian Federation 3 Department of Technical Cybernetics, Samara State Aerospace University (SSAU), 443086 Samara, 34, Moskovskoye shosse, Russian Federation

Received  August 2014 Revised  April 2015 Published  June 2015

A mathematical or computational model in evolutionary biology should necessary combine several comparatively fast processes, which actually drive natural selection and evolution, with a very slow process of evolution. As a result, several very different time scales are simultaneously present in the model; this makes its analytical study an extremely difficult task. However, the significant difference of the time scales implies the existence of a possibility of the model order reduction through a process of time separation. In this paper we conduct the procedure of model order reduction for a reasonably simple model of RNA virus evolution reducing the original system of three integro-partial derivative equations to a single equation. Computations confirm that there is a good fit between the results for the original and reduced models.
Citation: Andrei Korobeinikov, Aleksei Archibasov, Vladimir Sobolev. Order reduction for an RNA virus evolution model. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1007-1016. doi: 10.3934/mbe.2015.12.1007
##### References:

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2018 Impact Factor: 1.313

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# Closest Pair Algorithm

I want to find the most optimal algorithm for finding closest ordered pairs and link them.

For example: Go thru the vector, one by one

For each one set the next according to a set of various conditions

For example, something like:

A B X C Y Z D

First we will link A to B Then when we have a look at B, we will link B to C Next when we consider X, we will link X to Y Next when we consider C, we will link C to D ... and so on

Currently someone has completed this task with some nested loops that are genuinely sub-par and it completes in O(n^2) time. Not to mention it doesn't work. What might be a better algorithm?

Here is what the current algorithm is:

for(i in list)
while (i and j) is not linked
increment j
if j is not the end
while ++j is not the end
if link exists (i and j)
break
end while
end if
end for

I was going to write my own but I have a feeling I've seen this problem before at university and there was a simple solution to it..? Can anyone point me in the right direction please?

• – Rob Sep 13 '18 at 1:55
• @Rob Thats it, closest pair algorithm I think, thanks – solarflare Sep 13 '18 at 2:07
• @Rob thanks for that. I'm going to wait in case someone else has a better solution as a divide and conquer algorithm with 2 quicksorts might in fact be overkill in my situation when there's going to be 20 conditions for the sort. – solarflare Sep 13 '18 at 2:30
• If you could specify which algorithm you wanted to use in your question it would narrow down the answer to a speed/complexity appropriate answer. I suggested that algorithm because you wanted an improvement over $O(n^2)$. – Rob Sep 13 '18 at 3:03

An alternative approach could be

1. Scan the vector and insert all elements into a self-balancing BST (AVL or Red-Black Tree etc etc, your pick) along with its index in the vector. $Node := (Value,VectorIndex)$

2. Then you have two choices

• Scan the vector again, and for each element, search for it in the BST, and find its successor. Link the element and its successor in the vector.
• Do an inorder traversal of the BST and for every pair of consecutive nodes visited in the BST, link their corresponding elements in the vector.

Either way, step 1 and 2 each take $O(n\space log \space n)$, hence the total algorithm ends up with a time complexity of $O(n \space log \space n)$ and space complexity of $O(n)$.

• I like it. But as I have been informed by the business analysts the use case will never have more than around 7 or 8 items in it so I'm going with brute force. – solarflare Sep 14 '18 at 0:03
• "X will never happen" might be the quickest way to shoot yourself in the foot though. – Aristu Sep 14 '18 at 0:36
• @solarflare just like Aristu said, never trust the fact that the use case once specified will still be enforced many years later. Its always safer to use a condition like ' if (sizeof(dataset)<threshold) then doBruteForce() else doEfficientAlgo()'. – RandomPerfectHashFunction Sep 14 '18 at 4:44
• I don't want to reveal the industry or use case but trust me if it did happen it would be an incredibly bizarre (and somewhat laughable) rarity. If it happens, they can wait an extra 2 seconds we don't mind. Thanks for all the tips guys, much appreciated. – solarflare Sep 14 '18 at 5:22


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# The Four Nines

$\Huge 9^{9^{9^9}}$

Find the last 2 digits of the number above.

×

Problem Loading...

Note Loading...

Set Loading...


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# What's the potential of the LHC's heavy ion experiment?

RHIC has been the dominant player in heavy ion physics, producing tantalizing evidence in support of the entropy/viscocity formula from AdS/CFT. What's the potential of the LHC's Pb ion collsions? What can it achieve which RHIC can't? What measurements will be improved?

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• The kinematic range of high-energy nucleon-nucleus collisions extends down to values of Björken $x$ as small as $10^{-6}$ (compared to $10^{-3}$ for RHIC), which expands our knowledge of the nuclear parton distributions (e.g. whether and how well they factorize into individual nucleon PDFs)


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×

# Finding the 'x'

How can we solve these type of problems:

e.g. $2^{x}=x$

or, $2^{x}=x^{2}$

To find the possible value(s) of x.

Note by Dinesh Nath Goswami
1 year, 11 months ago

Sort by:

use log in both this is the simplest way 1st one answer is no solution 2nd one has 2 solution I.e., 2&4 · 1 year, 11 months ago

actually it has 3 solutions · 1 year, 11 months ago

thanks... · 1 year, 11 months ago

Try using graphs and apply calculus. · 1 year, 11 months ago

thanks. I'll try that way. · 1 year, 11 months ago

Use logarithm too. That will make the problem easier. · 1 year, 11 months ago


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§ Use back-substitution and Gaussian elimination to solve a system of linear equations. Linear equation represents relations between two or more variables. Use the following steps to solve a non-homogeneous linear Diophantine equation. If a, b, and c are real numbers wherein (ax + by + c = 0) or (ax + by = c), and if a and b are not equal to 0 then the equation is said to be a linear equation having two variables.E.g. 2) Declare a matrix[n][n] and constant[n][1]. The detailed, step-by-step solutions will help you understand the concepts … They are represented by English letters like a, b, c, x, y, z etc. Posted 26 March 2009 - 06:43 AM The variables a and b are determined and x is computed by -b/a. System Of Linear Equations Involving Two Variables Using Determinants. sol = solve([eqn1, eqn2, eqn3], [x, y, z]); xSol = sol.x ySol = sol.y zSol = sol.z. and the variables denote the unknowns. A linear equation is any equation that can be written in the form $ax + b = 0$ where $$a$$ and $$b$$ are real numbers and $$x$$ is a variable. Say that we wish to solve for $x$. Just put in the coefficients of the variables and the equivalent sum to the right of the equation. In this case, the equation can be put in the form + =, and it has a unique solution = − in the general case where a ≠ 0.In this case, the name unknown is sensibly given to the variable x.. Step 1: Determine the GCD of a and b. In this case, we have one variable and one equation, so the situation is solvable (we simply subtract five from both sides): This equation, however, is not completely solved: We have two variables and only one equation. println(“Enter the number of variables in the equations: “); Scanner input = new Scanner(System. int n = input. Given the coefficients as input, it will solve the equation and output the roots of the equation. Program to solve a system of linear equations in C++. If a = 0, there are two cases.Either b equals also 0, and every number is a solution. The method for solving linear equations in one variable is quite simple. Please fill in all input boxes. The inputs to solve are a vector of equations, and a vector of variables to solve the equations for. It may take several steps to solve this equation, so we need a clear and organized strategy. Examples of n Variables, n Equations. This will clear students doubts about any question and improve application skills while preparing for board exams. \begin{align*}ax + by & = p\\ cx + dy & = q\end{align*} where any of the constants can be zero with the exception that each equation must have at least one variable in it. Just put in the coefficients of the variables and the equivalent sum to the right of the equation. We’ll start off the solving portion of this chapter by solving linear equations. Re: program that solves two linear equations in two variables. There are multiple ways to solve such a system, such as Elimination of Variables, Cramer's Rule, Row Reduction Technique, and the Matrix Solution. Solve the linear Diophantine Equations: $$ax+by=c, x, y \in\mathbb{Z}$$. In general, to solve for n variables, you need n distinct equations. ... (More systematically, the variables corresponding to leading $1$’s in the echelon form matrix are dependent variables, and the rests are independent (free) variables.) system of linear equations in two variables has graphs that are parallel, intersectin g, and coinciding 19. solves a system of linear equations in two variables by (a) graphing; (b) substitutio n; (c) eliminatio n. 20.. solves problems involving systems of linear equations in two variables. 22 thoughts on “ C++ Program for Gauss-Elimination for solving a System of Linear Equations ” Orest March 22, 2016 Дякую, те що треба! If equation (2) is multiplied by the opposite of the coefficient of $y$ in equation (1), equation (1) is multiplied by the coefficient of $y$ in equation (2), and we add the two equations, the variable $y$ will be eliminated. 10x – 3y = 5, 10x + 4y = 2, etc. Use the following steps to solve a non-homogeneous linear Diophantine equation. The number of operations for the LU solve algorithm is as .. Section 2-2 : Linear Equations. xSol = 3 ySol = 1 zSol = -5. solve returns the solutions in a structure array. Background ax+b = 0 is the format for one variable equations. Here we will use the recursive leading-row-column LU algorithm.This algorithm is based on writing in block form as:. To find the solution for linear equations in two variables with the help of cross-multiplication, we need to use the following equation: x(b 1 c 2-b 2 c 1) = y/(c 1 a 2-c 2 a 1) = 1/(b 2 a 1-b 1 a 2) What is the condition to get a unique solution? Viewed 14k times -3. One variable. Algorithm Begin 1) Take the no of variables n and the coefficients of each variable as input. char []var = {‘x’, ‘y’, ‘z’, ‘w’}; System. The basic skill learned in linear algebra course. 3) Make for loops i = 0 to n-1 and j = 0 to n-1 to take the coefficients of each variable as the elements of the matrix. In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same set of variables. This solve linear equation solver 3 unknowns helps you solve such systems systematically . I say distinct because these two are not: 1. Frequently the term linear equation refers implicitly to the case of just one variable.. Below is the program immediately after solving three equations. In nature, linear occur most often. The LU decomposition algorithm. The solution of triangle form of system is simple. Simultaneous Linear Equations Solver for Eight Variables: This calculator calculates for the eight unknown variables in eight linear equations. The solution to a system of equations having 2 variables is given by: Java Program to Solve any Linear Equations //This is a sample program to solve the linear equations. This is a C++ program to represent Linear Equations in matrix form. I need to programmatically solve a system of linear equations in C, Objective C, or (if needed) C++. Usually, when using the substitution method, one equation and one of the variables leads to … To access the solutions, index into the array. Substitute the value found into any equation involving both variables and solve for the other variable. A linear system of two equations with two variables is any system that can be written in the form. Please fill in all input boxes. The method I have implemented for solving linear equations in two variables is a formula which can be derived by operating on both sets of the equation. We eliminate one variable using row operations and solve for the other. Solve System of Linear Equations. public class Solve_Linear_Equation. ... You can solve this with a program exactly the same way you solve it by hand (with multiplication and subtraction, then feeding results back into the equations). out. Let the equations be a 1 x+b 1 y+c 1 = 0 and a 2 x+b 2 y+c 2 = 0. After substitution of x N into the next to the last equation we can find x N-1.And so on, and so on until we find x 1.This is the method of backward-sweep. a,b and c are real numbers and coefficients of x and y such that the solution for such an equation is a pair of … Hi Dani, yeah,but the fact is that I am using mathematics related programs like this for the first times.So don't know,what to do. 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 sexp for base (1/e)^e ~= 0.0660? sheldonison Long Time Fellow Posts: 639 Threads: 22 Joined: Oct 2008 03/06/2013, 11:57 PM (This post was last modified: 03/07/2013, 12:23 AM by sheldonison.) Can it have a Tetration superfunction with sexp(-1)=0, sexp(0)=1, sexp(1)=b, sexp(2)=b^b? This is a base on the shellthron boundary, with one fixed point $L=\frac{1}{e}$ where $b^L=L$ and $b={(\frac{1}{e})}^e\approx 0.065988$ This base has a rationally indifferent multiplier=-1, which has a pseudo period=2. Because the multiplier is an exact root of unity, rather than an irrational root of unity, this base doesn't have a Schroeder function. $b^{L+x}\approx L-x+1.35914x^2-1.23151x^3+...$ Iterating this function starting with z=1 doesn't lead to behavior that seem much like Tetration. I was curious if it was possible to generate an sexp(z) solution given that I think there is only one fixed point, and that the fixed is rationally indifferent, in this case with a pseudo period of 2. As I understand it, the abel function for this function, would be generated from b^b^z. If you iterate the b^z function twice, b^b^z would be a parabolic case with an Abel function for each of the four leaves on the Leau-Fatou flower. But I don't know how you can combine these four Abel functions together into a single inverse abel function for b^z. Note in the equation below, that there is no x^2 term. $b^{b^{L+x}}\approx L+x-1.2315x^3+0.836897x^4+...$ At the real axis starting close to the fixed point and iterating twice always gets you a little bit closer to the fixed point, but the convergence is very slow. For example, starting with z=0, iterating z=b^z oscillates towards the fixed point. I generated some results and graphs for a complex base on the shellthron boundary with a rationally indifferent fixed point with a multiplier of $\exp(\frac{2\pi i}{5})$, with a pseudo period=5, which also lacks a Schroeder function because the multiplier is a 5th root of unity; the other fixed point is repelling. See post #10 of this thread, http://math.eretrandre.org/tetrationforu...hp?tid=729 This result was generated via the conjectured merged complex tetration solution, using both fixed points of the base. It seems that if there is a solution for other rationally indifferent roots of unity on the Shell Thron boundary, than there could be a solution for this base as well. In the upper half of the complex plane, such a solution for base (1/e)^e might only be well behaved near the imaginary axis, exponentially converging towards the fixed point as $\Im(z)$ increases much like the conjectured solution with pseudo period=5. Such a solution would become increasingly chaotic as real(z) increased or decreased. I have no idea what the solution could look like in the lower half of the complex plane, since I think there is only one fixed point. - Sheldon « Next Oldest | Next Newest »

 Messages In This Thread sexp for base (1/e)^e ~= 0.0660? - by sheldonison - 03/06/2013, 11:57 PM RE: sexp for base (1/e)^e ~= 0.0660? - by tommy1729 - 03/07/2013, 12:01 AM RE: sexp for base (1/e)^e ~= 0.0660? - by mike3 - 03/08/2013, 06:18 AM RE: sexp for base (1/e)^e ~= 0.0660? - by sheldonison - 03/08/2013, 02:30 PM RE: sexp for base (1/e)^e ~= 0.0660? - by mike3 - 03/09/2013, 06:00 AM RE: sexp for base (1/e)^e ~= 0.0660? - by sheldonison - 03/09/2013, 08:25 AM RE: sexp for base (1/e)^e ~= 0.0660? - by sheldonison - 03/10/2013, 06:35 AM RE: sexp for base (1/e)^e ~= 0.0660? - by sheldonison - 11/13/2013, 01:08 AM RE: sexp for base (1/e)^e ~= 0.0660? - by sheldonison - 11/13/2013, 06:45 PM

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# 3n+1 problem and cycles

Just to make sure I am up to date with this problem. I know (or I think I do) that it is not yet proven that there are no non-trivial cycles for the collatz sequence (please correct me if I am wrong). But have we already reduced this problem to finding a non-trivial cycle? i.e. if we suppose that there are no non-trivial cycles at all, is it already proven that with this assumption the Collatz conjecture holds? As far as I can see in the literatures, even this is not shown (i.e. we can reduce the problem to proving whether there are no non-trivial cycles). But I may be wrong, so please correct me if I am wrong.

Edit: Most of my knowledge are on par with Lagarias annotated bibliography and I believe it has been a few years since this literature review.

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A priori, the Collatz conjecture could fail in two ways: a non-trivial cycle or a Collatz sequence tending to infinity. As far as I know, both possibilities are still open. –  Andreas Blass Aug 3 at 22:40
I can only confirm what Andreas Blass has written. The Wikipedia article on the Collatz conjecture is usually kept up-to-date: en.wikipedia.org/wiki/Collatz_conjecture. –  Stefan Kohl Aug 3 at 22:47
With respect to Lagarias' bibliography: on February 12, 2012, when he updated his bibliography for the last time so far, Lagarias has written to me: "I am happy to hear about other articles but I have run out of energy for updating the annotated bibliography from 2010 -- on." -- I think this is quite a pity, and it would certainly be good if the further maintenance of this valuable bibliography would be passed on to somebody else, rather than just letting it become a document of purely historical interest by the time. –  Stefan Kohl Aug 4 at 9:16
@Stefan agreed! –  Jose Capco Aug 4 at 11:38

The page http://www.ericr.nl/wondrous/index.html lists numerous records and data related to the Collatz problem, and was updated recently (only about two month ago).

It explictly lists as open the problem:

Conjecture 1a : (weak 3x+1 conjecture) No integer N is divergent.

(There divergent has the natural meaning means that the sequence of iterations diverges.)

It is thus possible that there is no other cycle but the known one, yet still the conjecture fails as there might be divergent elements. Thus, the reduction asked about in OP is not known.

(To be very precise let me add that it could be theoretically possible there is some result that shows: if there were divergent elements there'd be other cycles, so that the above could be open while the reduction in OP still known, yet neither do I know such a result nor does it seem natural.)

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# Definition:Completed Dirichlet L-Function

## Definition

Let $\chi$ be a primitive Dirichlet character to the modulus $q \ge 1$.

Let $\kappa = \dfrac 1 2 \paren {1 - \map \chi {-1} }$.

Let $\delta = 1$ if $\chi$ is the principal character, and $0$ otherwise.

The completed Dirichlet $L$-function for $\chi$ is defined to be

$\map \Lambda {s, \chi} = \dfrac {1 + \kappa} 2 \paren {s \paren {1 - s} }^\delta \map \Gamma {\dfrac {s + \kappa} 2} \map L {s, \chi}$

where $\map L {s, \chi}$ is the Dirichlet $L$-function for $\chi$.


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osula9a

2022-01-19

Proving limit with $\mathrm{log}\left(n!\right)$
I am trying to calculate the following limits, but I dont

turtletalk75

You can easily show that The first inequality is a very standard induction proof, and the second inequality is straight-forward (you're comparing
From there, since $f\left(n\right)=\mathrm{log}n$ is an increasing function, you have that
$n\mathrm{log}\left(2\right)\le \mathrm{log}\left(n!\right)\le n\mathrm{log}\left(n\right)$
This tells you basically everything you will need. For example, for the first one:
$\underset{n\to \mathrm{\infty }}{lim}\frac{3\sqrt{n}}{n\mathrm{log}n}\le \underset{n\to \mathrm{\infty }}{lim}\frac{3\sqrt{n}}{\mathrm{log}\left(n!\right)}\le \underset{n\to \mathrm{\infty }}{lim}\frac{3\sqrt{n}}{n\mathrm{log}\left(2\right)}$

Andrew Reyes

Stirlings

RizerMix

For the first one, use that $\mathrm{log}\left(n!\right)\sim n\mathrm{log}$ n as $n\to \mathrm{\infty }$ to conclude that $\underset{n\to \mathrm{\infty }}{lim}\frac{3\cdot \sqrt{n}}{\mathrm{log}\left(n!\right)}=\underset{n\to \mathrm{\infty }}{lim}\frac{3\cdot \sqrt{n}}{n\mathrm{log}n}=\underset{n\to \mathrm{\infty }}{lim}\frac{3}{\sqrt{n}\mathrm{log}n}=0$


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# Generalizing Foundations of Decision Theory II

As promised in the previous post, I develop my formalism for justifying as many of the decision-theoretic axioms as possible with generalized dutch-book arguments. (I’ll use the term “generalized dutch-book” to refer to arguments with a family resemblance to dutch-book or money-pump.) The eventual goal is to relax these assumptions in a way which addresses bounded processing power, but for now the goal is to get as much of classical decision theory as possible justified by a generalized dutch-book.

I made some predictions about what I would and wouldn’t be able to justify with generalized dutch-book. These turned out to be only partially correct. I was able to get a version of Jeffrey’s evidential decision theory, but with nonstandard utility and probability functions and with less assumptions about the structure of the boolean algebra. Nonstandard probability and utility functions are not a very big generalization in themselves. This makes it a little less interesting, supporting Scott’s comment. I still have some hope that more interesting things can be developed from this going forward, though.

I wouldn’t be too surprised if this has all been done before and I just haven’t been able to find it. Justifying all of the axioms in terms of generalized dutch-book arguments seems like an obviously desirable thing to do. Yet, I haven’t seen such a complete justification of common axioms via generalized dutch-books; that is part of why I initially assumed it would not be possible. The closest thing I’ve heard about is Stuart Armstrong’s post showing that immunity to a generalization of the money-pump argument is necessary and sufficient for the VNM axioms minus continuity. So, although the result here doesn’t give as much of a generalization of decision theory as I might have hoped for, it’s the nicest justification of evidential decision theory that I know of.

##Agents

The basic representation of an agent’s preference is as follows:

• The agent has a set of possible beliefs, which is a Boolean algebra. Elements of will be called propositions.

• The agent has preferences over , which can be revealed by asking it to choose from a non-empty set of propositions . (These need not be mutually exclusive; we can ask things like “Would you rather be a bank teller or a feminist bank teller?”) The choices of the agent are represented by a choice function , such that .

• will be abbreviated . will be abbreviated . means that either or .

I’m representing preferences on sets only so that I can argue that this reduces to binary preference. Forcing the choice function to return at least one item is like assuming the completeness of binary preference. This deprives me of the chance to apply Vanessa’s money-pump argument for completeness, but I think “the agent has to make a choice” is a decent justification for completeness. I allow the agent to return more than one choice because to do otherwise would be equivalent to assuming anti-symmetry; this seems too strong. We want an agent to be able to be indifferent between propositions. (You can imagine that the agent would choose randomly when confronted with such choices, but the choice function encodes the set of choices they might make.)

##Money and Contracts

In order to make the money-pump argument, I’ll need a concept of money. I assume that money comes from an ordered field . To make dutch-book arguments, I also need contracts which pay money conditional on certain propositions. These can be represented as a pair, . The agent holds a multiset of contracts, since it’s meaningful to hold several copies of the same contract. In fact, for one proof I’ll need counts of contracts to come from , so that we can have fractionally many, negatively many, and whatever else provides. (This is a somewhat unfortunate complication, but I didn’t find a way to do without it.) However, to avoid headaches, I consider only multisets with finitely many non-zero counts. So, putting these together, we will extend the preferences of the agent to triples giving a proposition, an amount of money, and a multi-set of contracts: . Such triples will be called prospects. I will still use the abbreviations , , and . I’ll also write to indicate the addition of money to a prospect. Multi-set addition will be written , so for example means that is like except that the count of contract is one greater.

Lemma 1 (constraints on value of money). We can extend on non-empty sets of propositions to a function on non-empty sets of prospects, in a way which satisfies the following requirements:

• Like choices over propositions, choices over prospects return only elements from the given set, and return at least one such element: for a non-empty set of prospects , we have and .

• For a set of propositions , let be the set of prospects where . Then we require . In other words, decisions are the same as before when no money or contracts are involved.

• All else being equal, more money is better: for real numbers , .

• For a set of prospects , let be the set derived by adding to each prospect in . We require .

• We require that if the choice set contains more than one prospect, subtracting any positive amount of money from a single item removes it from the choice set. For example, if , then for , .

• For each set of prospects , there is a quantity of money such that we can subtract it from all prospects in (leaving the rest of the same) without changing the choice set. That is, . (This is a way of saying that a strict preference implies a preference we’re willing to pay for—money has small enough units to be less important than anything else.)

• For each set of prospects such that , there is a quantity of money such that we can subtract it from all the prospects in (leaving the rest of the same), all those elements would be excluded from the new choice set. That is, . (This is a way of saying that any preference can be overcome with enough money.)

• Contracts are valued independently of each other and of the amount of money the agent has; IE, then for all .

Proof. Let be the ordered field defined by rational-coefficient polynomials in , considering integer powers of and taking and for all .

For a set of prospects , let be the set chosen by the following process: order the prospects first by the negative-power elements of the money alone, discarding any prospects which are non-maximal in this respect, to yield ; Apply to the set of propositions of prospects in , keeping all prospects whose propositions would be kept by ; finally, order the remaining set by money (considering non-negative powers in the polynomials this time), keeping only the maximal elements. In other words, we first consider the infinite amounts of money; then, we consider what would do; finally, we consider the finite and infinitesimal amounts of money. This definition of satisfies all properties above.

Note that the given construction of is not necessarily the one the agent will use; for example, most agents typically considered would be better-represented by taking and assigning real values to propositions. Rather, this lemma is just to show that my assumptions about how the agent’s preferences are extended to prospects aren’t restricting the agent’s native preferences on propositions at all. Any preferences on propositions can be extended to prospects in a way which obeys these rules. The same could not be said of other similarly natural assumptions, for example assuming all of the above while requiring real-valued money. We don’t want to sneak in restrictions on agent’s preferences via assumptions about money; the money should only serve to facilitate dutch-book arguments.

Henceforth, refers to some function which obeys the properties from lemma 1.

As for the question of why these particular assumptions are made rather than some other set, this is at the level of asking why an agent should care about dutch-book or money-pump arguments, and I’ll make little attempt to answer that here. The assumptions are made with the express purpose of facilitating the proof. One way to look at it is that we’re asking the agent hypothetical questions, augmenting the propositions the agent understands with totally imaginary objects called “money” and “contracts”. The possibility of our assumptions about these being “wrong” is meaningless. Instead, the question is whether arguments employing such hypotheticals to conclude various principles of rationality are compelling.

##Games

The agent plays games with the bookie, as follows:

• The game proceeds in rounds. At each round, the agent is “holding” some prospect. In the initial round, the agent holds a prospect given by the bookie (the “starting prospect”).

• The bookie can move the game to the next round in two ways:

• The bookie can offer the agent a set of prospects (“choice round”). The set must include the prospect which the agent is currently holding. The agent chooses one of the prospects, according to . If contains several prospects, we let the bookie choose. The agent holds the chosen prospect in the next round.

• The bookie can test a proposition (“test round”). This refines what the agent is holding according to whether the test comes out true or false. In addition, contracts pay out if they are triggered by the new proposition. More specifically: if the agent is holding in this round, and the bookie tests , then the agent either holds in the next round or . is minus any contracts such that ; the quantity is the sum of the payouts of those contracts. (The multi-set payout of copies of a contract is .) Similarly, subtracts the contracts activated by the other case, whose payouts are summed up in . We can never move to a prospect whose proposition is ; so, sometimes there will only be one option.

• The bookie can end play at any round.

• I restrict games to have all offers before all tests, so that games proceed in two phases; in phase one, the bookie presents the agent with a sequence of choices, while in phase two, the bookie tests propositions in order to pay out contracts. I don’t think this restriction makes any difference except to simplify matters.

(Letting the bookie choose when the agent is indifferent is a convenience; we could consider the agent’s response to be random, but we wouldn’t want to consider an agent immune to dutch-book if it sometimes escaped them via random choice. We’d condemn the agent if it had a chance of being dutch-booked. So, it’s simpler to just say that the bookie chooses.)

Fixing a particular bookie strategy and outcomes for each test round, we get a play-out. Suppose the agent starts off holding . Call the prospect which the agent actually holds at the end of the play-out . Then an agent experiences regret if, in every situation, is as bad or worse than it would have if it had stuck with the starting prospect. More formally, regret is defined in two cases:

• If the last prospect held before any test rounds begin is (IE, we hold the starting proposition), then take the starting prospect through the same sequence of test rounds (if any) with the same outcomes to get a comparison prospect . This is the end-condition which would have resulted if the agent had stuck with the initial prospect, but the bookie had run the same tests and they had come out the same way. If , the agent has regret. If , the regret is strict.

• If the last prospect before test rounds doesn’t have proposition , then instead, take the starting prospect and generate the set of all possible results if the agent had stuck with the starting prospect, and the bookie had performed the same tests, but the tests could go either way within the limits of logical possibility. In this case, the agent is only said to experience regret if is less than or equal to every element of . The regret is strict if is strictly less than any element of .

A bookie’s strategy is a generalized dutch-book if the agent regrets every possible play-out, and strictly regrets at least one play-out.

My definition of regret is a bit clunky. The intuition behind splitting up the cases in this way can be understood by a couple of examples.

Suppose the agent starts with prospect . The bookie offers the choice of trading this for , which the agent accepts. This is interpreted as paying $10 for a$1 bet on . The bookie then tests for , which comes up true. The agent now holds . Clearly, the agent should regret this outcome. If it had stuck with , it would have after a successful test for , which is strictly preferred by my assumptions concerning money.

On the other hand, suppose that the agent starts with and is offered the trade of , which it prefers. The bookie then tests , which can only come out true. The agent now has . In this case, it doesn’t make sense to say that the agent could have had if it had only declined the trade. came out true because of the trade. To establish regret in this case, we would want both and .

Still, I’m not totally happy comparing to the starting prospect in this way. It would be more natural to compare the agent to agents with differing preferences: an agent has (strict) regret if some other preferences would have given it (strictly) better outcomes, given the bookie’s strategy. We then deem a system of preferences irrational if some other system of preferences achieves superior results even by the judgement of . However, that definition wouldn’t work here; bookies could reward irrationality, making every preference system dutch-bookable. The definition I use avoids this.

##Arguments for Necessity

Here, I prove a number of properties on the assumption that the agent is not susceptible to generalized dutch-books.

We can show that all preferences reduce to binary preference (a version of Independence of Irrelevant Alternatives):

Theorem 1 (Reduction to Binary Preference). If is not susceptible to a generalized dutch-book, then is exactly the set of prospects such that for all .

Proof. Suppose not. Then either there is a with a such that , or otherwise, there is a with no such but .

In the first case, the bookie can start the agent with , present the agent with the set of choices , and make the agent choose . At this point, the bookie can offer , with sufficiently small so as to be favorable. The agent is now strictly worse off than it started, contradicting the assumption that it has no generalized dutch-books.

In the second case, start the agent with , and present the choice of . The agent will choose something other than . Now the bookie can charge to switch back, offering . Again, this should be impossible.

Here’s the classic money-pump:

Theorem 2 (Transitivity). If is not susceptible to a generalized dutch-book, then whenever and , .

Proof. Suppose not. Then we can find such that and , but . The bookie can start the agent holding and offer the choice to switch to and then . The agent either strictly prefers these or is indifferent; in either case, the bookie can get the agent to switch. Since , the bookie can now offer . By the properties of money, there exists an small enough to ensure this trade is favorable. The agent is now strictly worse off than it started.

Note that binary preference is complete by definition, so we have complete, transitive preferences. Also note that transitivity of implies transitivity of and of .

Now we prove a version of Jeffrey’s law of averaging:

Theorem 3 (Averaging). Suppose is not susceptible to a generalized dutch-book. If , then implies .

Proof. Suppose not. Then by transitivity, either the disjunction is strictly preferred to both disjuncts, or both disjuncts are strictly preferred to it.

In the first case, suppose wlog that . Start the agent with . Offer the choice to switch to with small enough to be appealing. The agent accepts. Then, test whether . The agent either ends up with or . In both cases, it experiences strict regret.

In the second case, suppose again that wlog. Start the agent with and charge to switch to . Then, test for . This creates strict regret, since the agent would have ended up in either or if it did nothing, both of which are better than .

Next, we want to make classic dutch-book arguments to establish probability laws. To do this, however, we need to establish some properties of . Specifically, we can show from transitivity and completeness that prospects act like values which extend the ordered field .

Definition. Suppose that is isomorphic with a subfield of the surreal numbers. Fix one such subfield to identify with . Also fix an arbitrary well-ordering of prospects, , which will act like the surreal ordering of stages. Then the relative value of prospect relative to prospect , denoted , measures the prices at which the agent will accept offers to switch: it is defined as the surreal number where is the set of monetary values such that , plus those values where and . is the set of monetary values such that , plus those values where and . The value (non-relative) shall be defined as .

(There are broad conditions under which can be embedded into the surreal numbers as required here; in NBG set theory, this is always possible.)

Lemma 2. If is isomorphic to a subfield of the surreal numbers, then the relative value given above is well-defined, and the closure of all prospect values under the field operations is an ordered field. Furthermore, if and only if .

Proof. Due to completeness, any prospect splits the set of values so far (IE, plus all ) into disjoint sets worth less than, more than, and (possibly) the same as the trade of for . Call the first set and the second . Due to transitivity, it must be that any element from is less than any element from . We can conclude that is a well-formed surreal number. We can therefore take the closure under field operations to get a new subfield of the surreals, which will itself be an ordered field.

Suppose , with and . By the definition of ordering on surreal numbers, there must either be an such that , or an such that . If , let ; otherwise, is for some . So, we have either or . In either case, by transitivity.

Now, suppose . Either or . In the first case, must have in its left set; in the second case, must have in its right set. Either way, .

Ordinarily, after getting a notion of value like this we’d like to show that it is unique up to linear transformations or something similar. This type of result is very unlikely here, since the arbitrary choice of can result in significant differences in (on top of the already non-unique choice of to represent a given ). Nonetheless, we have the first half of a representation theorem. We can use this to define probability.

For notational convenience, define the value of a contract to be the value of adding that contract to the contract set of prospect , with . We define probability via the value of $1 bets: Definition. The probability of a propositions is defined as the value$V ([A, $1])$.

Theorem 4 (Probability Laws). If has no generalized dutch-books,

1. (Non-Negativity).

2. if is a tautology (Normalization).

3. If , then (Finite Additivity).

4. If , $Pr(A)>0 (Non-Dogmatism). Proof. (Non-negativity.) Suppose . This means . But since a strict preference implies a preference we’re willing to pay for, for some . A game which offers that trade and then tests is a generalized dutch-book. The agent is worse off by in the case where comes out false, and worse off by if comes out true. So, this can’t happen. (Normalization.) Suppose is a tautology and . This is a preference it’s willing to pay for, so$(\top, 0, { [A,$1] } ) \prec (\top, 1-n, \emptyset)$ for some . But this implies that the agent can be general-dutch-booked by starting it out with $(\top, 0, { [A,$1] } )$, offering it the trade for , and then testing for (which will certainly come true); the agent could have had$1 rather than \$1-n. Similarly, if , we can start the agent with for some , and get the agent to trade for , ultimately leaving it with only after testing . So .

(Finite Additivity.) Suppose . Since and are mutually exclusive, the agent must be indifferent between and the pair of contracts , . (Holding that pair of contracts pays out in exactly the same way as holding , so if they aren’t valued equally, then the bookie can charge a small amount for switching from one to the other to make a dutch book.) However, the pair of contracts must be valued as ; otherwise, the amount of money the agent is willing to pay for a direct offer of the pair would be different than what the agent would be willing to pay if offered first and then . In the case where , the bookie can start the agent out holding both and , pay the agent an amount of money slightly more than but less than (a quantity of money whose existence is assured by order-density) to give them up, and then charge individually to give them back. In the reverse case, the bookie would pay for their removal individually and then charge to give them back jointly. Either way, we have a dutch book.

(Non-Dogmatism.) Suppose and . The bookie can start the agent with and offer , which the agent is willing to take, and then test . If , the agent is worse off by one dollar; otherwise, the outcome is the same as it would have been. So, this is a generalized dutch book.

So, we’ve got our probability and utility functions. We just need to do a little more work to connect them to each other.

Lemma 3 (Exchange Rate). Probabilities are exchange rates between money and conditional bets: .

Proof. Remember that by a property of money, if and only if for all . This implies that depends on only and . In particular, the value is independent of the number of copies of the contract already in . Since copies of a contract will have equivalent payouts to one copy of a contract , it must have the same value to avoid a generalized dutch book. Since the multi-set of contracts allows the number of copies of a contract to be anything in , this argument applies for any . Since has multiplicative inverses, for any but 0 we can take , proving the result. The result also holds for , since the value of a contract paying zero must be zero.

In what follows, I will abbreviate as .

Theorem 5 (Expected Utility). If propositions and are mutually exclusive and has no generalized dutch-books, .

Proof. Suppose . Then for any such that , , we have . Taking the negative of both sides and applying the exchange rate lemma, we have . This means there is a price, , which the agent is willing to pay in order to trade the two contracts and for the contract . Choose so that , , and . Now, an agent who pays for the trade (starting from and being offered ) is strictly worse off in every outcome after and have been tested, as compared with an agent who didn’t. This contradicts our assumption that there are no generalized dutch-books.

Now suppose . By a similar application of the lemma, we get and make the agent pay for the trade in the opposite direction. This also contradicts our assumption. So we must have the desired result.

This completes the proof that an agent immune to generalized dutch books must have preferences which can be represented by (nonstandard) probability and utility functions obeying the law of expected utility. This in itself does not tell us much, because it may be that I can argue all sorts of constraints from the assumption that no generalized dutch book exists—the condition could even be unsatisfiable. Next we want to see that preferences being representable as probability and utility functions is sufficient.

##Argument for Sufficiency

Theorem 6 (Sufficiency). If can be represented as taking the highest- option according to a (possibly nonstandard) on propositions (except ) which obeys the law of expected utility as mediated by a (possibly nonstandard) function which follows the probability laws, then there exists a which is immune to generalized dutch books.

Proof. Define a value on prospects, , where gives the multiset count of the contract in . Let choose the subset with maximal . This definition of obeys a generalized expected value rule, in which the value of a prospect is the expected value of all possible outcomes after applying any combination of test rounds to the prospect. Now, if is willing to trade during any choice rounds, it must be for a prospect with higher or equal expected value. This makes a generalized dutch-book impossible. There are two cases to consider, based on the definition of regret. Either the agent has traded for a prospect whose proposition is the same as the starting proposition, or the agent has traded for one which is different. In the first case, the probabilities of different test outcomes all remain the same. It is not possible that each outcome is the same or worse as a result of the trade, with at least one being worse; the expected utility would have been lower, and no trade would have been made. In the second case, the probabilities may change. However, due to non-dogmatism, all possibilities still have positive probability. It is therefore not possible that every outcome after trade is the same as or worse than every possible outcome of the prospect before trade, with at least one of the new worse than every old; this would assure a lower expected utility, preventing trade.

Now, it follows from results in the previous section that a which is immune to generalized dutch-books implies a and following the law of expected utility, which represent the original preferences . So, combined with theorem 6, we’ve got necessary and sufficient conditions for such a representation.

##Discussion

I still feel uncomfortable with certain aspects of this, especially the assumptions which I had to edit as I went to make the arguments go through. The most glaring case is the long list of assumptions about money. Other bits I feel uneasy about are the M-valued multisets of contracts and the definition of regret. It’s not a flaw in the formal argument that worries me, but rather, the question of how much contrivance we can throw into the setup. With a few different choices in definitions, would I get a significantly different decision theory?

The overall structure of the argument is that we propose a set of thought experiments to an agent whose choice function (on non-thought-experiments) is . The agent is asked to construct a function representing what it would do under the conditions of the thought experiments. It can construct any it wants, so long as it obeys some assumptions we make as part of the thought experiment. However, if it gives us a which violates some rule of classical decision theory, we give it back a generalized dutch-book illustrating a case where seems “inconsistent”: sub-optimal by its own standards.

From these thought experiments, the agent is not only supposed to arrive at a “consistent” ; it is supposed to be motivated to revise if necessary to do the job. Presumably, this is because “inconsistencies” in the thought experiments are supposed to indicate real inconsistencies in (where the system of preferences itself judges some other system to be better, in some sense). Of course, this isn’t directly true.

Perhaps a better way of thinking about it is not that we’re trying to convince an agent to change its own preferences, but that we’re trying to convince an agent’s designer to give it coherent preferences in the first place. Still, it’s unclear why the designer should pay special attention to this class of hypothetical scenarios.

I did the whole argument this way to avoid “structural” assumptions, where the agent’s beliefs are assumed to already contain things like fair coins in order to define probability, and so on. In the end, though, an agent may only find these hypotheticals convincing to the extent that they resemble the structure of situations it actually believes in.

Perhaps arguments based on money can be seen as making transitivity and other decision-theoretic properties “contagious”: if there is something you care about approximately independently of everything else, which has properties of money, then you can argue for clean decision-theoretic properties for everything else based on that one thing.

However, some aspects seem strange even with this kind of thinking. One of the basic aspects of money-pump arguments, which I’ve faithfully replicated here, is that the agent apparently must “forget” previous choices, and decide only on the basis of the current choice in front of it. I think there is a strong case to be made that by requiring a choice to be made only as a function of the prospects on offer in a given round, we’re not letting the agent fully understand the game it is playing. This leads to objections such as “you can refuse further transactions when you notice a money-pump” (for a discussion, see Money Pumps, Diachronic and Synchronic, Yair Levy).

In any case, these are questions which I hope to answer more deeply in the next stage. Now that I have a version of classical DT justified exclusively by generalized dutch-book, I’ll be thinking about how to adapt this kind of thing to logical uncertainty and perhaps other MIRI problems. I won’t promise another post in the series; it might not go anywhere, or I might decide that the expected value is low and I should do other things for a while. Hopefully this post has some value.

• I think you meant ?

• Ah, yes, corrected.

• Regret Theory with General Choice Sets by John Quiggen is a generalization of DT of more the sort I was initially hoping to produce. It doesn’t try to justify probability theory (it assumes it). Like me, it considers sets of options rather than only binary choices. Unlike me, it requires that if the bookie makes sequential offers, the bookie must keep all previously-given offers in the set (where I only require the bookie to keep the option which the agent chooses). This blocks the money-pump argument for transitivity, but still allows significant constraints on preferences to be argued by money-pump.

The result of this modification to the setup is that the agent can have many different utility functions which are used for different choice-sets. The condition on this is that the utility function must stay the same whenever the “best achievable outcome” is the same. (Best see the paper for that notion.)

• This isn’t too related to your main point, but every ordered field can be embedded into a field of Hahn series, which might be simpler to work with than surreals.

That page discusses the basics of Hahn series, but not the embedding theorem. (Ehrlich, 1995) treats things in detail, but is long and introduces a lot of definitions. The embedding theorem is stated on page 23 (24 in the pdf).


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# Economic surplus

(Redirected from Marshallian surplus)
Graph illustrating consumer (red) and producer (blue) surpluses on a supply and demand chart

In mainstream economics, economic surplus, also known as total welfare or total social welfare or Marshallian surplus (after Alfred Marshall), is either of two related quantities:

• Consumer surplus, or consumers' surplus, is the monetary gain obtained by consumers because they are able to purchase a product for a price that is less than the highest price that they would be willing to pay.
• Producer surplus, or producers' surplus, is the amount that producers benefit by selling at a market price that is higher than the least that they would be willing to sell for; this is roughly equal to profit (since producers are not normally willing to sell at a loss and are normally indifferent to selling at a break-even price).[1][2]

## Overview

In the mid-19th century, engineer Jules Dupuit first propounded the concept of economic surplus, but it was the economist Alfred Marshall who gave the concept its fame in the field of economics.

On a standard supply and demand diagram, consumer surplus is the area (triangular if the supply and demand curves are linear) above the equilibrium price of the good and below the demand curve. This reflects the fact that consumers would have been willing to buy a single unit of the good at a price higher than the equilibrium price, a second unit at a price below that but still above the equilibrium price, etc., yet they in fact pay just the equilibrium price for each unit they buy.

Likewise, in the supply-demand diagram, producer surplus is the area below the equilibrium price but above the supply curve. This reflects the fact that producers would have been willing to supply the first unit at a price lower than the equilibrium price, the second unit at a price above that but still below the equilibrium price, etc., yet they in fact receive the equilibrium price for all the units they sell.

## History

Early writers of economic issues used surplus as a means to draw conclusions about the relationship between production and necessities. In the agricultural sector surplus was an important concept because this sector has the responsibility to feed everyone plus itself. Food is notable because people only need a specific amount of food and can only consume a limited amount. This means that excess food production must overflow to other people, and will not be rationally hoarded. The non-agricultural sector is therefore limited by the agricultural sector equaling the output of food subtracting the amount consumed by the agricultural sector.

William Petty [3] used a broad definition of necessities, leading him to focus on employment issues surrounding surplus. Petty explains a hypothetical example in which there is a territory of 1000 men and 100 of those men are capable of producing enough food for all 1000 men. The question becomes, what will the rest of the men do if only 100 are needed to provide necessities? He thereby suggests a variety of employments with some remaining unemployed.[4]

David Hume approached the agricultural surplus concept from another direction. Hume recognized that agriculture may feed more than those who cultivate it, but questioned why farmers would work to produce more than they need. Forceful production, which may occur under a feudal system, would be unlikely to generate a notable surplus in his opinion. Yet, if they could purchase luxuries and other goods beyond their necessities, they would become incentivized to produce and sell a surplus. Hume did not see this concept as abstract theory, he stated it as a fact when discussing how England developed after the introduction of foreign luxuries in his History of England.[3]

Adam Smith's thoughts on surplus drew on Hume. Smith noted that the desire for luxuries is infinite compared to the finite capacity of hunger. Smith saw the development in Europe as originating from landlords placing more importance on luxury spending rather than political power.[3]

## Consumer surplus

Consumer surplus is the difference between the maximum price a consumer is willing to pay and the actual price they do pay. If a consumer is willing to pay more for a unit of a good than the current asking price, they are getting more benefit from the purchased product than they would if the price was their maximum willingness to pay. They are receiving the same benefit, the obtainment of the good, at a lesser cost.[5] An example of a good with generally high consumer surplus is drinking water. People would pay very high prices for drinking water, as they need it to survive. The difference in the price that they would pay, if they had to, and the amount that they pay now is their consumer surplus. The utility of the first few liters of drinking water is very high (as it prevents death), so the first few liters would likely have more consumer surplus than subsequent quantities.

The maximum amount a consumer would be willing to pay for a given quantity of a good is the sum of the maximum price they would pay for the first unit, the (lower) maximum price they would be willing to pay for the second unit, etc. Typically these prices are decreasing; they are given by the individual demand curve, which must be generated by a rational consumer who maximizes utility subject to a budget constraint.[5] Because the demand curve is downward sloping, there is diminishing marginal utility. Diminishing marginal utility means a person receives less additional utility from an additional unit. However, the price of a product is constant for every unit at the equilibrium price. The extra money someone would be willing to pay for the number units of a product less than the equilibrium quantity and at a higher price than the equilibrium price for each of these quantities is the benefit they receive from purchasing these quantities.[6] For a given price the consumer buys the amount for which the consumer surplus is highest. The consumer's surplus is highest at the largest number of units for which, even for the last unit, the maximum willingness to pay is not below the market price.

Consumer surplus can be used as a measurement of social welfare, shown by Robert Willig.[7] For a single price change, consumer surplus can provide an approximation of changes in welfare. With multiple price and/or income changes, however, consumer surplus cannot be used to approximate economic welfare because it is not single-valued anymore. More modern methods are developed later to estimate the welfare effect of price changes using consumer surplus.

The aggregate consumers' surplus is the sum of the consumer's surplus for all individual consumers. This aggregation can be represented graphically, as shown in the above graph of the market demand and supply curves. The aggregate consumers' surplus can also be said to be the maxim of satisfaction a consumer derives from particular goods and services.

### Calculation from supply and demand

The consumer surplus (individual or aggregated) is the area under the (individual or aggregated) demand curve and above a horizontal line at the actual price (in the aggregated case, the equilibrium price). If the demand curve is a straight line, the consumer surplus is the area of a triangle:

${\displaystyle CS={\frac {1}{2}}Q_{\mathrm {mkt} }\left(P_{\max }-P_{\mathrm {mkt} }\right),}$

where Pmkt is the equilibrium price (where supply equals demand), Qmkt is the total quantity purchased at the equilibrium price, and Pmax is the price at which the quantity purchased would fall to 0 (that is, where the demand curve intercepts the price axis). For more general demand and supply functions, these areas are not triangles but can still be found using integral calculus. Consumer surplus is thus the definite integral of the demand function with respect to price, from the market price to the maximum reservation price (i.e., the price-intercept of the demand function):

${\displaystyle CS=\int _{P_{\mathrm {mkt} }}^{P_{\max }}D(P)\,dP,}$

where ${\displaystyle D(P_{\max })=0.}$ This shows that if we see a rise in the equilibrium price and a fall in the equilibrium quantity, then consumer surplus falls.

### Calculation of a change in consumer surplus

The change in consumer surplus is used to measure the changes in prices and income. The demand function used to represent an individual's demand for a certain product is essential in determining the effects of a price change. An individual's demand function is a function of the individual's income, the demographic characteristics of the individual, and the vector of commodity prices. When the price of a product changes, the change in consumer surplus is measured as the negative value of the integral from the original actual price (P0) and the new actual price (P1) of the demand for product by the individual. If the change in consumer surplus is positive, the price change is said to have increased the individuals welfare. If the price change in consumer surplus is negative, the price change is said to have decreased the individual's welfare.[5]

### Distribution of benefits when price falls

When supply of a good expands, the price falls (assuming the demand curve is downward sloping) and consumer surplus increases. This benefits two groups of people: consumers who were already willing to buy at the initial price benefit from a price reduction, and they may buy more and receive even more consumer surplus; and additional consumers who were unwilling to buy at the initial price will buy at the new price and also receive some consumer surplus.

Consider an example of linear supply and demand curves. For an initial supply curve S0, consumer surplus is the triangle above the line formed by price P0 to the demand line (bounded on the left by the price axis and on the top by the demand line). If supply expands from S0 to S1, the consumers' surplus expands to the triangle above P1 and below the demand line (still bounded by the price axis). The change in consumer's surplus is difference in area between the two triangles, and that is the consumer welfare associated with expansion of supply.

Some people were willing to pay the higher price P0. When the price is reduced, their benefit is the area in the rectangle formed on the top by P0, on the bottom by P1, on the left by the price axis and on the right by line extending vertically upwards from Q0.

The second set of beneficiaries are consumers who buy more, and new consumers, those who will pay the new lower price (P1) but not the higher price (P0). Their additional consumption makes up the difference between Q1 and Q0. Their consumer surplus is the triangle bounded on the left by the line extending vertically upwards from Q0, on the right and top by the demand line, and on the bottom by the line extending horizontally to the right from P1.

### Rule of one-half

The rule of one-half estimates the change in consumer surplus for small changes in supply with a constant demand curve. Note that in the special case where the consumer demand curve is linear, consumer surplus is the area of the triangle bounded by the vertical line Q = 0, the horizontal line ${\displaystyle P=P_{\mathrm {mkt} }}$ and the linear demand curve. Hence, the change in consumer surplus is the area of the trapezoid with i) height equal to the change in price and ii) mid-segment length equal to the average of the ex-post and ex-ante equilibrium quantities. Following the figure above,

${\displaystyle \Delta CS={\frac {1}{2}}(Q_{1}+Q_{0})(P_{0}-P_{1}),}$

where:

• CS = consumers' surplus;
• Q0 and Q1 are, respectively, the quantity demanded before and after a change in supply;
• P0 and P1 are, respectively, the prices before and after a change in supply.

## Producer surplus

Producer surplus is the additional benefit that the owners of production factors and product providers bring to producers due to the differences between production, the supply price of the product, and the current market price. The difference between the amount actually obtained in a market transaction and the minimum amount it is willing to accept with the production factors or the products provided.

### Calculation of producer surplus

Producer surplus is usually expressed by the area below the market price line and above the supply curve. In Figure 1, the shaded areas below the price line and above the supply curve between production zero and maximum output Q1 indicate producer surplus. Among them, OP1EQ1 below the price line. This indicates that the total revenue is the minimum total payment actually accepted by the manufacturer. The area OPMEQ1 below the S curve is the minimum total revenue that the manufacturer is willing to accept. In Figure 1, the area enclosed by the market price line, the manufacturer's supply line, and the coordinate axis is the producer surplus. Because the rectangle OP1EQ1 is the total revenue actually obtained by the manufacturer, that is, A + B, and the trapezoid OPMEQ. The minimum total profit that the manufacturer is willing to accept, that is, B, so A is the producer surplus.

Producer surplus

Obviously, the manufacturer produces and sells a certain quantity of Q1 goods at the market price P1. The manufacturer has reduced the quantity of goods for Q1, which means that the manufacturer has increased the production factors or production costs equivalent to the amount of AVC·Q1. However, at the same time, the manufacturer actually obtains a total income equivalent to the total market price P1·Q1. Since AVC is always smaller than P1, from the production and sales of goods in Q1, manufacturers not only get sales revenue equivalent to variable costs, but also get additional revenue. This part of the excess income reflects the increase in the benefits obtained by the manufacturers through market exchange. Therefore, in economics, producer surplus is usually used to measure producer welfare and is an important part of social welfare.

Producer surplus is usually used to measure the economic welfare obtained by the manufacturer in the market supply. When the supply price is constant, the producer welfare depends on the market price. If the manufacturer can sell the product at the highest price, the welfare is the greatest. As part of social welfare, the size of the producer surplus depends on many factors. Generally speaking, when other factors remain constant, an increase in market price will increase producer surplus, and a decrease in supply price or marginal cost will also increase producer surplus. If there is a surplus of goods, that is, people can only sell part of the goods at market prices, and producer surplus will decrease.

Obviously, the sum of the producer surplus of all manufacturers in the market constitutes the producer surplus of the entire market. Graphically, it should be expressed as the area enclosed by the market supply curve, the market price line and the coordinate axis.

## References

• Bade, R., & Parkin, M. (2017). Essential Foundations of Economics (8th ed.). Pearson.
• Corporate Finance Institute. (2021, March 26). Consumer Surplus and Producer Surplus.
1. ^ Boulding, Kenneth E. (1945). "The Concept of Economic Surplus". The American Economic Review. 35 (5): 851–869. JSTOR 1812599.
2. ^ "Consumer and producer surplus|Microeconomics|Khan Academy". Khan Academy.
3. ^ a b c Brewer A. (2008) Surplus. In: Palgrave Macmillan (eds) The New Palgrave Dictionary of Economics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-349-95121-5_2208-1
4. ^ Petty, William (1899-11-11). "The Economic Writings of Sir William Petty". Cambridge: Cambridge University Press. 2 (98): 30. doi:10.1093/nq/s9-iv.98.409b. hdl:2027/hvd.li4qq6. ISSN 1471-6941.
5. ^ a b c Slesnick, Daniel T. (2008). "Consumer Surplus". The New Palgrave Dictionary of Economics. pp. 1–7. doi:10.1057/978-1-349-95121-5_626-2. ISBN 978-1-349-95121-5.
6. ^
7. ^ Willig, Robert D. (1976). "Consumer's Surplus Without Apology". The American Economic Review. 66 (4): 589–597. ISSN 0002-8282. JSTOR 1806699.


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Tamagawa number conjecture

I heard somewhere that the above formulation of conjecture is for predicting the exact leading term of a L-function at an integer. But i didnt find any reference about how it is stated, anyone please give me any reference or either state the conjecture precisely ,thank you

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