Conway’s Game of Life

The Game of Life is a cellular automaton devised by the British mathematician John Horton Conway in 1970. It is the best-known example of a cellular automaton – Wikipedia

The universe of the Game of Life is an infinite two-dimensional orthogonal grid of square cells, each of which is in one of two possible states, live or dead. Every cell interacts with its eight neighbours, which are the cells that are directly horizontally, vertically, or diagonally adjacent. At each step in time, the following transitions occur:

1. Any live cell with fewer than two live neighbours dies, as if by loneliness.
   
2. Any live cell with more than three live neighbours dies, as if by overcrowding.
   
3. Any live cell with two or three live neighbours lives, unchanged, to the next generation.
   
4. Any dead cell with exactly three live neighbours comes to life.

Dowload Excel Game of Life

The Excel document uses macros to simulate the birth and death of the cells according to the rules outlined above. There is a macro that will seed the game with a random configuration of cells. You can then increment this a single generation at a time or run ‘life’, where you will see the evolution of the cells over one hundred generations. There are also some interesting configurations stored on other worksheets. You can cut and paste these into the start generation and run life from there. There are a numbert of interesting ‘creatures’ outlined in Martn Gardiner’s Scientific American article from 1970.

Here are two screenshots of the population when life is running.

The macro also records and plots the population change over time. The plots for xbar and ybar give the average location of all the cells – when looking as single creatures such as a ‘glider’ you can see the aggregate direction they take.

St. Petersburg Paradox

Extracted from Wikipedia article – The St. Petersburg paradox is a paradox related to probability theory and decision theory. It is based on a particular (theoretical) lottery game (sometimes called St. Petersburg Lottery) that leads to a random variable with infinite expected value, i.e. infinite expected payoff, but would nevertheless be considered to be worth only a very small amount of money.

In a game of chance, you pay a fixed fee to enter, and then a fair coin will be tossed repeatedly until a tail first appears, ending the game. The pot starts at 1 dollar and is doubled every time a head appears. You win whatever is in the pot after the game ends. Thus you win 1 dollar if a tail appears on the first toss, 2 dollars if on the second, 4 dollars if on the third, 8 dollars if on the fourth, etc. In short, you win 2^k−1 dollars if the coin is tossed k times until the first tail appears.

What would be a fair price to pay for entering the game? To answer this we need to consider what would be the average payout: With probability 1/2, you win 1 dollar; with probability 1/4 you win 2 dollars; with probability 1/8 you win 4 dollars etc. The expected value is thus

This sum diverges to infinity, and so the expected win for the player of this game, at least in its idealized form, in which the casino has unlimited resources, is an infinite amount of money. This means that the player should almost surely come out ahead in the long run, no matter how much they pay to enter; while a large payoff comes along very rarely, when it eventually does it will typically far more than repay however much money they have already paid to play. According to the usual treatment of deciding when it is advantageous and therefore rational to play, you should therefore play the game at any price if offered the opportunity. Yet, in published descriptions of the paradox, e.g. (Martin, 2004), many people expressed disbelief in the result. Martin quotes Ian Hacking as saying “few of us would pay even $25 to enter such a game” and says most commentators would agree.

Download the St Petersburg Paradox Excel Model

The model simulates the coin flipping and payout for a streak of heads. You can get a feel for what you would pay to take the bet; you can run it with a Monte Carlo package to get the payoff distribution. The screenshots below show three different runs where there is a streak of 4, 5 and 6 heads with a payout of $15, $31 and $63 respectively.

Interactive Gompertz Model

In high tech start-ups the development cycle can last for a period of several years. We can capture new product introduction where the pre-revenue start-up phase is anticipated to be long using a Gompertz curve. There is a full description on Wikipedia.

Sales Function

Cumulative Sales Function

m	500	Ultimate market potential (m)
b	0.4	Scale Parameter (b)
η	30	Shape Parameter (n)

Click here to launch Interactive Gompertz Model

The Xcelcius dashboard allows users to interactively vary the parameters of the model. This is useful when doing ‘what if’ analysis during product portfolio planning stage. The sales profile is more realistic and can be embedded into the interactive portfolio or we can create a Monte Carlo income statement with distributions to describe uncertainty.

Interactive Net Present Value (NPV)

Net present value (NPV) is defined as the present value of net cash flows. It is a standard method for using the time value of money to appraise long-term projects. See full description at Wikipedia.

Each cash inflow/outflow is discounted back to its present value (PV). Then they are summed. Therefore

Where

t – the time of the cash flow

N – the total time of the project

r – the discount rate (the rate of return that could be earned on an investment in the financial markets with similar risk)

C_t – the net cash flow (the amount of cash) at time t

Interactive NPV Calculation – Click here to launch NPV chart

The following dashboard allows you to change the discount rate and see the time value of a dollar over a period of 5 years.

At a 10% discount rate a dollar in year 5 is worth 62 cents in today’s money

At a 50% discount rate a dollar in year 5 is worth 13 cents in today’s money

We can also demonstrate an arbitrary project with an initial cash outlay and an increasing yearly income stream

At a higher discount rate of 50% the future income is heavily discounted and the overall project NPV is significantly reduced.

Index of Interactive Dashboards

Interactive Portfolio Model – Posting / Dashboard

Product Requirements Capture and Competitive Benchmarking – Posting / Dashboard

Economic Value Model – Posting / Dashboard

Bass Diffusion Model – Posting / Dashboard

Value at Risk – Posting / Dashboard

Net Present Value – Posting / Dashboard

Gompertz Model – Posting / Dashboard

Intuitive Bass Diffusion – Posting / Dashboard

 

Interactive Bass Diffusion Model

Click here to launch Interactive Bass Diffusion Model

The Bass diffusion model was developed by Frank Bass and describes the process how new products get adopted as an interaction between users and potential users. The model is widely used in forecasting, especially product forecasting and technology forecasting. Click here for the full description on Wikipedia.

The function describing Sales S(t) is given by

m	500	Ultimate market potential
p	0.02	Coefficient of imitation
q	0.4	Coefficient of innovation

Click here to launch Interactive Bass Diffusion Model

The Xcelcius dashboard allows users to interactively vary the parameters of the diffusion model. This is useful when doing ‘what if’ analysis during product portfolio planning stage. The sales profile is more realistic and can be embedded into the interactive portfolio or we can create a Monte Carlo income statement with distributions to describe uncertainty in ultimate market potential (m), coefficient of imitation (p) and coefficient of innovation (q).

Market Requirement and Product Fit

In a high value technology product there are typical three types of stakeholder that have separate concerns who each ’speak a different language’. These types of buyer are highlighted in Crossing the Chasm. The economic buyer, technical buyer and user.

When building a sales proposition and message it is common to make the mistake of trying to force your perception of benefits, which may not be shared by your customer. It is also easy to convolve messages, which may not be relevant to the buyer you are talking to, for example, focusing on price or ROI with a technical buyer.

We want to deliberately and separately address the concerns of the technical buyer and the economic buyer. We can use an interactive dashboard to capture market requirements and support the sales process from the technical buyer perspective.

Product Requirements Capture and Competitive Benchmarking – Click here to launch the product dashboard

Instead of pitching a solution to a perceived problem we can work with a technical buyer to capture their ‘total’ requirement and the key points of pain. The feature or performance space has many dimensions, for example, size, weight, speed etc. Different requirements will also have different levels of importance, some are essential and some are ‘nice to haves’. With the dashboard the technical buyer can work though the feature list and select their target requirement and associated importance. They build up a map of the total requirement, which is then plotted on a spider diagram. They can drill down in a gap analysis chart to compare specific features across multiple products.

Target real needs in sales process – Instead of ranting about ‘guessed’ requirements, you allow the customer to fully articulate their need. There will be fewer objections as you spend more time understanding the problem instead of pitching a boxed solution. As they work through the requirements and weighting you can engage them to better understand the reasoning and problem implications. For example, if speed is an essential requirement you may drill down to find that the implication of delay is under utilisation of another costly resource etc. This implication knowledge will be valuable in future sales where there are common problem sets.

Map the need to your product – There is no algorithm for sales! However if the customer builds up the map of total requirement and there is a good match with your product this can be compelling when making the decision. In addition the document is permanent; often you can bounce between objections on specific features, with this representation you keep a global focus. The technical buyer can forward this dashboard to colleagues and they will have an audit trail of the decision process.

Competitive Benchmarking – once the requirement has been mapped you can compare it not only to your own offering but also competitive products. Ideally you are demonstrating significant and compelling differentiation. Even if other products are comparable in meeting the requirement you are acting more in the role of an honest information broker as opposed to just hawking goods by saying anything. This may be valuable in building trust in the customer relationship, which is often a larger determining factor in the sales process.

Interactive and Engaging – Most people have sat through interminable power point pitches. The dashboard changes the process from a passive information dump to an interactive conversation. Also dashboards are new and different; people find them fun and ‘every little helps’ when you are trying to get the product and company noticed in the crowd.

Market Analysis – In addition to using a dashboard in a sales context they can also be used when doing market analysis. They are a lot more interesting than a questionnaire. I’ve previously described two principle uncertainties in new product introduction 1)’Likelihood of market entry’- a binary event; whether we can enter the market at all with a product and subsequently 2) ‘Market penetration and growth rate’ –the income over time for a product. The risks can be split into further levels of granularity. The requirements capture can be used to make a detailed assessment of likelihood of market entry. If your product has a very poor fit with requirement then the ability to enter the market at all is greatly diminished. For example if the ‘fit metric’ was 20% we could use this directly in our Monte Carlo portfolio analysis as the threshold for market entry. You can also spot requirement trends that can have a strong influence on the direction of your technology roadmap.

Interactive Economic Value Models – Click here to launch the value model

The language of an economic buyer is very different from a technical buyer. They use words like, cost savings, ROI, payback period, utilisation, efficiency, payment terms etc. Again we often fall into the trap of guessing points of pain instead of listening to the customer who actually feels them. We can use interactive dashboards to capture costs and metrics on the fly and automatically calculate saving, ROI etc.

We don’t talk about technical advantages at this stage; we assume we have addressed the needs of technical buyer at the level of requirements capture and we are now speaking to the person who writes the cheques (in reality we probably speak to both in a parallel fashion).

The example value model above was developed for the Lonestar chemical process monitor. A factory processing an arbitrary number of samples will have a proportion of faulty goods. If these are not detected then the faulty goods can be shipped to customers. The question is whether we can save money by employing a detection system to identify the out of spec goods. The user can input the number of samples they process and how many defects occur. They can then input the associated costs, for example if a batch of out of spec goods was shipped then there may be a direct monetary cost in refunding / paying compensation as well as potential for significant brand damage. If there are false alarms then we may end up scraping product which was actually good. The model takes these inputs and calculates the cost of not using detection vs the cost of using a detection system. They get metrics including ROI and payback period.

 

 

Monte Carlo Portfolio Optimisation

Download the excel model here

The portfolio representation described previously included project costs and returns that were purely deterministic. In reality there were be large uncertainties in both, for example a 6 month project overrun will increase overall project costs and is also likely to reduce returns. We can use Monte Carlo analysis to model the effect of these uncertainties on the discrete portfolio options.

A real model can have different layers of granularity and detail, specific to the individual projects. For example, uncertain returns will be dependent on volumes of sales over time, pricing, competition etc. To illustrate the principle we use a simple model where the individual project are independent and returns and costs are triangular distributions in which we specify the minimum, maximum and average.

Project		Return			Cost
-	Min	Average	Max	Min	Average	Max
A	$5.00	$8.00	$9.00	$3.80	$4.00	$6.00
B	$0.50	$2.50	$5.00	$0.80	$1.10	$1.50
C	$1.00	$4.00	$4.80	$1.00	$1.30	$2.00
D	$1.00	$3.00	$6.00	$0.70	$0.75	$1.30
E	$1.50	$2.00	$3.00	$1.00	$1.25	$2.00
F	$0.30	$1.60	$1.90	$1.70	$2.00	$2.40

Simulating Uncertain Portfolio Returns and Costs

When we run a simulation we resample from the distributions and realise a particular outcome. Below are two simulated instances; we can run a full simulation of thousands of samples with a package such as @Risk or the Solver risk optimiser.

In this instance Project A has a return of $7.15M and cost of $5.24M, Project D had a return of $3.95M and cost of $0.78M; they are independent so the portfolio selection of Project A and D has an overall return of $11.4M and cost of $6.02M. This turns out to be the optimum portfolio for this level of investment.

In this instance Project A has a return of $7.36M and cost of $3.91M, Project D had a return of $2.41M and cost of $0.94M; they are independent so the portfolio selection of Project A and D has an overall return of $9.77 and cost of $4.85M. In this instance the combination of A and D is no longer the optimum portfolio for this level of investment.

What becomes clear is that when we add uncertainty there will not necessarily be a unique optimum that we can select. The optimum portfolio can be different but we don’t know how the uncertainties will be resolved in advance.

Visualisation of Portfolio Uncertainties

When we run the simulation each discrete portfolio point will have an associated cost and return distribution. We can add this distribution information to the portfolio representation by adding errors bars. The y error bars are the 95% limits of the distribution of return and the x error bars are the 95% limits of the distribution of cost. We now have representation that embeds risk, which has a very high information density to help us make optimum portfolio decisions with regard to return, cost and risk.

Optimisation of an Uncertain Portfolio

MinMax Rules

The uninitiated executive may have no knowledge of a probability distribution or understand how to use it. What are more common are questions like, “well what is the worst case?”, “what happens if the costs are high and the return is lower than we expect?” We can add extra interactivity to the dash board to allow the executive to explore these possibilities and see how that may affect the optimum portfolio selection as a function of their own appetite for risk.

The first representation is what we have already seen previously; we plot the average cost and return and select the optimum as normal.

We now re-plot the portfolio to show the worst case of maximum cost and minimum return. In most cases there would be a degree of interdependency in cost and return and we may have to re-simulate to obtain this plot. One approach would be to extract the 5% and 95% percentiles for return and cost respectively.

The best case would be the maximum return for minimum cost.

Between these extremes we can continuously vary the plot of return and cost. It is also very straightforward to add likelihood figures to the plots to show the executive that the extreme values are less likely than the average values; they are getting all the information about the underlying distribution but the representation and output is more accessible.

What is apparent is that the pareto efficient frontier will change. The optimum portfolio will be different for those who have a differing preference for risk. Some of the portfolio options may have a very large upside but also a large downside; if I was risk averse and was operating a max cost minimum return decision rule this portfolio option would be plotted in the lower region of the chart and would not be an optimum choice for me given the actual portfolio and my risk appetite. The reverse would be true if I was risk seeking. We have a representation that allows us to revise the optimal profile as a function of risk behaviour.

Semantic Zooming

Each portfolio combination point has an associated cost and return distribution. Points in the lower region of the plot are clearly suboptimal, the returns are low and costs are high, however with points close to each other on Pareto frontier it can sometimes be difficult to decide which one is better given their distribution profiles.

We can use a drill down or semantic zoom to look at distributions of adjacent points on the efficient frontier.

There are a few interesting situations where it is difficult to decide the optimum; typically the resolution boils down to a question of appetite for risk. In the example below the portfolio combinations A and B are very similar; the costs are comparable and the average returns are almost the same. However when we drill down to the return distribution we can see that even though the mean is similar there is a big difference in range of possible returns. Portfolio A has both a higher upside and lower downside; if I was risk averse I may prefer portfolio B, which has a slightly lower return but with more chance of hitting.

Interactive Value at Risk (VAR) chart

The VAR chart is useful in the sense that it describes values such as NPV a distribution as opposed to a single point estimate. No amount of modelling will ever fully anticipate what lady fortune has in store but it is still worth trying to gain a deeper understanding of risk drivers so you can make better decisions or create managerial options that help to exploit future uncertainty.

In the vein of trying to make the concept generally accessible to the uninitiated, busy executive we can add some useful refinements. The general shape of the chart itself conveys some information quickly e.g. if it is bimodal.

Personally I have these specific problems with the representation

1. A cumulative plot is not as intuitive as a probability density function – especially if the distribution is bimodal or more complex (in a cumulative plot people look at the flat region of constant probability and get confused, whereas when you see two humps in a pdf it is obvious that some values are not allowed)
   
2. When I want to extract numeric values (in a presentation when showing power point slides) it is quite tiresome to try and read off the chart axis every time; especially if it is interval data
   
3. Not everyone is familiar with likelihood or probability but odds on they’ll be familiar with the concept of odds.
   

Interactive VAR Chart – Click here to launch VAR chart

The output of a Monte Carlo model will be table of numbers describing the distribution. We can easily take these values and create an interactive VAR chart dashboard.

1. The user still gains the same insight from the overall shape
   
2. They can control a lower limit slider that allows them to look at the downside part of the distribution to answer questions such as “what are the odds we’ll lose money overall on this project”
   
3. A second slider allows them to get upside and interval information e.g. “what are the odds the NPV will be between $5m and $10m”.
   

Some of the newer Monte Carlo packages (Risk Solver at www.solver.com) allow you to generate these charts in near real time; however with Xcelcius I can embed this chart into power point and pdf documents and distribute the model.

Key Features of US SBIR

Here are the key features highlighted in David Connells report “Secrets” of the World’s Largest Seed Capital Fund

1. Regular solicitations at fixed dates during the year;
   
2. Awards directed at the best submissions from across the US; no state or regional quotas;
   
3. Complete transparency in terms of topics, awards winners and amounts;
   
4. Standard contracts; companies own the intellectual property developed;
   
5. Clear linkage to agency R&D interests and priorities; strong focus on commercialisation;
   
6. Companies do not have to be established until awards have been won;
   
7. 100% funding of all contract costs plus a profit element;
   
8. Flexible mechanisms to encourage involvement of academics and support academic spin-outs and technology transfer;
   
9. Phased awards to manage risk, typically with $100k for a Phase I feasibility study and 50% of Phase I award winners going on to win a $750k Phase II development award;
   
10. Phase III SBIR awards funded from mainstream (i.e. non SBIR budgets), and adding probably as much again to overall federal R&D expenditure on SBIR projects;
    
11. Phase III projects bring businesses the opportunity to win valuable sole supplier contracts with federal agencies;
    
12. Prime contractors are encouraged to take up SBIR developed products.