The Ultimate Game of Strategy
Chapter 3
Choosing in conditions of uncertainty

Many small risks or one big one?

The second interesting response to the first two chapters came from Yvan Caron, a systems analyst working in Canada. He identified what would seem to be a paradox. He'd been following a discussion on an Internet advertising list serve discussion forum, where a question had been posed by a restauranteur who'd wanted to know what percentage of his gross revenue he should risk on Web advertising. After supplying a copy of the restauranteur's post Yvan Caron wrote...

Let's analyze this restauranteur's problem in terms of the roulette game where the Gambler *has* the *Advantage*

Suppose you've checked a wheel and discovered that through a *bias* you have a mathematical advantage over the casino. We know from chapter 2 that everything changes almost from black to white.

From the standpoint of capital requirements, we have two choices: do we minimize our risk or maximize our gain? As you may begin to guess this relates to the question: "What percentage of my gross revenues should I dedicate to my web strategies?"

What we seek is a compromise between the extremes of minimal (long-term) risk and maximal (short-term) gain. We need a wager optimally combining the greatest safety with the greatest growth rate of our capital or bankroll.

Given this, what should be our bank-to-bet ratio? What do you bet?

Yvan Caron

Yvan's problem highlights a typical gambler's problem. Do you play safe by spreading risk, or, do you go for broke in order to make a substantial gain?

This was a problem that came home to me when I spent a year in the City of London writing an educational course on Investment and Finance. To make the course interesting, I proposed that student's give themselves an imaginary amount of capital and during the twelve weeks of the course, pretend to be buying and selling equities with this imaginary money at the prices prevailing each day in the Financial Times newspaper. This would allow them to get some experience of market conditions without risking any real money.

With the student's, I joined in this imaginary game, with the hope that the methods I was describing would produce a handsome profit at the end of the twelve week period.

The methods I'd outlined in the course were the traditional methods that had evolved from hundreds of years of stockbroker experience. This involved trying to get what is known as a technical valuation of the investments (based upon the anticipated earnings per share of the equities: comparing the ratio of earning to equity price with the current prices being quoted for fixed annual incomes i. e., annuities).

Of course, there is no straightforward comparison that can be made between the relatively safe and certain income from annuities and the uncertain income that can be derived from equities; all kinds of risks and uncertainties have to be discounted. But, they can give a rough guide, which often exposes anomalies.

Using a Game Theory approach, a fundamental investment strategy rule was established that gave preservation of capital a top priority. To satisfy this rule, the investment of the capital had to be split up as much as possible so as to spread the risk - just in case any of the investments failed. This gave an overall investment result that was effectively the average gains and losses of many different separate investments.

To my disappointment, the result of this investment strategy produced very little improvement on the average for all equities in the market. Any deviation to try to invest more heavily in volatile offerings compromised the safety of the capital. It was a problem I never resolved and led to my abandonment of investing in the Stock market as a route to riches.

The reader might like to ponder here on the similarity between playing the Stock Market and playing the game of roulette as discussed in the last chapter. In a rising market, the game is the same as if zero wins for the players. In a falling market, it is as if zero wins for the house. Notice too, that playing the Stock Market is really a zero sum game. There is no value introduced by the investors, they are merely winning from one another (minus the stockbroker's fees).

Playing the Stock Market, however, is quite different from the situation where investors invest in a company by taking up equity to provide a company with working capital. This is a game where the capital can be put to profitable use and so create real wealth to make it a win-win, non zero sum game. It is this second way of investing in equities that can be more appropriately related to the strategies discussed in this book.

When I later applied Game Theory strategy to entrepreneurial business situations, I came up with a similar problem to that pin pointed by Yvan Caron. Do you go for high gains by concentrating all your resources on favoured projects, or, spread the risk by dividing available resources between as many different options as possible?

The reader might consider here that the term "resources" does not necessarily relate to capital. It applies as much to time. In many cases time might be an even more critical factor, maybe even the only one to many under capitalised start ups. In which case the problem revolves around "Do I invest all of my time in one favoured project, or, spread the risk by dividing my time between several". This is a critical decision for all entrepreneurs where projects involve risk.

This is also true for anyone involved in e-Business: even sub contracting experts and specialists run a risk with their time, just as much as any entrepreneur. How many specialists are there that have devoted months, even years to a particular niche speciality (authoring package, hardware, database system or computer platform) only to see that speciality area be superseded or disappear altogether?

When Yvan Caron came back with his own answer to the problem he'd posed, this was his post:

Hi All,

Remember the question that came from a local restauranteur where he asked: "What percentage of my gross revenues should I dedicate to my web strategies?"

Then I told you that what we were seeking was a compromise: between the extremes of minimal (long-term) risk and maximal (short-term) gain. What we seek is a wager , optimally combining the greatest safety with the greatest growth rate of our capital or bankroll. Given this, what should be the bank-to-bet ratio?

So here is my second part to this post ...

I did not invent this strategy, but took it from an old book on my book shelf. I must confess though, that I don't know exactly how we could apply it to the web.

One of the scholars of proportional betting is an American mathematician, John L. Kelly, Jr., who proposed an optimal ratio in 1956. In simplified form the Kelly system states that, when the game is *favorable to us* (whether on red or on one or more numbers), at every spin we should wager an amount equal to our current bankroll divided by a number based upon the amount of risk involved.

Simple isn't it?

Mathematicians consider the Kelly proportional betting system the best mathematical strategy, constituting, as it does, the perfect compromise between betting, in relation to our capital, too little (overly time-consuming) versus too much (overly risky).

In explaining the Kelly proportional betting system, for simplicity I used an even chance (say red), but it's just as valid for any kind of investing enterprise.

The problem here, is that I have not yet resolved how to know or evaluate business advantage to be able to give a value to the constant multiplier.

Yvan

This problem of what proportion of available resources to allocate to a risk, is not something that can be formalised in the real world of business. There is no optimum or universal percentage that can be applied because all risks in business come with different probabilities of success. There is also the problem that business risk is usually impossible to estimate. This was pointed out by another reader in the cafe, Bonnie Austin, from Texas, USA.

(Note: probabilities are usually expressed such that one (1) equals certainty and nought (0) equals absolute impossibility. Anything in between is expressed as a decimal fraction of one (1) e. g., a fifty percent chance of success would be expressed as a probability of 0.5; a ten percent chance of success would be expressed as a probability of 0.1; an eighty percent chance of success would be expressed as a probability of 0.8)

Bonnie wrote:

In my experience, with my own businesses and from discussing strategies with other business owners, determining one's present position, including probabilities of various outcomes, is the most difficult (and often impossible) task for the entrepreneur.

I can't think of any example in the real world where a person can say with certainty, "The probability of my getting the job I'm bidding on is .2"

Rather, there's more like a range of probabilities. The best one can do is to say, "Well, I think I have a pretty good chance of getting the contract." Meaning that the probability is somewhere between .15 and .3.

If an entrepreneur has several irons in the fire (that's an old Texas expression meaning that a person's working several possibilities at the same time) it gets easier to say that there's a certain probability that at least one of the deals will work out. Even so, one has to keep in mind that a model is not a perfect representation. Sometimes, I say that the entrepreneur calls upon intuition to bridge the gap between the model and reality; but, maybe there's a large element of luck.

Bonnie Austin

As Yvan Caron's post pointed out, a high risk situation will warrant risking a lower proportion of available resources than a less risky situation. But, how can this rule be applied if the risk is difficult or impossible to estimate? This conundrum makes a game of roulette an inappropriate model with which to consider this problem. A more appropriate game with which to contemplate this problem might be the game of poker - even though this is a pure zero sum game.

I'd spent a short period of my life as a professional poker player. This had involved the continuous use of strategies that allocated sums of money for bets according to perceived risks. The need to keep in the game, surviving over the course of many losing hands, necessitated that bets were made with due consideration to the amount of money available to play with. After a winning run, and with accumulated winnings, higher bets could be made - and bets made on riskier situations. After a losing run, when funds were low, bets were smaller, less frequent and made with much more care.

However, whether winning or losing, the basic betting strategy remained more or less the same: bets were always only a small fraction of my capital if I was playing just to stay in a hand and where I was not particularly hopeful of winning (I'd be staying in the hand for a small bet because there was always a chance for a favourable fall of the cards to turn the hand to my advantage).

I'd play in many hands where the cards wouldn't fall my way, but, as soon as a hand came along where the cards turned in my favour, I'd up the stakes, increase the amount of my bets and play to the limit of my capital.

NOTE:

The hypothetical man in chapter 3

Because this book will be read by many different kinds of people it will be impossible to cover individual niches. However, Game Theory strategies, whoever uses them and for whatever purposes, will have many fundamental similarities.

For this reason it will be convenient to invent a hypothetical businessman who will be the focus of attention. This the reader is asked to identify with, even if at times this person will not exactly coincide with their own personality or relate directly to their own unique situation.

Ladies, please excuse the choice of gender, but, I'm sure you will understand that to keep having to refer to "businessman or businesswoman" every time will be highly inconvenient. As "businessman" is shorter to write than "businesswoman" I opted to make the hypothetical business person a man. But, I'm sure you will understand.

This person will seem at most times to have an entrepreneurial viewpoint. This is not meant to imply that this book is only for entrepreneurs. It's just that entrepreneurs are more likely to be associated with freedom of action, which is what this book is about.

In fact most people, will make decisions of an entrepreneurial kind in the world of e-business so it shouldn't make much difference. Where there are obviously differences in strategies (say for employees, or, contractors) the differences in strategy will be covered.

This hypothetical "businessman in chapter 3" will now be introduced. We will be developing his strategy as we progress through the book.