Sequential investment
Ryan O. Murphy’s Homepage

    “It takes money to make money” is a well known saying in financial circles. Often money must be risked in order to make more money. Knowing how much money to risk, and when to risk it, is necessary for the successful management of resources over time.

    Consider a stylized problem of a decision maker who starts with 100 units of currency in a multistage investment environment. The house flips a biased coin with the chances of it coming up heads equal to 0.65. Before each flip of the coin, the decision maker (DM) may invest some fraction (including the option of nothing) of her money on the flip of the coin. If the coin comes up heads, whatever the decision maker has invested will be doubled and returned to her. If the coin comes up tails, then the investment is lost to the house. This process ends after 20 stages or sooner if the DM loses all her money. Whatever money the DM has at the end of 20 stages is what she earns for the whole interaction.

    What sort of betting policy maximizes the decision maker’s long term expectations? How should a smart decision maker manage her money over these independent trials? How much risk should a DM take on in order to maximize her long term expectations?

    This is a highly stylized problem, but it has at its core the same dilemma that individuals face as they invest for retirement, or as a venture capital firm has when deciding how much money to risk sequentially on independent ventures. Risking too much of one’s holdings could lead to bankruptcy in only a few stages. Whereas not risking enough over stages, means that a decision maker would have a flat growth rate over time, yielding a relatively small total payoff. There exists some optimal level of investing that decision makers should use to maximize their long term earnings.

    If you would like to try this decision problem out, click here or on the image below.  The picture shows a screen shot of an online version of the sequential investment problem that anyone online can play.


    From the normative perspective, the Kelly criterion is a formula that yields the optimal size for a series of bets in a situation with a consistent positive expectation. When followed, a fixed proportion of a DM’s capital is risked at every stage. The result is that this optimal policy maximizes the expectation of the growth rate for the decision maker. It was described first by Kelly (1956; see also Poundstone, 2005) and has been demonstrated as practically useful by Edward Thorp in both the contexts of gambling (1961) as well as the stock market (1967).