Friday, February 27, 2015

Pure and Mixed Strategies

In a few recent posts, I suggested how game theory might be used to gain a different perspective on the Social Security claiming decision (Game Theory and Social Security Benefits) and why updating your sustainable spending amount periodically (Dominated Strategies and Dynamic Spending) will always perform better than spending a fixed amount based on your initial portfolio balance in retirement (SWR-Fixed), or by spending a fixed percentage of remaining portfolio balance each year but ignoring other determinants of portfolio survival like decreasing life expectancy (SWR-Variable).

The benefit of the spending strategy analysis it that is allows us to winnow out inferior strategies when we choose our retirement income plan. SWR-Fixed and SWR-Variable are dominated strategies. Game theory tells us never to play a dominated strategy, which only makes common sense.

I admit two motives for these posts. The first is that I am fascinated by game theory and believe it provides valuable perspective on the retirement planning problem and the second is that I'm convinced we can simplify retirement planning. 

How does this simplify the retirement income strategy choice? By eliminating dominated strategies as game theory recommends, and eliminating other strategies that aren't logically sound, we can winnow a dozen or more proposed strategies to a significantly smaller number of truly valuable strategy choices. 

In this post, I'll consider another concept of game theory, pure and mixed strategies, and how they might be useful for analyzing retirement income strategies.

According to, a pure strategy defines a specific move or action that a player will follow in every possible attainable situation in a game. A mixed strategy is created by playing members of a set of available pure strategies at some proportion of each.

Assume a tennis player has two pure strategies available: serve to her opponent's forehand or to her opponent's backhand. She might also attempt to keep her opponent guessing with a mixed strategy by randomly serving to her opponent's backhand or to her opponent's forehand.

Game theory will use the server's success rate serving and the opponent's success rate returning serve from both sides to calculate the optimum proportion of serves to each. Based on probabilities of success for both pure strategies and responses, game theory might tell her, for instance, that the optimum strategy is to randomly serve to a particular opponent's forehand 30% of the time. This is a mixed strategy.

Let's consider some pure retirement income strategies including sustainable withdrawal rates (the dynamic kind, since game theory tells us that SWR-Fixed and SWR-Variable are dominated), a Social Security benefits strategy, an annuity strategy, a time-segmentation strategy and a TIPS bond ladder strategy. Other strategies have been proposed, but let's go with this shorter set of pure strategies for now.

Why isn't the floor-and-upside strategy on the list? Glad you asked. Because floor-and-upside is a mixed strategy consisting of some mixture of pure floor strategies and pure upside strategies.

The floor strategy could consist of life annuities, TIPS bonds held to maturity, Social Security benefits or some combination of these.

The upside strategy contains risky assets like stocks and bonds. SWR portfolios typically recommend something like 50% stocks and 50% bonds. Jason Scott's and John Watson's floor-leverage rule (download a PDF) recommends 15% of assets be invested in a triple-leveraged ETF of derivatives. Zvi Bodie and Nassim Taleb have recommended an upside portfolio of 10% of assets invested in long term index options (LEAPS).

Note that a mixed strategy can allocate zero percent to some available pure strategies, so for instance, an SWR strategy can be considered a floor-and-upside strategy allocated 100% to the upside portfolio and 0% to the floor strategy. More importantly, because nearly all Americans have some Social Security income or public pension income, it will be very rare that a retiree plays a pure upside strategy.

An exception to this observation is retirees who postpone claiming Social Security benefits and spend from a stock and bond portfolio until those benefits start, but by age 70 at the latest, they will likely have a floor-and-upside strategy, though they may not think of it that way.

While it will be rare for a retiree to implement a pure upside strategy with no floor, it is easy enough to implement a pure floor strategy with no upside portfolio. A retirement income plan based solely on pension or Social Security income would qualify as a 0% upside/100% floor portfolio, as would any strategy comprised solely of Social Security benefits, TIPS bond ladders and life annuities.

In other words, nearly all of us will have a floor. Those of us with adequate retirement savings can choose to add more floor, add an upside strategy, or implement some combination of the two. This is the first decision in choosing a retirement income strategy. It answers the question, "how much of your retirement savings are you willing to risk in the stock market in hopes of being able to spend more?"

For those who answer that they wish to take no risk with their standard of living, the next step will be to determine how to most effectively build a floor of income. For the rest, the next step will be to determine how much of their desired standard of living should be locked in with a floor portfolio, with the remainder put at risk in the market.

Viewed from this perspective, sustainable withdrawal rates is a floor-and-upside mixed strategy with a floor consisting of Social Security or pension benefits. A TIPS Bond Ladder strategy is a floor-and-upside mixed strategy of Social Security or pension benefits and a TIPS Bond Ladder with zero percent upside portfolio strategy. Floor-leverage rule is a floor-and-upside mixed strategy with a floor consisting of 85% of our portfolio plus Social Security or pension benefits and an upside portfolio strategy consisting of investing 15% of assets in a triple-leveraged derivatives portfolio.

Most strategies can be viewed as a form of a mixed floor-and-upside strategy and understanding this may simplify your decision of which strategy to implement.

Pure upside strategies will be rare, because most Americans will have Social Security benefits or public pension income at some point. That leaves a floor strategy or a mixed floor-and-upside strategy as the options available to most retirees.

This, of course, is the root of the "safety first" versus "probabilities" divide, but I don't see the divide so much as a disagreement on whether or not to put standard of living at risk as one of how much of our standard of living we should bet in the market. Because most of us are going to have a floor and probably a mixed strategy, the big question is, "how much floor?"

I think this is a far more reasonable approach than having retirees read about a dozen or so strategies to pick the one with which they feel most comfortable.

If you're interested in game theory, William Spaniel has an outstanding series of tutorials on YouTube entitled Game Theory 101.  If the academics of the subject interest you, Yale filmed Professor Ben Polak teaching Econ 159 Game Theory. He is an amazing professor and, although it doesn't use modern on-line teaching technology, it is probably the best on-line class I have ever taken.

Made me wish I'd gone to Yale. Go figure.