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.


  1. Hi Dirk, I am enjoying the thought process here because I have always wondered just what you are asking - 'how much floor.' For example, once I have covered my needs (needs floor say) with some combination of pension, Soc Sec and a portion of my assets then how do I determine how to split the rest of my assets between additional floor vs upside potential. I realize that this to a certain extent is a personal choice, but I wondered whether you had any thoughts on once you have your basic needs floor established what percent of your remaining assets should be devoted to additional floor vs upside potential? Is there a decision making process (like game theory) that would aid in answering this question? Thanks, Brad.

    1. Brad, I struggle with this, too, and I'm unaware of a decision-making process that helps.

      I struggle with the entire discretionary vs. non-discretionary categorization. I could say that my house payment, food, medical and clothing expenses are non-discretionary, for instance, but I won't consider this my floor because I would not be happy if that's all I could spend. I would want to guarantee more than subsistence as a standard of living.

      When I then look at the amount that would maintain my standard of living, I find that it is pretty much all of my planned spending. So, discretionary spending for me becomes spending on extensive travel, for example, which I might not do in any event.

      If someones asks, "How much of your current standard of living could you give up in bad times and still be happy?", my honest answer would be "not very much." (I could give up Showtime.)

      Instead of discretionary versus non-discretionary spending, I tend to think of maintaining my current standard of living (which becomes my floor) and "fun money" (which is my upside portfolio).

      This means that my floor is my current standard of living and upside is any excess. I don't feel any need to increase my standard of living at this point in my life, so buying more floor doesn't interest me. Given the financial challenges, I think improving one's standard of living after retiring is a long shot.

      If I couldn't build enough floor to maintain my current standard of living, my personal preference would be to build the highest floor possible while maintaining an emergency portfolio of liquidity and then forgoing upside.

      I suspect this is a personal preference issue, but I would be interested in your perspective and I hope others will chime in.

      Thanks, Brad.

  2. Thanks Dirk for your perspective. I think to a certain extent for me it is all about expectations. For most of my life (single) I lived well below my means with the expectation that when I retired I would live much better (trips etc), and be more secure. I have been blessed in that for the most part my expectations have come true. Because I have been fortunate to have enough assets to live above subsistence I am hesitant to gamble too much on upside potential especially without having a good idea of the decision rules I would employ. Seems to me that I would need to know what sort of guaranteed income say 100K in assets would buy me (additional floor) versus what I could likely expect those assets to earn or lose were they in some stock/bond combination. If, for example, 100K might buy me 5K in annual income in a SPIA or TIPS, or say 10K (100% in stocks) in stocks with a downside potential of 50K then this gives me a way to trade-off the additional floor versus upside potential choice. I realize there are many combinations of stock/bond for the upside potential in the example, but I just chose 100% stocks to make the contrast more apparent. I will have to think about this more, but thanks as always for your valuable thoughts. Brad

    1. The first two are easy. $100,000 in a life annuity with inflation protection would pay out $3,750 a year right now. A TIPS ladder would pay out $3,650.

      Wade Pfau's current estimate is that a SWR strategy would pay out $2,900 on average, but it might pay out more and it might pay out less. David Blanchett thinks you might start out at $4,000, but that's a huge range for SWR. Unfortunately, SWR is just risky and that means a lot of uncertainty. It won't become more certain until you're late in retirement, and not very certain even then.

      That should tell you how much floor you can buy today, about 3.7% of your retirement savings. You have to find the point where you are comfortable saying, "I'll give up this much certain income to bet on an even higher standard of living if my investments perform well. And, if they don't, I'm happy with my floor."

      It isn't possible to know how much you will actually be able to spend from a volatile portfolio in the future.

  3. Thanks Dirk. I think I have validated these on Wade Pfau's new dashboard. Do you agree that I am reading the dashboard correctly?

    I like the new dashboard where I think I see the 3.75% for a life annuity (SPIA CPI-U for 65 yo couple with 100% spouse benefit as of 1/15), the 3.65% for the 30 yr TIP ladder with CPI-U, and from the SWR dashboard I see about 2.88% for a moderate portfolio (50/50 stock bond) so that jives with your 2,900 figure. So I plan on going to Wade's dashboard periodically to look for these three items. Based on this comparison I like the TIPS option for both the security and liquidity vs the other two options. My task is to get to age 65 (five yrs for me and about 11 for my wife) at least in as good a shape as now so these tables apply. Thanks again, Brad.

  4. Thanks so much Dirk!

  5. Dirk, Nice job on the pure versus mixed strategies. For the allocation on the upside or "fun" money, I think in terms of spending flexibility. Those who want high average discretionary spending and are flexible about making year-by-year adjustments are candidates for higher stock allocations. Someone who wants to take the same vacation every year would likely be a candidate for a higher bond allocation. One can apply a bit of utility theory to get at specific allocation recommendations, answering questions like--"What level amount of discretionary spending would you accept in trade for spending that bounced randomly between $20,000 and $30,000 each year?" The type of utility function assumed makes a difference. In the above trade-off exercise I assume what economists call time-separable utility--each year's consumption stands on its own. Scott and Watson assumed habit forming utility and a strong aversion to drops in consumption from one year to the next. Their floor/leverage approach starts at a low level, but prevents decreases. I realize trying to apply utility theory can get a bit murky, but, like game theory, it can sometimes help frame the practical considerations.

    1. Thanks, Joe. Good points.

      I'm actually fine with the floor part of Floor-leverage Rule, but I don't think the upside strategy is viable. I wouldn't want to invest mine with that much risk and, even if I did, leveraged ETFs aren't the same as a triple-leveraged stock portfolio.

      I think "spending flexibility" makes a lot more sense as a perspective than a discretionary/non-discretionary distinction. (No one wants to live on non-discretionary expenses alone, even as a worst case.)

      Utility is clearly a consideration in this choice, though one I hadn't thought about a lot. Lot to think about here. Thanks for contributing to the conversation.