Practical analysis for investment professionals
17 February 2021

Bernoulli’s Prisoner’s Dilemma: A Goals-Based Perspective

In 1738, the Swiss mathematician and physicist Daniel Bernoulli proposed a simple thought experiment:

“A rich prisoner who possesses two thousand ducats but needs two thousand ducats more to repurchase his freedom, will place a higher value on a gain of two thousand ducats than does another man with less money than he.”

Let’s continue to play this out and place Bernoulli’s prisoner within the context of modern markets and ask him to evaluate various investments. What becomes immediately clear is that his ducats are dedicated to one objective: getting the heck out of prison!

Our prisoner has a goal for his money, just like we do.

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Our prisoner can invest his ducats as he sees fit, and because he wants to maximize his chances of release, we can describe his use for various investments with goals-based portfolio theory.

We don’t need to bother too much with the details right now, but clearly our prisoner will evaluate both the expected returns and expected volatility of a given security over time through the prism of achieving his freedom. His willingness to trade off return and volatility is presented in the following graphic. The line is the minimum return he requires for any given level of volatility. As volatility, or the X axis, increases, our prisoner requires ever-higher levels of return, as depicted by the Y axis. This is hardly a revelation: It is exactly what traditional theory would expect.


The Prisoner’s Dilemma: Return and Volatility

Chart depicting Return and Volatility Chart

But what if we build a stock exchange in our prison and let our wealthy prisoners trade shares among themselves? This is where things get interesting.

In the second graphic, we plot three different prisoners, A, B, and C, each of whom has different starting wealth, required ending wealth, and time horizon. For the sake of simplicity, we’ll suppose each has the exact same view of a security’s future volatility and return, which are labeled as s and m in the figure.


Three Prisoners’ Dilemma: Return and Volatility

Chart depicting Three Prisoners' Dilemma: Return and Volatility

Here’s the thing: Each investor is willing to accept completely different returns for the same security!

Moreover, if the security’s price is simply the inverse of return — 1/m, a simple but not unreasonable model — then each investor is willing to pay a completely different price for the exact same security!!

There is no difference of opinion about the characteristics of the security driving differing acceptable prices, but rather a difference in investor needs.

When we place these three prisoners in the marketplace, we would expect Prisoner A and Prisoner B to sell their shares to Prisoner C at the price of 1/c until Prisoner C exhausts his liquidity or Prisoner A and Prisoner B exhaust their inventory. Then, the price drops to 1/b, and Prisoner A continues to sell to Prisoner B. From there, the price drops to 1/a, and Prisoner A would buy, but no one would be willing to sell.

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Prisoner C is an enigma. Traditional utility models would not expect anyone to accept lower returns in response to higher volatility. But goals-based investors can be variance-seeking when their initial wealth is low enough. Behavioral finance characterizes their goals as “aspirational.” This is why people buy lottery tickets and gamble: Increasing the volatility of outcomes is the only way of increasing their chance of achieving life-changing wealth.

Of course, all this is more than a simple thought experiment: It reveals some critical lessons about markets.

First, when setting capital market expectations or target prices for stocks, analysts would do well to assess the marketplace of buyers and sellers to determine how their needs and liquidity will influence the coming price. This is more complicated than our example, of course, because in addition to different needs, everyone also has a different outlook for a given security.

This is no surprise to practitioners. Markets dominated by institutional buyers look vastly different than those dominated by aspirational investors and “YOLO” traders.

A very present example is our current regime of ongoing quantitative easing (QE) from central banks around the world. For investors befuddled by sky-high stock valuations, the difference between Prisoner A and Prisoner B is illuminating. They are exactly the same except for one thing: Prisoner B is wealthier today.

In general, then, this means that adding cash to financial markets creates investors who are willing to pay more for the exact same security. Conversely, when excess liquidity is drained from markets, prices should drop, all else equal, because investors with less cash today require higher returns. Thus line B moves back to line A.

Book jackets of Financial Market History: Reflections on the Past for Investors Today

Second, and most striking: There is no “correct” market price. No security has a “fair value” or “fundamental value.” Rather, price emerges from a security’s characteristics interacting with the needs of the investors in the marketplace.

Another key component of price: each investor’s relative liquidity in the marketplace. If enough aspirational investors, or Prisoner Cs, deploy their cash into a security market, prices can remain elevated or spike until their liquidity is exhausted. Sound familiar, GameStop?

This may seem obvious, but it is not the traditional perspective on markets. The efficient market hypothesis asserts that securities always trade at their fair value and that market timing cannot work. Of course, predicting the development of the fundamentals of a security is a difficult task. But that is only half the equation. As our hypothetical prison stock market demonstrates, understanding the marketplace of investors and their behavior can yield insights that are just as valuable.

What’s even crazier: Every investor in the market is acting rationally. Prisoner C is offering a perfectly rational price for the security even if it is the highest bid in the marketplace! Prisoner A is acting just as sanely despite having the lowest buy price.

And this is some of the promise goals-based portfolio theory offers. Behavioral finance would describe the price action of our prison market as irrational albeit predictable investor behavior, and traditional theory would dismiss it as nonexistent. But goals-based investors can more clearly see what is really happening.

Goals-based portfolio theory may, in fact, be a helpful bridge between normative and descriptive theories.

Investment Professional of the Future report graphic

Like the prisoner in Bernoulli’s thought experiment, we have specific objectives to achieve with our money. And like the prisoner, we interact with public markets with those objectives in mind.

Those objectives influence prices in ways that traditional theory might not expect. And while behavioral finance offers some models to predict irrationality, goals-based theory suggests that people may be more rational than initially thought.

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All posts are the opinion of the author. As such, they should not be construed as investment advice, nor do the opinions expressed necessarily reflect the views of CFA Institute or the author’s employer.

Image credit: ©Getty Images / erlobrown

About the Author(s)
Franklin J. Parker, CFA

Franklin J. Parker, CFA, is founder and chief investment officer of Directional Advisors in Dallas. He is a CFA charterholder, international speaker, and author of numerous peer-reviewed papers and articles. In 2017, Parker was awarded the NAAIM Founder’s Award for Investment Research for his work on merging active investment management and goals-based investing. Though raised on the family cattle ranch in central Texas, Parker now lives in Dallas with his wife and three children.

9 thoughts on “Bernoulli’s Prisoner’s Dilemma: A Goals-Based Perspective”

  1. Jim Hamilton says:

    Total nonsense. Fair value or intrinsic value may not be precise and will change over time but it is the most important factor when assessing investment merit. Estimating intrinsic value is a complicated process, particularly when valuing intangible assets and risks, but it is preferable to the madness of crowds.

    1. Bruno says:

      Not sure if you have actually read the article Jim.

      The main point is that the prices flutuate according to people’s analysis, and different people think differently, which makes price move, sometimes in unexpected ways.
      This is not the equivalent of “Crowds be mad”.
      And also does not negate the validity of value investing.

      Let’s say:

      Investor A is investing for the objective of maximizing return in 50 years – He wants the highest return possible, and probably does not care so much for volatility.
      Investor B is investing for maximizing the chance of reaching $10k for this mom’s surgery. If he has $5k right now and 2-year horizon, he needs high volatility. That’s a mathematical need.

      Both are rational. And both will price the same assets with different prices, because they want different things.

  2. Michel Charbonneau says:

    Interesting article. Thanks for sharing. Would it correct to say that competitors in an oligopolistic market bidding on the same acquisition target are close real life examples of these prisoners?

    However, in the dilemna, the prisoners are few and their resources (liquidity and inventories) are limited in which case the notion of an objective “fair value” seems irrelevant. But for a market with a very large number of participants, in the long run, will not price converge towards “fair value”? Is this prisoner’s dilemna not a sort of illustration of Ben Graham famous statement: “In the short-run, the market is a voting machine, but in the long-run, the market is a weighting machine”. Thanks.

  3. Kirk Cornwell says:

    Fair value is out there with stock analysts’ “target price”. (It might actually BE the price occasionally.) In options, its definition as the midpoint between “bid”and “asked” serves as a beginning of negotiation based on demand.

  4. Crispy says:

    The article is saying that demand and supply meet to determine price. Nothing new there. What makes this article unusual is marketplace of only three investors.

  5. chris says:

    Very interesting.

    But the analysis needs to be extended to …. what happens when the players and the player’s goals change? Does that not reduce this argument to the simple reality of ‘market sentiment’ over and under-reacting?

    So C buys lottery tickets until the day when he realizes he hasn’t the cash to put food on the table for the rest of the week, and changes his money into a safe savings account. Or good market leave ‘normal’ investors having accomplished their goals early with extra to ‘play with’.

    Would there not be a reversion to some mean – the intrinsic value?

  6. brc says:

    The article brings some sense of reality to an otherwise crazy market.

    I am “retired” (well at least beyond retirement age). My two sons are 30+ years younger than I. My financial objectives are far different then both. Whereas, I would like to make money on my stock investments, more importantly, I do not want to lose money. That is, I can live on what I have. Many of my son’s friends have little in retirement saving, but are willing to bet on a “greater fool” willing to buy a security or bond at a much higher price. They have no need to look at P/E, S/P, or any other the statistics that I would look at before buying a security. All that matters to them is perception that a “greater fool” will purchase the security at a higher price in the next day, week or coming month(s). If they lose $2000 on bet, no big thing. Try again.

    As the Feds give this population of younger betters more money to bet with, their chances increase on having a winning bet simply because these “prisoners” are given free resources to continue their endeavor. If they lose a significant portion, so what? The money was free and unearned in the first place. Whereas, my resources were hard-earned over a lifetime.

  7. Pat Moran says:

    Utility is what those Bernoulli experiments led up to. The part about increasing variance for a chance at more potential upside, when we don’t care about the potential downside, is known, and is moral hazard.
    The borrowing/lending constraints used importantly here are not generally present. Assuming borrowing and lending of money and shares in real life is generally a good assumption (but not in Gamestop’s case, granted).
    Also, ‘prisoner’s dilemma’ is a term already used in game theory for a long time to describe a different thing. Probably should call this something else to avoid possible confusion. There’s no dilemma if all prisoners’ optimally desired choices can be accommodated in a single outcome.

    1. Steve says:

      In economics, moral hazard occurs when an entity or person has an incentive to increase its exposure to risk because it does not bear the full costs of that risk. “not caring” is not quite the same thing…

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