Investing Games

Ed Thorpe invented the strategy to beat blackjack, a method known today as “card counting”.  I first learned of Ed in college when my friends and I spent one summer at the local casinos trying to copy his card counting techniques (I wasn’t very successful).  A few months ago, while researching the viability of my investment strategy, I learned that he is also an incredible investor. 

Later in life, while working as a math professor, Mr. Thorp became interested in investing and ultimately started a hedge fund.   Over almost 20 years, his hedge fund returned 19% annually with only 3 down months, results so phenomenal I almost question their accuracy.   I was drawn to Mr. Thorp’s ideas; they echo mine and similarly originate from experiences outside the investing world.  In one interview, he espouses that “understanding gambling games is one of the best training grounds for getting into the investment world” (min 49:30-58:20)”.  I agree.

In that light, I offer the following games which demonstrate the philosophical framework of my strategy.  The games are purposely exaggerated and simplified to highlight the essence of the logic.

Game #1

Bet $100.  Flip a coin, heads you win 50% of the bet (receive $150 back, for a gain of $50), tails you lose 40% of your money (get $60 back, for a loss of $40).  Do you play this game?

The game is in your favor with a 50% chance to receive $150 (win $50) , and a 50% chance to receive $60 (lose $40).  The expected outcome is a gain $5 per game.  On average, you will win money.  Most people recognize this.  A large percentage of people would play this game. 

Game #2

Now let’s add a wrinkle to the game.  After the first round, the game repeats with the entire payout of the first round.  You get to play repeatedly as long as you desire, but each time you have to bet everything.    Example:

“You start the game with $100, and win the first round.  You now have $150.  You play again putting up $150 and win the second round leaving you with $225.  You play again with the $225 and lose the third round, falling to $135.  You decide to stop.”

Do you also play this game?  It is subtly different than the first game.  If you do play, when do you stop?

Game #3 

This game is the same as the prior one, with the one difference being I decide when the game ends, not you.  You will bet $100, and the game will repeat with the payout of each round risked, until I decided to stop.  Do you play this game?

Think hard about what the differences of these games.

The correct answer, in my opinion, is to never play.  You will lose everything nearly every time.  Actually, if allowed to go on forever, the game will always trend toward total loss.  

Strange isn’t it.  A game works in your favor when played once, and against you when repeated?

Repeated games of chance have very different odds of success than single games.  The odds of a series of bets – specifically a series of products (multiplication)- are driven by, and trend toward, the GEOMETRIC average.  Single bets, or a group of simultaneous bets -specifically a series of sums (addition)-, are driven by the ARITHMETIC average.   The arithmetic average for the game is $1.05, as seen in game #1. The geometric average of game #3 is $0.949 per game ( √ {1.5*0.60} ).  A loss of over 5 percent per play.  

Let us work through an example to see this clearly.  Playing the game twice yields the following possible results:

On the surface, the game is still averaging in your favor, but you can already see 3 out of 4 times you receive less than $100, losing money.  The most likely results is one win and one loss:  150% x 60% = 90%.  A 10% loss.   Everyone understands that a coin flip game will trend toward  half heads, and half tails over time (law of large numbers), therefore this game will also trend toward losses given enough time. 

Let us look at the game played 20 times:

The average is still positive, but risky with a 75% chance of losing money.  The most likely outcome is a loss of $65. Forty percent of the time you will leave with less than $14.  The repetition of bets has  greatly skewed the results, with a large percentage of losers and a few winners (the logarithmic scale on the graph makes the chart look symmetrical). 

Playing 100 times leads to a 86% chance of losing money and an 54% change of having less than a dollar.  The game is more like a lottery ticket now, with a small chance of a nice pay off, but a high chance of losing mostly everything.  With more repetitions, the closer the percentage of a loss will approach 100%.

A few lucky people will become filthy rich, but most will be broke.   It’s not crazy to play this game with $100.  But would you play it with 10% of your wealth?  With 50% of your wealth? Would anyone call this type of game an “investment”?  I would not.

This is a strange phenomenon and it may take a while to fully grasp.  Don’t worry if it hasn’t fully clicked yet, because we will revisit this concept may times, from different vantage points.  Every game with a negative geometric return will trend toward losses over time, even if the game has a positive return for a single play.  The same effect happens with less variable games (i.e. heads to win 4%, tails to lose 3.9%), only requiring more repetitions for the effect to grow. 

Now ask yourself, do the people investing your wealth understand this concept? Are you invested in any “lottery tickets”?

Game #4

Same rules as game #3, except each game is a bet of $100.  You have an $10,000 to play the game.  Do you play?

You are repeating the game, which in light of the prior game should give you pause.  However, the payout is not being rolled into the next round.  Each round is only worth $100.  So the winning and losses are additive, not multiplicative as in game 2 and 3.  When the game adds and subtracts winnings the arithmetic average is followed.  The arithmetic average is in your favor by $5 per game (game 1).  Therefore you should play.  This is the game most people think of when they think of game 3.

So what can we learn from this game?  Games that are additive in wins and losses are more likely to be profitable than games that are multiples of their wins and losses.  These games will produce expectations closer to the arithmetic average, not the geometric average(The geometric average is always less than arithmetic average).  The key difference in this game and game 3 is you limited the amount you bet.

Game #5. 

Start with $100 and play with the same rules as game #3, but this time, you get to decide what percent of your payout is wagered.  I still decide when to quit.  Do you play? Example:

“You start the game with $100, and win the first round.  You now have $150.  You play again and decide to put up $120 and win the second round leaving you with $210 ($120*1.5+$30).  You play again and decide to put up $80 and lose the third round, falling to $ 178 ($80*.6+$130).  I decide to stop.”

Game #3 is a big loser and this game is very similar.  It is still a product of a series game, driven by a negative geometric average.

However, if you bet the correct amounts, Game #5 is a good game to play.  Keep your bets below 50% of your total cash account and you will be favored to win money.  Keep your bets to 25% of your cash account, and you will be favored to win the most money, making 0.6% each game.

In Summary:

• Game 3 –  Rolling all your money – Guaranteed Losses
• Game 5  – Rolling a portion of your money – Likely small gains.

So what happened here?  Well, with a cash “buffer” of 75% of your funds, you are limiting the effect of the negative geometric average and increasing the effect of the positive arithmetic average.  The actual profile of the game is:

A positive geometric average!  You can play this game all day long and be confident the winnings will trend in your favor!

This concept is very powerful.  Your bet size changes the long term odds of the game.

Game #6 

Same rules as game #5, except there are two games going simultaneously, and you get to decide how much to invest in each game.  How do you play this game?

You play by putting 25% on game 1, and 25% on game 2.  Re-balance the bets after each round, so you always put 25% of your cash on each game.  This strategy produces a geometric average of 101.24% each play.  The diversification of two games running simultaneously increases the geometric return above that of a single game, even though the arithmetic average is still 1.05.  The more simultaneous games played, the closer the return moves toward the arithmetic average.

Game #7 

Same rules as game #6 with two games going simultaneously, except you get to decide how much to invest in each game only at the start.  Each game is then played with the full payout from the previous game rolled into the next round.  How do you play this game?

You don’t. The game is the same as game #3, except there are two games going at the same time.  Each game will end up trending toward ruin.  Starting with two games will slow the degrading process, but over time, it still occurs.   Re-balancing prevents these games from collapsing.

Universal Nature of Math

As stated before, these games are more exaggerated and much simpler than in the investing world.  However, the math is the math no matter the size and complexity.  The same principles found in these 7 games exist in the risk/return environments of every investment. Geometric averages will drive long term results. Bet size will influence long term payouts. Re-balancing increases the geometric average. So, let’s think about various popular investment strategies in reference to these games. 

Which game represents buy and hold? 

Which game(s) represents index investing? 

Which game(s) represents traditional portfolio construction? 

Which game(s) represents trend following? 

Which game represents Geometric Balancing?

Which game represents the best strategy?