Financial Mathematics Text

Monday, May 19, 2014

Ode to Lady Luck

In Roman mythology, Fortuna was the goddess of luck and fortune.

Some say she is blind passing out good and bad luck indiscriminately.

While I offer no good luck charms, I think it's worthwhile to consider the role chance events play in our world (and how we interpret them.)

Can't read my poker face

Poker is a fascinating game and it's even achieved a sportslike status appearing on ESPN. Many investors and traders find the game fascinating sometimes comparing their activities with a game of poker (the metaphor is overused in my opinion and breaks down rather quickly as poker is a zero-sum game.) Hedge fund manager David Einhorn even competed in the World Series of Poker for charity.

The game has many aspects which include a behavioral/psychological component that is quite important and involves looking out for "tells" and making sure you don't have any.

But like most casino games, there is something very fixed about it. In a game of Texas Holdem, there are 52 cards in the deck. Each player is dealt 2 cards. A round of betting ensues and then there are three rounds in which cards are dealt on the table (which are community cards). After each round there is more betting until either all (but one) player folds are everyone is done betting.

Which cards one is dealt is entirely due to chance; it's outside your control. What you do with those cards, however, can give you an advantage. Do you fold them? Do you play them? How much do you bet? If another bets do you call? raise? fold?

There are a large number of possibilities that depend on a number of factors. How many people are at the table? Are they all calling or do some fold? What are they playing? Are they drawing a straight? a flush? Do they have a pocket pair?

But even if you knew all of this information (you don't), the averages will win out over many hands. How many? Well, that all depends. For example, consider the following set up (odds calculated here):

If you're playing the pocket 2's (where one 2 has the same suit as the suited AK), your odds of losing are only 49.37%. So if you knew for a fact that your (only) opponent had a suited AK, you should bet, right? But statistically there are plenty of scenarios in which you could lose money. It could very well take thousands of hands before those averages are likely to work out in your favor.

If you're curious, I created a spreadsheet that pays \$1 if the pocket twos win, \$0 if it's a tie and a loss of \$1 if the AK win. There are 25 trials with 100 hands each. It's possible to lose quite a bit of money even though betting on the twos is a good bet.

Which baseball team is the best?

In his book Moneyball (which was also made into a movie), Michael Lewis tells the tale of Billy Beane, General Manager of the Oakland Athletics (A's). Faced with the fact that his team would never be able to compete with teams that had a much larger budget to hire the best talent, he decided to use a different strategy. Beane used statistical methods (referred to as sabermetrics) to find players that would add value to his team at discounted prices.

And that strategy worked!

For example, in 2002, the A's won 103 games during the regular season. This was the same number of wins as the New York Yankees. This was all done on a \$41 million budget (versus the Yankees' \$125 million budget).

But that did not give them the World Series. Beane observed that his strategy worked well during regular season when many games are played. But in the playoffs, chance takes a stronger role.

Consider the following question: How many games would one have to win in a best of 7 series for us to conclude that Team A is better than Team B?

To answer that question, we would need to be confident that Team A could beat Team B over 50% of the time. The common practice is to look for p-values that are at least lower than 10%. So Team A would have to win within 5 games (if not 4) for us to be really confident that they were better than Team B.

The situation is even worse in football or NCAA basketball where teams face off in one game. Is the Superbowl winner really the best team in the NFL?

Chance and Success (or Failure)

We attribute quite a deal to people's successes and failures. In the above examples, the rules of the games make these relatively easy to quantify. With repetition, it becomes clear which strategies, individuals, teams, etc are better than others.

But what about in normal situations? We talk about good (or bad) managers, CEOs, presidents, politicians, engineers, entrepreneurs, investors, etc. How much of their good (or bad) fortune is due to their skills, talents, determination, processes, etc? How much is due to chance?

This becomes less clear to quantify. But we attribute a good deal of success to hard work and determination. Some of that praise may be appropriate but some (perhaps a good deal) is misplaced. While some may have skills at identifying and exploiting opportunities, a good portion of that may be due to the fact that they just happened to be at the right place, at the right time and their bets just happened to win.

Actually quantifying how much is due to chance and how much is due to skills (etc) is not straight forward (and may, in principle, not be possible.) But it's something one should reflect upon. Having a good process, developing skills, working hard, etc are all good things and will likely improve your odds. But there's still quite a bit of variability in what results will be obtained.

Allow me to quote from Nassim Taleb's preface to Fooled by Randomness:
Of course chance favors the prepared! Hard work, showing up on time, wearing a clean (preferably white) shirt, using deodorant, and some such conventional things contribute to success - they are certainly necessary but may be insufficient as they do not cause success. The same applies to the conventional values of persistence, doggedness and perseverance: necessary, very necessary.

[. . . ]

No, I am not saying that what your grandmother told you about the value of work ethics is wrong! Furthermore, as most successes are caused by very few "windows of opportunity," failing to grab one can be deadly for one's career.
Much of our life we don't get to play 1,000 hands or 100 regular season games; we only have a few shots. This leaves open the possibility that some of our success and failure can be attribute to a random element. There can be a lot of variability in the outcomes.

So here's to you, Lady Luck. May you show me some good fortune!

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