Roulette - Rational Play

Roulette is a classic casino game that derives its popularity from both a rich tradition and also its comparatively simple game mechanics. The simplicity of Roulette can be used with exactly the right risk/reward-tradeoff that any player could desire and it is possible to determine from different situations what the risk/reward- tradeoff could possibly be.

Typically a Roulette player will quickly pass through three separate stages during gaming. The first stage we call fascination during which time the player gambles on a whim, trying to outguess fate. Many play their favorite numbers, "hot" numbers or simply make a big bet on red. Roulette is a very forgiving game to play like this, the expected value of no strategy at all is exactly the same as for any other strategy (typically 97.3% but variations of this rule exist).

Ironically it is the second stage that typically turns the Roulette player into a worse player; this stage is named the system-stage. This occurs when the player thinks they can affect the expected value by using a systematic way of betting. There are a multitude of different systems that according to their creators will provide the player with unlimited wealth at no risk. Stated like this it should be obvious to most people that such systems are both fraudulent and impossible, however the lure of wealth is a proven way to get people to surrender logic. While fundamentally, all these systems have the same expected return value as the naïve random betting schemes used in the first stage, in reality they are typically worse in that they tend to endanger the players entire bankroll as they encourage the idea of winning back losses through successively larger bets.

When the player finally sees through the system and realizes that there is nothing that can be done regarding the intrinsic house-edge in roulette he or she enters the third state, indifference. At this stage the player leaves the Roulette table and moves on to other games, possible returning occasionally just to experience the atmosphere around the table but with no illusions about the ability to make any money besides by chance.

Does this mean it is always irrational to play Roulette if you discount the entertainment value? Well, to answer this question we must first define rational, the definition that best suits us is the one used in the rational choice theory, one of the foundational theories behind economics. It defines rational as acting in such a way as to maximize your expected utility. In plain English this means that a rational individual will always act in a way that they think will be the best for them. So now the question becomes: Is there a way for you to maximize your utility by playing Roulette?
The correct answer to this is sometimes. Put in economic terms it comes down to a reality when the utility-value of the money won at the table isn't linear and therefore there becomes a need to select a betting strategy that will maximize your expected value measured in utility rather than money. This kind of situation typically occurs for one of two reasons, one is that your real life situation presents you with a very real nonlinear utility and the other one is when side-bets or tournaments change the monetary expected value outside the game. Let me illustrate the first situation with an example.

Example 1: The stolen luggage and the wedding.
It is the last day of your vacation and you arrive at the airport in plenty of time. It is very important for you not to miss your flight back as this would lead to you missing your nieces' wedding if you had to take a later flight. As you climb out of the taxi, the driver gets out to help you unload your luggage. At this moment someone jumps into the taxi and drives off at high speed! To your horror you realize that you have lost your ticket home, your mobile phone and your credit card. All you have left is $50 in cash. As it is late in the evening all banks are closed until the morning and unless you can find a way to turn your $50 into $180 to buy a new ticket you will have to stay at the airport until the next day.
As you sit and contemplate your situation you notice a Roulette table in one of the restaurants and you immediately try to use rational choice theory to determine your next action. This is how you do it.
First, you realize money is good, in fact usually you would use expressions like "a dollar saved is a dollar earned" and within reasonable bounds it is safe to say you perceive that money does have linear utility value. In order to be able to carry out a mathematical analysis you use this as the base unit in which to measure your utility. Next you consider the fact that unless you have $30 for the cheapest room available at the Motel that you can see a short walk away from the airport, you are going to have to sleep on a bench in the airport, with all the pain and risk that comes with it! To assign a utility value to how much it is worth to have a room for the night you consider how much money you would be prepared to pay for the room if you had access to your bank account for this purpose only. You value your comfort at a modest $300. Finally you do the same exercise for how much you would value being able to attend to your niece's wedding (and enjoying the comforts of your own bed), knowing how disappointed everyone would be if you couldn't make it you value this to $1300. Now you have the information you need in order to make a rational decision. Below you can see the money/utility graph of your current situation.

If we avoid the Roulette table, we have a guaranteed utility of 350, now this wouldn't be much of an example if that was the best option. Gambling on a row of the Roulette table would give you 2:1 on your money if you won (as you can expect to do 32.43% of the times) you would have $180 and enough to go home. The expected utility of this play would be 1180*0.3243 + 0 * 0.6757 = 480. Can you do better? Yes, in fact from the situation it should be easy to spot the optima, you need at least $30 to avoid disaster and you need $180 for it being worth playing a negative expectancy game such as Roulette. This leaves you $30 to gain $180; that is you are looking for a 1:6 gamble. The easiest way to accomplish this in Roulette is to bet six numbers. The value if this play becomes 1480 * 0.162 + 330 * 0.838 = 516, much better! I leave this as an exercise to the advanced reader to prove that betting a single number with $5, six times will give you a slightly worse result than he alternative .The improvement in expected utility that comes from the potential money you could save does not make up for the decreased chance of reaching $180. Armed with this knowledge you walk up to the table and bet 1, 9, 8, 0, 6, 4 with $5 on each hoping to get some extra utility and appreciation from you niece for you betting her birthday.

Conclusions: So what have we learned today? We have demonstrated that it pays of to have a good understanding of your own personal situation before sitting down at a Roulette wheel. If you have a thorough understanding of your own money-utility curve you can make the best gambling decisions.