Gambler’s Fallacy & Game Theory — A-to-Z Blog Challenge, Literary Terms

GHere we are at G and…wait a minute…what the hell is going on. Gambler’s Fallacy and Game Theory are not literary terms! One is a principal of Mathematics and the other is Psychology. That’s cheating…

Not really. I firmly believe that when we write, we write about life, and literary terms should borrow from art, economics, law, science, psychology, philosophy, math and sociology. We can’t limit ourselves to just one set of words, we have to embrace it all.

While I did have terms I could have used that are more directed to literary terms, I chose these terms the same way as I chose the other terms, what I thought was the most important for my visitors to know. However, as I get to harder letters, I will have to rely on terms outside of a literary text book to accomplish my goal, which I feel I should do anyways.

Trust me when I say that this does apply to writing, even if it doesn’t seem like it does.

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Gambler’s Fallacy

Gambler Fallacy is a rather interesting principal. It is the false notion that the more something happens than what is normal, the more likely it won’t happen in the future.

Allow me to demonstrate this. First we need a fair coin. What is that? Simply a coin that will randomly land on either side. This way we take out any special flips or added weight on either side. Now, we start flipping it and after 21 times, it has come up heads each time. Now the question is, what is your prediction of what it will come up next; heads or tail?

If you said tails…you are wrong.
If you said heads…you are wrong.

Wait a minute, how can you be wrong on both accounts. It has to come up as one of them. That is true, it does have to come up heads or tails. The reason someone would be wrong choosing either heads or tails is that they are using past observation to determine an outcome of a truly random occurrence.

The Gambler’s Fallacy states that people would choose tails, and would likely keep choosing tails the more flips come to head. The more times it comes to head, the more likely the individual will believe it comes to tails.

Aside from understand patterns or understanding odds, the human mind is a machine of balance. And if it sees an imbalance, then it has a desire to want something to balance out, wanting to see the coin come up tails 21 times just so the universe is in balance. So rarely is the universe in balance.

Don’t worry if you answered head or tails to the outcome of the coin, it is human nature to second guess a lucky occurrence. I’m sure a few of you saw through the problem and when you analyzed it, realized that you can’t use the past coin flips to predict the next one.

One of the reasons I believe that we second guess an outcome like that, especially if there is a hot streak is subconsciously we are working out the odds. Chance a coin will come up heads after flips:
1: 1/2 or 50%
2: 1/4 or 25%
3: 1/8 or 12.4%
4: 1/16 or 6.25%
5: 1/32 or 3.125%
21: 1/2,097,152 or .000048%

Our mind may not be doing the actual math here, but it can estimate the chance of an occurrence to be so improbable that it is practically nil.

Now here is the part that will really explain this. So the chances of getting heads 21 times is 1 chance in 2,097,152, however, that’s not what was asked. We simply asked you to predict the next outcome. The fact that it come out to 21 heads is useless information. When it comes right down to it, the chance of it coming up head is 1 chance in 2, the same exact chance it will come up tails. Just because 21 times before it came up heads, doesn’t mean the outcome of the coin is affected.

This fallacy is telling us one thing: the human mind is tricking itself. This fallacy is demonstrating our making a decision on intuition rather than logic. Sometimes intuition can tell us things logic cannot, but often, pure logic is usually the best course.

This is also called the Monte Carlo fallacy. On August 18th, 1913 at a roulette table at the Monte Carlo casino, the ball fell on black 26 times in a row. This was a rather uncommon occurrence, for it to land on just black, however, gamblers lost millions of francs betting against black. They assumed that that the more times it came up black, the more likely it would come up red.

Now there is a reverse to this, called the Reverse fallacy or the Reverse Gambler’s Fallacy (not to be confused with the Inverse Gambler’s Fallacy). Now we’ve kind of covered this, but I will state it here, that just because it came up heads 21 times, doesn’t not mean that the coin is more likely to come up heads.

This will be discussed in the final segment, but sometimes we make decisions on useless information, and often don’t stop to think about what information means. It’s not always easy to figure out what is good or bad information, but when we really look, sometimes we can figure out good and bad information. This can run parallel to superstitions.

Game Theory

You may have heard this term before. The simple explanation is the study of games. However, that definition leaves out much of what it actually is.

A better explanation is the logical analysis of situation of conflict and cooperation. Game theory looks at games that have at least two players that are given options of moves and choices in which a strategy can be derived by different players with an outcome that comes off as a huge payoff to the winning player. The outcome is not entirely up to the players choice as there can be the element of randomness and dealing with the other player’s strategy.

Players may choose to work against each other, or to work for themselves, and it is entirely possible that they work together for an optimum outcome.

With that definition, it seems to take the fun out of gaming, but in actuality, what we learn from games can be used in a variety of places, including: economy, business, politics, biology, computers, philosophy, sociology, and even writing.

Now that you have a basic understanding of what game theory is, at least the definition, I am going to introduce you to two games that demonstrate Game theory. Those two games are Prisoners Dilemma and the Monty Hall Paradox (or Problem).

The Prisoners Dilemma is perhaps the most famous of games for Game Theory. The basic premise is this:

Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. The police admit they don’t have enough evidence to convict the pair on the principal charge. They plan to sentence both to a year in prison on a lesser charge. Simultaneously, the police offer each prisoner a Faustian bargain. Each prisoner is given the opportunity either to betray the other, by testifying that the other committed the crime, or to cooperate with the other by remaining silent.


Say you are one of the prisoners. In the traditional game, if you can trust you partner to remain silent, then both of you serve 1 year in prison. Knowing that, you can then decide to betray your friend, in which he would serve 3 years and you go free. However, if your friend also decides to betray you counting on you remaining silent, then you both serve 2 years.

What is your choice? To betray or not to betray. While you figure that out, let me give you another hypothetical.

In the early 2000’s, there was a game show called Friend or Foe?. At the end of the game, two players remaining would need to convince each other to be friends. They would then choose which option they wanted, either to choose friend or foe. If both chose friend, they would split the money, if one betrayed the other who voted friend, then the betrayer (or foe) would get all the money. If they both betrayed each other, neither gets any money.

Friend_or_Foe _(TV_gameshow)_titlecard

How do you choose? The obvious solution is that you go for friend. However, if you know the other person is 99% certain to choose Friend, could you resist the temptation of choosing Foe? What if you know they will betray you, will you betray them so no one gets the money?

Now, in the first example, there have been contests to determine the most optimum circumstances in the prisoners’ dilemma, with it running in iterations. One of the most successful solutions was a method called tit for tat. Tit for tat works by one agent (or prisoner) choosing to immediately cooperate. However, if the opposing agent chooses to betray, then in the next round, the first agent mirrors the result. This does have its flaws.

If I simply retaliate based on your last result and you do the same, then we may always lose or always betray each other. Or simply you try to change your strategy, but I’m still punishing you from the last round. In order to get past this problem, you have to allow for forgiveness. That if you decide to forgive me and I still betray you, go friend another time to show that you are ready to be friends. If I choose your last response, then we are both friends again.

When this works, it is what we call reciprocal altruism or superrationality in biology. An example of this is from the movie Longest Yard. When the MC is told to lose the game or else he faces murder charges, he betrays his team. When he decides to try to get back in the game and win, the team betrays him. Several times. He finally pleads with them and convinces them to work with him.


A tit for tat system can also be used when a villain and a hero must come to an agreement in order to work with each other. They know that if one person betrays them, the other may retaliate.

Now, let us quickly discuss the Monty Hall Paradox:

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

Now you might be saying, “This isn’t a two player game.” In actuality, it is, as the host is a player. He is working against the contestant, hoping you choose a goat and not a car.

Prior to the 1990’s, the belief was that there was equal chance of door 1 having the car as door 2. 50/50. In the 90’s, Marlyn vos Savant in Parade magazine stated that it was in your best interest to switch your answer. Her statement caused a lot of controversy, with many academics disagreeing with her. However, she was actually right.


There is a lot to understand this, and I will try to keep it simple. Prior to opening one of the 3 door, the chance of choosing the right door is 1/3, or 33% chance. You choose Door #1. The host opens up door #3. It’s a goat. Why didn’t he open up Door #2? Perhaps it was a random choice…or perhaps it was a car. The thing to remember, is that the host has to open up a door with a goat behind it. With the door open, your  original choice still only nets 33% chance that you will be correct, however, Door #2 has 66% chance of being correct.

If you are still confused by this, then I recommend watching Mythbusters. In the episode Wheel of Mythfortune, they tackled this problem and explained it much better than I can (I’m sure someone in the comments will tackle this). But they did two experiments:

1 Will people always stick to their choice
2 Are your odds doubled by switching your choice

In the second experiment, they proved that this is confirmed. You do have a 66% chance that changing your answer will reveal the car. But for the first one, they found that 20 out of 20 people didn’t change their answer. It wasn’t about if they won or lost (which I think 1 person won).

The thing to take away from this, is that even when given new relevant information, people are likely to remain with their first choice. Likely because it was based on intuition rather than logic, and when given a chance to change, they maintain their intuition. Human intuition is a very powerful force, but it can often times be wrong.

Gambler’s Fallacy & Game Theory

The common theme here is character decisions. Not just how our characters make a decision, but why. It is too easy for us as writers for our MC or Protagonist (as they may be different characters) to choose the right course of action because the plot demands it. A conflict between two characters can be seen as an element of Game Theory, as we have two people (or groups) working against each other for their own outcome.

Sometimes when we make up our mind about something, it is extremely difficult for us to change our thoughts on it, even when given evidence to the contrary. This is the human condition. Often times we choose what feels right rather than using logic to guide us. It can be a good thing and a bad thing.

For example, in the Star Trek Voyager episode Rise, Neelix uses his intuition to believe there is something on top of the orbital tether based on what a man said before he died. Tuvok however feels it is illogical to do so, as stopping could cause them to die. Eventually Tuvok gives in and Neelix was proven correct.

But often in our decision, we use the wrong information to help us make a decision, that if we sat down and actually analyzed the situation, we may come to a different conclusion. But acting on gut or instinct or intuition, can easily cause us to over look something.

More than that, how are our character with altruism. While they might save children from a burning fire, would they betray their friends? How sweet of a pot would it have to be for someone to betray their friends? As they saying goes, everyone has a price. How do they handle betrayal and can they forgive?

There is much more to Game Theory to which can provide you insights on how your characters make decisions. Rather than them making the right choice right away, understand them as human and flawed and see them make the wrong choice initially before coming to the right decision. See them reflect on why their initial choice was wrong.

Remember that some people, especially men, feel that to change our answer is an admission of being wrong, which can be seen as a sign of weakness. People don’t like to admit they’re wrong and will choose to maintain what they think is right no matter what.

Consider all of this when you write, there are no hard fast rules to good characterization. What they look like, how they act, what is their background, what are their flaws…all vitally important. But so is their decision making process.

Writing is life, and life is very complex. Looking into Mathematics and Psychiatry is essential for us writers to do an effective story. There are vast resources to learn from and this contest allows me to open you up to the possibilities. But I can only give you a glimpse, you must go the rest of the way on your own.

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  1. Wow. What a fascinating post. I’ve heard of Gambler’s Fallacy (though I didn’t know the name for it), but Game Theory is completely new to me. And you’re right in that it can be used for characterization and decision making. I’ve got lots to think about now. I love it.

    I also love the picture of the goat behind the door. 🙂

  2. […] have so far discussed Game Theory a few times and for our last article, we discuss it once more. Zero-Sum Game is a type of game that […]

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