Game Theory: The Mathematics of Game Theory — Part 1

When attempting to locate an optimal strategy in any game, there is some information that could aid us in choosing the optimal strategy. We need to calculate the probabilities of a specific event occurring. For example, the probability in a two-player game of rock, paper, scissors of you winning is always ⅓, which is equal to 3/9.

Scenario Analysis:

You Choose: Rock, Opponent Chooses: Rock = Draw

You Choose: Rock, Opponent Chooses: Paper = Loss

You Choose: Rock, Opponent Chooses: Scissors = Win

You Choose: Paper, Opponent Chooses: Rock = Win

You Choose: Paper, Opponent Chooses: Paper = Draw

You Choose: Paper, Opponent Chooses: Scissors = Loss

You Choose: Scissors, Opponent Chooses: Rock =Loss

You Choose: Scissors, Opponent Chooses: Paper = Win

You Choose: Scissors, Opponent Chooses: Scissors = Draw

This is a simple example. Calculating probabilities get more difficult as the complexity of the game increases. Let’s consider Texas Hold’Em Poker.

In Texas Hold’Em, each player is dealt two cards, and five cards are dealt by the dealer onto the table. Each player uses the best five-card hand out of the seven cards. To calculate the total potential number of different 7-card combinations, we use the binomial coefficient. This can be used in any game to calculate the total number of scenarios.

On a scientific calculator, if you have one, the binomial coefficient is represented by the nCr button. In this example, there are 52 cards, and we draw 7. So the input is 52 nCr 7, also written in a vector as (52 7). This provides 133,784,560 possible 7-card arrangements.

Take note: (n k) = n!/[k!(n-k)!], where ! refers to factorial, where you multiply every number including that number and before it together. For example, 5! = 5 * 4 * 3 * 2 * 1. More detail on this mathematical topic revolves around the binomial theorem and Pascal’s triangle if you want to explore further. Still, we won’t need to for our exploration of mathematical game theory.

Figure 1 — Note: The probabilities are approximate.

Stick with me, right? I’m aware this looks chaotic. So let me explain every term so we all can go away happy that we understand the chances of winning a game of poker with each hand. You might learn something that could aid you in your next game!

To understand the binomial coefficients, we first have to understand that some events when drawing the 52 cards are mutually exclusive. This means both can’t happen at the same time. For example, no card can be the ace of spades and the ace of hearts at the same time. Also, the probability of getting an ace is ¼, and the probability of a spade is 1/13. The probability of the ace of spaces is 1/13 * 1/4 = 1/52. (When multiplying fractions just multiply the top two numbers and the bottom two numbers into one fraction). Other scenarios discussed in probability theory emerge such as independence, and probabilities given previously occurring events. If you want me to explain these I’ll include them in an appendix at the end of the entire game theory section.

Now back to the Texas Hold’Em table. A royal flush is a 10, Jack, Queen, King, and Ace of the same suit. There are four suits, hence the (4 1). coefficient. The (47 2) coefficient emerges from the fact that 5 cards will be on the table. So the deck has 47 cards remaining, and you have two of your own cards you are given that no other player has. Hence (47 2).

Some of the other winning hands offer rather gnarly calculations. But this table demonstrates it is possible to calculate the probabilities of each poker hand winning. Although in an actual game, calculating this in your head at speed the game moves are impossible. Online poker has seen the creation of calculators that figure out the odds of a hand winning a game. They can be used in online poker games where time is more abundant.

Expected Value

If we have an expected payout of an event, we can calculate an expected value. Say we have an event X. The probability of event X occurring is ½, and if you bet on event X happening, you receive a payout of £10. Your expected value would be:

E(X) = 1/2 * £10 = £5.

Let’s expand this scenario. You bet on event X occurring. If it doesn’t occur, you lose £7.50. This scenario unfolds as follows:

E(X) = (1/2 * £10) + (1/2 * (-£7.50)) = £5 — £3.75 = £1.25

So you have a positive expected value here. Over time, if you continued to take this 50/50 bet, you would walk away with more money. This is an example of a good expected value, and hence a good bet. If you lost £12.50 every time event X doesn’t occur:

E(X) = (1/2 * £10) + (1/2 * (-£12.50)) = £5 — £6.25 = -£1.25

This negative expected value would mean an average loss over time. An expected value of 0 occurs by not taking the bet at all. Not gaining anything is better than losing.

Poker expected values are a lot more complicated due to the difference in every round. So when using mathematics in poker from an expected value standpoint, you have to consider the earlier table. If you draw two 7s, and the first two cards down on the table are two more 7s, you have a higher probability of winning. This is because there only exists a 0.031% chance of a straight flush or royal flush occurring. But in this situation, there is an even smaller chance of a straight flush or royal flush being drawn because two cards are already on the table.

This is described as the probability of event A given event B has happened.

P(A|B) = The probability of a royal flush being drawn, given that two 7s have already been placed on the table and you are drawn two 7s.

Event A = Royal flush drawn.

Event B = Two 7s are drawn in the first two cards on the table and you are drawn two 7s.

We can calculate the probability of this with Bayes’s Theorem (I’ll cover this later). If you enjoy this mathematical stuff, let me know! It was all topics I learnt in my first degree, so I love to flash back to it!

It all boils down to wanting to maximize your expected value. Either by increasing the probability of your success, or your payoff if the bet comes off. To think about this in any situation you find yourself in, do the following things:

1. Consider all the potential events involved in your situation. Give each event a specific way of identifying it, for example, events A, B etc.
2. Assign or calculate the probabilities of these events occurring.
3. If you take a bet on this event, what is your expected value? Is it a positive expected value? If so it could be a good bet to take.

Apologies that I didn’t post yesterday. I had my Masters’s graduation and couldn’t get this piece finished in time.