Game Theory: Beyond the Basics

Using game theory, you can design real scenarios for pricing competition and product launches (and more), and predict the outcome. Companies that use (and stick to) this method to determine the Nash Equilibrium see huge benefits in their budget strategy.

Whose turn is it?

Although sequential games are played in turns, simultaneous games are made by each player at the same time. For synchronous games, we no longer use the common introduction method of reverse induction. Proponents of game theory often list different outcomes in so-called matrices (below).

Player one/player two remain right
up (1, 3) (4, 2)
down (3, 2) (3, 1)

This matrix is ​​called a normal form. The choice of player one is displayed on the vertical axis on the left, and the choice of player two is displayed on the horizontal axis at the top. The income of each player is in their corresponding intersection and is displayed as follows (player one, player two).

Nash Equilibrium

The Nash equilibrium is an achieved result. Once reached, it means that no player can unilaterally change the decision to increase revenue. It can also be considered as “no regret”, that is, once a decision is made, considering the consequences, the player will not regret the decision.

In most cases, the Nash equilibrium will be reached over time. However, once the Nash equilibrium is reached, it will not deviate. After we have learned how to find the Nash equilibrium, let’s see how unilateral actions will affect the situation. Does it make any sense? It shouldn’t, which is why the Nash equilibrium is described as “no regrets.”

Finding Nash Equilibrium

Step 1: Determine the player’s best response to player two’s behavior.
When examining options that maximize player spending, we must look at how player one responds to each option that player two has. An easy way to do this visually is to obscure the choice of player two. When we apply this method, please consider the matrix described at the beginning of this article.

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Player one/player two remain right
up (1, -) (4, -)
down (3, -) (3, -)

Player one has two possible choices: “up” or “down”. Player two also has two choices: “Left” or “Right”. In this step of determining the Nash equilibrium, we look at the reaction to the player’s two actions. If player 2 chooses to play “left”, we can play “up” with a profit of 1, or play “down” with a profit of 3. Since 3 is greater than 1, we will bold 3 to indicate the choice to play “down” here.

If player 2 chooses “correct”, we can choose “up” to get 4’s profit or “down” to get 3’s profit. Since 4 is greater than 3, we bolded 4 to indicate that the option is playing “up” here. The results in bold are shown in the complete matrix below.

Player one/player two remain right
up (1, 3) (4, 2)
down (3, 2) (3, 1)

Step 2: Determine Player Two’s best response to Player One’s behavior.
As we did with player one’s player two benefits, we will hide player one’s benefits when determining the best response for player two.

Player one/player two remain right
up (-, 3) (-, 2)
down (-, 2) (-, 1)

Just like looking at the number one player, every player has two choices. If the player chooses “up”, we can “go to the left” with a profit of 3, or “go to the right” with a profit of 2. Since 3 is greater than 2, we will bold 3 to show the option to play “Left” here. If the player chooses to play “downward”, we can play “leftward” with a gain of 2, or play “rightward” with a gain of 1. Since 2 is greater than 1, we will bold 2 to indicate that we choose to play “Leave” here. The results in bold are shown in the complete matrix below.

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Player one/player two remain right
up (1, 3) (4, 2)
down (3, 2) (3, 1)

Step 3: Determine which results have both benefits in bold. This particular result is the Nash equilibrium.
Now, we combine the bold options of the two players onto the complete matrix.

Player one/player two remain right
up (1, 3) (4, 2)
down (3, 2) (3, 1)

Look for the intersection where both returns are in bold. In this case, we find that the intersection of (Down, Left) and return (3, 2) meets our criteria. This shows our Nash equilibrium.

This method of finding Nash equilibrium is very suitable for finding equilibrium in simultaneous games, because we are studying how a player can react independently of the actions of others. This simultaneous game scene often appears in companies such as airlines. Here is an example of how airline pricing might work, similar to the game above. The payment amount is thousands of dollars. Remember, these are expenses, not prices. The method we applied before has been applied to show where the Nash equilibrium appears.

Airline One/Airline Two Low price High price
Low price (3,000, 3,000) (4,000, 2,000)
High price (2,000, 4,000) (3,500, 3,500)

Just looking at the selection of A1, we can see that if A2 chooses to play a low price, we choose between the low price of 3000 and the high price of 2000. We choose Low because 3,000> 2,000. We did the same thing with A2 playing high prices, and saw that we played low prices, because 4,000> 3,500. Conversely, just looking at the selection of A2, we can see that if A1 chooses to make a low price, we choose between the “low price” of 3000 and the “high price” of 2000. Since 3,000> 2,000, we choose the low price option here. If A1 plays high price, we can low price 4000 or high price 3,500. Since 4000>3500, we choose to hit a low price here.

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The Nash Equilibrium is that both airlines will charge a low price (shown when the choices of the parties are highlighted). If both airlines charge high prices, their respective conditions will be better than the Nash equilibrium.

So why don’t they agree to do this? First, collusion is illegal. Secondly, if this happens, it would be beneficial to act unilaterally on behalf of an airline to make lower prices, so that the airline in turn can make more money. This logic also explains how the Nash equilibrium is reached, and why it is not good to deviate from it once it is reached.

Multiple Nash equilibrium

Generally speaking, there can be multiple equilibriums in a game. However, this usually happens in games where the elements are more complex than two players with two choices. In a game that is repeated over time, one of these multiple equilibriums is reached after trial and error. In the business world, when two companies are determining the price of highly interchangeable products (such as airline tickets or soft drinks), this is the most common situation where different choices are made over time before equilibrium is reached.

Bottom line

Using these advanced methods, more real-world situations can be modeled and solved. The different types of Nash equilibriums we discussed are the most common solutions in real-world modeling games. Practical knowledge of game theory can help you develop a strategy, whether it is playing tic-tac-toe or fighting for maximum profit.


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