# Game Theory- Unveiling Strategic Decision-Making through Secondary School Maths

Among the many models and usage of maths, one of its purposes can be found in game theory. Game theory is defined as the study of the ways in which interacting choices of economic agents produce outcomes with respect to the preferences of those agents, where the outcomes of the question might have been intended by none of the agents. (Stanford Encyclopaedia of Philosophy).

In simplicity, the statement can also be rephrased as “ways in which a player can strategize and win the game among competing players.”. While that sounds simple, it can get complex and deeply mathematical. Hence, this article will teach you more about how that probability class learned in Secondary 4 may be your foundation for a gaming career!

## Decisions and Outcomes

An action or decision always leads to an outcome. Players of Elden Ring or Dark Souls are most familiar with the effects of a wrong decision costing significant setbacks. In these games, a calculated attack promises a set of damage values, while a risky move implies possible self-inflicted, or boss-inflicted damage.

With this as an example, probability is taught to represent the likelihood of an event occurring which ranges from 0 (Impossible event) to 1 (certain event).

If a player chooses to attack, probabilities can be affected by considering factors such as the type of attack being ranged or melee, information available, and assumptions about other player strategies.

Ultimately, this is applicable to any type of decision in games including poker, role-playing games, or even guessing games.

## Payoff Matrix and Probability

The outcome of a game and its rewards based on the actions of the player are also calculable through a payoff matrix. The explanation of how the matrix works is as follows:

The payoff matrix is a fundamental concept in game theory that illustrates the outcomes and associated rewards or payoffs for each player’s strategy choices. Probability, in the context of the payoff matrix, is used to assign likelihoods to different outcomes based on the players’ strategies. Here’s an explanation of the payoff matrix and its relationship with probability. There are five main steps in using a payoff matrix.

Firstly, the structure of the matrix:  A payoff matrix is typically presented as a grid, with rows representing the strategies of one player and columns representing the strategies of another player. Each cell in the matrix represents a specific combination of strategies chosen by the players, while the numbers within the cells represent the payoffs or rewards received by the players based on the outcome of the game.

Secondly, probabilities in the matrix: Probabilities can be assigned to different outcomes to represent the likelihood of those outcomes occurring. These probabilities can also be affected by subjective assessments made by the players or can be based on statistical analysis, historical data, or assumptions about the game, and it ranges from 0 to 1.

Thirdly, calculating the probabilities: For every choice that is made, multiply each payoff in the player’s row by the corresponding probability assigned to that outcome.  Sum up these products to calculate the expected payoff for that strategy.

Fourthly, Strategy choice: Use the expected payoffs to guide the strategy selection. Players can choose to maximize their expected payoffs, but it is important to note that the choice of strategies is independent, i.e each player’s strategy affects the other player’s payoff.

Lastly, Strategical Analysis: Through these outcomes, players can assess the stability of strategies, identify dominant strategies, or explore mixed strategy profiles.

## How Does Secondary School Tuition Help With Probability?

Before diving further into game theory, it is critical to address that the formulations and explanations above are enough to intimidate one from using the payoff matrix. This is also particularly difficult for students or any individual that is not familiar with the rules of a matrix and probabilities.

Therefore, this is where Secondary School math tuition can help. Tutors teach the basics of these mathematical formulas such as the addition rule, which determines the probability of one event OR another event occurring.

Through this example, students can see the relationship between game theory and mathematics. Very often, teachers are inclined to impart the mechanisms of mathematics, however, it is also important to understand that the foundational building block of mathematics can achieve real-world applications.

Now that these probability ideas have piqued your curiosity, feel free to explore more about mathematics and ask questions in this topic as well as the articles to come.