# Game Theory Behind Shop Competition

## Abstract

This paper covers game theory at a basic level and looks at an elementary application of game theory, the prisoner’s dilemma. The paper goes on to use the example of competitive pizza shops and how game theory can help to make the best business decisions when it comes to pricing slices of pizza. This is an interesting topic that is quite new in the mathematical world. Since John von Neumann and Oskar Morgenstern invented the idea in 1944, there has been a lot of progression in this field with lots of applications. With John Nash being one of the most well-known game theorists too date.

## Introduction

Individuals make decisions on a daily basis. Game theory, also called the strategy theory or the theory of interactive decision making, analyses how individuals and companies make decisions by comparing what they get or the outcome. Even though it has applications in a lot of fields, it is mainly applied in economics to determine the outcome of economic transactions by applying the utility theory to model the interests of the players. The theory states that players are most likely to use a strategy that will maximise their utility. Among the most renowned applications of game theory is the prisoner's dilemma. It is mainly used in determining the choices that different parties make when subjected to strategic situations or conditions with similar rules and outcomes by applying mathematical models. The term strategic situations refer to a situation in which two parties try to make decisions that will favour them by trying to anticipate the reaction of the other party in a similar situation. It is applied in a lot of fields, but more so in economics to analyse any strategic interaction between two parties. This paper will introduce the readers to the game theory by enlightening them on some common terms that will enhance their understanding of game theory, assumptions, the prisoner’s dilemma in more detail, looking at an example including two different pizza shops and conclude with some of the math’s behind finding the ‘Nash equilibrium’.

## Terms

It is important to familiarise with the terms that are frequently used in game theory. “The game” refers to a set of circumstances whose outcome is dependent on the actions of two or more parties that are making the decisions. “Players” are the parties that are involved in a game, and they make decisions that are within the rules and outcomes of the game. A “strategy” refers to the plan of action that any given player in the game will take given the rules and circumstances provided in the game. “The payoff” refers to the payout, or what any player receives after making certain choices within the context of the game. The payoff may be in a monetary form or in the form of a utility if it is in a quantifiable form. The “information set” refers to the data available to each player at a given point in the game. “Equilibrium” is another term that is used to refer to the final outcome that is arrived at after all the parties involved make a decision. The Nash equilibrium is a concept in which the players have no motive to deviate from their chosen strategy even after considering the choice taken by the opponent since they cannot gain any additional benefit.

Just like every other theory, game theory also has its assumptions. The first assumption is that the parties involved in the game practice instrumental rationality that assumes that each player tries to maximise their utility under the given rules. Additionally, it assumes the following: that the player can adopt multiple strategies to solve a problem, that the players are well informed about the rules and the outcomes of other players and that there are pre-determined outcomes for the players. The theory also assumes that there is a conflict of interest among the players.

## Elementary Applications

Game theory has a lot of applications in modern life. There are numerous games. However, the applications are mainly based on the three fundamental categories of games. The first fundamental category is the sequential games in which the players decide after each other and they at least have some knowledge on the previous move made by the other party. trust games, which are a subset of sequential games are widely used when it comes to where to place and honour trust. Common examples of their application in the modern world are when it comes to delaying payments and sending defective goods and taxi drivers who risk being deceived or assaulted by bad customers.

The second fundamental category is games with either perfect or imperfect and complete or incomplete information. In games with complete or perfect information, the players are provided with all the information they need to decide their next move. Games with incomplete information are characterised by a move in nature in which the types of players are determined, or the game structure is determined.

There are also cooperative or non-cooperative games. Cooperative games are characterised by pre-determined joint action agreements that are enforceable through a third party. Non-cooperative games, on the other hand, allow individuals to act on their self-interest as there is no enforcement of the laws of the game. The last category is the zero-sum and positive sum games. In a zero-sum game, one player’s gains or losses are equally balanced by what the other player loses or gains. Positive-sum games, also called “win-win” games are constructed in such a way that every player can benefit from the transactions.

In the modern world, the prisoner's dilemma can be used by individuals who have feelings for each other, but they fail to muster up the courage needed to confess their feelings to one another. In the above example, partner one chooses the column while partner two chooses the row. The words in each cell are used to represent the payoff for each choice of strategy by either of the two partners. In the first column, if both confess their feelings, then the two partners will go ahead to develop a relationship that will bring each of them a lot of joy. In the second column, one partner confesses his/her feelings while the other one is too shy to confess their feelings, or they don’t have the feelings. The payoff for this choice by the first partner will lead to a feeling of rejection that is not desired in any way. The same happens if the two partners switch roles and one of the partners decides to express themselves while the other partner chooses to remain silent. The one partner who decided to confess their feelings will end up feeling rejected, an outcome that is rarely desired by anyone. In the last scenario, both partners decide to remain silent about their feelings and therefore no one ends up being rejected. Such is the dominant strategy since none of the partners will feel rejected regardless of the choice that the other partner would have taken. Even though the option of not confessing is the dominant strategy according to game theory, it rarely brings the most desired outcome since one only gets to protect their feelings from being hurt just for a while but then they get to settle for less by allowing the person they really had feelings for to go. In that scenario, the best option is to confess one's feelings instead of settling for less.

## Competitive Pizza Shops

Based on the above example, one can also develop an example of how game theory can be applied by business entities. Since pizza can be said to be a favourite meal for many people, applying the theory to determine how various restaurants come up with the prices of pizza can serve as a good example in addition to offering insight to pizza lovers on how the prices of pizza are determined. Let us take, for instance, there are two pizza restaurants that charge different prices for a piece of pizza, either £1, £2, or £3. In a given month, it is expected that 10,000 pieces of pizza are taken by tourists while the locals take 8,000 pieces and they prefer to go to the restaurant with lower prices. In case the prices are equal, then they split evenly between the two restaurants.

If restaurant A charges £1 and B charge £2, then all the locals will buy their pizza from restaurant A. therefore, it will serve 5000 pieces to tourists and 8000 pieces to the locals that will result in 13000 pieces of pizza sold. As a result, the restaurant will earn £13000 while B will only serve 5000 pieces to tourists and therefore the restaurant will earn £10000. The payoff matrix gives the money earnt with it reading B, A, meaning that the value to the left of the comma is the payoff for restaurant B and to the right of the comma is the payoff for restaurant A.

For each restaurant, move 2 strictly dominates over move 1 and therefore move 1 can be eliminated for both restaurants. Eliminating the move will lead to a reduced game. Upon removing move 1 and remaining with moves 2 and 3, move 2 will dominate move 3. Therefore, move 3 can be eliminated to remain with move 2 that will lead to an income of £18,000 for each restaurant and therefore they will choose the price of the pizza to be £2.

The game theory is also applied in Cournot competition, in which two competing companies producing the same goods within an industry determine the number of goods to produce independently and simultaneously. If they produce too much, the market will be flooded and therefore the prices of their goods will fall that will lead to a reduction in their profits. Taking the price equation as and the firms objective is to maximise profits. For both firms, the profits will be equal to the quantity they supply in the market. The market price is determined by the quantity supplied by both firms. Therefore, the profit function of firm A will be:

to maximise the profits, the firm will have to satisfy the first order condition which is the derivative of the profit function with respect to q. profit is maximised when the first order condition is equal to 0.

Consequently, both firms should produce 1/3 of the quantity that the other firm produces to maximise their profits. Since 1/3 is the most optimal strategy and both firms are comfortable with it and they are not willing to change regardless of what the other player chooses, then it can be said to be the ‘Nash equilibrium’.

## Conclusion

Game theory has lots of applications in the day to day lives of individuals. In the social setting, game theory can be used to determine the best strategy to choose, for instance when it comes to deciding whether to dump one’s waste locally or to hire a truck to take the waste away. Through the application of game theory in such scenarios, the dominant strategy is always the optimal solution since it always yields a higher payoff regardless of the strategy that the rival adopts. In business situations, the game theory can also be used to guide business decision-making process to ensure that the organisation remains competitive in the market place, like in the competitive pizza shops scenario. Individuals can also apply game theory in making crucial decisions, for instance, while playing rock paper scissors. Additionally, using the game theory, players can arrive at win-win situations, in which both the players gain. It is therefore important to learn and understand game theory and learn to apply it in the day to day activities to gain the most from each interaction that individuals engage in.

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