Game Theory and its Connection in Real Life Decisions - Coursework Example

Game Theory and its Connection in Real Life Decisions Carlo Avati Math Exploration Introduction Game theory refers to the use of mathematical modelsin the analysis of rational and strategic decision making in social situations where the outcome is dependent on the actions of the participants. Such games or hypothetical situations normally involve two or multiple decision makers who may have different objectives but are sharing the same resources or acting on the same system. According to Turocy and Von Stengel (2001, p.4), game theory concepts provide the language required to formulate, analyse, structure as well as understand strategic scenarios. The principles of game theory are normally based on five key conditions: 1. Each of the decision-makers or players has two or more choices, 2. All the possible combinations of the decisions leads to clear outcome such as a win or lose, 3. The scenarios are well defined, 4. Each of the decision-makers understands the rule of the game 5. The players are rational. The use of game theory in the analysis of decision-making dates back to 1944 when the concept was first invented and published by John von Neumann, in his book titled “The Theory of Games and Economic Behaviour.” Since then, a number of improvements have been made on the theory. Although it only represents an abstract decision making model, it is significantly connected to real life decisions, as it provides the needed mathematical setting to analyse a wide range of real life situations. Some of the important real life decision-makinginstances in which game theory has always been applied include: war, economics, crime, ethics, international relations, biology and law. The use of game theory in these real life situations is justified by the fact that its mathematical assumptions can provide important clues on the potential outcomes of each of the possible decisions. This paper provides a mathematical exploration on probability based on the application of games, such as the Monty Hall challenge and the Prisoner’s dilemma in various real life decision-making situations. Potential Applications of the Game Theory in Real Life Decisions Game theory primarily involves the use of the mathematical concepts of probability and statistics, in order to analyse strategic decision-making and its potential outcomes. The n-events represents a set of payoffs whereby n is the number of players in a particular game where the strategies are varied and the set is equal to the number of all the possible outcomes. For example, when one flips in coin toss probability, the first 4 flips, the relative frequency estimate is 0.25. However, the probability normally gradually stabilizes around 0.5 with more flips as shown in the figure below. The probability aspect of game theory offers the rules and language required for dealing with uncertainties, while statistics provides the necessary tools required for the application of game theory in real world situations. The use of probability distribution in game theory is based on two important conditions: each of the events is mutually exclusive, (they cannot occur simultaneously), and the probability of the occurrence of each of the event is 0≤P(e)≥1 The Prisoner’s Dilemma Game The prisoner’s dilemma is one of the games that have been widely used in the application of game theory in several areas of everyday life: politics, philosophy, economics, psychology and many others. The game is based on two suspects who are taken to custody and held separately. Although they are guilty of some crime, there is no sufficient evidence to convict them to trial. During interrogation, they are told that they will be set free if they testify against their partner, while the partner would get a 10-year jail sentence. It is also acknowledged that they will each get 5 years in jail if they both testify against each other. However, if both remain silent, they will be jailed for 6 months, as shown in the diagram below. The above problem involving Prisoner A and Prisoner B can be mathematically represented using the following matrix: Based on this analogy, the dilemma implies the fact that while cooperation may be the best strategy for both prisoners; the best mathematical strategy for any of the prisoner would be to testify against their partner, even if innocent, as this would lead to the most favourable outcome. Although this may seem selfish, it is the option with the least risks due to the incentive of being set free,combined with the fear of being betrayed by the partner that is questionedseparately, leading to a potential 10 year sentence (Vakil, 2008, p.56).The above Prisoner’s dilemma, non-cooperative game matrix, can be represented using a 2 by 2-probability payoff matrix as shown below: The equation above indicates that each of the cells of the game matrix contains a payoff for each of the players and every player would be expected to rationally attempt to maximize their favourable outcomes. In this regard, regardless of what player B does, Player A would be better off making a move that is not favourable to player B. In this regard, the payoff relationship: i. Implies that mutual cooperation between the players has superior outcomes for both the players ii. implies that non cooperation is the best decision strategy for each of the players from a self centred point of view. Real life application: Decision making during Cold War One of the best examples of real life application of the Prisoner’s dilemma game, in real life situations, was decision making during the Cold War, particularly during the nuclear standoff between the United States and Russia. There was no incentive for either of the two sides to strike, due to the threat of Mutually Assured Destruction (MAD) and therefore not striking resulted in the best possible outcome. The mathematical explanation of the strategy can be represented as shown in the matrix below. Monty Hall Challenge Another probability-based game that has a wide range of applications in real life decisions is the Monty Hall Challenge. It is a common multi-stage probability puzzle in which the solution primarily relies on conditional probability. The mathematical puzzle starts with an individual invited to a game show, which is given the choice to open any of three doors, being told that a fantastic car is behind one of them, while behind the two remaining ones there are goats (Griffin, 2012, p.10). The main question, around which all the challenge is based, is the following: if the participant has chosen door 1 and the host opens door 3, which has a goat, assuming that the prizes are randomly assigned behind the doors, is it in the interest of the contestant to switch his choice to door 2 in order to win the car? It is argued that it is in the advantage of the contestant to switch, as it increases the chance of winning by 2/3 from the initial 1/3 (Gnedin, 2011, p.3). The stages of the decision-making involved in the Monty Hall probability puzzle are as shown below: Generally, the Monty Hall challenge is a game that has three outcomes. Each outcome is represented by a door. The host of the challenge has people choosing one of the three doors. Behind one door there is a car and behind the remaining two doors there is a goat. A letter was once written to the Monty Hall, which brought many controversies. It asked if a caller selected one door and the host who knows where the car is, opens one door and asks the caller to change their choice, should the caller actually change it?At this point, the player has selected a door, but the host has not chosen one yet. The host remains with two choices. If the player choses door A and the car is behind door B, then the host must open door C. The possibility of this outcome is represented by a third layer of branches to the existing tree. The question in the letter is vague and so we have to make a few assumptions before modelling the game. The first assumption is that the car has an equal chance of being behind any of the three doors. The second assumption is that the player has an equal chance of choosing any of the three doors ,irrespective of where the car is. Another assumption is that if a player chooses a door, the host must open another door, which has a goat behind it and offer the player the opportunity to change their choice (Taylor, 2014, p.102). The final assumption is that if the host has to open a door, then he has an equal chance of opening either of the two. The question to answer now is therefore the following: what chance does a player who changes his choice have of winning the car? There are three possible outcomes in this game represented by each door. Specifically: the door that hides the car, the door chosen by the guest player, and the one opened by the host, which hides a goat. For example, suppose one was doing a multiple question exam with three different answers (ABC). He has chosen choice A as the answer, but suddenly realizes that one of the answers is wrong (choice B is wrong), the probability of the student getting the correct answer can increase from 25% to 75% by switching to choice C as shown below: The Based on the above analysis, both intuition and the math problem in the Monty Hall diagram suggests that the more we know, the better the decision. For example, it would be the best interest for the players in certain situations to switch their initial choices during decision making after they have been shown a non-prize which was not initially chosen. This is particularly because it improves the favourability of their outcomes by increasing the chances of winning from 1/3 to 2/3. Real Life Application of Monty Hall Game One of the potential applications of Monty Hall is the strategy of the kicker (penalty shooter and goalkeeper (goalie) during penalty kicks in soccer games. For example, during a penalty kick, the kicker has the choices regarding the direction of the shoot while the goalkeeper has the choices of throwing himself either left or right. However, there is usually no sufficient time for the goalkeeper to see the direction of the ball and subsequently decide on the direction to go in order to intercept it. As a result, the choice of the direction for the goalkeeper is independent of the kicker’s choice. Although most penalty kickers often prefer to use their stronger side (either left or right). However, knowing the goalie may exploit this by always going in the same direction, the penalty kickers will most likely counter-exploit this by changing the direction of their kick. In this regard, both the goalie and the penalty shooter can use the principles of Monty Hall challenge by switching or altering their choices in order to optimize their winning strategy and enhance their chances of winning. Generally, the application of the Monty Hall Challenge in such a situation is largely based on the presumption that it is always advantageous to switch choices when one has been shown a non-prize which was not initially chosen, as it increases the chances of winning from 1/3 to 2/3 as shown below: The expected payoff for the shooter’s strategy SS can be given as: This means that the payoff for the shooter if he does not change the direction of the penalty shot is 50% and 95% if he changes to the other side. Similarly, the goalie’s expected payoff can be given as However, since the shooter is indifferent when the E(SS) = E(SW), the entire equation for the payoff of the goalkeeper can be given as: The Battle of Sexes Game The Battle of Sexes is a two-player puzzle game, where a couple wants to meet for a date in the evening, but there are only two forms of entertainment available in town: a ball game and an opera. This game was first studied by Dr.Luce and Raiffa in 1957. It involves two players who are a couple: a man and a woman. The two have discussed during the day about spending the evening watching either a football match or an opera, without reaching a conclusion. Both of them would like to go together to either the match or the opera.In particular, the woman would prefer the opera, while the man prefers going the ballgame. The dilemma is in finding the most rational way to solve the situation, so that the couple can go back home, both happy. It is argued that the best answer would be for each of the players to stick to their choices while anticipating that the other partner will do the same, as described in the theory of Nash Equilibrium (López, 2012). Generally, Nash theory refers to a concept in which the best outcome of the game is achieved when non of the players has an incentive to change their strategy after knowing the choice of their opponents.For example, there is no beneficial outcome in changing actions when other players remain constant in their strategies. The battle of sexes goes that If the woman goes to the opera with her boyfriend, then she will have a 3, but if she goes to the opera alone, she will have a 0. If she goes to the football match with her boyfriend, she has instead a 1. This is because she prefers the opera to the football match. The boyfriend, on the other hand, prefers the football match over the opera. If he goes to the football match with her, he consequently gains a 3, but if he goes alone it is a 0. If he goes to the opera with her then he has a of1. The payoffs matrix of the game of sexes can be given as: The expected payoff value for both the players lies between 1 and 4and the resultant graph for the cooperation between the the woman and her boyfriend can be given as shown below: Real life Application In real life applications, the game of the Battle of Sexes can be used to analyse the decision making of two drivers approaching each other on one side of the road. Although each of them would like to drive on the right side of the road, the best mathematically correct strategy for each of the drivers, would be to drive on the opposite sides of the road in order to avoid an accident. Each of the drivers will therefore benefit from deviating their lane. Rationale/Aim of the Exploration The primary aim of this mathematical exploration is to investigate the basic concepts of probability based on games such as the Monty Hall challenge and the Prisoner’s Dilemma, in various real-life decision-making situations. I have always had an interest in the potential use of mathematics in modelling real life situations. I therefore exploited this coursework to gain more understanding and knowledge on this wide topic. The results of this exploration will therefore give me an opportunity to understand more in depth the use of mathematical models to explain real life phenomena. Conclusion In conclusion, this exploration has allowed me to understand various basic concepts of game theory, particularly through the evaluation of the results of the different examples and applications of the aforementioned games games in real life decision making. Based on game theory, I have also learnt that although some of the strategies suggested are not always morally right and may seem selfish, they are the most mathematically correct strategies and that they are likely to bring to the best possible outcomes. References Gnedin, A. 2011.The Monty Hall Problem in the Game Theory Class.The Mathematical Intelligencer pp. 1-17.Available at http://arxiv.org/pdf/1107.0326.pdf Griffin, C. 2012.Game Theory: Penn State Math 486 Lecture Notes. Available at http:// www.personal.psu.edu/cxg286/Math486.pdf López, J.M. 2012. Battle of the Sexes: A Quantum Games Theory Approach. Quantum Information and Computation 385-409. Available at http://cdn.intechopen.com/pdfs/29135.pdf Turocy, T.L., Von Stengel, B. 2001. “Game theory”, London School of Economics, Research Report LSE-CDAM-2001-09. http://www.cdam.lse.ac.uk/Reports/Files/ cdam-2001-09.pdf Read More