IndexIntroduction: The Prisoner's DilemmaZero-Sum GameSurvey 1: The Prisoner's DilemmaConclusionIntroduction: I first learned about game theory in my economics course, as an introduction to oligopolies and cartels. Since then I have been fascinated by the prisoner's dilemma: how sometimes the most logical decision is not the option with the highest payoff. After researching about it, I also discovered other forms of game theory such as the Hawk-Dove game and the zero-sum game. When our Economics teacher ran the simulation, I was surprised by how much our class deviated from what was considered rational. After that lesson and watching “A Beautiful Mind,” I wanted to investigate the mathematics behind it. Say no to plagiarism. Get a tailor-made essay on "Why Violent Video Games Shouldn't Be Banned"? Get an Original Essay Game theory is the use of mathematical models and analysis to describe and predict economic and psychological behavior. It is primarily used to make decisions in conflict scenarios involving at least two players, where one player's surrender will depend on other players. Therefore, his actions are completely based on his judgment of what the other players will decide: there is interdependence. In addition to economics, it can also be applied in biology, computer science, political science and psychology. According to MIT, Nash equilibrium occurs when “players guess the strategies of other players and choose the most rational option available.” Rationality, therefore, is when a player maximizes his utility and profits. When the Nash equilibrium is reached, there is no incentive to change: it is stable. In the exploration, the Nash equilibrium is highlighted for each scenario. This exploration will consider 3 types of game theory (prisoner's dilemma, hawk-dove and zero-sum) examining 6 scenarios: the basic prisoner's dilemma, cartels, entering a monopoly market, investing in technology based to the size of the business and determining where to locate. To study the reliability of game theory in predicting real-life behavior, I also conducted an investigation covering 4 of the above scenarios: the basic prisoner's dilemma, technology investment based on two company sizes, and the location. popular example of game theory. This game includes at least two players and imperfect information. It can be applied in many fields such as economics, psychology, biology and politics. Below is the scenario I used in the survey: Example 1: You (A) and a friend (B) are arrested for a crime and sentenced to 2 years. You are also both suspected of a much more serious crime (which you didn't commit), but the police don't have enough evidence to prove it. You are in isolation and have no way to talk to each other. They make you a deal: you could confess to committing the greatest crime, betray your friend, or deny it. These sentences are modeled in the pa-off table below – Figure 1 shows the number of years you will receive for each possible outcome (i.e. if you confess and they deny, you will get 1 year and they will get 10 years). Initially, both choose to deny seems the most optimal return (2,2). However this is not the case; for both players there is an incentive to confess. If player A confesses, he will receive 5 years if B also confesses or 1 year if B denies. However, if A denies, he will receive 10 years if B confesses or 2 years if B denies. The most rational decision would be to choose the option with the least repercussions: confess. This is because the worst possible scenario if A confesses would be 5 years, while theworse if A denies it would be 10 years. This also applies to Player B. Another incentive to confess is the fear of being betrayed, thus receiving 10 years, which is the worst possible scenario. This reasoning is called backward induction. However (2,2) seems to be the optimal yield, in this game, if the players were rational, they would both choose to confess (5,5). This stable profit is called the Nash equilibrium. It can be deduced that there are 4 possible combinations from the table above and using the combinations. A= (_12)CA=2B= (_12)CB=2A×B=4A general prisoner formula The dilemma can be found in the payoff table below, where w > x > y > z (w is the most favorable (when one betrays the other), x is the most optimal return (when both cooperate), y is the Nash equilibrium and z is least favorable (when one is betrayed by the other all cases, it would be better to confess). . can be used to decide whether a firm should invest in technology, advertising, research and development, join a cartel, etc., given the size of its competitor. Example 2: You are firm A in a cartel with the. Firm B. You are the same size as Firm B. You must decide whether to follow the price set by the cartel or lower prices (thus benefiting you), betraying the cartel rules below showing the consequences of all possible outcomes on existing profits. Figure 3: AFollow CheatB Follow $20m, $20m $50m, -$10mCheat -$10m, $50m $0, $0Figure 3 shows both firms in the short run. Following the backward induction reasoning used above, although following the cartel agreement it seems that the maximum Nash equilibrium is obtained when both firms cheat – the worst possible profit is -$10 million if firm A follows and $0 if company A cheats (also applicable to company B). There is also the temptation to cheat; the best possible profit is $20 million if Firm A follows and $50 million if Firm A cheats. A general formula for the profit table above can also be found in the table below, where w>x>y>z.Figure 4:AFollow CheatB Follow x, xw, zCheat z, wy, yHowever, if this were the case, no cartels would exist to the world. In reality, there would be punishment for defrauding a cartel in the form of a price war. Figure 5 below shows both firms in the long run after a price war due to punishment (which reality and economic theory say will occur when at least one firm cheats). Figure 5: AFollow CheatB Follow 20, 20 -∞, -∞Cheat - ∞, -∞ 0, 0 Since firm A is the same size as firm B, a price war would be mutually destructive, leading to a profit of -∞. Since the worst possible profits are equal, the highest possible profit will determine whether a firm remains in the cartel or cheats. Since $20 million is greater than $0, it is in the interest of both firms to remain in the cartel, in contrast to the previous conclusion derived from the short-run profit table. A general formula can also be found in the table below, where x>y>t.Figure 6:AFollow CheatB Follow x, xt, tCheat t, ty, yIn conclusion, when there is punishment, it would be better to stay in the cartel. This is an example of a repeated "eye for an eye" prisoner's dilemma, which can also be represented by a tree diagram as shown below. Figure 7: Example 3: Company B is debating whether to enter an industry controlled by a monopoly. Firm A could maintain current production levels (allowing Firm B to enter), or increase production by investing in expensive machinery, a barrier to entry that would harm Firm B if it decides to enter. TherePaytable shows all possible additions/deductions to profit below: Figure 8: A Increase (1-p) Equal (p) B Enter $80 million, -$50 million $40 million, $40 million Stay out $100 million, $0 $50 million, $0p is the probability that Firm A will maintain its level of production (because it will not have access to machinery). 1-p is the probability that firm A will increase its production (since it has the ability to invest in machinery). When you include probability, it is easier to represent possible outcomes in a tree diagram. Figure 9: In this scenario, if B were to enter the industry when A increases its production by investing in machinery, A would decrease prices to ease competition - ultimately, B would lose profits: A would receive $80 million while B would lose $50 million . If firm B were to enter the industry but A was unable to invest in machinery, the industry would become a duopoly and profits would (theoretically) be distributed equally between the two: each would receive $40 million. However, if B stayed out and A increased its production, A would be more productive and would not face competition, so it would get the most profit at this point: A would receive $100 million while B receives $0. If the firm A remained the same and firm B stayed out, firm A would continue to earn profits from the lack of competition but not as much due to inefficiency x: A receives $50 million while B receives $0. In this scenario, A It would be better to increase its machinery: the lowest profit is $80 million if it increases in size and $40 million if it remains unchanged. Furthermore, the highest profit is $100 million if it increases in size and $50 million if it stays the same – there is no incentive not to stay the same size. Furthermore, whether B chooses to enter or stay out, the best possible outcome in each case occurs when A increases in size. Theoretically, Firm B would be better off staying out of the industry since the worst possible profit if it entered would be -$50 million, and $0 million if it chose to stay out. Therefore, the Nash equilibrium would be at (100, 0). However, this does not take into account diseconomies of scale (disadvantages with increasing size) and the probability (p) that firm A will not be able to increase its production by investing in machinery – if A will not be able to do so, the Nash equilibrium will shift to (40, 40). The value of p could influence whether it would be rational for B to enter or stay out. P can be found by creating a general formula for the payoff that firm B receives in each scenario. This is done by multiplying the probability by the profit values, as shown in Figure 9. If B stays out, the profit B receives would be 0, since there is no production. If B enters the market, the general formula for the profit he would receive would be: win=40p+-50(1-p) win=90p-50The win should be greater than 0; otherwise there is no incentive for B to enter the market.90p-50>090p>50p>5/9 Therefore, B will enter the market if the probability that A cannot invest in machinery and has to maintain the same production levels is greater higher than 5/9. Firm B can also calculate its possible profits with a known value of p. Example 3 can be represented by a general formula where u > y > w > x > z > v. Figure 10: A Increase (1-p) Equal (p) B Enter u, vw, wLeave out x, zy, zA general formula for calculating p can be found where the symbols used are from Figure 9: payoff=w(p) +v(1-p)payoff=(wv)p+v (wv)p+v>z(wv)p>z-vp>(zv)/(wv)Hawk-DoveExamples 4 and 5 are game examples hawk-dove. A Hawk-Dove game, also known as Chicken, yesoccurs when two players compete for a good of known value (v) and there are two possible options “Hawk” or “Dove”. “Falco” is considered the strongest and riskiest strategy while “Dove” is considered the safe strategy. Players choose simultaneously. This began as a biological game, but can be applied to economics as it is used to model scenarios involving resources. Example 4: Company A chooses to invest in technology. Its competitor, Company B, is also considering doing so. Firm A is the same size as Firm B. The payoff table below shows the resulting profits for all possible combinations: Figure 11: AInvest Don't InvestB Invest 20, 20 0, 50Don't Invest 50, 0 25, 25In the example above, the technology profit for both A and B is $50 million, the investment cost is $10 million, and firm A is the same size as firm B (so the resulting profits are the same). The optimal outcome appears to be $25 million for both, when both do not invest in technology. However, the Nash equilibrium, and the most rational strategy, requires both firms to invest in technology because the lowest possible return if they choose to invest is $20 million, while the lowest possible return if they choose not to invest is of $0. Example 5: Company A chooses to invest in technology. Its competitor, Company B, is also considering doing so. Firm A is twice the size of Firm B. The payoff table below shows the resulting profits for all possible combinations: This can also apply to firms of different sizes. If company A is twice the size of company B, the profit from the technology investment for company A will be 100 million dollars, the profit from the technology investment for company B will be 50 million dollars and the investment cost will be 10 million dollars. The resulting profits are displayed in Figure 4 below. Again, although the optimal decision appears to be one in which both firms do not invest in technology, both firms have incentives to invest: for firm A, the lowest possible profit is $45 million if it chooses to invest , and $0 if you do so. It does not choose to invest, while for company B the lowest possible profit is $20 million if it chooses to invest and $0 if it does not. Therefore, the Nash equilibrium is equal to (45, 2) when both firms invest. The general formula for examples 4 and 5 is found below: V is the value of the resource, while C is the cost incurred by struggling to obtain the resource. If the resource is shared between two, its value is halved, but when they end up fighting for it, each incurs a cost of C/2. When both choose the hawk, each has a ½ chance of winning. The game is considered a kind of prisoner's dilemma when V>C. Although the optimal decision appears to be when both firms choose the weakest and least aggressive option (Dove), game theory shows that both firms should choose the most aggressive option (Hawk), even though this would entail costs because the profit is greater than 0. However, when the cost incurred is greater than the value received (VZero-sum game According to Investopedia, a zero-sum game is a situation in which one player's gain equals another player's loss, so the net change in the advantage for both players is zero. It is a non-cooperative game. Example in economics is the futures market, while examples in other fields are poker and chess a beach– left or centre. or on the right side. 60 customers are evenly spaced along the beach. The profit table shows the number of customers each company would have for each outcome. However, having two companies selling the same product next to each other may seem counterintuitive and wasteful. of resources, the Nash equilibrium is reached when both firms choose to position themselves in the center. The highest possible profit for both companies when they choose to position themselves in the middle is $40 million, while the lowest possible profit is $30 million. When they choose to position themselves to the left or right, the highest possible profit is $30 million and the lowest possible profit is $20 million. Although (30,30) can also be seen in 4 other cases (LL, LR, RL, RR), these are not equilibrium positions since there is still an incentive to betray the other by choosing the middle path to gain 40 million dollars. It can be deduced that there are 9 possible combinations from the table above and using the combinations. Survey I conducted a survey using scenarios 1, 4, 5 and 6 on a sample of 48 year 12 students (24 of whom were from company A and the other 24 were from company B) who had never done any of game theory first to investigate how well the real-life data fit what is theoretically correct and rational (the theoretically correct option for each scenario is highlighted in Figures 14 and 15 below). I chose to do this because I was fascinated by how far the results of a prisoner's dilemma simulation conducted by my economics teacher during an economics class deviated from the rational. The survey questions (for Company A) are reproduced below: In all of the following scenarios, you are Company A and Company B is your competitor. 1: The Prisoner's Dilemma You and a friend are arrested for a crime and sentenced to 2 years. You are also both suspected of a much more serious crime (which you didn't commit), but the police don't have enough evidence to prove it. You are in solitary confinement with no means of speaking to each other. They make you a deal: you could confess to committing the greatest crime, betray your friend, or deny it. The paytable below shows the number of years you will receive for each possible outcome: A Confess Deny B Confess 5, 5 10, 1 Deny 1, 10 2, 2 What you would do: Confess Deny 2: Technology You are choosing to invest in technology . Your competitor is thinking about doing it too. You are the same size as company B. The payoff table below shows the addition/deduction to your current profits: Would you invest in technology? Yes No3: TechYou are choosing to invest in technology. Your competitor is thinking about doing it too. Your size is double that of firm B. The payoff table below shows the addition/deduction to your current profits: Would you invest in technology? Yes No4: LocationYou are choosing to position yourself on a beach: on the left, center or right side. Customers are evenly spaced on the beach. The payoff table shows the number of customers each company would have for each outcome. Where would you place yourself? Left Center RightThe results of the survey I carried out for the 4 scenarios above can be seen below:Before the analysis carried out previously (so if each subject had guessed an option randomly) the options in scenarios 1, 4 and 5 would each have probabilities of 0.5, while the options in scenario 6 would each have probabilities of 0.33. However, the data in Figure 15 does not reflect this. There are 24 possible combinations that each subject could adopt when answering the survey. Possible combinations = (_12)C × (_12)C × (_12)C ×.