Table of ContentsIntroduction:The Prisoner's DilemmaZero-Sum GameInvestigation1: The Prisoner's DilemmaConclusionIntroduction:I first discovered game theory during my economics class, in as an introduction to oligopolies and cartels. Since then, I have been fascinated by the Prisoner's Dilemma: sometimes the most logical decision is not the one that pays the most. After researching this, I also discovered other forms of game theory such as the Hawk-Dove game and Zero-Sum. When our economics teacher ran the simulation, I was surprised to see how far our class deviated from what was deemed rational. After this 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 should not be banned”?Get the original essayGame theory is the use of mathematical modeling and analysis to describe and predict economic and psychological behavior. It is primarily used to make decisions in conflict scenarios involving two or more players, where one player's performance will depend on the other players. Therefore, his actions are based entirely on his judgment of what the other players will decide – there is an interdependence. Besides economics, it can also be applied in biology, computer science, political science and psychology. According to MIT, Nash equilibrium occurs when "players guess other players' strategies and choose the most rational option available." Rationality is therefore when a player maximizes his utility and his gains. When the Nash equilibrium is reached, there is no incentive for 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, Dove and Zero Sum) by examining 6 scenarios: the basic Prisoner's Dilemma, cartels, entry into a monopolistic market. market, investing in technology based on company size and determining where to locate. To study the reliability of game theory in predicting real-world behavior, I also conducted a survey covering 4 of the above scenarios: the fundamental prisoner's dilemma, investing in technology based on two company sizes and of location. The Prisoner's DilemmaThe Prisoner's Dilemma is the most popular example of game theory. This game includes two or more players and imperfect information. It can be applied in many fields such as economics, psychology, biology and politics. The scenario I used in the investigation is given below: Example 1: You (A) and a friend (B) are arrested for a crime and sentenced to 2 years in prison. You are also both suspected of a much more serious crime (that you didn't commit), but the police don't have enough evidence to prove it. You are in solitary confinement with no way to talk to each other. They give you a good deal: you can confess to having committed the most serious 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 have 1 year and them 10 years). At first, both choosing to opt out seems to be 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 either 5 years if B also confesses, or 1 year if B denies. However, if A denies, hewill receive either 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. Indeed, the worst possible scenario if A confesses would be 5 years, while the worst if A denies would be 10 years. This also applies to player B. Another incentive to confess is the fear of being betrayed and therefore receiving 10 years in prison, which is the worst possible scenario. This reasoning is called backward induction. Although (2,2) seems to be the most optimal return, in this game, if the players were rational, they would both choose to confess (5,5). This stable gain is called Nash equilibrium. It can be deduced that there are 4 possible combinations from the table above and using combinations. A= (_12)CA=2B= (_12)CB=2A×B=4A general formula for the prisoner's 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 the least favorable (when one is betrayed by the other).In any case, it would be better to admit it can be used to decide whether a company should invest in technology, advertising, R&D, join a cartel, etc., given the size of its business. competitor Example 2: You are company A in a cartel with company B. You are the same size as company B. You must decide whether to follow the price set by the cartel or lower your prices (this. which will benefit you), thus betraying the cartel rules The payoff tables below show the consequences of all possible outcomes on existing profits. $50 million, -$10 million Cheat - $10 million, $50 million $0.0 Figure 3 displays the two companies in the short term. Following the backward induction reasoning used above, even though the cartel agreement appears to be the Nash equilibrium best achieved when both firms are cheating – the worst possible payoff is -$10 million if Company A follows and $0 if Company A cheats (also applicable to Company B). ). Additionally, there is a temptation to cheat; the best possible win is $20 million if Company A calls and $50 million if Company A cheats. A general formula for the paytable 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 such were the case, there would be no cartels in the world. In reality, a cartel's deception would result in retaliation in the form of a price war. Figure 5 below shows the two companies in the long run after a price war due to retaliation (which reality and economic theory suggests will occur when at least one company cheats). Figure 5: AFollow CheatB Follow 20, 20 -∞, -∞Cheat - ∞, -∞ 0, 0 Since Company A is equal in size to Company B, a price war would be mutually destructive, leading to a gain of -∞. Since the worst possible payoffs are equal, the highest possible payoff will determine whether a company should stay in the cartel or cheat. Since $20 million is greater than $0, it is in the interest of both companies to remain in the cartel, contrary to the previous conclusion drawn from the short-term payoff 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 retaliation, it would be best to remain in the table below. cartel. This is an example of a repeated prisoner's dilemma, whichcan also be represented by a tree diagram as shown below. Figure 7: Example 3: Company B is considering whether to enter an industry controlled by a monopoly. Company A could either maintain current production levels (allowing Company B to enter) or increase production by investing in expensive machinery, a barrier to entry that would harm Company B if it decided to enter. The payoff table shows all possible additions/reductions to profit below: Figure 8: AIncrease (1-p) Same as (p)B Enter $80 million, -$50 million $40 million, $40 million Stay out of $100 million, $0 $50 million, 0p is the probability that company A will maintain its level of production (because it will not have access to machines). 1-p is the probability that firm A increases its production (because it has the capacity to invest in machinery). When probability is included, it is easier to represent possible outcomes in a tree diagram. Figure 9: In this scenario, if B were to enter the industry while A increases production by investing in machinery, A would lower prices to lessen competition – ultimately B would lose profits – A would receive $80 million while B would lose $50 million. If Company B were to enter the industry but A was unable to invest in machinery, the industry would become a duopoly and the profits would (theoretically) be split equally between the two – each would receive 40 million of dollars. However, if B stayed out and A increased production, A would be more productive and would not face competition, meaning he would make the most profit at this point: A would receive $100 million while B would receive $0 million. If Company A stays the same and Company B stays out, Company A will continue to make profits due to lack of competition, but not as much due to inefficiency x – A receives $50 million while that B receives $0. In this scenario, A would be better off increasing its machines – its lowest payoff is $80 million if it increases in size and $40 million if it stays the same. Additionally, the highest gain 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's size increases. Theoretically, Company B would be better off staying out of the industry, because the worst possible gain if it entered was -$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 of increasing size) and the probability (p) that firm A will not be able to increase its production. by investing in machines – if A fails to do so, the Nash equilibrium will move to (40, 40). The value of p could determine 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 payoff values, as shown in Figure 9. If B remains absent, the payoff B receives would be 0, because there is no production. If B enters the market, the general formula for the payoff he would receive would be: payoff=40p+-50(1-p)payoff=90p-50The payoff must be greater than 0; otherwise, B has no incentive to enter the market.90p-50>090p>50p>5/9Therefore, B will enter the market if the probability that A does notcannot invest in machinery and must maintain its production levels unchanged is higher. than 5/9. Firm B can also calculate its possible gains with a known value of p. Example 3 can be represented by a general formula where u > y > w > x > z > v. Figure 10: Increase (1-p) Same as (p) B Enter u, vw, wStay outside x, zy, zA general formula for calculating p can be found where the symbols used come 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 examples Hawk-Dove game instances. A Hawk-Dove game, also known as Chicken, occurs when two players compete for a good of known value (v) and there are two possible options "Hawk" or "Dove". “Hawk” is considered the strongest and riskiest strategy, while “Dove” is considered the safest strategy. Players choose simultaneously. It is originally 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. Company A is the same size as company B. The payoff table below shows the resulting profits for all possible combinations: Figure 11: AInvest, do not investB, invest 20, 20 0, 50Do not invest 50 , 0, 25, 25In the above example, the technology benefit for A and B is $50 million, the investment cost is $10 million, and company A is of equal size company B (the resulting gains are therefore the same). The most optimal outcome appears to be $25 million for both, when both are not investing in technology. However, the Nash equilibrium, and the most rational strategy, is for both companies to invest in the technology, because the lowest possible return if they choose to invest is $20 million, while the return the lowest possible if they choose not to invest is $0. Example 5: Company A choosing to invest in technology. Its competitor, Company B, is also considering doing so. Company A is twice the size of Company B. The payoff table below shows the profits resulting from all possible combinations: This can also apply to companies of different sizes. If company A is twice as large as company B, the benefit of investment in technology for company A is $100 million, the benefit of investment in technology for company B is $50 million and the investment cost is $10 million. The resulting gains are displayed in Figure 4 below. Again, even though the most optimal decision seems to be when both companies do not invest in the technology, both companies have an incentive to invest: for Company A, the lowest possible gain is $45 million. dollars if she chooses to invest, and $0 if she does so. I do not choose to invest, while for Company B the lowest possible gain is $20 million if it chooses to invest and $0 if it does not. Therefore, the Nash equilibrium is at (45, 2) when both firms invest. The general formula for Examples 4 and 5 is below: V is the value of the resource, while C is the cost incurred in fighting for the resource. If the resource is shared between two people, its value is halved, but when they end up fighting for it, they each incur a cost of C/2. When both chooseHawk, everyone has a winning probability of ½. The game is considered a sort of prisoner's dilemma when V>C. Although the most optimal decision seems to be when both companies choose the weaker and less aggressive option (Dove), game theory shows that both companies should choose the more aggressive option (Hawk), even whether this would incur costs because the gain is greater than that of the most aggressive option. 0. However, when the cost incurred is greater than the value received (VZero-sum gameAccording to Investopedia, a zero-sum game is a situation in which one player's gain is equal to another player's loss, therefore the net change in profit for both The number of players is zero This is a non-cooperative game An example in economics is the futures market, while examples in other fields are poker and chess. Example 6: Company A chooses to locate on a beach – either on the left, in the middle or on the right side, 60 customers are evenly distributed on the beach. each firm would have for each outcome Although having two firms selling the same product next to each other may seem counterintuitive and a waste of resources, Nash equilibrium is achieved when the. two firms choose to locate in the middle The highest possible gain for both firms when they choose to locate in the middle is $40 million, while the lowest possible gain is $30 million. . When they choose to settle left or right, the highest possible gain is $30 million and the lowest possible is $20 million. Even if (30,30) can also be observed in 4 other cases (LL, LR, RL, RR), these are not equilibrium positions because there is always an incentive to betray the other by choosing the middle for win $40 million. We can deduce that there are 9 possible combinations from the table above, and using combinations.SurveyI carried out a survey using scenarios 1, 4, 5 and 6 with a sample of 48 students 12th graders (24 of whom were from Company A and the other 24 were from Company B) who had never done game theory before to study how well the actual data agreed with 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 the results of a Prisoner's Dilemma simulation my economics professor ran 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: Prisoner's DilemmaYou and a friend are arrested for a crime and sentenced to 2 years in prison. You are also both suspected of a much more serious crime (that you didn't commit), but the police don't have enough evidence to prove it. You are in solitary confinement with no way of speaking to anyone else. They give you a good deal: you can confess to having committed the most serious crime, betray your friend, or deny it. The payoff table below shows the number of years you will receive for each possible outcome: AConfess DenyB Confess 5, 5 10, 1Reny 1, 10 2, 2What would you do: Confess Deny2: TechYou choose to invest in technology. Your competitor is also considering doing this. You are the same size as Company B. The payoff table below shows adding/reducing your current profits: Would you investin technology? Yes No3: TechYou choose to invest in technology. Your competitor is also considering doing this. You are twice the size of Company B. The payoff table below shows your current profit increase/decrease: Would you invest in technology? Yes No4: LocationYou choose to locate on a beach – either on the left side, in the middle or on the right. Guests are evenly distributed across the beach. The payoff table shows the number of customers each business would have for each outcome. Where would you fit in? Left Middle Right The results of the survey that I carried out for the 4 scenarios above are visible below: Before the analysis carried out previously (so if each subject guessed an option at random), the options of 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 can make when answering the survey. Possible combinations = (_12)C × (_12)C × (_12)C × (_13)C= 2 ×2 × 2 × 3= 24If each subject guessed at random for the 4 scenarios, each combination would have a probability of 1/ 24 or 0.042 and a frequency of 2. However, the data in Figure 16 does not reflect this either. Given the analysis done on the above situations, theory says that rational decisions (highlighted) should have probabilities of 1 and total frequencies of 48, while the other options would have probabilities and frequencies of 0 Therefore, the theory also says that the combination admit -invest-invest-middle would have a probability of 1 and a frequency of 48, unlike the other options. It is clear from the above data in both figures that this is not the case in the real world. Reasons for deviation from the expected result included trusting the opponent (even without communication), being misled by the highest possible payoff (applicable to scenarios 1, 4 and 5), etc. Company A's results in scenario 5 deviated the most from the expected result. (by 0.5), perhaps because people were misled by the change in size relative to the opposing firm.ConclusionOverall, in this exploration, I studied three forms of theory of games (prisoner's dilemma, dove, zero sum) by examining 6 scenarios: the fundamental prisoner's dilemma, cartels, entry into a monopolistic market, investment in technology based on company size and determining where to go (left, middle, or right) through modeling by constructing payoff tables and tree diagrams, then arriving at a general formula for each scenario. Some assumptions include company size, payoff values, number of options available, all else equal (other factors remain unchanged), and payoff values ​​constant (they will not change in the long run). By conducting an investigation to obtain real data and comparing it to theory, I discovered that game theory has many limitations. The biggest assumption underlying game theory is that humans are rational – players are rational and know that their opponents are as well. In reality, humans are very unpredictable and irrational, so real data almost always differs from theoretical results. The degree of uncertainty, a key element of all these scenarios, cannot be predicted in real life: some may be more confident than others and this would therefore have an impact on their decision-making. Moreover, in.