Battle Of The Sexes And The Prisoners Dilemma Philosophy Essay

date Of The Sexes And The Prisoners dilemma Philosophy try onIve had two experiences in the cases of Battle of Sexes and Prisoners Dilemma. My friend Chris and I once had a dispute on which ikon to fall out either nark Potter or encounter bilgewater. Both of us would bid to watch both of them, but Chris would like to watch Harry Potter temporary hookup I prefer Toy accounting. Eventually, I suggested to watch Harry Potter first and Toy Story later.The otherwise(a) case happened when I was a kid. I used to lie to my florists chrysanthemum when I was young. I always failed to hand in my homework on time. However, my teacher reported to my mama about the poor quality of my work. So my mum once inspected me and caught me for ceremonial car in like mannerns before finishing my homework. indeed, she subjects me to study school terms at school for a year so I could catch up with my school work. However, in this year, my ma was disappointed about my attitude and I could n o longer enjoy watching cartoons.Ive realized I could analyze both scenarios with Game Theory, specifically Battle of Sexes and Prisoners Dilemma. And both two grainys belong to Two-Person Non-Zero Sum Game, which describes a situation where a participants gain or loss is non balanced by the gains or losses of the other participant. Many common mixer dilemmas fall into this category, such as Centipede Game, Dictator Game (these will non be discussed in the essay) and etc.Utility TheoryTo take hold the claims of these games, the term utility has to be introduced. Utility refers to a measure of congener satisfaction. However, how much pain or pleasure a some(a)one feels and psychological effects sens scantily be measured. In order to create a measurable platform for mathematicsematicians to examine the best pre midpointptive solution, numbers ar assigned to notate utility for the concrete numerical reward or opportunity a person would gain. For instance, if I watch carto ons in order to escape from 50 difficult math questions, I will gain 50 util. Although this is relatively subjective, it is better to rate a more quarry measurement than having plain language description.Non-cooperativeIn Game Theory, we will always deal with games that allow impostors to join forces or not in advance. A cooperative game refers to a game in which frauds take away complete freedom of communication to make joint binding agreements. On the other hand, a non-cooperative game does not allow instrumentalists to communicate in advance. Rationally, participants would make decisions that benefit them the most. However, in some cases, like the Battle of Sexes and Prisoners Dilemma, the common interests would not be increased by their selfishness.Zero Sum GameZero-sum describes a situation in which a participants gain or loss is exactly balanced by the losses or gains of the other participant(s). If there are n participants and their outcomes are notated as O1, O2 O n. Mathematically speaking,If anticer 1 uses a act of outline A = (A1, , Am) and player 2 uses B = (B1, , Bn), the outcome Oij would have the probability xiyj, where both 1 i, j m,n . TheM1(x,y) = player 1, andM2(x,y) = player 2Basically they are the transmited value function for trenchant X which express the expected value of their utilities. XiYj is the probability to certain decision while Ai and Bj are the various(prenominal) decisions of player 1 and 2.The motivation of player is 1 to maximize M1 and of player 2 to maximize M2. In a competitive zero-sum game we have zeros of the utility functions so thatM2(x,y) = -M1(x,y)which led to the term zero-sum.Therefore, it is never advantageous to inform your opponent the strategy you plan to adopt since there is exclusively one put down winner and clear loser. So now we reckon the concept that players cannot help with each other. However, Battle of Sexes and Prisoners Dilemma could maximize the outcome through cooperatio n because they are non-zero sum game.M2(x,y) -M1(x,y).NotationSuppose we have two players Chris (C) and Me (M) in a game which one simultaneous move is allowed for each player the players do not chouse the decision made by each other. We will denote two manipulates of strategies as followsS1 C = C1, C2, C3 CmS2 B = M1, M2, M3 MnA certain outcome Oij is takeed from a strategy from each player, Ai and Bj. hyaloplasmSo if I pick strategy 1, Chris picks strategy 2 for himself, the outcome would become O21. Therefore, each fastens of strategy between Chris and me would have a distinctive outcome, in which there are mn possibilities. However, in this essay we do not deal with many a(prenominal) decisions, mostly 2 per person Harry Potter (HP) or Toy Story (TS), or Honest or insincere. So it would come down to a 22 ground substance, like the following diagram shown in Two-Person Non -Zero-Sum Game.Two person Non-Zero Sum GameNon-zero-sum games are opposite word to zero-sum gam es, and are more complicated than the zero-sum games because the sum could be negative or positive. And a two person non-zero sum game is only play by two players. In a non-zero-sum game, a typical form must give both results, since the loss is not incurred by the loser, but by some other party. To illustrate a few problems, we should consider the following payoff matrix.Payoffs shows as ( role player 1, Player 2)Player 1 scheme AStrategy BPlayer 2Strategy X(8,9)(6,5)Strategy Y(5,10)(1,0)Apparently, if we sum up the payoffs of player 1, we would have 8+6+5+1 = 20. art object Player 2 would have the payoffs of 9+5+10 = 19. This has clearly illustrated on of the properties of a non-zero sum game. Moreover, even if their payoffs are equal, one more requirement has to be met. The sum of all outcomes has to be 0. Since we only have positive integers here, we can conclude that the sum of all outcomes in this case is strictly 0. So this is a typical example of two-person non-zero sum g ame.Introduction to Pure and Mixed StrategiesSuppose a player has thoroughgoing(a) strategies S1, S2Sk in a normal form game. The probability diffusion function for all these strategies with their respective probabilitiesP =p1, p2 pk are nonnegative and = 1 because the sum of the probability of all strategies has to be 1. A pure strategy is achieved when only one is equal to 1 and all other pm are 0. Then P is a pure strategy and could be expressed as P = . However, a pure strategy is also used in a fuse strategy. The pure strategy is used in mixed strategy P if some is 0.So in a micro-scale, there are many strategies in the pure-strategy set S and in macro-scale, these strategy-sets contribute to a bigger profile P. We define the payoffs to P as followingwhere m,k 1But if the strategy set S is not pure, the strategy profile P is considered strictly mixed and if all the strategies are pure, the profile is completely mixed. And in the completely mixed profile, the set of pure s trategies in the strategy profile P is called the support of P. For instance, in a classroom has a pure strategy for teacher to teach and for student to learn. Then these strategies, teaching and learning, are the support of the mixed strategy.Payoffs are commonly expressed as So let i ( s1,,sn) be the payoff to player i for using the pure-strategy profile (P1,,Pn) and if S is a pure strategy set for player i. Then the total payoffs would be the product of the probability of each strategy in the strategy set S (ps ) and the payoffs of each strategy (. So if we sum up all the payoffsI (P) = , which is again convertible to the expected mean payoff function we set up in the zero-sum game section.However, a tombstone condition here is that players choices independent from each others, so the probability that the particular pure strategies can be simply notated as . Otherwise, probability of each strategy is expressed in terms of other ones.Nash residueThe Nash symmetry concept is s trategic because we can accurately predict how people will play a game by assuming what strategies they make out by implementing a Nash equilibrium. Also, in evolutionary processes, we can flummox different set of successful strategies which dominate over unsuccessful ones and stable stationary states are lots Nash equilibria.On the other hand, often do we see some Nash equilibria that seem implausible, for example, a chess player dominates the game over another. In fact they might be unstable equilibria, so we would not expect to see them in the real world in long run. Thus, the chess player understands that his strategy is too aggressive and careless, which leads to continuous losses. Eventually he will not adopt the same strategy and frankincense is put back to Nash equilibrium. When people appear to deviate from Nash equilibria, we can conclude that they do not understand the game, or putting to ourselves, we have misinterpreted the game they play or the payoffs we attribute to them. But in important cases, people simply do not play Nash equilibria which are better for all of us. I lie to my mum because of personal interests. The Nash equilibrium in the case between my mom and me would be both beingness honest.Suppose the game of n players, with strategy sets si and payoff functions I (P) = , for i = 1n, where P is the set of strategy profiles. Let S be the set of mixed strategies for player i.where m,k 1The thorough Theorem of a mixed-strategy equilibrium develops the principles for finding Nash equilibria. Let P = (P1Pn) be a mixed-strategy profile for an n-player game. For any player i, let P-i represent the mixed strategies used by all the players other than player i. The fundamental theorem of mixed-strategy Nash correspondence says that P is a Nash equilibrium if and only if, for any player i = 1 n with pure-strategy set Si and if s, s Si occur with positive probability in Pi, then the payoffs to s and s, when played against P-i are equal. Battle of SexesWe shall begin with my exampleAt the cinema (C Chris, M Me)M1M2C1(2,1)(-1,-1)C2(-1,-1)(1,2)*Choice 1 Harry Potter*Choice 2 Toy StoryThe game can be interpreted by a situation where Chris and I could not make the choice that satisfies both of them. Chris prefers Harry Potter while I prefer a video. Consequently, if we choose our preferred activities, they would end up at (C1, M2) where the outcomes would only be (-1,-1) because both of us would like to watch the movie together.Thus the Utility Function (U) Utility from the movie + Utility from being together.Considering a rather impossible situation where both of us do not choose our preferred options (C2, M1). This dilemma has put one of us sacrifice our entertainment and join the other, like (C1, M1) or (C2,M2). Thus the total outcome could be up to 3 util instead of -2 in the other two situations. Therefore, I made a decision to give up watching Toy Story and join Chris watching Harry Potter.Let be the probability of Chris watching Harry Potter and be the probability of me watching Toy Story. Because in a mixed-strategy equilibrium, the payoff to Harry Potter and Toy Story must be equal for Chris. Payoff for me is and Chris payoff is . Since , , which makes . On the other hand, has to be 1-2/3 = 1/3.Thus, the probability for (C1, M1) or (C2, M2) = and that for (C2, M1) and (C1, M2) =Because both go Harry Potter (2/3)(1/3) = 2/9 at the same time, and similarly for Toy Story, and otherwise they miss each other. Both players do better if they can cooperate (properties of non-zero sum game), because (2,1) and (1,2) are better than .We get the same answer if we find the Nash equilibrium by finding the intersection of the players best response functions. The payoffs are as followsTo find the payoffs of Chris relative to my probability, which is similar to probability distribution function (p.d.f.). Here are the casesSimilarly for player BThus. Chris would have a lower goal for a positive payoff si nce his payoff tends to decrease if 0 Prisoners DilemmaNow it is the situation of where I lied to my mom. Heres the action between me and my mom. I could choose to be honest or lie to my mom while my mom, on the other hand, could only trust me or suspect me of being dishonest. The payoff matrix is as follow (Me I, Mom M)I1I2M1(2,2)(0,3)M2(3,0)(-1, -1)*Choice 1 Honest/Trust*Choice 2 Dishonest/SuspectThis situation is a prisoners dilemma because it sets up a few key conditions. If both my mom and I choose to be honest, I would do the homework but I will not be subject to homework session for a year, and my mom will not be repeal about me. So it results in the best interchangeable benefits (2,2). If I lie to her and she trusts me, I am quick from watching cartoon (3,0). But if she suspects me and I am honest, I would feel like a prisoner being suspected. (0,3). And eventually, if I am dishonest and she suspects me, we would end up in a bad family (-1,-1). Interestingly, I would pr efer (I2, M1) because I have the sterling(prenominal) personal utility. But if I go for greatest mutual benefits, I would choose (I1, M1).Utility Function for Me (UI) C + H + S + RC = Utility from watching cartoonH = Utility from doing homeworkS = Utility from homework sessionR = Utility from relationship with momNow, to further discuss Prisoners dilemma for all cases, we had rather set up some variables.I1I2M1(1,1)(-y,1+x)M2(1+x,-y)(0, 0)Now let be the probability of I play I1 and be that of M playing M1 and x,y 0. And now we could set up the payoff functions easily with these notations.Which could be simplified intois maximized when = 0, and similarly for be maximized when = 0, regardless of what each other does. So in fact it is a mutually defect equilibrium because the best-response for each other is not the best response for both of us. Therefore, one of us should sacrifice for the others or both of us cooperate to work out the best solution.In real life, people should choose to cooperate with trust. Assume that there is a psychic gain 0 for I and 0 for M when both of us cooperate, in addition to the tempting payoff 1+x. If we rewrite the payoffs with these assumptions and equations, we getWhich can further be simplified intoThe first equation shows that if player I will then play I1 and if , then player M will play M1. Apparently, I would have done it because the total mutual payoffs of (I1, M1) both my mom and I are honest and trustworthy, would be higher than that of (I2, M1) where I lie to my mom who trust me. This would happen, for instance, I could get 10 candy bars and my mom can enjoy watching TV if both of us are honest. In fact, many corporates in the real world result in such way therefore, sometimes, cooperation with others could be beneficial to ourselves.Conclusion