Battle Of The Sexes And The Prisoners Dilemma Philosophy Essay

Battle Of The Sexes And The Prisoners dilemma Philosophy EssayIve had two experiences in the cases of Battle of Sexes and Prisoners Dilemma. My friend Chris and I once had a dispute on which movie to stay either Harry Potter or toy dog business relationship. two(prenominal) of us would like to watch somewhat(prenominal) of them, but Chris would like to watch Harry Potter while I prefer Toy Story. Eventu aloney, I suggested to watch Harry Potter first and Toy Story later.The separate case happened when I was a kid. I used to lie to my mum when I was young. I always failed to hand in my homework on time. However, my teacher reported to my mammary gland about the poor quality of my work. So my mum once inspected me and caught me for ceremony vignettes before finishing my homework. accordingly, she subjects me to study sessions at school for a year so I could catch up with my school work. However, in this year, my florists chrysanthemum was foiled about my attitude and I could no longer enjoy observation cartoons.Ive realized I could analyze both scenarios with plot of ground supposition, specifically Battle of Sexes and Prisoners Dilemma. And both two patchs belong to Two-Person Non-Zero junction Game, which describes a situation where a participants gain or loss is not balanced by the gains or losses of the oppositewise participant. Many common social dilemmas fall into this category, such(prenominal) as Centipede Game, Dictator Game (these leave behind not be discussed in the essay) and etc.Utility TheoryTo support the claims of these games, the term utility has to be introduced. Utility refers to a measure of relative satisfaction. However, how much pain or pleasure a person feels and psychological effects can hardly be measured. In order to create a measurable platform for mathematicians to examine the best probable solution, numbers atomic number 18 assigned to notate utility for the concrete numerical reward or probability a person would gain. For instance, if I watch cartoons in order to ladder from 50 difficult math questions, I will gain 50 util. Although this is relatively subjective, it is better to stage a more objective measurement than having concentrated language description.Non-cooperativeIn Game Theory, we will always deal with games that book players to assist or not in advance. A cooperative game refers to a game in which players read complete freedom of communication to make joint binding agreements. On the new(prenominal) hand, a non-cooperative game does not allow players to communicate in advance. Rationally, players 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 maximised 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 atomic number 18 n partic ipants and their outcomes argon notated as O1, O2 On. Mathematically speaking,If player 1 uses a set of system 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 expected value function for discrete X which submit the expected value of their utilities. XiYj is the probability to certain decision while Ai and Bj are the respective 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 protest your opponent the system you plan to adopt since there is only one clear winner and clear loser. So now we understand the sentiment that players cannot cooperate with each other. However, Battle of Sexes and Prisoners Dilemma could maximi ze the outcome through cooperation 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 know the decision made by each other. We will denote two sets of strategies as followsS1 C = C1, C2, C3 CmS2 B = M1, M2, M3 MnA certain outcome Oij is resulted from a strategy from each player, Ai and Bj.MatrixSo if I pick strategy 1, Chris picks strategy 2 for himself, the outcome would become O21. Therefore, each sets 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 decisions, mostly 2 per person Harry Potter (HP) or Toy Story (TS), or Honest or Dishonest. So it would come down to a 22 matrix, like the following diagram shown in Two-Person Non -Zero-Sum Game.Two person Non-Zero Sum GameNon-zero-sum games are opposite to zero-sum games, and are more co mplicated than the zero-sum games because the sum could be negative or positive. And a two person non-zero sum game is only played by two players. In a non-zero-sum game, a normal form must give both payoffs, 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 (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. While 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 stop that the sum of all outcomes in this case is strictly 0. So this is a typical example of two-person non-zero sum game.Introduction to Pure and Mixed StrategiesSuppose a player has everlasting(a) strategies S1, S2Sk in a normal form game. The probability distribution 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 unadulterated 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 mixed 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 strategies 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 similar to the expected hold still for payoff function we set up in the zero-sum game section.However, a key condition here is that players choices independent from each others, so the probability that the bad-tempered pure strategies can be apparently notated as . Otherwise, probability of each strategy is expressed in terms of other ones.Nash EquilibriumThe Nash equilibrium concept is important becaus e we can accurately predict how people will play a game by assuming what strategies they film by implementing a Nash equilibrium. Also, in evolutionary processes, we can model different set of successful strategies which dominate over unsuccessful ones and stable stationary states are often 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 runny equilibria, so we would not expect to see them in the real world in long run. thereof, the chess player understands that his strategy is too aggressive and careless, which leads to endless losses. Eventually he will not adopt the same strategy and thus 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 case s, people simply do not play Nash equilibria which are better for all of us. I lied to my mom because of personal interests. The Nash equilibrium in the case between my mom and me would be both being 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 fundamental Theorem of a mixed-strategy equilibrium develops the principles for ascertaining 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 Equilibrium 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 wit h my exampleAt the cinema (C Chris, M Me)M1M2C1(2,1)(-1,-1)C2(-1,-1)(1,2)* plectrum 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 movie. 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(predicate) are the casesSimilarly for player BThus. Chris would have a lower tendency for a positive payoff since his payoff t ends 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/ devote*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 upset about me. So it results in the best mutual benefits (2,2). If I lie to her and she trusts me, I am happy 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 relationship (-1,-1). Interestingly, I would prefer (I2, M1) because I have the greatest 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 th at 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