The Story of Marx and Anna - Essay Example

Marriages and Game Theory The Game Theory is a knowledge-based technique used in everyday life as a means of devising strategies. Although the game is mostly used in the field of Economics to ensure that players are able to maximize decisions given a set of choices, the mathematical and logical applications of the techniques of game theory have been applied in other diverse fields and provide useful merits on which different people can make logical and rational decisions. In this respect, the techniques involve situations known as games, where the players are faced with problems on which strategies to choose in order to maximize their returns or escape trouble. It is interesting to note that game theories can also be applied to resolve relationship problems, or provide the rational moves for couples in relationships. In context, Marx and his wife Anna are faced with critical decisions in their life. Given that their marriage has been on the rocks for a period, each of them separately relies on the game theory as the route towards getting preferable choices in their marriage, which is one of the top techniques to resolve marital problems (Szuchman). In addition to being a couple, the two operate separate but similar businesses, on which they have had fights because of their stiff competition. Their profiled problems start on Valentine’s Day, regarding the gifts they want to buy for each other and even proceed to resolution of their business woes, whose best propositions are determined by the game theory. The couple requires solutions that will salvage their marriage and prosper their businesses using the best strategies. Scenario 1 Marx wants to buy a comb for Anna this Valentine’s Day so that she can tend to her long hair. On the other hand, Anna would love to buy a wrist strap for her husband’s pocket watch, who has difficulties in keeping time at his place of work. Despite their perfect love for each other, neither has enough money to purchase the gift they would want to give. We may want to assume quite a different scenario, where the two lovers are in an uncertain short-term relationship and are self-centered. In this perspective, they do not care about each other, but are more concerned on the gains or losses they hope to make in the term of their relationship. Using the tools of strategic thinking and action, the lover may choose to buy a gift for the other person on this day, but is skeptical about whether the other will give a gift. If the other person does not give a gift, then this person is better off not buying one. If conversely the person does not give a gift, but receives a gift from the other, he would still be better off. In such a situation, no one buys a gift because the two players in the relationship think in the same way and this equilibrium may not be the best according to relationship counselors. A loving relationship is indeed self less, with mutual caring attributes in the two parties in the relationship. The story of Marx and Anna is an assumption of the best of lovers, who mind about each other. Indeed, the most precious thing about Anna according to Marx is her hair, while Anna thinks that what Marx needs most is the wrist strap of the pocket watch. The payoffs for such a scenario may be analyzed as follows: If both Marx and Anna do not buy the gifts (N), then each person retains their items as they were and both get a payoff of zero. On the other hand, if one does not buy (N) and the other buys (B). The person who buys the gift and received nothing has a payoff equal to -2, for his purchase loss and no return. The other selfish person has a payoff of 1. Solution by the Nash equilibrium Indeed, if both buy gifts for each other, they demonstrate true love for each other by the wife auctioning her hair to buy Marx the wrist strap, while Marx sells his watch to buy Anna the comb. The payoffs are illustrated in the matrix below: B1 N1 B2 -1,-1 -2,1 N2 1,-2 0,0 Where B2 is a decision to buy a gift by Anna, Marx will choose between buying a gift with a payoff of -1 and not buying the gift at all which gives a better payoff of 1. On the other hand, if Marx decides to buy the gift given by B1 on the table, the Anna will choose between buying the gift, which gives -1 and not buying the gift, given by the preferable outcome of 1. If Marx and Anna decided to play the payoff matrix game above, it is evident that both lovers will not buy anything for each other on Valentine’s Day, because solution by the Nash equilibrium illustrates that both will prefer not to give any gift in this situation. The withdrawal of gifts is the dominant strategy, which gives each player a preferable outcome. The prisoner’s dilemma approach It would be imperative to note that the payoffs are illustrated strategically, which postulates a scenario of selfish lovers. With the initial assumption that the two lovers are in perfect love, it would be plausible to state that their utilities are functions of the other’s well being and are more concerned about the other, rather than their own self. A mathematical representation of the utilities of Marx and Anna may be given as (Marx, Anna) = (c, d). With the assumption of perfect love, the utility function above is different, with each utility element interchanged for each other to explain the above situation. Therefore, perfect love (c, d) = (d, c). On this perspective, Marx’s utility is dependent on how well Anna feels, while Anna’s payoff is hoe good Marx feels. We rewrite the matrix payoffs to include this situation of perfect love, and a transformation of the matrix to represent such a situation. The perfect love payoffs are illustrated as below: B1 N1 B2 -1,-1 1,-2 N2 -2,1 0,0 A closer look at the payoffs above reveals a rather interesting situation, where both players are obliged to take one strategy, which is deemed dominant from the available options. It would be plausible to state that this is the prisoner’s dilemma situation, in which the (0, 0) is the most preferable outcome after solution of the problem under perfect love. In this kind of scenario, Anna auctions her hair to by the wrist strap, and Marx sells his watch to buy his wife a comb. The game played out implies that they do what is best for sacrifice, but end up at the worst point ever because an analysis of the outcome may imply that the gifts are just signs of love but practically useless. Scenario 2 After Valentine’s Day woes, the two lovers embark on their daily businesses of selling cakes. Despite being husband and wife, both are competitors in the business and sell exactly the same product in three markets X, Y and Z. Jim has a monopoly while selling to market X, Anna has a monopoly when selling to market Z, while both Jim and Anna compete to sell their products in market Y. The nature of this competition is illustrated as below: While both Jim and Anna are suppliers of the products, they can charge prices or supply the products depending on the market in which they are focusing on. Because Jim has a monopoly in market X, he will charge the price that will maximize the profits in his market, as well as structure his supply in the way that seeks to maximize his outcome. On the other hand, Anna will use her monopoly powers to influence price in market Z, by charging a price that will maximize the size of her returns in the market. The competition between the two competitors in the business becomes exciting in market B, where both Marx and Anna operate in perfect competition. None of them has control over the price, leading to a Bertrand competition if their customers are only concerned with the prices of the snacks. On this perspective, the two will want to twist the prices of the products, in a way that gives them a comparative advantage in both their dominant and competitive markets. If the people buying the products from Jim and Anna realize that the snacks sold in market Y are much cheaper than those in X and Z, they will shift their focus and only buy products from the cheaper market. This will have the result of driving prices down in the two monopoly markets. To this end, the two individuals are faced with the threat of arbitrage, which forces them to reconsider their decisions and strategies. In this perspective, we assume that the forces in the market have made both Marx and Anna only have a single price that is charged for their two markets, since it is evident that they can no longer operate their monopolized markets as before. The Nash equilibrium for the Scenario We will assume that the market sizes equal to 1, and are of the same magnitude. In addition, we assume that a good can be taken from market Y to Z or X, but it is illegal for the same practice between X and Z. For purposes of convenience, the prices in the market are assumed to range between 0 and 1, with a marginal cost of zero for every additional product. If market B remains as the Bertrand competition situations, the rational customers will always go for the lowest prices and the market may experience splits. The solution to this game is interesting to come up with, and we assume that Marx will charge a price of 1 in both market X and Y. IF Anna decides to charge the same price in both of her markets, stiff competition is bound to occur and settle the scores. If Marx is able to capture all the sales in market 1, he earns a profit of 1 in market X, receiving just a half portion of the proceeds of market Y that is a mere 0.5. Indeed, both Marx and Anna will earn equal profits of 0.5, which are identical. Whether such a situation is equilibrium is a matter of careful analysis of the situation, which reveals that there is no equilibrium. Anna realizes that she could have got more sales and profits if she reduced her price. If she decides to charge a lower price of 0.99, she will earn the same profits in her monopoly market, by capturing all the sales to a value of 1. In addition, selling at a lower price in both markets implies that the customers in the competitive market Y will make them prefer to buy goods from her, making her capture the whole market from Marx. Indeed, the lower price produces a better comparative payoff of 1.98 as compared to the previous rating of 1.5 derived from both markets. Marx will not stand the defeat from his wife on the strategies and will therefore lower his prices to try beating her in her own game. Instead of charging a price of 1, he will lower his price to 0.98, way below that which Anna has to offer for the market. This will raise his profits to a level of 1.96 in both markets and this trend will continue; as long as more profits are anticipated with a reduction in price. This game experiences a twist when Anna reaches a price of 0.5 because of continuous lowering of prices. It is tricky on what strategy is best for Marx, as the situation is trivial to the extent that he could easily make losses with unwise decisions. He has a choice of lowering his price to 0.49, which will yield an overall profit of 0.98, which loses ground on an aspect of rationality. It is not rational to make such a choice, when he can sell in his monopolized market and get a profit of 1. As a rational businessperson, Marx will charge a higher price of 1, because using this price he could still earn enough profits in the market where he is a monopolist. When Jim is charging a price of 1 and earning better profits in only one market, Anna realizes that she is a fool to charge a lower price of 0.5 in two markets and get the same magnitude of profits. She will therefore respond to such competition by raising her prices to 0.99, a situation that implies net profits of 1.98. Using the Nash equilibrium, therefore, we derive two approaches that serve as the best strategies for the two competitors. 1. Condition 1 implies that if Anna charges, any prices between 0.5 and 1, then Marx should place his prices just slightly below her price. 2. Condition 2 implies that if Anna decides to charge prices less than or equal to 0.5, Jim should retaliate back to his monopoly market and charge a price of 1, which in that situation will give him more profits and success. It is imperative to note that the game at this point becomes conditional, with the Nash equilibrium giving no clear-cut strategy that can resolve the situation. Marx and his wife are held in constant battles in situations where the two conditions are violated and not met by the parties. It is apparent that with the absence of dominant strategies, the above conditions would help resolve the problems in the business, where a closer look at the conditions implies that the competition may be very stiff in the short run, with both parties making losses or profits as the conditions dictate. Marx and Anna may decide to find other ways to discriminate their products in the competitive market in an aim to beat the competition and take a better share of the profits. Scenario 3 Jim and Anna carried out the business for a period and utilized the strategies that were given by the Nash equilibrium. However, the sharp losses incurred when one of them reduced the prices led to disagreements and constant feuds at home. In this respect, Anna got tired of the relationship and applied for a divorce. Before the finalization of the processes leading to a formalized divorce, they separate and live in separate houses. However, Jim elopes with Anna’s beautiful necklace that she inherited from her grandmother, while Jim forgets a precious ring that he had bought with his savings at college. The couple agrees to meet in a secret place and exchange the two things in well-wrapped boxes. For fear that, Jim may cheat her, she substitutes his ring with a fake bracelet and puts in the box. In addition, Jim suspects that Anna may not be honest with her promise and substitutes her necklace with an old piece of wire. At the place of meeting, they exchange the boxes and each speeds off away, each thinking that he had cheated the other. This is the prisoner’s dilemma, which can be solved by the backward induction. The payoffs for the two individuals can therefore be analyzed as below in the payoff matrix Marx Anna Ring Fake bracelet necklace 1,1 -1,2 Old wire 2,-1 0,0 In the payoff matrix above, a value of 1 is given where the exchange is successful and they both get whatever they wanted. On the other hand, 2 is the payoff for successfully cheating the other and taking your valuable belonging and retaining the other’s valuable product. Zero is the result when both cheat and retain the valuable products of the other, there is no ultimate winner or loser in the situation. On the other hand, -1 is the grading when one of the parties has been cheated and got a fake product after giving out the right product. The backward induction method In the figure above, it is apparent that a payoff of 1 would be the most suitable for the two parties as a rational gainful venture that would be mutually benefiting. However, analysis using the backward induction reveals that there are two dominant strategies, which make the parties want to trick each other and be left at the best possible position of equilibrium. It is important to note, from the dominant strategy, that each of them would lose on an equal measure if they suffer the trick posed by the other player. Indeed, using the above decision metrics as a guide towards establishing the best strategy, the two parties find it too risky to be cheated by the other and decide not to bring the real items and they trick each other into bringing the wrong items at the place of meeting. Solution using the Nash equilibrium Marx Anna Ring Fake bracelet Necklace 1,1 -1,2 Old wire 2,-1 0,0 In the Nash equilibrium analysis, it is evident that if Marx brought her the necklace without getting his ring back, he suffers a loss of two. The same case applies for Anna, if she brings the rings without getting her necklace back, she loses two as her net payoff derived using the Nash equilibrium. The Nash equilibrium is therefore a combination of two strategies, either that they all give the right products to each other, or they cheat or give fake items to each other. To the extent that the couples cannot trust each other because of the tough conditions that their marriage finds itself, they cannot deliver the right products for fear of mistrust and they both lose. Their problem is best solved when each of them remains with a property of the other, until they decide to resolve their marriage conflict, accept each other and find the heart to trust and give back the items or resolve their marriage. Without trust, every bond in any relationship is broken and only leads to losses. In conclusion, it is clear that the game strategy techniques are useful given their diverse approach to issues. In addition to determining the best strategies for people, they are also useful in resolving problems and are critical in decision-making processes. Hence, they can be used to give solutions as well as advice to situations such as the one given above. Works Cited Szuchman, Paula. "Marriage and the Art of Game Theory." The Daily Beast [New York] 13 June 2012: Print. Read More