Question

In some game theoretic scenarios, there is no Nash equilibrium while players play ‘pure strategies’. This results in players constantly switching their strategies to gain an advantage over their opponents. However, a balance can be achieved by players stochastically mixing the pure strategies, resulting in mixed strategy equilibriums. A new synthetic material has been introduced to the shoe-making industry in Mongolia recently, which is suitable to the local weather. MoShoo and Newbie are two shoe-making companies which are planning to launch women’s shoes using the new material. MoShoo has always been a trend setter in women’s fashion in Mongolia, whereas Newbie is a relatively new firm. Both companies are trying to decide whether the new shoe will be high-heel or flat. Both companies know that the users will perceive whatever MoShoo comes up with as trendy. Therefore, MoShoo tries to make sure that their design is different from Newbie, to maximize their market share, while Newbie tries to make sure their design matches MoShoo’s, to ensure some buy-in. If both produce high-heel shoes, then Newbie’s expected income from the project will increase by 5M, while MoShoo’s will be reduced by the same amount. If MoShoo goes with flat shoes while Newbie produces high-heels, newbie’s expected income will reduce by 4M while MoShoo’s will increase by the same amount. If both produce flat shoes, then both companies will get their expected income without any increase or decrease. If MoSoo produces high heels while Newbie produces flat shoes, Newbie will lose 2M from its 3 expected income, while Moshoo will gain only one Million from its expected income. What their expected incomes are for the project is immaterial, as we can just consider the increase or decrease.

i) Given that each company has the option of producing either High-heel (H) or Flat (F) shoes, represent the scenario in a pay-off matrix. (6 marks)

ii) Is the game a zero-sum game? Justify your answer.(2 marks)

iii) Is there a dominant strategy for either party? Explain your answer (2 marks) iv) Is there a pure strategy Nash equilibrium? Explain your answer. (3 marks)

v) Assume that each company can change its’ pure strategy at the 1st of each month, and continue to do so for a year. Compute the ‘mixed strategy’ Nash equilibrium for the scenario, which indicates how often they should play each strategy in terms of probability. Given that there are twelve decision points (at the beginning of each month), state clearly, to maintain equilibrium, how many times MoShoo must produce High heel shoes, and how many times Newbie must do so. (12 marks)

Answer 1

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