advertisement

TOPIC 4. GAME THEORY AND APPLICATION 1. IDENTIFYING NASH EQUILIBRIA Consider the following game, where two firms (Firm A and Firm B) must decide simultaneously whether to expand business in the West or in the South: Firm B Firm B Expand in the West Expand in the South Firm A Expand in the West Firm A Expand in the South 10, 60 50, 90 20, 80 40, 50 A) there is one pure strategy Nash equilibrium: for both firms to expand in the West. B) there is one pure strategy Nash equilibrium: for both firms to expand in the South. C) there are two pure strategy Nash equilibria: either firm can expand in the West, and the other expands in the South. D) there is only a mixed strategies equilibrium. Answer: C Diff: 3 2. PRISONER’S DILEMMA TYPE GAMES Consider the game below: Player R Strategy R1 Player R Strategy R2 Player C Strategy C1 Player C Strategy C2 600, 600 100, 1000 1000, 100 200, 200 True or False: This game is not an example of the Prisoners’ Dilemma game. Answer: False. It is an example of the Prisoners’ Dilemma game. A Prisoners’ Dilemma game is one where each player has a dominant strategy and the Nash equilibrium does not coincide with the outcome that maximises the collective payoffs of the players in the game. In this game the dominant strategies are R2 for Player R and C2 for Player C. The Nash equilibrium is (R2, C2) but both players would be better off by cooperating and selecting (R1, C1). 3. MAXIMIN STRATEGIES Consider the game below about funding and construction of a dam to protect a 1,000-person town. Contributions to the Dam Fund, once made, cannot be recovered, and all citizens must contribute £1,000 to the dam in order for it to be built. The dam, if built, is worth £70,000 to each citizen. If each player choses a maximin strategy, the outcome would be: One citizen Contribute to dam One citizen Don’t contribute to dam Other 999 citizens Contribute to dam 69000, 69000 -1000, 0 Other 999 citizens Don’t contribute to dam 0, -1000 0, 0 A) £69,000, £69,000. B) £0, -£1000. C) -£1000, £0. D) £0, £0. E) a mixed strategy equilibrium. Answer: D Diff: 2 By not contributing to the dam, the citizens minimise their potential losses – or in other words, they maximise the minimum gain that can be earned. 4. REPEATED GAMES/SEQUENTIAL GAMES Consider the game below where a firm (Entrant) is considering entering into the digital camera business, and can decide to do so on either a small scale or a large scale. The Incumbent firm must decide whether the Accommodate the new firm or start a Price War. The payoffs (profits in millions of dollars) to each firm are represented below: Entrant Small scale Entrant Large scale Incumbent Accommodate Incumbent Price War 4, 20 1, 16 8, 10 2, 12 If the game is played sequentially and the Entrant moves first, the equilibirum would be: a) (Small scale, Accommodate) b) (Small scale, Price War) c) (Large sclae, Accommodate) d) (Large scale, Price War) Answer: A 5. MIXED STRATEGIES Consider the Matching Pennies game: Player A - heads Player A - tails Player B - heads 1, -1 -1, 1 Player B - tails -1, 1 1, -1 Suppose Player B always uses a mixed strategy with probability of 3/4 for head and 1/4 for tails. Which of the following strategies for Player A provides the highest expected payoff? A) Mixed strategy with probability 1/4 on heads and 3/4 on tails B) Mixed strategy with probability 1/2 on heads and 1/2 on tails C) Mixed strategy with probability 3/4 on heads and 1/4 on tails D) Pure strategy in which Player A always selects heads Answer: D Diff: 3