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- Consider the following price game: Firm 1 Firm 2 High Low High 20, 20 12, 24 Low 24, 12 14, 14 Remark: In simultaneous move games (games with rows and columns) theconvention is to write the row player’s payoff first and the column player’spayoff second. (a) What is the Nash equilibrium of this game? Recall that for each playeryou should find the best response to each of the opponents’ strategies andunderline the associated payoff. Then look for a cell where both strategiesare best responses to each other. This is a Nash equilibrium. (b) Does either firm have a dominate strategy (a strategy that is always abest response)?NASH EQUILIBRIUM Consider the following game in strategic form. bị b2 a1 10,2 4,20 a2 6,28 9,14 аз 8,6 12,4 1-T-0 1-T The Equilibrium Existence Theorem guarantees that this game possesses at least one Nash equilibrium of the form (T*, o*,T*). Which values of a* are part of such an equilibrium? (Mark all values for which an associated equilibrium exists.) Select one or more: O 0.00 O 0.60 O 0.30 O 1.00 O 0.75 O 0.10 O 0.80 O 0.2520.38 C 1 Search il 4G O Microeconom... 1. In the sequential games such as the sequential Battle of the Sexes, why does the Nash equilibrium allow for outcomes with noncredible threats? 2. Left Right Up 1, 3 3, 1 Down 2, 2 4, 4 a. Find the Nash equilibrium or equilibria b. Which player, if any, has a dominant strategy? c. Does A benefit from changing the game by reducing his or her payoff in this way? 3. Noah and Xaviera love to play "Hide and Seek." It is a simple game, but it continues to amuse. It goes like this. Xaviera hides upstairs or downstairs. Noah can look upstairs or downstairs but not in both places. If he finds Xaviera, Noah gets one scoop of ice cream and Xaviera gets none. If he does not find Xaviera, Xaviera gets one scoop of ice cream and Noah gets none. a. Fill in the payoffs in the matrix below! Is this a zero-sum game? What are the Nash equilibria in pure strategies? Hide and Seek Xaviera Upstairs Downstairs Upstairs Noah Downstairs b. Find a Nash equilibrium in…
- 2. For the game illustrated in the figure below, find all pure-strategy Nash equilibria and mixed- strategy Nash equilibria (Hint: apply IDSDS first) Player 2 W y 0,20 А 5,30 10,15 Player 1 В 0,5 15,5 5,10 0,5 0,30 -10,5The payoff matrix below describes a one-shot game in which two farmers ("Player A" and "Player B") choose between planting corn or planting wheat. The payoffs in this matrix represent thousands of dollars of income, so more is better. Pla yer A Plant Corn Plant Wheat What kind of equilibrium is this? Plant Corn Dominant Strategy Equilibrium Nash equilibrium 3, 3 Player B 5,0 Both a Nash and a Dominant Strategy Equilibrium O No equilibrium Plant Wheat 0,5 1, 11. From the New York Times: In a "noncooperative game [...] [players] cannot convey intentions to each other." Do you agree? Discuss briefly. 2. In any competitive game there exists a unique strategy profile that is a Nash equi- librium. True or false? 3. Consider a normal-form game. Let al such that Va-Ja : w (a), a-) < w (à, a-). Prove or disprove that a is never a best-response. 4. A subgame perfect equilibrium of the Stackelberg game is also a Nash equilibrium of the game in which players choose their quantities simultansously. True or false? 5. Consider an infinitely repeated normal-form game. Formulate a condition under which all players playing grim trigger strategies constitutes a subgame perfect equi- librium. 6. Provide an example where grim trigger strategies do not constitute a subgame per- fect equilibrium.
- In the 3 x 3 matrix below, what is the Nash equilibrium? Р2 N S E U 6,4 7,3 5, 5 P1 H7,3 3,7 4,6 L 8,2 | 6,4 | 3, 7 Select one: a. U,E b. 8,2 О с. 5,5 O d. L,N1) What are the Nash equilibria? Which one is unreasonable/non-credible threat? 2) What are the subgame perfect Nash equilibria? Does SPNE concept eliminate the unreasonable Nash equilibrium?age Which of the following statements is the best description of a Nash equilibrium? Select one: O a. An equilibrium outcome achieved by cooperation between players in the game b. An outcome that is achieved when players in the game have jointly maximized profits and divided those profits according to market share of each player O c. An outcome where each player's strategy depends on the behaviour of its opponents d. An outcome where each player's best strategy is to maintain its present behaviour given the present behaviour of the other players Next page
- NASH EQUILIBRIUM Consider the following game in strategic form. bị b2 a1 16,12 4,20 a2 6,28 9,14 a3 8,6 12,4 1-T-0 1-T The Equilibrium Existence Theorem guarantees that this game possesses at least one Nash equilibrium of the form (7* , o*, 7*). Which values of T are part of such an equilibrium? (Mark all values for which an associated equilibrium exists.) Select one or more: 0.00 0.60 0.30 0.80 0.25 0.75 0.20 O 1.00fnan421 WWord Gozden Geç r Gorunum Varc m Ne yaomak steci gnz soy evn 1) Two firms, X and Y, are planning to market their new products. Each firm can develop either TV or Laptop. Market research indicates that the resulting profits to each firm for the alternative strategies are given by the following payoff matrix: FIRM Y TV LAPTOP FIRM X TV 30, 30 50, 35 LAPTOP 40,70 20, 20 A) If both firms make their decisions at the same time and follow maximin (low-risk) strategies, what will the outcome be? B) Suppose both firms try to maximize profits, but Firm X has a head start in planning, and can commit first. Now what will the outcome be? What will the outcome be if Firm Y has a head start in planning and can commit first? IWhat is the Nash Equilibrium in a game?A. A situation where all players cooperate for maximum gainB. A situation where no player can improve their outcome by changing their strategy unilaterallyC. A situation where players always choose the same strategyD. A situation where players randomly select strategies
