There are three players who must each choose an “effort” level from 1 to 7, that is, Si = {1, 2, 3, ..., 7}. The payoff for each player i is ui(si, s−i) = 10 max{s1, s2, s3} − si. How many pure- strategy Nash equilibria are there? Select one: a. 2 b. 4 c. none of the other answers d. 3 e. 1
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There are three players who must each choose an “effort” level from 1 to 7, that is, Si = {1, 2, 3, ..., 7}. The payoff for each player i is ui(si, s−i) = 10 max{s1, s2, s3} − si. How many pure- strategy Nash equilibria are there?
Select one:
a.
2
b.
4
c.
none of the other answers
d.
3
e.
1
Step by step
Solved in 2 steps
- 1.a) If the three executives of a fraudulent organization report nothing to the authorities, each gets a payoff of 100. If at least one of them blows the whistle, then those who reported the fraud get 28, while those who didn’t get -100. Suppose they play a symmetric mixed-strategy Nash equilibrium where each is silent (does not report fraud) with probability p. What is p?A, 0.1B, 0.28C, 0.5D, 0.8 b) In a two-player game, with strategies and (some known and some unknown) payoffs as shown below, suppose a mixed-strategy equilibrium exists where 1 plays C with probability 3/4, and Player 2 randomizes over X, Y, and Z with equal probabilities. What are the pure-strategy equilibria of this game? A, (A, Y) and (B, X)B, (A, Z) and (C, Y)C, (B, X) and (C, X)D, (C, X) and (C, Y)Describe the game and find all Nash equilibria in the following situation: Each of two players chooses a non-negative number. In the choice (a1, a2), the payoff of the first player is equal to a1(a2 - a1), and the payoff of the second player is equal to a2(1 – a1 – a2).4. You have probably had the experience of trying to avoid encountering someone, whom we will call Rocky. In this instance, Rocky is trying to find you. It is Saturday night and you are choosing which of two possible parties to attend. You like Party 1 better and, if Rocky goes to the other party, you get a payoff 20 at Party 1. If Rocky attends Party 1, however, you are going to be uncomfortable and get a payoff of 5. Similarly, Party 2 gives you a payoff of 15, unless Rocky attends, in which case the payoff is 0. Rocky likes Party 2 better, but he is likes you. He values Party 2 at 10, Party 1 at 5, and your presence at either party that he attends is worth an additional payoff of 10. You and Rocky both know each others strategy space (which party to attend) and payoffs functions.
- Consider a game in which two players simultaneously pick an integer from 1 through 10. Let s1 denote choice of player 1, and let S2 denote the choice of player 2. If s1 7 82 then player 1's payoff is: Si + 82 2 and player 2's payoff is 10 – ,*. Si + 82 2 If s1 = 82 then each player gets 5. How many pure strategy Nash equilibria does the game have (you will get full points if you are within 2 of the correct answer)? Numerical answera) Find the Nash equilibria in the game (in pure and mixed strategies) and the associated payoffs for the players. b) Now assume that the game is extended in the following way: in the beginning Player 1 can decide whether to opt out (this choice is denoted by O) or whether to play the simultaneous-move game in a) (this choice is denoted by G). If Player 1 opts out (plays O) then both Player 1 and Player 2 get a payoff of 4 each and the game ends. If Player 1 decides to play G, then the simultaneous-move game is played. Find the pure-strategy Nash equilibria in this extended version of the game. (Hint: note that Player 1 now has 4 strategies and write the game up in a 4x2 matrix.) c) Write the game in (b) up in extensive form (a game tree). Identify the subgames of this game.Consider two players in the game-theoretic setting below: Player 2 Choice A Choice B Choice 20 |-30 A -20 30 Player 1 Choice |-30 40 B 30 -40 What is the (pure-strategy) Nash equilibrium? There are more than one (pure-strategy) Nash equilibria. O (P1, P2) = (A, B) is the only (pure-strategy) Nash equilibrium. O (P1, P2) = (A, A) is the only (pure-strategy) Nash equilibrium. O There is no (pure-strategy) Nash equilibrium. O (P1, P2) = (B, A) is the only (pure-strategy) Nash equilibrium. (P1, P2) = (B, B) is the only (pure-strategy) Nash equilibrium.
- See the extensive form game in the image attached (the payoffs of player 1 are written on top and the payoffs of player 2 are on the bottom). a) Write this game in normal form (a player's strategy is a complete contingent plan that tells them what to play at each of their information sets) (b) Find all the Nash equilibria of the normal form game from part (a)3. Two individuals are planning to start a business. The earnings from a grocery store business would be $30,000 for each, but the business requires two people (one to watch over the stock and the other to act as a cashier). Each individual could instead start an mpesa business. The take from mpesa business is only $3000 but can be done with one person acting alone. i) Write down the payoff matrix for this game ( ii) Solve the Nash equilibriumA and B are competitors in the mobile phone industry. Both A and B have to decide whether to participate or not to participate in a Phone for the Future Trade Fair next month. The matrix payoff below shows the profits (USD million) corresponding to their actions. a) What is the Nash equilibrium of the above game? b) Is the Nash equilibrium Pareto Optima? Explain. c) Suppose B is pessimistic of A's rationality, what is B's strategy? Compare and comment on B's strategy in (a) and (c). A Participate Do not participate B Participate Do not participate 400,1000 200,200 500,500 1000,400
- Two individuals are bargaining over the distribution of $100 in which payoffs must be in increments of $5. Each player must submit a one-time bid. If the sum of the bids is less than or equal to $100, each player gets the amount of the bid and the game ends. If the sum of the bids is greater than $100, the game ends and the players get nothing. Does this game have a Nash equilibrium? What is the most likely equilibrium strategy profile for this game?Three politicians are voting on whether to allow themselves a salary increase of$3,500per a year. If they vote in favor of a raise, then they lose some public support, costing them each$1,500 per year. The salary increase passes if two or more politicians vote in favor of it. What is this game’s Nash equilibrium (or equilibria)? Explain. There is no need to draw a payoff matrix.Problem 2. Consider the partnership-game we discussed in Lecture 3 (pages 81-87 of the textbook). Now change the setup of the game so that player 1 chooses x = [0, 4], and after observing the choice of x, player 2 chooses y ≤ [0, 4]. The payoffs are the same as before. (a) Find all SPNE (subgame perfect Nash equilibria) in pure strategies. (b) Can you find a Nash equilibrium, with player 1 choosing x = 1, that is not subgame perfect? Explain.
