30. In a simultaneous-move game of incomplete information, a player's payoff function is determined by their what? a. Probability b. Type c. Action d. Strategy
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- 1. Each of two players simultaneously announces either Rock, Paper, or Scissors. Paper beats Rock, Rock beats Scissors, and Scissors beats Paper. The player who names the winning object receives $1 from her opponent. If both players make the same choice, no payment is made. Each players's preferences are represented by by the expected amount of money she receives. a) Formulate the situation as a strategic game and find all its mixed strategy equilibria. Make sure that you consider all different supports. b) Find all the mixed strategy equilibria of the modified game in which player 2 is prohibited from announcing Rock.Game theory Consider a simultaneous move game with two players. Player 1 has three possible actions (A, B, or C) and Player 2 has two possible actions (D or E.) In the payoff matrix below, each cell contains the payoff for Player 1 followed by the payoff for Player 2. Player 2 7. Player 1 A B C D -3, -3 0, -11 -4, 3 -11, E 0 -7, -7 -12, 0 (a) Identify any dominated strategies in this game. If there are none, state this clearly. (b) Identify any pure strategy Nash Equilibria in this game. If there are none, state this clearlyGame theory Consider a simultaneous move game with two players. Player 1 has three possible actions (A, B, or C) and Player 2 has two possible actions (D or E.) In the payoff matrix below, each cell contains the payoff for Player 1 followed by the payoff for Player 2. Player 2 7. Player 1 ہے A B C D -3, -3 0, -11 -4, 3 -11, E 0 -7, -7 -12, 0 (a) Identify any dominated strategies in this game. If there are none, state this clearly. (b) Identify any pure strategy Nash Equilibria in this game. If there are none, state this clearly.
- A st SW B st 2,1 5, 15 2. Consider the game of chicken above. SW 11,4 3,5 a. Find all Nash equilibrium in pure strategies. b. Say that player A uses a mixed strategy, where she picks st with prob. 0.55 and sw with prob. 0.45. What is B's expected payoffs from choosing pure st and pure sw? c. Player B uses a mixed strategy and wants A to choose sw. What is the minimum prob. p, for which B needs to play st so A will choose sw.i. ii. QUESTION ONE A. A Nash equilibrium is a strategy profile such that every player's strategy is the best response to all the other players. It requires that each player makes a best response and that expectations regarding the play of other players are correct. Below is the table showing strategies and payoff for Player 1 and Player 2. PLAYER 1 R1 R2 R3 R4 C1 0,7 5,2 7,0 6,6 C2 2,5 3,3 2,5 2,2 PLAYER 2 C3 7,0 5,2 0,7 4,4 CA 6,6 2,2 4,4 10,4 REQUIRED; Transform the normal form game above into an imperfect extensive game form Find the Nash equilibrium for the game above using iterative deletion of strictly dominated strategies. Find the Nash equilibrium using brute force or cell by cell inspection.on 8.1 Consider the following game: Player 1 A C D 7,6 5,8 0,0 Player 2 E 5,8 7,6 1, 1 F 0,0 1,1 4,4 a. Find the pure-strategy Nash equilibria (if any). b. Find the mixed-strategy Nash equilibrium in which each player randomizes over just the first two actions. c. Compute players' expected payoffs in the equilibria found in parts (a) and (b). d. Draw the extensive form for this game.
- 1. Use the following game tree to answer parts a & b. At a chance node (Nature's turn) or when a player is indifferent between strategies, each branch has equal likelihood (50% chance they take either path). Nature receives no payoff. Assume players are mistake proof. Use a complete strategy when discussing the results of this game and provide a payoff, even if that payoff is an expected one. Recall that an expected payoff is the "average" payoff: for example if with 50% chance we receive a payoff of 5 and with 50% chance we receive a payoff of 2, the expected payoff is .5 x 5+ .5 x 2 = 3.5 a. What is the outcome of the game? Assume players are risk neutral (i.e., they see no difference in getting a payoff of 5 vs an expected payoff of 5). Red 0, 1, 3, 3 3, 2, 3, 2 Dorothy Blue Top 1, 4, 3, 2 Scarecrow 50% Green Nature Right Bottom 0, 2, 3, 1 Lion 50% 1, 3, 4, 2 3, 4, 3, 2 50% Left Nature 3, 2, 1, 2 Up 50% Red. 1, 3, 4, 2 Down Tinman Blue 0, 3, 3, 3 Yellow Dorothy 4, 3, 4, 2…Subgames • • • At each node where a player makes a decision, we can think about the "subgame" starting at that node A subgame is the portion of the tree following a particular node How many subgames does our first sequential move example have? • How many subgames does our entry game have? • How many subgames does the game on the right have? • What are its Nash equilibria? Player 1 (1,4) T Player 2 U B (5,2) (3,3) р Player 1 D L 9 (2, 0) Player 2 R (6, 2)Use the following payoff matrix for a simultaneous-move one-shot game to answer the accompanying questions. Player 2 Strategy C D E F Player 1 A 9, 8 14, 14 18, 25 12, 19 B 23, 13 10, 18 14, 26 19, 21 a. What is player 1’s optimal strategy? multiple choice Strategy B Strategy A Player 1 does not have an optimal strategy. b. Determine player 1’s equilibrium payoff. Prev
- 2. Consider the following Bayesian game with two players. Both players move simultaneously and player 1 can choose either H or L, while player 2's options are G, M, and D. With probability 1/2 the payoffs are given by "Game 1" : GMD H 1,2 1,0 1,3 L 2,4 0,0 0,5 and with probability 1/2 the payoffs are according to "Game 2" : G |M|D H 1,2 1,3 1,0 L 2,4 0,5 0,0 (a) Find the Nash Equilibria when neither player knows which game is actually played. (b) Assume now that player 2 knows which one among the two games is actually being played. Check that the game has a unique Bayesian Nash Equilibrium.8.1 Consider the following game: Player 1 A B C D 7,6 5,8 0,0 Player 2 E 5,8 7,6 1.1 E 0,0 1,1 a. Find the pure-strategy Nash equilibria (if any). b. Find the mixed-strategy Nash equilibrium in which each player randomizes over just the first two actions. c. Compute players' expected payoffs in the equilibria found in parts (a) and (b). d. Draw the extensive form for this game..1. Identify the Nash Equilibria and Subgame Perfect Nash Equilibria in pure strategy of this game. 2. Using beliefs (p, 1−p) at P2's decision nodes in their information set, show that one of the NE is not sequentially rational.
