We consider the following two-player strategic form game, where Alice's strategies are U and D, and Bob's strategies are L and R. The payoffs are given in the Table below. LR 3.3 2-6 D2,-2 4, 6 Find the probability with which Alice plays U in the mixed strategy equilibrium (round your answer to the 4th decimal).
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- We consider the following three-player strategic form game, where Alice's strategies are U, C, and D, and Bob's strategies are L, M, and R, and Carol's strategies are A and B. Carol's strategy consists of choosing which table will be used for the payoffs, Table A or Table B.Table A is below, where for each cell the first number is Alice's payoff, the second number is Bob's payoff and the third number is Carol's payoff.. L M R U 8,11,14 3,13,9 0,5,8 C 9,9,8 8,7,7 6,5,7 D 0,8,12 4,9,2 0,4,8 Table A Table B is below, where again, for each cell, the first number is Alice's payoff, the second number is Bob's payoff and the third number is Carol's payoff.. L M R U 14,1,0 13,2,11 1,3,2 C 0,0,2 7,2,3 14,3,2 D 7,12,11 12,12,0 2,11,2 Table B This game may not have any Nash equilibrium in pure strategies, or it may have one or more equilibria.How many Nash equilibria does this game have?Consider the following two-player game.First, player 1 selects a number x≥0. Player 2 observes x. Then, simultaneously andindependently, player 1 selects a number y1 and player 2 selects a number y2, at which pointthe game ends.Player 1’s payoff is: u1(x; y1) = −3y21 + 6y1y2 −13x2 + 8xPlayer 2’s payoff is: u2(y2) = 6y1y2 −6y22 + 12xy2Draw the game tree of this game and identify its Subgame Perfect Nash Equilibrium.Consider the following game. There are two payers, Player 1 and Player 2. Player 1 chooses a row (10, 20, or 30), and Player 2 chooses a column (10/20/30). Payoffs are in the cells of the table, with those on the left going to Player 1 and those on the right going to player 2. Suppose that Player 1 chooses his strategy (10, 20 or 30), first, and subsequently, and after observing Player 1’s choice, Player 2 chooses his own strategy (of 10, 20 or 30). Which of the following statements is true regarding this modified game? I. It is a simultaneous move game, because the timing of moves is irrelevant in classifying games.II. It is a sequential move game, because Player 2 observes Player 1’s choice before he chooses his own strategy.III. This modification gives Player 1 a ‘first mover advantage’. A) I and IIB) II and IIIC) I and IIID) I onlyE) II only
- Suppose Edison and Hilary are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that Edison chooses Right and Hilary chooses Right, Edison will receive a payoff of 3 and Hilary will receive a payoff of 7. Hilary Left Right Left 4, 6 6, 8 Edison Right 7, 5 3, 7 The only dominant strategy in this game is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Edison chooses and Hilary choosesConsider a two-player game in which the players take turns, with player 1 moving first. When it is a player's turn, she must announce a number between 1 and 3. The announced number is added to the previously announced numbers. The player who announces the number such that the sum of all announced numbers is 6 wins (receives 1) and the other loses (receives 0). Please indicate whether or not each of the following sequences of announcements is a Nash equilibrium of the game. Hint: Think about how one verifies whether or not a pair of strategies is a Nash equilibrium. P1 says 3, then P2 says 2, then P1 says 1 P1 says 1, then P2 says 3, then P1 says 2 P1 says 2, then P2 says 3, then P1 says 1 P1 says 3, then P2 says 1, then P1 says 2Consider a sequential game where there are two players, Jake and Sydney. Jake really likes Sydney and is hoping to run in to her at a party this weekend. Sydney can't stand Jake. There are two parties going on this weekend and each player's payoffs are a function of whether they see one another at the party. The payoff matrix is as follows: Sydney Party 1 Party 2 Party 1 6, 18 18, 6 Jake Party 2 24,8 0,24 a) Does this game have a pure strategy Nash Equilibrium? b) What is the mixed strategy Nash Equilibrium? c) Now suppose Sydney decides what party she is going to first. Her roommate is friends with Jake and will call him to tell him which party they go to. Write the extensive form of this game (game tree). d) What is the subgame perfect Nash equilibrium from part c?
- The centipede game, first introduced by Robert Rosenthal in 1981, is an extensive form game in which two players take turns choosing either to take a slightly larger share of an increasing pot, or to pass the pot to the other player. In other words, player 1 chooses between D (Down) and A (Across), where D is pocketing the pot and A is passing the pot to the player 2. Similarly, player 2 chooses between A and D. The payoffs are arranged so that if one passes the pot to one's opponent and the opponent takes the pot on the next round, one receives slightly less than if one had taken the pot on this round. A 2 A A 2 A 1 A (3,5) Ꭰ D D D D (1,0) (0,2) (3,1) (2,4) (4,3) 1. Find the subgame perfect Nash Equilibrium using backward induction.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).The childhood game of Rock–Paper–Scissors is shown in the accompanying figure. Show that each player’s assigning equal probability to his or her three pure strategies is a symmetric Nash equilibrium.
- in game theory, when are there player 1 and palyer 2 and the payoff function is as follow (if player 1 get to choose first) (u1, u2). my question is if the order change like player 2 get to choose first, will the order of the payoff change as well like (u2, u1)?Consider the following representation of a hockey shootout. The shooter can shoot on their forehand, or deke to their backhand, and the goalie can anticipate either move. The number in each cell in the table below represents the percentage chance that the shooter scores for each pair of pure strategies. Anticipate Forehand Anticipate Backhand Shoot Forehand 20 40 Deke Backhand 40 10 In the mixed strategy Nash equilibrium of this game, what is the percentage chance that the player scores? (ie. An 80% chance should be recorded as 80)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)