Consider the following simultaneous game: Player 1 U D Player 2 L 30,20 -10, -10 R -10, -10 20,30 Suppose player 1 plays a mixed strategy in which she plays U 25% of the time and D 75% of the time, and player 2 plays a mixed strategy in which she plays L 25% of the time and R 75% of the time. This pair of strategies ✓a Nash equilibrium. Player 1's expected payoff from playing U (when player 2 plays the mixed strategy above) is
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- Consider 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 2Let's think about a traditional game theory principal agent game. In this game we have two players, Lauren and Jason Lauren will choose to invest $500 dollars with Jason If she doesn't invest the money, she keeps the $500 and Jason gets nothing If she does invest the money, then Jason has a choice-he can cooperate by investing Lauren's money and then splitting the profit of another $500. This would give Lauren $750 (her original $500 investment plus $250 of profit) and would give Jason $250. Jason's other choice is to appropriate, or to steal Lauren's money and not invest it. If he does this, Lauren ends up with nothing and Jason has the entire $500. The following diagram outlines this game. Principal Agent Game Jason (750.250) Invest appropriate 10. 500) doninve What will happen in this game? Lauren will not invest Lauren will invest and Jason will cooperate Lauren will invest and Jason will appropriate Next pagein 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)Consider the following sequential game. Player 1 plays first, and then Player 2 plays after observing the choice of Player 1. At the bottom of the decision tree, the first number represents the payoff of Player 1, while the second number represents the payoff of Player 2. For player 2, A stands for Accommodate and F stands for Fight. Player 1 Enter Player 2 A F A Stay out Player 2 F (16,30) (-6,18) In the Nash equilibrium of this game, player 2 earns Player 2 player 1 would play (0,40) (0,20) ✓an incentive to threaten F because, if player 1 believed him then so that player 2 would earn ✓. However, this threat isSolving for dominant strategies and the Nash equilibrium Suppose Felix and Janet 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 if Felix chooses Right and Janet chooses Right, Felix will receive a payoff of 3 and Janet will receive a payoff of 7. Attached the table The only dominant strategy in this game is for (Janet / Felix) to choose (Left / Right). The outcome reflecting the unique Nash equilibrium in this game is as follows: Felix chooses (Left / Right) and Janet chooses (Left / Right).
- 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.Refer to Table Left-Right-ZigZag. How many pure strategy Nash equilibria does this game have? Table: Left-Right-ZigZag Player 1 Drive left Driver right Zigzag 1) 1 2) 2 3) 3 4) 4 5) None of the above. Drive left (1,1) (-1,-1) (0,0) Player 2 Driver right (-1,-1) (1,1) (0,0) Zigzag (0,0) (0,0) (0,0)7. Solving for dominant strategies and the Nash equilibrium Suppose Dmitri and Frances 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 if Dmitri chooses Right and Frances chooses Right, Dmitri will receive a payoff of 7 and Frances will receive a payoff of 6. Frances Left Right Left 4, 3 6, 4 Dmitri Right 6, 7 7, 6 to choose The only dominant strategy in this game is for and Frances chooses The outcome reflecting the unique Nash equilibrium in this game is as follows: Dmitri chooses v
- PLAYER B LEFT RIGHT UP 5 FOR A, 30 FOR B 10 FOR A, 12 FOR B PLAYER A DOWN -2 FOR A, 10 FOR B 8 FOR A, 15 FOR B In the above game, the players are seeking to maximize the number they recieve. They choose at the same time. What is the Nash equillibrium? Player A will choose UP and player B will choose LEFT Player A will UP and player B will choose RIGHT Player A will choose DOWN and player B will choose LEFT Player A will choose DOWN and player B will choose RIGHT Player A will choose LEFT and player B will choose UP Player A will choose LEFT and player B will choose DOWN Player A will choose RIGHT and player B will choose UP Player A will choose RIGHT and player B will choose DOWNThree 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.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 answer