Jesse and Janie are playing a game. On the table in front of them, there are 50 coins. They take turns removing some of these coins from the table. At each turn, the person moving can remove either 1, 2, 3, 4, or 5 coins (they cannot remove 0 or 6 etc). The person to remove the last coin wins. Suppose Janie is the first player to have a turn. 1) In the subgame perfect equilibrium, Janie begins by taking 2) Suppose Janie makes a mistake in play, and removes a number of coins that leaves 32 coins on the table when it is Jesse's turn. To have a chance at winning, Jesse should remove coins. coins.
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- Jesse and Janie are playing a game. On the table in front of them, there are 50 coins. They take turns removing some of these coins from the table. At each turn, the person moving can remove either 1,2,3,4,5,6, or 7 coins (they cannot remove 0 or 8, etc.). The person to remove the last coin wins. Suppose Janie is the first player to have a turn (a) In the subgame perfect equilibrium, Janie begins by taking _________coins (b) Suppose Janie makes a mistake in play and removes a number of coins that leaves 27 coins on the table when it is jesses turn. To have a chance at winning, Jesse should remove _________ CoinsExercise 6.1Suppose that two airlines decide to collude. Analyse the game between these two companies. Suppose that each of them can charge for tickets a high price or a low price. If one of them charges 100 euros, it gets few profits if the other also charges 100 euros and high profits if the other charges 200 euros. On the other hand, if the company charges 200 euros, it obtains very little profit if the other charges 100 euros and an average profit if the other also charges 200 euros. a) Represent the matrix of results of this game. b) What is the Nash equilibrium in this game? Explain your answer. c) Is there an outcome that would be better than the Nash equilibrium for the two airlines? How could it be achieved? Who would lose out if it were reached?. Find the Nash equilibrium of the following modified Rock-Paper-Scissors game: • When rock (R) beats scissors (S), the winner’s payoff is 10 and the loser’s payoff is −10. • When paper (P) beats rock, the winner’s payoff is 5 and the loser’s payoff is −5. • When scissors beats paper, the winner’s payoff is 2 and the loser’s payoff is −2. • In case of ties, both players receive 0 payoff. You are suposed to create a system of equations and then solve for them and find 3 probabilities- please show how to do that
- Consider the following simultaneous move game where player 1 has two types. Player 2 does not know if he is playing with type a player 1 or type b player 1. Find the all the possible Bayesian Nash Equilibriums (BNE) of this game.John and Jane usually vote against each other’s party in the SSC elections resulting to negating or offsetting their votes. If they vote for their party of choice, each of them gains four units of utility (and lose four units of utility from a vote against their party of choice). However, it costs each of them two units of utility for the hassle of actually voting during the SSC elections. A. Diagram a game in which John and Jane choose whether to vote or not to vote.Two individuals each receive fifty dollars to play the following game. Independently of each other, they decide how much money to put in a common pot. They keep the rest for themselves. As for the money in the pot, it is increased by 80% and then distributed equally among the two individuals. For instance, suppose that the first individual puts $10 in the pot while the second individual puts $20. Increasing the total pot of $30 by 80% gives $54 to share equally between the two individuals. So the first individual’s payoff in this case is $(40 + 27) = $67, while the second individual’s payoff is $(30 + 27) = $57. (a) Compute the Nash equilibrium. (b) Is the Nash equilibrium Pareto efficient? Explain
- Two players bargain over $20. Player 1 first proposes a split of(n, 20 - n), where n is an integer in {0, 1, ..., 20}. Player 2 can either accept or reject this proposal. If player accepts it, player 1 obtains $n and player 2 obtains $(20 - n). If player 2 rejects it, the money is taken away from them and both players will get $0. Question: Find two subgame perfect Nash equilibria of this game and state clearly each player's equilibrium strategies (recall that in a dynamic game, a player's strategy is a complete-contingent plan). Explain why the strategy profiles form a subgame perfect equilibrium.Please no written by hand Two players bargain over how to split $10. Each player i ∈ {1, 2} choose a number si ∈ [0, 10] (which does not need to be an integer). Each player’s payoff is the money he receives. We consider two allocation rules. In each case, if s1 + s2 ≤ 10, each player gets his chosen amount si and the rest is destroyed. 1. In the first case, if s1 + s2 > 10, both players get zero. What are the (pure strategy) Nash equilibria? 2. In the second case, if s1 + s2 > 10 and s1 6= s2, the player who chose the smallest amount receives this amount and the other gets the rest. If s1 + s2 > 10 and s1 = s2, they both get $5. What are the (pure strategy) Nash equilibria? 3. Now suppose that s1 and s2 must be integers. Does this change the (pure strategy) Nash equilibria in either case?In a classroom election, two presidential candidates, namely, Lisa and Teddy, both garnered the same number of total votes. As such, they decided to play a custom dice game to determine the winner of the election. In this game, a player needs to roll a pair of dice. Teddy will win the game if the sum is odd whereas Lisa will be declared winner if the sum results to even. However, prior to the start of the game, Teddy complained that the custom dice game is biased because according to him, the probability of an even result is 6/11, and for odd - only 5/11. Verify Teddy's claim and evaluate the fairness of the game. ANALYZE IT CLEARLY AND SOLVE
- Alice and Betsy are playing a game in which each can play either of two strategies, leave or stay. If both play the strategy leave, then each gets a payoff of $400. If both play the strategy stay, then each gets a payoff of $800. If one plays stay and the other plays leave, then the one who plays stay gets a payoff of $C and the one who plays leave gets a payoff of $D. When is the outcome where both play leave a Nash equilibrium? a) never, since 800 > 400 b) when 400 > C and D > 800 but not when 800 > D c) whenever 400 > C d) when D > C and C > 400 e) whenever d < 800Consider 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. For example, a payoff 100/200 would mean Player 1 receives 100 and Player 2 receives 200.What is [are] the Nash Equilibrium [Equilibria] of this game?A) (10/10) and (20/20)B) (30/30)C) (10/20) and (20/10)D) (20/20)E) (30/30)Consider the strategic voting game discussed at the endof this chapter, where we saw that the strategy profile (Bustamante, Schwarzenegger,Schwarzenegger) is a Nash equilibrium of the game. Show that (Bustamante, Schwarzeneg-ger, Schwarzenegger) is, in fact , the only rationalizable strategy profile. Do this by firstconsidering the dominated strategies of player L. (Basically, the question is asking youto find the outcome of the iterative elimination of strictly dominated strategies)