Problem 3 Consider the following simultaneous normal form game: P.2 R P R 0,0 -1,1 1, -1 P.1 P 1, -1 0,0 -1, 1 -1, 1 1, -1 0,0 Now assume the game is not simultaneous and player 1 gets to play first. Create a game tree that represents the sequential game. Also, name each node, define the set of information sets for each player, show which nodes are in which information set and add define action sets at each info set.
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- Q56 A Nash equilibrium is an outcome... a. Achieved by cooperation between players in the game. b. That is achieved by collusion where no party has an incentive to change their behaviour. c. Where each player's strategy depends on the behaviour of its opponents. d. That is achieved when players in the game have jointly maximized profits and divided those profits according to market share of each player. e. Where each player's best strategy is to maintain its present behaviour given the present behaviour of the other players.Splitting Pizza: You and a friend are in an Italian restaurant, and the owner offers both of you a free eight-slice pizza under the following condition. Each of you must simultaneously announce how many slices you would like; that is, each player i ∈ 1, 2 names his desired amount of pizza, 0 ≤ si ≤ 8. If s1 + s2 ≤ 8 then the players get their demands (and the owner eats any leftover slices). If s1 + s2 > 8, then the players get nothing. Assume that you each care only about how much pizza you individually consume, and the more the better.What outcomes can be supported as pure-strategy Nash equilibria?A strategy for player 1 is a value for x1 from the set X. Similarly, a strategyfor player 2 is a value for x2 from the set X. Player 1’s payoff is V1(x1, x2) =5 + x1 - 2x2 and player 2’s payoff is V2(x1, x2) = 5 + x2 - 2x1.a. Assume that X is the interval of real numbers from 1 to 4 (including 1and 4). (Note that this is much more than integers and includes such numbers as 2.648 and 1.00037). Derive all Nash equilibria.b. Now assume that the game is played infinitely often and a player’s payoff is the present value of his stream of single-period payoffs, where dis the discount factor.(i) Assume that X is composed of only two values: 2 and 3; thus, aplayer can choose 2 or 3, but no other value. Consider the followingsymmetric strategy profile: In period 1, a player chooses the value 2. In period t(≥2), a player chooses the value 2. In period a player chooses the value 2 if both players chose 2 in all previous periods; otherwise, she chooses the value 3. Derive conditions which ensure…
- You and a rival are engaged in a game in which there are three possible outcomes: you win, your rival wins (you lose), or the two of you tie. You get a payoff of 50 if you win, a payoff of 20 if you tie, and a payoff of 0 if you lose. What is your expected payoff in each of the following situations? (a) There is a 50% chance that the game ends in a tie, but only a 10% chance that you win. (There is thus a 40% chance that you lose.) (b) There is a 50–50 chance that you win or lose. There are no ties. (c) There is an 80% chance that you lose, a 10% chance that you win, and a 10% chance that you tie.A clothing store and a jeweler are located side by side in a shopping mall. If the clothing store spend C dollars on advertising and the jeweler spends J dollars on advertising, then the profits of the clothing store will be (36 + J )C - 2C 2 and the profits of the jeweler will be (30 + C )J - 2J 2. The clothing store gets to choose its amount of advertising first, knowing that the jeweler will find out how much the clothing store advertised before deciding how much to spend. The amount spent by the clothing store will be Group of answer choices $17. $34. $51. $8.50. $25.50.Two friends, Khalid and Mahmood, are going to a watch a world cup football match. They play a simple game in which they hold out one or two fingers to decide who will pay for the other's ticket. Khalid wins if the fingers held out add up to an even number; Mahmood wins if the fingers held out add up to an odd number. The price of the ticket is 25 OMR. Construct a payoff matrix for the game. Is there a unique Nash equilibrium in this game? Which strategy should a player use to maximize her chances of winning the game?
- 4 Consider an extensive game where player 1 starts with choosing of two actions, A or B. Player 2 observes player 1’s move and makes her move; if the move by player 1 is A, then player 2 can take three actions, X, Y or Z, if the move by player 1 is B, then player 2 can take of of two actions, U or V. Write down all teminal histories, proper subhistories, the player function and strategies of players in this game.6 Two people will select a policy that affects both of them by applying a "veto" in a sequential and alternate manner, that is: person 1 begins to veto a policy and then person 2 exercises his "veto" with the remaining policies; the process repeats until only one policy remains. Assume that there are 3 policies: X,Y,Z, and that person 1 prefers X to Y to Z and person 2 prefers Z to Y to X. a. Represents the game extensively b. Give the number of subgames C. Indicate the total strategies of the players d. find all subgame perfect nash equilibria e. Find all Nash Equilibriums.14. Company A and Company B are each telecommunications manufacturers. Both companies manufacture the same products, and they make their decisions based on the other's actions. Both companies are considering opening retail outlets to increase their profits. The payoff matrix shows the profits of the companies in millions of dollars if they choose to open retail outlets. The government imposes a new $5 million tax to open retail outlets. What is the expected outcome of the new payoff matrix, given the tax? The Nash equilibrium is for Company A to not open retail outlets and for Company B to open retail outlets. The Nash equilibrium is for Company A to open retail outlets and for Company B to not open retail outlets. The Nash equilibrium is for both Company A and Company B to open retail outlets. The Nash equilibrium is for both Company A and Company B to not open retail outlets. There is no Nash equilibrium after the change given in the scenario.…
- E3 Bayesian Game]. Consider a Bayesian game described by a following payoff matrix. Please solve (show your solution). 1. Enumerate all pure strategies for each player. 2. Suppose that player 1 observes his type ?1 = 3. How does player 1 think of the probability of ?2? 3. Find a (pure strategy) Bayesian Nash equilibrium.Consider the following coordination game: Player 2P1 Comedy Show Concert Comedy Show 11,5 0,0 Concert 0,0 2,2 a. Find the Nash equilibrium(s) for this game.b. Now assume Player 1 and Player 2 have distributional preferences. Specifically, both people greatly care about the utility of the other person. In fact, they place equal weight on their outcome and the other person’soutcome, ρ = σ = ½. Find the Nash equilibrium(s) with these utilitarianpreferences.c. Now consider the case where Player1 and Player2 do not like each other. Specifically, any positive outcome for the other person is viewed as anegative outcome for the individual, ρ = σ = -1. Find the Nashequilibrium(s) with these envious preferences.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.