c.ass. Sup- 2. Consider the following sequential variant of the public goods game w pose that there are 2 consumers, 1 and 2. First consumer 1 chooses a quantity ₁ ≥ 0 to provide of the public good. After observing 1's choice, 2 chooses a quantity 2 ≥ 0 to provide. When the price of the public good is p, 1's payoff is u₁(1, 2) = a√√x₁ + x₂-p²₁ where a > 0 and 2's payoff is u₂(x₁, x2) = √√x1 + x2 − px2. (a) Suppose that a = 1. Show that this game has a Nash equilibrium in which 1 contributes a positive amount. Solution: There are many such Nash equilibria. One is for 1 to contribute and for 2 to contribute 0 regardless of how much 1 contributes.
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- 6. The owner of an antique piece of furniture is looking to sell their good to a known buyer. The seller has a reservation value r whereas the buyer has valuation v > r. Suppose the buyer incurs a one-off transportation cost of t from travelling to the seller to purchase the good. Assume that r, vand t are known to both parties. The game proceeds in two stages: First, the buyer decides whether or not to travel to the seller's location in order to purchase. Second, if they travel then the seller makes a take-it-or-leave-it offer (ultimatum) of price p to the buyer, which the buyer then can either accept or reject. (a) Represent this game in extensive form (b) Find the seller's optimal offer in Stage 2 of this game given that the buyer has already travelled. Is it accepted or rejected? (c) Find a Subgame Perfect Equilibrium of this game. (d) Are there any Nash equilibria which are not subgame perfect? Give an example if one exists. (e) Suppose the seller could offer free delivery at a…4. An auctioneer holds a second-price auction for two bidders, Ann (A) and Bonnie (B), who have independent private values of the good 0, and e, If a bidder wins, her payoff is her value 0 minus the price she pays, and if she loses, her payoff is 0. The values are independently and identically distributed, but otherwise you don't need to know the specific distributions to solve the problem. Ann and Bonnie's respective strategies are to bid some value b (0.). that is, bid given their privately-known value (type). a. Explain what a second price auction is, who wins given some pair of bids b, and bg. and what the winner pays. b. Why is a strategy where Ann bids above her own value 0, weakly dominated by a strategy where she bids her value? c. Why is a strategy where Ann bids below her own value e, weakly dominated by a A strategy where she bids her value? d. Applying the ideas from (b) and (c) to both Ann and Bonnie, what is the Weakly Dominant Strategy Equilibrium for this game?4. An auctioneer holds a second-price auction for two bidders, Ann (A) and Bonnie (B), who have independent private values of the good 0, and 0g If a bidder wins, her payoff is her value 0 minus the price she pays, and if she loses, her payoff is 0. The values are independently and identically distributed, but otherwise you don't need to know the specific distributions to solve the problem. Ann and Bonnie's respective strategies are to bid some value b0), that is, bid given their privately-known value (type). e. Suppose the good had one true value for both bidders equal to the average of 0, and e, (signals that are still i.i.d.); hence, the good's true value has a common component. Suppose Ann knows Bonnie is going to bid her own evaluation 0, no matter what, but like normal, Ann doesn't know 0g. Explain why bidding 0, is now a strictly dominated strategy for Ann.
- 2 Consider Anna and Joe, who value a certain good by the same amount v and can choose to either contribute (C) e to get the good or not (N) where v > e> 0. Obtaining the good only requires c from one person. The game is summarized in the payoff table below: Joe V-c,V-c V-c,V v,V-c 0,0 Find a pure strategy equilibrium and a mixed strategy equilibrium. Anna O ZDescribe the Nash equilibrium/a and the mutually-preferred outcome. Finally, consider how people are sometimes able to resolve this social dilemma in the real world. Clearly explain how this method is successful in moving the players to a socially optimal outcome.L 0 4 15.1 X₂ A M R 0 1 4 0 2 B L 4 0 с X3 M R 0 1 3 FIGURE 15.5 Exercise 15.1. Equilibrium Selection: Consider the extensive-form game in Figure 15.5. a. Find all the Bayesian Nash equilibria of this game. b. Which of the Bayesian Nash equilibria are also perfect Bayesian equi- libria? Why?
- Consider the following game: Steffi N K 14, 6 7,6 8, 7 Jim 5, 8 8, 8 13, 6 M 1,5 8, -6 6, 6 What is Jim's best response to Steffi playing P? (you can select more than one answer) V a. K O b. M O c. N O d. L V e. 14 O f. 8Describe the payoffs for an R&D game of chicken and contrast them with the payoffs in a prisoners' dilemma game. The outcome is that In the prisoners' dilemma R&D game, a firm can use the R&D O A. if either it or its competitor conducts the R&D; one firm conducts R&D; B. only if it conducts the R&D; one firm conducts R&D; OC. If either it or its competitor conducts the R&D; neither firm undertakes R&D; D. only if it conducts the R&D; both firms undertake R&D; In an R&D game of chicken, a firm can use the R&D The outcome is that A. only if it conducts the R&D; one firm conducts R&D B. only if it conducts the R&D; both firms undertake R&D C. if either it or its competitor conducts the R&D; one firm conducts R&D D. If either it or its competitor conducts the R&D; both firms undertake R&DPlease 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 game theory, a dominant strategy is the best strategy to pick, no matter which moves are chosen by the other player. O to make the exact same move that was made by the other player. the choice that causes the payoff for the other player to be minimized, regardless of the payoff it earns the best strategy to pick, assuming the other player makes his or her best possible choice. to allocate all personnel resources towards defensive talent in order to dominate opposing offenses. 46°F3. Consider the following game with nature: 6, 8 X Y 4, 5, 0 X 4, 6 Y 2 (p) L (1-P) High Low L' H 1 M (1/2) (1/2) 2 M' (9) (1-q) X' 3, 3 Y 10,7 X' 3, 0 Y' 8, 4 Does this game have any separating perfect Bayesian equilibrium? Show your analysis and, if there is such an equilibrium, report it (only one is required).Suppose there is a second price sealed bid auction in which the players have the following values: v1=15, v2=4, v3=6, v4=8, v5=10, v6=6. In the symmetric equilibrium, what bid will bidder 4 submit? a. 10 b. 15 c. 4 d. 8