7. In a first price sealed bid auction with 2 bidders, let v; be the valuation of bidder i and b; be her bid. The bids are made simultaneously and the payoff function of player i is given as follows: Vi - - bi ui (b1,b2; V1, V2) = (v₁ - bi)/2 0 if bi > bj, if b₁ = bj, if bi < bj. The optimization problem of player i is given by (а) тахь; (v; — b;) (b) maxь; (vi — b₁) Prob{b; > b;(vj)} - (c) maxь, (vi — b₁) Prob{b; > bj(vj)} + ½ (vi − b;) Prob{b; = b;(vj)} (d) maxь; (vi — b₁) Prob{b; > b;(v;)} + ½ (v₁ − b;) Prob{b; < b;(vj)}
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- 4. Consider a first-price, sealed-bid auction in which the bidders’ valuations are independently and uniformly distributed on [0,1]. Suppose that each bidder uses a strategy of b(vi) = avi. What is the symmetric Bayesian Nash equilibrium of this game when there are n bidders?P1| P2 Left Middle Right Left 4,2 3,3 1,2 Middle 3,3 5,5 2,6 Right 2,1 6,2 3,3 Consider the simultaneous move game represented by this payoff matrix. Suppose that the game is repeated for two periods and the players know that the game will end at the end of two periods. They observe the first period outcome before they move to the second period. Assume that there is no discounting, i.e. 2nd period payoffs are not discounted, or the discount factor is equal to 1. Which of the following outcomes could occur in some subgame perfect equilibrium (SPE) of this repeated game? Choose True if you think the outcome can be a SPE, otherwise choose False. a) (Left, Left) is played in both periods. b) (Right, Right) is played in both periods. c) (Middle, Middle) is played in both periods. d) (Middle, Middle) is played in the first period, followed by (Left, Left). e) (Middle, Middle) is played in the first period, followed by (Right, Right).(9) Suppose the following game is played an infinite number of times. Denote the discount factor 8. Assume higher payoffs are more desirable. Player 1 C D Player 2 C D 6,3 0,10 3,0 10,0 (a) What is the payoff to each if both players always play C? (b) What is the payoff to each if both players always play D? (c) Suppose both players employ a Grim Trigger strategy. What value of d is required for this to be a Nash equilibrium?
- 2, 2 6,1 1,1 Player 1 B 1,6 5,5 1,1 C 1,1 1,1 4,4 Consider the simultaneous move game represented in normal form by this payoff matrix. Suppose that the game is repeated for two periods and the players know that the game will end at the end of two periods. They observe the first period outcome before they move to the second period. Assume that there is no discounting, i.e. 2nd period payoffs are not discounted. Which of the following outcomes could occur in some subgame perfect equilibrium (SPE) of this repeated game? Choose True if you think the outcome can be a SPE, otherwise choose False. i. (B, B) is played in the first period, (C, C) is played in the second period. ii. (A, A) is played in both periods. iii. (C, C) is played in both periods.1. Consider a sequential continuum game with two players, in which Player 1 first chooses a continuous variable £₁, and then Player 2 selects £2, with payoff functions ₁ and 7₂. Find the best response for P2, as the function î2 = BR₂(x1), for each given #₂: (a) (b) (c) T₂(x1, x₂) ) = (1400 — x1 — x2)x2 − 13x² T₂(x1, x₂) T₂(X1, X2₂) = : 100x2 + 3x1x2 − 14x1 – 30x1x² = 15x2 x² + x² +1(The All-Pay Auction). The seller has an item for sale. The valuations of the bidders are independently and identically distributed on R+ with a c.d.f. F. Find the symmetric equilibrium of an auction with two bidders in which both bidders pay their own bids but only the highest bidder wins the object. Show that each bidder’s expected payment is the same in this auction and in the first-price auction.
- - First-Price and Second-Price Auctions - Consider an auction of a single indi- visible object with 5 bidders, 1, 2, 3, 4 and 5, whose personal valuations (willingness to pay) for the object are v₁ = 10, V₂ = 8, 03 7, 04 5 and 5 3. The bidders simultaneously submit their bids and the winner is the one with the highest bid. Suppose that, when the highest bidder is not unique, the bidder with the smallest number (highest valuation) wins. (a) Suppose that this is a second-price auction, where the winner pays the highest bid among those from her opponents, determine whether each of the following bidding profiles is a Nash equilibrium. Explain. (i) (b₁,b2, b3, b4, b5) = (10, 8, 7, 5, 3) (ii) (b1,b2, b3, b4, b5) = (8,8, 0, 0, 0) (iii) (b₁,b2, b3, b4, b5) = (10, 0, 0, 0, 10) (b) Now suppose that this is a first-price auction, where the winner pays their own bid, determine whether each of the bidding profiles above is a Nash equilibrium. Explain. = =Assume that two collectors, X and Y are in a first prize sealed bid auction for a batch of vintage comic books. X and Y have different valuations (V) for this batch of comic books e.g. VX And VB are between $2000 and $4000. Both collectors know their own V but does not know the V of the other collector. All they know is that the other collector’s V is a uniformly distributed number between $2000 and $4000. Assume risk neutrality for X and Y e.g. expected payoff for X is: (VX – bX)Pr(bX) and expected payoff for Y is (VY – bY)Pr(bY). These collectors will make their bids strategically. Show how X’s bidding strategy is bX = ½ Vx + 1 and Y’s is bY = ½ Vy +1 in a Nash equilibrium.Players 1, 2, and 3 are playing a game in which the strategy of player i isdenoted yi and can be any nonnegative real number. The payoff function for player 1 is V1(y1,y2,y3) = y1 + y1y2 - (y1)2,for player 2 is V2(y1,y2,y3) = y2 + y1y2 - (y2)2,and for player 3 is V3(y1,y2,y3) = (10 - y1 - y2 - y3)y3.These payoff functions are hill shaped. Find a Nash equilibrium. (Hint: Thepayoff functions are symmetric for players 1 and 2.)
- 2. Suppose we have 2 players in a first price auction. More specifically, suppose player i values the object being auctioned at v, where v > vz > 0. These valuations are known to both players. They simultaneously choose bids b;. Whoever bids the most gets the object and pays what she bid. In case of a tie, each player has probability of receiving the good and paying her bid and probability of a payoff of 0. Suppose the strategy set (set of all possible bids) for each player is [0, 0). Show that there is no pure strategy Nash equilibrium.P1| P2 Left Middle Right Left 1,1 5,0 0,0 Middle 0,5 4,4 0,0 Right 0,0 0,0 3,3 Consider the simultaneous move game represented in normal form by this payoff matrix. Suppose that the game is repeated for two periods and the players know that the game will end at the end of two periods. They observe the first period outcome before they move to the second period. Assume that there is no discounting, i.e. 2nd period payoffs are not discounted. Which of the following outcomes could occur in some subgame perfect equilibrium (SPE) of this repeated game? Choose True if you think the outcome can be a SPE, otherwise choose False. a. (Middle, Middle) is played in the first period, followed by (Right, Right) b. (Left, Left) is played in both periods. c. (Right, Right) is played in both periods.5 Consider a first-price sealed-bid auction in which bidders valuations are independently and identically distributed according to the Uniform distribution on the interval [0, 1]. Explain what the rules of the First Price Sealed bid auction are. Set it up as a Bayesian game. Compute a symmetric Bayesian Nash equilibrium for the two bidder case.