First-Price Auction with Two Bidders - Consider a first-price sealed-bid auction with only two bidders, 1 and 2. Assume that v₁ = V₂ = 10 and player 1 is chosen as the winner whenever b₁ b₂. Find all the Nash equilibria of this game and justify your answer. (Note that the game here does not fit the framework we introduced in lecture, because the two players have the same valuation. However, the rationale behind the necessary conditions for an action profile to be a Nash equilibrium still applies.)

First-Price Auction with Two Bidders - Consider a first-price sealed-bid auction with only two bidders, 1 and 2. Assume that v₁ = V₂ = 10 and player 1 is chosen as the winner whenever b₁ b₂. Find all the Nash equilibria of this game and justify your answer. (Note that the game here does not fit the framework we introduced in lecture, because the two players have the same valuation. However, the rationale behind the necessary conditions for an action profile to be a Nash equilibrium still applies.)

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.9P

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Exercise 6.8. Consider the following extensive-form game with cardinal payoffs: 1 R O player pay 000 2 1 M 3 b 010 O player 3's payoff 1 2 221 2 000 0 0 (a) Find all the pure-strategy Nash equilibria. Which ones are also subgame perfect? (b) [This is a more challenging question] Prove that there is no mixed-strategy Nash equilibrium where Player 1 plays Mwith probability strictly between 0 and 1.
For the following simultaneous move game, we want to find a mixed strategy Nash equilibrium. ROWENA Safe 4*q + 4*(1-q) = Risky Safe 1 4,4 1,4 COLIN Risky 4,1 We denote Colin's mixture probability on Safe by q.. To find q in Colin's mixed strategy Nash equilibrium, we apply indifference property to player Rowena 's expected payoffs from pure strategies in mixture: 6,6 +q+6 *(1-q)
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Consider a modified Traveler’s Dilemma. In terms of strategy options that the players have and the dollars they earn, it is like the standard Traveler’s Dilemma, but the players do not have endless appetite for money. Up to 100 dollars, each dollar feels like a dollar. But any moneybeyond 100 is psychologically like 100 dollars. Assuming that players are maximizers of ‘psychological’ dollars instead of real dollars, describe all the Nash equilibria of this modified Traveler’s Dilemma.
2. Consider the following Bayesian game with two players. Both players move simultaneously and player 1 can choose either H or L, while player 2's options are G, M, and D. With probability 1/2 the payoffs are given by "Game 1" : GMD H 1,2 1,0 1,3 L 2,4 0,0 0,5 and with probability 1/2 the payoffs are according to "Game 2" : G |M|D H 1,2 1,3 1,0 L 2,4 0,5 0,0 (a) Find the Nash Equilibria when neither player knows which game is actually played. (b) Assume now that player 2 knows which one among the two games is actually being played. Check that the game has a unique Bayesian Nash Equilibrium.
- 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. = =
(a) Design a two-player game with the property that "the payoff for Player1 is Playerl plays a strategy Sl and Player2 plays a strategy S2 is the same as the payoff for Player2 if Player2 plays Sl and Playerl plays S2". Express your game in terms of payoff matrices. Show the above property holds in your given payoff matrices. Give a case to show that there is one Nash equilibrium and such an equilibrium is also a Pareto efficient (justify your answers) in your given payoff matrices. Prove or disprove the above case that is unique. Give another example to show that there are two Nash equilibria in your given payoff matrices.
Section 1: Two Romantics Ann and Bob had their first date. Each either felt romantic chemistry (C) or no chemistry (NC) with the other person. Each person knows his/her own feeling but does not know the feeling of the other person. Assume a common prior belief that the other person felt chemistry with probability Pr(C) = p and no chemistry with probability Pr(NC) = 1-p. Ann and Bob are old-fashioned romantics and they made the following rule after the first date: No texts/calls/DMs. Instead, they can choose whether to appear (A) or not appear (NA) under the USyd Quadrangle clock tower at sunset on the next day. Their payoffs are given as follows: (From a first-person perspective) If I felt chemistry (C) and I appear (A) under the clock tower, my payoff is 100 if the other person also appears (A) and -100 if the other person doesn't (NA). If I felt chemistry (C) and I choose not to appear (NA) under the clock tower, my payoff is -30 regardless of the other person's action (because I…
(a) Find all (pure or mixed) Nash equilibria in the following two-player game: L C R U 2,4 -2, 2-2,0 D-2,0 -1,6 0,8 (b) Find all (pure or mixed) Nash equilibria in the following three-player game: player 1 chooses the row, player 2 chooses the column, player 3 chooses the matrix. X Y X Y X 3,-2, 4 0,-3,2 Y 0,-3,2 0,-3,2 X0,-3, 2 Y 0, -3,2 0,-3, 2 12, 1, 10 X Y
Consider the location game we covered in Lecture 3. Now assume there arethree players (vendors). As we assumed in the lecture, consumers in each area choosethe closest vendor and if there are multiple closest vendors then these vendors receiveequal share of consumers in the area. Notice Si = {1, 2, 3, ...., 9} for i = 1, 2, 3. Here aresome examples of payoffs: u1(1, 1, 1) = 3, u1(1, 1, 9) = u2(1, 1, 9) = 2.25, u3(1, 1, 9) =4.5, u1(1, 5, 9) = u3(1, 5, 9) = 2.5 and u2(1, 5, 9) = 4. (a) Is s′1 = 1 strictly dominated by s′′1 = 2 for player 1?(b) Is s′1 = 1 weakly dominated by s′′1 = 2 for player 1?(c) Can you find a Nash equilibrium in pure strategies?
5. Consider a simultaneous game in which player A chooses one of two actions (Up or Down), and B chooses one of two actions (Left or Right). The game has the following payoff matrix, where the first payoff in each entry is for A and the second for B.(8 points) B Right Left 3,3 5,1 Down 2,2 4,4 a. Find the Nash equilibrium or equilibria. b. Which player, if any, has a dominant strategy? A Up
GAME UUU B1 Player B B2 A1 7,13 5, 10 A2 3,8 9,16 Player A A3 5,8 4,7 In Game UUU (see table above), assuming players move simultaneously, Player A choosing A1 and Player B choosing B3 is a Nash equilibrium. Player A choosing A3 and Player B choosing B2 is a Nash equilibrium. Both Player A choosing A1 and Player B choosing B1 and Player A choosing A2 and player B choosing B2 are Nash equilibria in pure strategies Player A choosing A1 and Player B choosing B2 is a Nash equilibrium.