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Pennsylvania State University**We aren't endorsed by this school

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ECON 402

Subject

Economics

Date

Dec 16, 2024

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Uploaded by DeaconMantis4741

Econ 402, Take-homeDecember 12, 2024““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““““work you submit must be your own. No consultation with anyone and no copyingfrom anyone is permitted. Explain all answers. Your answers should be submit-ted on Canvas.All submissions must be by11 : 59am on Monday, December16.1

1.Suppose Players 1 and 2 are participating in a first-price sealed bid auction withprivate, independent valuations. Each player’s valuation of the object to be sold, which isassumed to be worth 0 to the seller, is drawn from a uniform distribution on [0,1].Eachplayer knows her own valuation but only the probability distribution on the other player’svaluation. Bids are restricted to be in [0,1]. Remember bids are chosen simultaneously, thehighest bidder wins and pays the amount of his bid. If two bidders bid the same amount,one of them gets the object with probability 0.5.(a) What is a strategy for a player in this game?(b) Bidder 2 decides to choose his bidb2=.8v2,wherev2is his value. Consider bidder1 and suppose her value is 0.6. Which is the best bid for her, knowing Bidder 2’s biddingfunction, among these three bids (i)b1=.64,(ii)b1=.4,(iii)b1=.3.Assume she wants tomaximize her expected payoff from the auction where she gets.6−b1if she wins and 0 ifshe loses, so her expected payoff is given by (.6−b1)Prob(b1≥b2).(c) What would Player 1 bid with three bidders, with Bidders 2 and 3 bidding.8v2and.8v3respectively? Check forv1=.6 and the three bids listed in (b) above.2

2.Consider the following game that is playedTtimes. First, players move simultane-ously and independently. Then each player is informed about the actions taken by the otherplayer in the first play and, given this, they play it again, and so on.The payoff for thewhole game is the sum of the payoffs a player obtains in theTplays of the game.defA3,14,00,1B1,52,20,1C1,10,21,2(a) SupposeT= 2.Is the followingoutcomepath one that can be obtained from subgameperfect Nash equilibrium strategies?(If so, write down the strategies, if not explain whynot.)(B,d) in the first round, (A,d) in the second round.(b) SupposeT= 3 and you are checking whether it is possible for subgame-perfectequilibrium strategies to support an outcome of (B,e) in the first round. Describe briefly inwords the process you will follow to check this. Begin with: “A stage-game Nash equilibrium(I am confining myself to pure strategies) must be played in the last round, but players canco-ordinate on which one to play based on past outcomes.Therefore, I can sustain thefollowing outcomes in the second-last round...........”(c) Is it possible to sustain (as part of a SPNE) (B,e) as an outcome in the first round ofaT= 3 game? Write down the strategies that will do it, if the answer is “yes” and explainwhy not if the answer is “no”. (Hint: (B,e) is not a Nash equilibrium if the game is playedonly once-both the row player and the column player have incentives to deviate. So in orderfor each individually not to deviate, each must be rewarded sufficiently in what follows if heor she does not deviate, and get worse payoffs if they do. Use your answer to (a) above.)3

3. Consider the following game that is played with uncertain termination. That is, aftereach round the game continues with probabilityδand ends with probability 1−δ,whereuponeach player gets a payoff of 0.Players move simultaneously and independently. Then eachplayer is informed about the actions taken by the other player in the round and, given this,they play it again with probabilityδ. The payoff for the whole game is the expectation ofthe sum of the payoffs a player obtains.abcA3,25,01,1B2,24,41,2C1,20,22,3Is it possible forBandbto be played forever, in a subgame perfect equilibrium, eventhough (B, b) is not a Nash equilibrium in the payoff table above, ifδ= 0.9? What ifδ= 0?4

4.Suppose there are four players, named 1,2,3,4 moving sequentially (in the order 1,2,3,4)in the following game.At the beginning of the game, the players are all in the same room. Player 1 moves first,choosing an observable action C (stay in the room) or D (leave the room). Player 2 thenchooses likewise to stay or leave. Player 3, having observed Players 1 and 2, decides C orD, followed by Player 4, who has observed all three past players. The payoffs to each playerare as follows, withx= # of players who playC:Play C: 3xPlay D: 3x+2.What is the subgame perfect equilibrium in this game (remember strategies have to bedescribed)?5