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CSCI 1440/2440 Introduction to Game Theory - Homework 1: Nash equilibrium

USCSCI 1440CSCI 2440Introduction to Game TheoryNash equilibriumPure Strategy

In the game rock-paper-scissors, two players simultaneously se- lect one of three possible actions: rock, paper, or scissors. You might recall from your childhood, or The Simpsons, that rock beats scissors, scissors beats paper, and paper beats rock

CSCI 1440/2440 Introduction to Game Theory - Homework 2: Introduction to Auctions

USCSCI 1440CSCI 2440Introduction to Game TheoryIntroduction to AuctionsBayesian Prisoners’ Dilemma

Alice and Bob devise a plan to steal the jade monkey before the next full moon. They realize that it’s in a glove compartment, and decide to use a 3d-printed key to break in. However, Alice forgets to bring the key, so the pair gets caught and put into jail.

CSCI 1440/2440 Introduction to Game Theory - Homework 3: Myerson’s Lemma - Welfare Maximization

USCSCI 1440CSCI 2440Introduction to Game TheorySocial WelfareMyersons lemma

The key difference between optimization and mechanism design problems is that in mechanism design problems the constants (e.g., vi and wi) are not assumed to be known to the center / optimizer; on the contrary, they must be elicted, after which the optimization problem can then be solved as usual.

CSCI 1440/2440 Introduction to Game Theory - Homework 4: Myerson’s Theorem - Revenue Maximization

USCSCI 1440CSCI 2440Introduction to Game TheorySocial WelfareSponsored Search Aucktion

Recall the Bayesian (i.e., interim) formulation of the auction design problem from Homework 4. Since the interim constraints are weaker than the ex-post constraints presented in lecture, you might imagine that the welfare achieved by the welfare-maximizing auction in the interim case exceeds that of the ex-post case

CSCI 1440/2440 Introduction to Game Theory - Homework 5: Posted-Price Mechanisms

USCSCI 1440CSCI 2440Introduction to Game TheorySocial WelfareSponsored Search Aucktion

Bids are collected. A reserve price is chosen by removing an arbitrary bidder j from the auction, and setting the reserve price to be j’s bid. The auctioneer then allocates the good to the bidder with the highest bid iff their bid is at least this reserve, and charges the winner, if any, the greater of the second-highest bid and the reserve price.

CSCI 1440/2440 Introduction to Game Theory - Homework 6: Approximation Mechanisms

USCSCI 1440CSCI 2440Introduction to Game Theory Equal-Revenue DistributionApproximation Mechanisms

Recall the game that motivated the Prophet Inequality: Assume n independent random variables π with non-negative, continuous distributions Gi. The distributions Gi are known in advance, but the “prize” πi is not revealed until period i.

CSCI 1440/2440 Introduction to Game Theory - Homework 7: Vickrey-Clarke-Groves Mechanism

USCSCI 1440CSCI 2440Introduction to Game Theory Equal-Revenue DistributionVickrey Auction

Consider a single-good auction, and assume bidders’ values are drawn i.i.d. from a regular distribution F. Prove that the expected revenue of the Vickrey auction (second-price without a reserve) with n bidders,

CSCI 1440/2440 Introduction to Game Theory - Homework 8: EPIC Auctions

USCSCI 1440CSCI 2440Introduction to Game Theory Equal-Revenue DistributionEnglish Auctions

This problem concerns Japanese auctions, a variant of the classic English auction that poses demand queries rather than value queries, and that forbids bidders from re-entering after exiting (i.e., skipping even one round of bidding in) the auction.

INFR11020 Algorithmic Game Theory and Applications - Homework 1: Nash equilibrium and Farkas Lemma

University of EdinburghINFR11020Algorithmic Game Theory and ApplicationsNash equilibriumFarkas Lemma

One variant of the Farkas Lemma says the following: Farkas Lemma A linear system of inequalities Ax ≤ b has a solution x if and only if there is no vector y satisfying y ≥ 0 and yT A = 0 (i.e., 0 in every coordinate) and such that yT b < 0.

INFR11020 Algorithmic Game Theory and Applications - Homework 2: Network Congestion Game, Pareto Optimal and VCG Mechanism

University of EdinburghINFR11020Algorithmic Game Theory and ApplicationsNash equilibriumNetwork Congestion GamePareto Optimal

Recall that a Nash equilibrium in an extensive game is subgame perfect nash equilibrium (SPNE) if it is also a Nash equilibrium in every subgame of the original game

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