Answers to Questions from Three Introductory Lectures on Game Theory for Mathematicians

Here are answers to some of the questions from lectures 2 and 3

Lecture 2: Mechanism Design (April 20, 2022)

(1) Is No Veto Power the same as the “no dictator” condition in the Arrow impossibility theorem?

Answer: They are closely related but not exactly the same.

(2)

(A) Does the variable theta have to be common knowledge among the players of the mechanism?

Answer: In the mechanism constructed in the lecture, common knowledge of theta was assumed, but the mechanism can be modified so that players need know theta only approximately.

(B) If theta is common knowledge, why wouldn’t the mechanism designer know it too (in which case, why would a mechanism be needed)?

There are two answers:

(i) In reality, there are many circumstance where players in the mechanism know theta, but the mechanism designer does not. Think, for example, of the players as firms in an industry and the designer as a government agency that wishes to regulate the industry. In that example, the firms are likely to have much better knowledge of the state than the regulator does.

(ii) All the results in the lecture can be extended to the framework of the auctions lecture, in which each player i knows only his particular part, theta_i, of the state and that information is private. In this case, there is NO common knowledge of the state.

Lecture 3: Auction Theory (April 22, 2022)

(1) If a mechanism can only implement a social choice rule in Nash equilibrium but not in dominant-strategy equilibrium, does that mean it is more unlikely to be used in practice?

Answer: Mechanisms with dominant strategies are desirable because they dramatically simplify the strategic problem that a player faces, i.e., he need not take account of what other players will do, However, in many circumstance it may be impossible to find implementations in dominant strategies, in which case we have to settle for Nash implementation.

(2) Sometimes the number of bidders participating in an auction is unknown to the bidders themselves. Does this affect the results we covered in class?

Answer: A bidder’s equilibrium behavior in the second-price and English auctions is unaffected by how many other players there are, so the equilibrium equivalence between these auctions remains the same. And the Dutch and high-bid auctions are always strategically equivalent. The result connecting all four auctions generalizes to the case of an unknown number of bidders.

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