# Games of Incomplete Information

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1 Games of Incomplete Information Yan Chen November 16, 2005 Games of Incomplete Information Auction Experiments Auction Theory Administrative stuff

2 Games of Incomplete Information Several basic concepts: To say that a game is of complete or incomplete information is to say something about what is known about the circumstances under which the game is played. All the games we studied so far are games of complete information. Roughly speaking, enough information has been supplied to allow the game to be analyzed. For this to be true in general, various things are taken to be common knowledge. In online auctions, it is not realistic to assume that bidders know the other bidders valuations. Consider preferences: in the auction experiment, it is not realistic to assume that players risk attitudes are common knowledge. In a game like Russian Roulette, both players have an incentive to pretend to be more reckless than they actually are. In a Cournot duopoly game, it is unrealistic to assume that one firm knows the cost structure of the other firm.

3 Harsanyi s theory of incomplete information offers a means to get a handle on such matters: a technique for completing a structure in which information is incomplete. Main techniques: expected utility calculation. Solution concepts: a comparison Strategic Form Extensive Form Complete Info. DSE; NE SPNE Incomplete Info. Bayesian Nash Perfect Bayesian

4 Strategic Form Games of Incomplete Information Situation: some parties are uncertain about the characteristics of some other party. 1. A Bayesian game consists of A finite set of players, N; A set of states, Ω, each of which is a description of all the players characteristics. For each player i N: The set of actions available to each player i, A i ; the set of signals observed by i; a probability measure on Ω, which denotes the prior belief of player i. a preference relation, i. 2. A Bayesian-Nash equilibrium is a Nash equilibrium of the Bayesian game, in which each player chooses the best action available to him given the signal that he receives and his belief about the state and the other players actions that he deduces from this signal. I.e., each player chooses an action to maximize expected utility given the signals and his beliefs.

5 3. Three differences of Bayesian Nash equilibrium from Nash equilibrium: You can be more than one type; You do not know what types other players are; Strategies: plans of actions for every types you and others might be. 4. Note: The mention of Bayes here does not mean that Bayes Rule is necessarily going to be invoked somewhere along the way. It simply refers to the incomplete information part.

6 Auctions: Background What s so interesting about auctions? An alternative to bargaining for selling a fixed supply of a commodity for which there is no well-established on going market Applications Real estate, art, flower, oil lease Privatization and deregulation Government contract Electricity Airwaves: FCC spectrum auction Allocation of common resources E-commerce: ebay Auction Institutions English Dutch First price sealed bid Vickrey Many other kinds

7 Types of Auctions Private value auctions Bidders valuations for the auctioned item(s) are independent from one another and are their private information. E.g., flowers, art, antiques Common value auctions Bidders are uncertain about the ultimate value of the item, which is the same for all bidders E.g., oil leases, Olympic broadcast rights Affiliated (correlated) value auctions Bidders valuations for the auctioned item(s) are correlated, but not necessarily the same for all. In between private and common value auctions.

8 Auction Research Research Questions Efficiency comparison of auction institutions Revenue comparison Bidder earning comparison Collusion? Transparency? What do we know? Single item: well Multiple items: little Substitutes Complements Agenda for theoretical research Multi-item auctions Agenda for experimental research Test/discriminate among theories Design and test new institutions

9 English Auction Background oral auctions in English-speaking countries. Originally Roman. Commodities antiques, artworks, cattle, horses, real estate, wholesale fruits and vegetables, old books, etc. Rules for Experiment Auctioneer first solicits an opening bid from the group. Anyone who wants to bid should call out a new price at least \$1 higher than the previous high bid. The bidding continues until all bidders but one have dropped out. The highest bidder gets the object being sold for a price equal to the final bid. Winner s profit = Buyer Value - price; Everyone else s profit = 0. Your Buyer Value = Last two digits of your SSN.

10 Optimal strategy participate till price = buyer value, then drop out. Equilibrium Outcome The highest bidder gets the object at a price close to the second highest Buyer Value. Comparative statics As n increases, the winning bid is closer to the highest BV. The more spread-out the different bidders valuation are, the larger v max v 2nd. This means that if there is wide disagreement about the item s value, the winner might be able to get it cheaply. Problems collusion; bidding rings.

11 The Dutch Auction Background whole sale produce, cut-flower markets in the Netherlands. Commodities flowers in the Netherlands fish market in England and Israel tobacco market in Canada Rules Auctioneer starts with a high price. Auctioneer lowers the price gradually until some buyer shouts Mine! The first buyer to shout Mine! gets the object at the price the auctioneer just called. Winner s profit = Buyer Value - price; Everyone else s profit = 0. Your Buyer Value = Last two digits of your SSN.

12 First-Price Sealed-bid Auctions Background used to award construction contracts (lowest bidder), real estate, art treasures; Rules Bidders write their bids for the object and their names on slips of paper and deliver them to the auctioneer. The auctioneer opens the bid and find the highest bidder. The highest bidder gets the object being sold for a price equal to her own bid. Winner s profit = Buyer Value - price; Everyone else s profit = 0. Your Buyer Value = First and second number of the last four digits of your SSN.

13 Set up the problem In a sealed-bid, first price auction in a private values environment with n bidders, each bidder has a private valuation, v i, which is his private information. The distribution of v i is common knowledge. Let B i denote the bid of player i. Let π i denote the profit of player i. If v i u[0, 100], what is the Bayesian Nash equilibrium bidding strategy for the players? Optimal bidding strategies If B i v i, then π i 0. Therefore, B i < v i, which gives π i = { 0, if Bi max j {B j } v i B i, if B i = max j {B j } The question is how much below v i should his bid be? The less B i is, the less likely he will win the object, but the more profit he makes if he wins the object. n = 2 you are characterized by the strategy-type two tuple, (B, v). Suppose the other bidder s value is X, and she is characterized by (αx, X), where α (0, 1). Your expected profit is Eπ = P (Your bid is higher) (v B)+P (Your bid is lower) 0

14 With uniform distribution, P (X < B/α) = 1 Therefore, Eπ = 1 B (v B), 100 α assuming risk neutrality, you choose B to It follows that B = v/2. With n bidders max B B(v B) = Bv B P (Your bid is the highest) = ( 1 B 100 α )n 1. max B Bn 1 (v B) B = n 1 n v. Note: as n increases, B v. I.e., increased competition drives bids close to the valuations. Equivalence of Dutch and First-price, sealed-bid auctions: same reduced form. The object goes to the highest bidder at the highest price. A bidder must choose a bid without knowing the bids of any other bidders. Optimal bidding strategies are the same. B α.

15 Sealed-bid Second-price Auctions Background: Vickrey (1961). Commodities stamp collectors auctions US Treasury s long-term bonds Airwaves auction in New Zealand e-bay and Amazon Rules Bidders write their bids for the object and their names on slips of paper and deliver them to the auctioneer. The auctioneer opens the bid and find the highest bidder. The highest bidder gets the object being sold for a price equal to the second highest bid. Winner s profit = Buyer Value - price; Everyone else s profit = 0. Your Buyer Value = First and second number of the last four digits of your SSN.

16 Equilibrium bidding strategy It is a weakly dominant strategy to bid your true value. Let V be your Buyer Value, let B be your bid, and let X be the highest bid made by anybody else in the auction. We want to show that overbidding or underbidding cannot increase your profit and might decrease it. Let π t be your profit when B = V. Let π be your profit otherwise. Proof: First consider the case of overbidding, B > V. 1. X > B > V : You don t get the object either way. π = π t = B > X > V : π = V X < 0, but π t = B > V > X: π = V X = π t > 0. Next consider the case of underbidding, B < V. 1. X < B < V : π = V X = π t > B < X < V : π = 0, but π t = V X > B < V < X: You don t get the object either way. π = π t = 0. Equivalence of English and sealed-bid, 2nd-price The object goes to the highest bidder. Price is close to the second highest BV.

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