Game Theory Lecture 18 Introduction,Bayesian Nash Equilibria. Extensive form games of incomplete information,Perfect Bayesian Nash Equilibria. Game Theory Lecture 18 Incomplete Information,Incomplete Information. In the last lecture we studied incomplete information games where. one agent is unsure about the payo s or preferences of others. Examples abundant, Bargaining auctions market competition signaling games social. We modeled such games as Bayesian games that consist of. A set of players I, A set of actions pure strategies for each player i Si. A set of types for each player i i i,A payo function for each player i ui s1 sI 1 I. A joint probability distribution p 1 I over types or. P 1 I when types are not nite,Game Theory Lecture 18 Bayesian Games. Bayesian Games, Importantly throughout in Bayesian games the strategy spaces the. payo functions possible types and the prior probability distribution. are assumed to be common knowledge, A pure strategy for player i is a map si i Si prescribing an action. for each possible type of player i, Given p 1 I player i can compute the conditional distribution. p i i using Bayes rule where i 1 i 1 i 1 I, Player i knows her own type and evaluates her expected payo s. according to the conditional distribution p i i,Game Theory Lecture 18 Bayesian Games. Bayesian Nash Equilibria,De nition Bayesian Nash Equilibrium. The strategy pro le s is a pure strategy Bayesian Nash equilibrium if. for all i I and for all i i we have that,si i arg max. p i i ui si s i i i i,or in the non nite case,si i arg max. ui si s i i i i P d i i, Hence a Bayesian Nash equilibrium is a Nash equilibrium of the. expanded game in which each player i s space of pure strategies is. the set of maps from i to Si,Game Theory Lecture 18 Auctions. A major application of Bayesian games is to auctions. This corresponds to a situation of incomplete information because the. valuations of di erent potential buyers are unknown. We made the distinction between, Private value auctions valuation of each agent is independent of. others valuations, Common value auctions the object has a potentially common value. and each individual s signal is imperfectly correlated with this common. We have analyzed private value rst price and second price sealed bid. Each of these two auction formats de nes a static game of incomplete. information Bayesian game among the bidders, We determined Bayesian Nash equilibria in these games and compared. the equilibrium bidding behavior,Game Theory Lecture 18 Auctions. There is a single object for sale and N potential buyers bidding for it. Bidder i assigns a value vi to the object i e a utilityvi bi when he pays. bi for the object He knows vi This implies that we have a private value. auction vi is his private information and private value. Suppose also that each vi is independently and identically distributed on the. interval 0 v with cdf F with continuous density f and full support on. Bidder i knows the realization of its value vi and that other bidders values. are independently distributed according to F i e all components of the. model except the realized values are common knowledge. Bidders are risk neutral i e they are interested in maximizing their. expected pro ts, This model de nes a Bayesian game of incomplete information where the. types of the players bidders are their valuations and a pure strategy for a. bidder is a map,Game Theory Lecture 18 Auctions, With a reasoning similar to its counterpart with complete information we. establish in a second price auction it is a weakly dominant strategy to bid. truthfully i e according to II v v,Proposition, In the second price auction there exists a unique Bayesian Nash equilibrium. which involves, For rst price auctions we looked for a symmetric increasing and. di erentiable equilibrium,Proposition, In the rst price auction there exists a symmetric equilibrium given by. I v E y1 y1 v,Game Theory Lecture 18 Auctions, We also showed that both auction formats yield the same expected revenue. to the seller, Moreover we established the revenue equivalence theorem. Any symmetric and increasing equilibria of any standard auction such that the. expected payment of a bidder with value 0 is 0 yields the same expected revenue. to the seller,Game Theory Lecture 18 Common Value Auctions. Common Value Auctions A Simple Example, Common value auctions are more complicated because each player has to. infer the valuation of the other player which is relevant for his own. valuation from the bid of the other player or more generally from the fact. that he has one, The analysis of common value auctions is typically more complicated So we. will just communicate the main ideas using an example. Consider the following example There are two players each receiving a. signal si The value of the good to both of them is. where 0 Private values are the special case where 1 and. Suppose that both s1 and s2 are distributed uniformly over 0 1. Game Theory Lecture 18 Common Value Auctions,Second Price Auctions with Common Values. Now consider a second price auction, Instead of truthful bidding now the symmetric equilibrium is each. player bidding, Given that the other player is using the same strategy the probability. that player i will win when he bids b is,Pr i s i b Pr s i b. The price he will pay is simply i s i s i since this is a. second price auction,Game Theory Lecture 18 Common Value Auctions. Second Price Auctions with Common Values continued. Conditional on the fact that b i b i e winning we can compute. the expected payment as, Next let us compute the expected value of player i s signal. conditional on player i winning With the same reasoning this is. Game Theory Lecture 18 Common Value Auctions, Second Price Auctions with Common Values continued. Therefore the expected utility of bidding bi for player i with signal si. Ui bi si Pr bi wins si E s i bi wins, Maximizing this with respect to bi for given si implies. establishing that this is a symmetric Bayesian Nash equilibrium of this. common value auction,Game Theory Lecture 18 Common Value Auctions. First Price Auctions with Common Values, We can also analyze the same game under an auction format. corresponding to rst price sealed bid auctions, In this case with an analysis similar to that of the rst price auctions. with private values we can establish that the unique symmetric. Bayesian Nash equilibrium is for each player to bid. It can be veri ed that expected revenues are again the same This. illustrates the general result that revenue equivalence principle. continues to hold for common value auctions, Game Theory Lecture 18 Perfect Bayesian Equilibria. Incomplete Information in Extensive Form Games, Many situations of incomplete information cannot be represented as. static or strategic form games, Instead we need to consider extensive form games with an explicit. order of moves or dynamic games, In this case as mentioned earlier in the lectures we use information. sets to represent what each player knows at each stage of the game. Since these are dynamic games we will also need to strengthen our. Bayesian Nash equilibria to include the notion of perfection as in. subgame perfection, The relevant notion of equilibrium will be Perfect Bayesian Equilibria.

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