Question

Consider a first price auction for selling one item. There are n bidders. Each bidder i has a valuation vi for the item, which is privately known and drawn independently from a uniform distribution of interval [0,50]. Each bidder i bids a non-negative real number bi. The bidder who bids the highest number wins and if more than one bidder bid the highest, the winner is chosen uniformly at random. The winner gets the item and pays her bid. All other bidders pay 0.

(a) Assume that every bidder except i uses a linear strategy, i.e., bi(vi) =a+cvi, where the constants a and c are the same for all bidders. Find the expected payoff of bidder i when she bids bi. (Hint: For simplicity of calculations, assume that bidder i wins only when all the other bids are less than bi.)

(b) Find a Bayesian-Nash equilibrium for this auction.

(c) Under your equilibrium in part b, what is the expected revenue for the seller when n= 10? What happens to the expected revenue as n→∞?

Answer 1

Answer a) :

We give two guides to outline games in vital structure. 1 . The first game is the game of Prisoner’s Dilemma. Suppose N = {1, 2}. These players are prisoners. Because of lack of evidence, they have been questioned in separate rooms and made to confess their crimes. If they both confess, then they each achieve a payoff of 1. If both of them do not confess, then they can achieve higher payoffs of 2 each. However, if one of them confesses, but the other one does not confess, then the confessed player gets a payoff of 3 but the player who does not confess gets a payoff of 0. What are the strategies in this game? For both the players, the arrangement of techniques is {Confess (C), Do not admit (D)}. The payoffs from the four strategy profiles can be written in a matrix form. It is shown in Table 1. 2. Now, consider an example of an auction. There are two bidders in an auction. Every bidder reports an offer in the closeout. The most noteworthy bidder wins and pays a sum equivalent to his offer - if there should be an occurrence of ties, both win the article with equivalent likelihood. The result of the bidder from winning.

C (1, 1) (3, 0) D (0, 3) (2, 2) Table 1: The Prisoner's Dilemma is his worth less his offer - if there should be an occurrence of ties, 1 2 times this worth. The result of the bidder from losing is zero. The methodology of every player in this game in any non-negative genuine number. On the off chance that technique bi is utilized by player I ∈ {1, 2}, at that point fi(bi , b−i) be the likelihood of winning for bidder I - this is 1 if bi > b−i and zero if bi < b−i and 1 2 in any case. The utility of the bidder I is simply (vi −bi)fi(bi , b−i) at a system profile (bi , b−i). Notice that the arrangement of procedures for every bidder is the arrangement of all non-negative genuine numbers for this situation - a boundless set.

Answer b) :

A  Bayesian Nash equilibrium is a Nash equilibrium of the “expanded game” in which each player 's space of pure strategies is the set of maps from Θi to i . The ideas underlying these theorems and proofs are identical to those for the existence of equilibria in (complete information) strategic form games.Abstract: Perfect Bayesian equilibrium is not a subset of Nash equilibrium.

Flawless Bayesian harmony expects players to have convictions that are reliable with the balance systems of different players. Nash equilibrium does not explicitly specify the beliefs of the players.Finding the Nash equilibrium for any game involves two stages. It does not identify mixed-strategy Nash equilibria. An unadulterated system balance is the place every player plays one explicit technique. A blended methodology balance is the place in any event one player in the game randomizes over a few or the entirety of their unadulterated methodologies.

Answer c :

Expected revenue n =10/2 =5

Income is the pay created from typical business tasks and incorporates limits and conclusions for brought stock back. It is the top line or gross pay figure from which expenses are deducted to decide overall gain. Deals Revenue recipe. Income is otherwise called deals on the salary articulation.

For a business, salary alludes to net benefit for example , what remains after expenses and taxes are subtracted from revenue. Income is the aggregate sum of cash the business gets from its clients for its items and administrations. At the point when salary is spoken to as a level of income, it's called net revenue.

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