To incorporate this possibility, we introduce Bayesian Games, or games of incomplete information. Complete games are characterized by three things: players, actions and payoffs.Bayesian Games build on the static games of complete information.

In Bayesian Games, each player can be one of a certain number of types. Each player knows his or her own type, but cannot observe the type of the other players directly. From the perspective of a given player the others’ types are random draws from some given prior distribution.

The type of a player influences that player in two ways. First, the player’s payoff in the game can depend on both the actions chosen, and on the types of all players. Thus, the combination of types influences each player’s payoff. Second, each type of each player has its own set of possible actions in the game, but we frequently assume that the set of possible actions does not depend on the type of player. Given his type, each player in the game selects an action from those available.

Similar to dynamic games, in Bayesian Games a strategy for each player is a plan which specifies an action for each possible type. Knowing another player’s strategy in a Bayesian Game does not yet allow a player to predict the other’s action for certain, because other players’ type is unknown.

However, given the other player’s strategy and the player’s prior distribution each player can figure out his or her expected payoff from the actions that are available to him or her.

Of the actions available, each player will choose some particular action. Therefore, given his or her type, the strategy of each player and the prior distribution, each player can figure out his or her expected payoff from the actions that are available to him or her. Of the actions available, each player can therefore figure out a best possible action or actions for each of his or her own types. The best possible action for each of his or her types is a player’s best responses to each other then the strategies constitute a Bayesian Nash Equilibrium.

An Example of Bayesian Game: Take It Or Leave It Offer

As in the standard Take it or leave it offer game, there are two players: a buyer and a seller. Each player’s type is his value for the good, v_B; v_S.

Both values are independent draws form a uniform distribution on [0; 1]; so that Pr (vi ≤ x) = x. The buyer is given the opportunity to make a take it or leave it offer to the seller of a price p at which he or she will be willing to buy the good. If the price exceeds the seller’s value, then the seller accepts, if it isn’t the seller rejects the proposal. What price will the buyer choose? Given his or her value v_B the buyer selects her price to maximize her expected payoff

Max Pr (v_S > p) (v_B - p) =

max p (v_B - p)

p* = v_B/2

Thus the buyer “shades” her bid by half in order to trade o¤ the chance that her offer is rejected with her payoff if it is accepted. This leads to a outcomes that can be bad socially.

If the seller’s value were known, whenever the buyer’s value was larger the parties would transact; thus the party who values the good most would always receive it and the collective payoff is as large as possible.

When the seller’s value is unknown, the buyer shades his bid, so in order for trade to occur, the buyer’s value has to be larger than twice the seller’s value, which occurs less often. Thus, because a value is private, it is more difficult to trade and the collective good is harmed.

This is in continuation with our previous articles on Economics, Auctions, Prisoners’ Dilemma and Cutting a Cake

For more details you can visit our website at http://www.helpwithassignment.com/economics-assignment-help and http://www.helpwiththesis.com

We have discussed that there are 4 types of auctions

First Price Auction:

Second Price Auction:

All-Pay Auction:

Lottery:

In these type of auctions we have seen that in the First Price Auction, the item is sold to the bidder who quotes the highest price.

In the Second Price Auction, the item is sold to the bidder who quotes the highest amount but he will pay the second highest price that was quoted.

In All-Pay Auction, all the bidders will write down their numbers but, the item is sold to the highest quotation. But all the bidders will have to pay even though they have not received the item.

In Lottery, all the bidders will have an equal chance of winning the item. All of the bidders will pay their share of the price, but only one person will receive the item. So, there is a chance that anyone in the bid can win the lottery.

Here, we have discovered a fact: the first three types of auctions, all resulted in the same revenue for the seller, but the fourth auction type did not result in the same revenue. Furthermore, the first three auction types always sold the good to the buyer who had the highest value of an item, but in lottery, if both buyers enter, there is a chance that the buyer whose value is lower can also win the lottery.

We will use the technique of mechanism design and apply this technique for the design of auctions. We will use techniques to determine which auction format will raise the most revenues for the seller among all the conceivable auction formats.

We shall retain the assumptions of symmetric, independent and private values. Each player’s type, v_i represents his or her value for the seller’s good. Given her value for the good v_i, the quantity of the good that she receives, q_i, and the payment that she must make p_i, each buyer’s payoff was assumed to be:

U_i (q_i p_i/ v₁ v₂) = v_iq_i - p_i

The goal of mechanical design is to represent any auction in a simple way. The rules of any auction can be potentially very complicated, but ultimately all of the rules boil down to just a quantity of the goods sold to player i, q_i (b₁, b₂) given the bids of both players and a payment required of player i, p_i (b₁, b₂) given the bids of both players.

q₁ (b₁; b₂) = 1 if b₁ > b₂

= 0 if b1 < b₂

p₁ (b₁; b₂) = b₁ if b₁ &gt; b₂

= 0 if b₁ &lt; b₂

First Price Auction

so that the high bidder wins the auction and pays his or her bid. The lottery, for example, could be written

q₁ (b₁; b₂) = 1 if b₁ = 1 and b₂ = 0

= 1/2 if b₁ = 1 and b₂ = 1

= 0 if b₁ = 0

p₁ (b₁; b₂) = p if b₁ = 1

= 0 if b₁ = 0

Lottery

Another type of auction, which we have not considered is the “Tullock Lottery,” in which

your chance of winning the good is just your share of the total of the bids

q₁ (b₁; b₂) = b₁/b₁+b₂

p₁ (b₁; b₂) = pb₁

Tullock Lottery

This is in continuation with our previous articles on Economics, Auctions, Prisoners’ Dilemma and Cutting a Cake

For more details you can visit our website athttp://www.helpwithassignment.com/economics-assignment-help and http://www.helpwiththesis.com

The General Setup of the Game:

Starting with static game G, called the “stage game” one can construct a new game which is the repetition of G game with T rounds, where T can be either finite or infinite. In each round, all players simultaneously choose actions from their sets of possible actions inherited from game G. A player observes the outcome of the stage game before the next round of play. Players can thus observe all of the past outcomes of the stage game when they choose their action in any particular round. All of the past outcomes are referred to as ‘history of the game’. Players can choose different actions in the stage game depending on the history of play up to that point in the repeated game. A strategy for a player is therefore a plan which specifies a particular action of the stage game for each possible history of play. Taking account of all players’ strategies determines a sequence of outcomes associated with a payoff for each player. Suppose it is period          k <= T. The strategies chosen by all players lead to a sequence of payoffs for a given player. The player’s payoff in the repeated game in period

k = U_k + EU_k+1 + E²U_k+2 +…+ E^T-k U_T

In other words, a player’s payoff is discounted sum of the stream of payoffs from the stage game, with payoffs that are received in the future reduced by a factor of E<1. There are two equivalent ways to think of E, as a representation of impatience. Typically, if one is willing to pay some money to get some benefit in the future, one would be willing to pay more for that benefit today. Conversely, a monetary payoff today can be invested at the risk free interest rate until next year.  Therefore,  the same amount of money received next year is worth less than today than the same amount of money received today.

Consider the example of Prisoners’ Dilemma once more, this time with a repetition of the game. In this game C is a dominant strategy for both players and this game has a unique Nash Equilibrium (C,C) worse for either player than (S,S).

Let’s consider a new game: the repetition of this game two times. A strategy for each player specifies and action in the first period and an action in the second period for each of the four possible outcomes in the first period.

A sample strategy for player 1 is

Period 1: Play S

Period 2: Plays S if (S,S) otherwise C

If both players adopt this strategy, in period one the outcome will be (S, S) leading to a payoff of (-1, -1) for period one. In period 2, the strategy dictates that each player will play S again, leading to a payoff of (-1, -1). The period one payoff to each player is just the payoff in period one, plus N times the anticipated payoff in period 2.

-1 + (-1)N.

This article is in continuation with our previous articles on Economics and Game Theory which include Prisoners’ Dilemma, Battle of the sexes, Cutting a Cake.

For more details you can visit our website at http://www.helpwithassignment.com/economics-assignment-help and http://www.helpwiththesis.com