Game Theory - Auctions at HelpWithAssignment.com

Auctions are among the oldest methods of selling items, with recorded occurrences as early as 500 BC. While historically not as common as sale by fixed price or bargaining, the development and proliferation of the internet and information technology has made auctions into a common means of selling and buying items, both large and small. At its most basic, an auction is a set of rules by which an item is sold to one of several potential buyers. In this set of rules by which an item is sold to one of the several potential buyers. There are different types of auctions available.

• The First Price Auction: In a first price auction, each bidder simultaneously writes down his name and a number on a slip of paper and submits the paper to the auctioneer. The auctioneer turns over all the of the papers, the bidder who wrote down the highest number is given the item and pays the amount that is written on his slip. Under certain assumptions, this procedure is identical to the so-called Dutch auction, or descending price auction. The price begins at some incredibly high amount and drops at a predetermined rate, for example 10 cents per second. The auction ends when one of the bidders presses a button which keeps the price from falling further and pays an amount of money, equal to the posted price when he stopped the clock. The other bidders receive nothing and pay nothing.
• The second price auction: In second price auction, each bidder simultaneously writes down his name and a number on a slip of paper and submits the papers to an auctioneer. The auctioneer turns over all the papers and the bidder who wrote down the highest number is given the item and pays the second highest amount that was written by other bidders. Under certain conditions and assumptions the auction is similar to that of Button auction and auction of fish in Japan. The bidders will have their names on a board with lights next to their names. The auction begins with zero and ascends at a predetermined rate. As long as the lights next to the name of a bidder is still on, the participant is still in the auction. Each bidder can press a button which will turn off the lights. The last bidder to be in the auction with lights on will win the item and pays the price at which his previous bidder dropped out. This is similar to the selling procedure on eBay.
• The All-Pay Auction: In all-pay price auction, each bidder simultaneously writes downs his name and a number on a slip of paper and submits the papers to an auctioneer. The auctioneer turns over all of the papers, the bidder who wrote down the highest number is given the item but all bidders who wrote down the highest number is given the item but all bidders have to pay their bids.
• The Lottery: In a lottery, each bidder can write down only one of the two bids, “In” or “Out”. All those players that are “in” are required to pay a price p, and have an equal chance of winning the item. Those that are “out” pay nothing and do not have the potential to win anything. We can think of these situations as Bayesian Games. In these games, the set of players will consist of a number of bidders, usually two. Given their type, each player chooses a bid, that is what to write down on the slip of paper and submit to the auctioneer. Given this outcome, each player’s payoff is

Ui (Qi, Pi/ V1, V2) = ViQi – Pi

Thus, each player’s payoff is his or her value for the good multiplied by the quantity of the good that he or she receives, minus any payment she has to make. Crucially, each buyer’s payoff only on his or her own value, any other bidder’s value for the item is irrelevant to each bidder. If two buyers are interested in acquiring a new painting from a young artist. If one art dealer has a very high valuation, it may mean, for example, that he or she has a client lined up for this piece, and although the other dealer may not know the identity of the client for certain, the fact that the other dealer places a high value on the painting may be important to a given art dealer’s assessment of its value.

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, Solow's Growth Model

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

Game Theory - Repeated Games at HelpWithAssignment.com

Repeated Games are an important and simple category of dynamic games. As the name suggests, repeated games are dynamic games generated by the repetition of some static game a certain number of times, either finite or infinite. On the other hand, repeated games are a simple class of dynamic games because the actions and payoffs to players stay the same over time, at the same time, this feature makes repeated games interesting. Any differences between the equilibrium outcomes of the static game must be coming from the fact that there are multiple periods in the interaction.

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).

		Player 2
		C	S
Player 1	C	-5, -5	0, -12
	S	-12, 0	-1, -1

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, Solow's Growth Model

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

Game Theory - Economics Assignment Help

Game Theory - Economics Assignment Help

Game theory models are used by managers in taking better decisions regarding price and output. As managers work in interactive payoff environments, they need a framework that anticipates the actions of others.

Game theory was developed in the early 40s in connection with war strategies. Now it is used by managers in respect of rational decisions in strategic situations. As the payoffs are dependent on the actions (behavior) of all players, managers must form expectations about the behavior of others. Specifically, because payoffs are interactive, a manager’s optimal decision depends on what he expects others to do. This means that others are similarly guided so all are aware that each must form beliefs about what others believe you believe and you believe others believe and so forth. This is called chain of reciprocal expectations. Good managers attempt to influence the behavior of others by systematically evaluating the variables subject to their control and using these variables to influence expectations.

Game theory is extremely helpful for making competitive sense of some of the most common strategic situations in the business world- situations where conflict interfaces with mutual dependence. Such games are common both within the organization and in market actions between firms, for example, in price wars, new product introductions, strikes, negotiations and divisional relationships. In these situations players having conflicting objectives but also share a mutual dependence.

Two –Person Games
Each player- a player may be a single player or an organization- is a decision making unit with a certain amount of resources. The rules of the game describe how these resources can be utilized. For ex: the President of a corporation wants his R&D program to start and what should be done at subsequent times in response to various actions of competing firms. The game’s outcome clearly depends on the strategies used by each player. In these Two-Person games, there is dominant strategy for each player.

Nash Equilibrium
Nash equilibrium uses solution concept, where two or more players are involved. In the game all the players are assumed to know the equilibrium strategies of other players and no player gains anything by changing his or her own strategy unilaterally. One player if he changes his strategy while others keep their strategies unchanged, then the current set of strategy choices and corresponding payoffs constitute a Nash equilibrium. For ex: If A and B are in Nash equilibrium, A makes the best decision, taking into account B’s decision and B is taking a decision taking into account A’s decision.

Prisoner’s Dilemma
A specific type of game called Prisoner’s Dilemma is particularly useful in oligopolistic competition. To illustrate this type of game, consider a situation where two people A and B get arrested for doing some illegal activity. The police question them separately saying, if you confess and your friend does not then you will get 2 years and he will get 8 years. If they both confess, they will get 8 years because they have cooperated with the police. If neither confesses, then they will both get only four years as evidence against them is weak. Both A and B have two possible strategies: to confess or not to confess. The four possible outcomes, depending on which strategy each person chooses are shown in the table below. Since A will serve less time than he would if he did not confess. If B confesses, the better strategy for A is to confess, since A will serve 8 years than he will serve 10 years. Thus, regardless of which strategy B adopts, A is better off to confess than not confess. Similarly, B will also confess regardless of which strategy A adopts, B is better off to confess than not to confess. Consequently, both A and B will confess. This is the dominant strategy for both players.

However, it is important to understand that both are doing worse if neither of them is confessing. Even if they were to meet and decide on something, they would still go ahead and break the agreement.

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