10.1 Bipartite Graphs and Perfect Matchings

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "10.1 Bipartite Graphs and Perfect Matchings"

Transcription

1 10.1 Bipartite Graphs and Perfect Matchings In bipartite graph the nodes are divided into two categories and each edge connects a node in one category to a node in the other category. For example: Administrators of a college dormitory are assigning rooms to students, each room for single student. Students and rooms are categories. When there are an equal number of nodes on each side of a bipartite graph, a perfect matching is an assignment of nodes on the left to nodes on the right, in such a way that I) each node is connected by an edge to the node it is assigned to, and II) no two nodes on the left are assigned to the same node on the right A set of nodes are called constricted set when their edges constrict the formation of a perfect matching. In figure 10.2b Vikram, Wendy and Xin form a constricted set. Matching Theorem: If a bipartite graph (with equal numbers of nodes on the left and right) has no perfect matching, the it must contain a constricted set. Proof is presented in more detail in chapter 10.6.

2 10.2 Valuations and Optimal Assignments To extend this mode we will add weights to edges. In the last example we would tell how much a student likes each choice. We refer to these numbers as students' valuations. This way we can evaluate the quality of an assignment. For example in figure 10.3b we can calculate the quality of the assignment. The maximum possible quality is the optimal assignment Prices and the Market-Clearing Property Now lets enter the world of real estate where we have houses, sellers and buyers, where two different buyers may have very different valuations for the same house. Buyers payoff is her valuation for the house, minus the amount of money she had to pay. She will of course try to maximize her payoff by choosing between several sellers. We call the seller or sellers that maximize the payoff for buyer j, the preferred sellers for buyer j IF the payoff is positive. In figure 10.5b (preferred-seller graph) all buyers value the house of seller a the highest but for some reason each buyer ends up with different house, why? The answer is price (and payoff). We call such a set of prices market clearing. A set of prices is market-clearing if the resulting preferred-seller graph has a perfect matching. x's payoff for a: 12-5=7 y's payoff for a: 8-5=3 (but 6 for c) z's payoff for a: 7-5=2 (but 3 for b) Optimality of Market- Clearing Prices: A set of market-clearing prices, and a perfect matching in the resulting preferred-seller graph, provides the maximum possible sum of payoffs to all sellers and buyers. Check details from chapters Properties of market-clearing prices, pages

3 10.4 Constructing a Set of Market-Clearing Prices Lets look at auctions next. 1. At the start of each round, there is a current set of prices, with the smallest one equal to 0 (initially all sellers set their prices to 0). 2. We construct the preferred-seller graph and check whether there is a perfect matching. 3. If there is, we're done: the current prices are market-clearing. 4. If not, we find a constricted set of buyers S and their neighbors N(S). 5. Each seller in N(S) raises his price by one unit. 6. If necessary, we reduce the prices the same amount is substracted from each price so that the smallest price becomes zero. 7. We now begin the next round. We say that auction must come to an end. Only way auction can come to end is if it reaches a set of marketclearing prices. To ease our way there we define potential energy of the auction to be the sum of potential of all participants: potential of a buyer is the maximum payoff potential of a seller is the current prices his charging Auction starts with all sellers with potential of 0 and buyers equal to their maximum payoffs. When sellers in N(S) all raise their prices by one unit in phase 5, their potentials go up by one unit and the potential of each buyer in S goes down by one unit. Since S has strictly more nodes than N(S) does, this means that the potential energy of the auction goes down by at least one unit more than it goes up. This means that we start auction with potential P 0 and cannot drop below 0 so the auction must come to and end within P 0 steps. And when it does, we have market-clearing prices How Does this Relate to Single-Item Auctions? We need equal number of buyers and sellers so we fake additional sellers and give buyers a valuation of 0 for the item offered by each of these fake sellers. Which buyer ends up paired with the real seller in perfect matching is the winner. And from a set of market-clearing prices, we will see what the real item sells for Advanced material: A Proof of the Matching Theorem

4 You might want to read pages by yourself, since I will give a very short view of the material. Matching Theorem: If a bipartite graph (with equal numbers of nodes on the left and right) has no perfect matching, the it must contain a constricted set. Take a bipartite graph with perfect matching, and consider a matching that includes as many nodes as possible we will call this a maximum matching. We will then try to enlarge it. If the search for a larger matching fails, it produces a constructed set. Lets take a look at a simple bipartite graph with matching and non-matching edges (Fig 10.10). We start the search for a larger matching from node W and we continue the path alternating between matched and nonmatched edges as long as we can while never repeating any nodes. We call this a simple alternating path. If we can find an alternating path that begins and ends at an unmatched node, then we can swap the roles of all edges on this path. We call this an augmenting path, since it gives us a way to augment the matching. We're not interested in alternating paths that don't reach D like W-B-Y-C-Z or paths that are not alternating like W-B-Z- C-Y-D. There's a natural procedure we can use to search for an augmenting path. It works by simply adapting the breadth-first search (BFS) procedure to include the requirement of alternation. We call this procedure as alternating BFS. If alternating BFS fails to find and augmenting path, we can in fact extract from this failed search a constricted set that proves there is no perfect matching.

5 The set of nodes on all even layers, at the end of a failed alternating BFS, forms a constricted set. Claim: Consider any bipartite graph with a matching, and let W be any unmatched node on the right-hand ride. Then either there is an augmenting path beginning at W, or there is a constricted set containing W. This is indeed enough to prove there is no perfect matching. However, it does not mean that the current matching has maximum size. If there is no augmenting path beginning at any node on the right hand side, then in fact the current matching has maximum size. We can search for maximum size by making all the unmatched nodes on the right constitute layer 0 in the alternating BFS, and otherwise running it as before. The if an unmatched node on the left is ever reached in some layer, we can follow the path from the appropriate node in layer 0 down to it, producing an augmenting path.

6 Trade with Intermediaries (traders). In a wide range of markets, individual buyers and sellers do not interact directly with each other, but instead trade through intermediaries(traders) brokers, market makers, or middlemen who set the prices. To get a sense for how markets with intermediaries typically work, this chapter will uses examples of how buyers and sellers interact on the stock market. New terms: Bid - the highest outstanding offer to buy the stock/goods. (Trader offers bid to seller) Ask - lowest outstanding offer to sell the stock/goods. (Trader offers ask to buyer) Interesting/Further read: Dark pools - p 279 (short) order book/limit orders/market order ( ) A Model of Trade on Networks Our network model will be based on three fundamental principles that exist on the stock market: individual buyers and sellers often trade through traders not all buyers and sellers have access to the same traders and not all buyers and sellers trade at the same price The prices that each buyer and seller commands are determined in part by the range of alternatives that their respective network positions provide. Network structure used in this chapter For the simplest form of the model, we don t try to address the issue of multiple goods for sale, or multiple possible quantities; instead, we assume there is a single type of good that comes in indivisible units. Each seller i initially holds one unit of the good, which he values at vi ; he is willing to sell it at any price that is at least vi. Each buyer j values one copy of the good at vj and will try to obtain a copy of the good if she can do it by paying no more than vj. No individual wants more than one copy of the good, so additional copies are valued at 0. All buyers, sellers, and traders are assumed to know these valuations. Since we assume that the traders act as intermediaries for the possible seller buyer transactions, we require that each edge connects a buyer or seller to a trader.

7 Assumptions: We assume that buyers have the same valuation for all copies of a good. In contrast to chapter 10 the network here is fixed and externally imposed by constraints such as geography (in agricultural markets) or eligibility to participate (in different financial markets) 1 Sample trading network model In all of our figures depicting trading networks we will use the following conventions. Sellers are represented by circles on the left, buyers are represented by circles on the right, and traders are represented by squares in the middle. The value that each seller and buyer places on a copy of the good is written next to the respective node that represents them.

8 Prices and the Flow of Goods. The flow of goods from sellers to buyers is determined by a game in which traders first set prices, and then sellers and buyers react to these prices. Figure 1 Sample trade network Once traders announce prices, each seller and buyer chooses at most one trader to deal with each seller sells his copy of the good to the trader he selects (or keeps his copy of the good if he chooses not to sell it) each buyer purchases a copy of the good from the trader she selects (or receives no copy of the good if she does not select a trader) trader can only sell as many goods to buyers as he receives from sellers Because each seller has only one copy of the good, and each buyer only wants one copy, at most one copy of the good moves along any edge in the network.

9 Payoffs A trader s payoff is the profit he makes from all his transactions: it is the sum of the ask prices of his accepted offers to buyers minus the sum of the bid prices of his accepted offers to sellers For a seller i, the payoff from selecting trader t is b ti, while the payoff from selecting no trader is vi. In the former case, the seller receives b ti units of money, while in the latter he keeps his copy of the good, which he values at v i. For each buyer j, the payoff from selecting trader t is vj atj, while the payoff from selecting no trader is 0. In the former case, the buyer receives the good but gives up atj units of money. Figure 2 Flow of goods For example, with prices and the flow of goods as in Figure 2(b), the payoff to the first trader is ( ) = 0.6 while the payoff to the second trader is ( ) = 1.4. The payoffs to the three sellers are 0.2, 0.3, and 0, respectively, while the payoffs to the three buyers are = 0.2, = 0.3, and 1 1 = 0, respectively. Best Responses and Equilibrium. Because the game in this chapter is in 2 stages (traders set prices; buyers/sellers accept/reject them) this equilibrium is called a subgame perfect Nash equilibrium; in this chapter, we will simply refer to it as an equilibrium. detailed explanation on page 286 Monopoly and perfect competition Monopoly. Buyers and sellers are subject to monopoly in our model when they have access to only a single trader. Perfect Competition. Perfect competition - game condition when both seller and buyer have access to 2 same traders.

10 Whichever trader is performing the trade at equilibrium must have a payoff of 0: he must be offering the same value x as his bid and ask. Because if he offers greater ask or bid (makes profit) other trader can undercut him, by providing better deal to the buyer/seller. Using the previous example it is not hard to work out the equilibrium in it: Sellers S1 and S3, buyers B1and B3 are monopolized and S2&B2 getting benefit from perfect competition. Implicit Perfect Competition. it turns out that traders can make zero profit for reasons based on the global structure of the network, rather than on direct competition with any one trader. In any equilibrium, all bid and ask prices take on some common value x between 0 and 1, and the goods flow from the sellers to the buyers. So all traders again make zero profit.

11 Ripple Effects from Changes to a Network. Here we will explore how small changes to a network can produce effects that ripple to more distant parts of the network (a) (b) Picture 2 all sellers and buyers are monopolized except for B2 we use indifference to assume that B3 will not buy the good but B1 and B4 will Note that there cannot be an equilibrium in which buyer B2 buys from trader T2, since B2 can pay only 2 while trader T2 can sell the unit of the good he is able to buy at a price of 4. Once S2 and T2 form a link, creating the network in example (b), a number of things change. First, and most noticeably, buyer B3 now gets a copy of the good while B1 doesn t. There are other changes as well. Seller S2 is now in a much more powerful position and will command a significantly higher price (since y is at least 1 in any equilibrium). Moreover, the range of possible equilibrium asks to B2 has been reduced from the interval [0, 2] to the interval [1, 2]. So in particular, if we were previously in an equilibrium where the ask to B2 was a value x< 1, then this equilibrium gets disrupted and replaced by one in which the ask is a higher number, y 1. This indicates a subtle way in which B2 was implicitly benefitting from the weak position of the sellers, which has now been strengthened by the creation of the edge between S2 and T2. Social Welfare in Trading Networks Socially optimal solution - solution that maximizes social welfare (sum of payoffs of all players). The social welfare is simply the sum of v j v i over all goods that move from a seller i to a buyer j The maximum value of this quantity over all possible flows of goods the socially optimal value depends not just on the valuations of the sellers and buyers but also on the network structure. Networks that are more richly

12 connected can potentially allow a flow of goods achieving a higher social welfare than networks that are more sparsely connected, with bottlenecks that prevent a desirable flow of goods. If we look at previous case (Picture 2) in each case, the equilibrium yields a flow of goods that achieves the social optimum. In (a), the best possible value of the social welfare is = 7, since there is no way to use the network to get copies of the goods to both B3 and B4. However, when the single edge from S2 to T2 is added, it suddenly becomes possible for both of these buyers to receive copies of the good, and so the value of the social welfare increases to = 9. This provides a simple illustration of how a more richly connected network structure can enable greater social welfare from trade. More to read: Equilibria and Social Welfare (p294) tl;dr; Every equilibrium produces a flow of goods that achieves the social optimum. Trader Profits The examples we ve studied so far suggest the informal principle that, as the network becomes more richly connected, individual traders have less and less power, and their payoffs go down. This example is a little bit counter intuitive. Here, traders T1 and T2 both have monopoly power over their respective sellers, and yet their profits are zero in every equilibrium. We can verify this fact as follows. First, we notice that any equilibrium must look like one of the solutions in (b) or (c). The sellers are monopolized and will get bids of 0. For each buyer, the two asks must be the same; otherwise, the trader making the sale could slightly raise his ask. Now, finally, notice that if the common ask to either buyer were positive, then the trader left out of the trade on the higher one has a profitable deviation by slightly undercutting this ask. Therefore, in this example, all bids and asks equal 0 in any equilibrium, and so neither trader profits. Trader profit: for a given trader T in a network, there exists an equilibrium in which T receives a positive payoff. It turns out that there exists such an equilibrium precisely when T has an edge e to a seller or buyer such that deleting e would change the value of the social optimum. In such a situation, we say that e is an essential edge from T to the other node.

13 Bargaining and Power in Networks In chapter 12.1 the notion of power is introduced. Power is not a property of an individual as it is a property of a relation between two individuals. Figures 1. Social network of five peoples, with node B occupying an intuitively powerful position.

14 Experimental Studies of Power and Exchange 1. A small graph (such as the one in Figure 1) is chosen, and a distinct volunteer test subject is chosen to represent each node. Each person, representing a node, sits at a computer and can exchange instant messages with the people representing the neighboring nodes. 2. The value in each social relation is made concrete by placing a resource pool on each edge let s imagine this as a fixed sum of money, say $1, which can be divided between the two endpoints of the edge. We will refer to a division of this money between the endpoints as an exchange. Whether this division ends up equal or unequal will be taken as a sign of the asymmetric amounts of power in the relationship that the edge represents. 3. Each node is given a limit on the number of neighbors with whom she can perform an exchange. The most common variant is to impose the extreme restriction that each node can be involved in a successful exchange with only one of her neighbors; this is called the 1-exchange rule. Given this restriction, the set of exchanges that take place in a given round of the experiment can be viewed as a matching in the graph: a set of edges that have no endpoints in common. However, it will not necessarily be a perfect matching, since some nodes may not take part in any exchange. For example, in the graph in Figure 1, the exchanges will definitely not form a perfect matching, since there are an odd number of nodes. 4. Here is how the money on each edge is divided. A given node takes part in simultaneous sessions of instant messaging separately with each of her neighbors in the network. In each, she engages in relatively free-form negotiation, proposing splits of the money on the edge, and potentially reaching an agreement on a proposed split. These negotiations must be concluded by a fixed time limit; and to enforce the 1-exchange rule defined above, as soon as a node reaches an agreement with one neighbor, her negotiations with all other neighbors are immediately terminated. 5. Finally, the experiment is run for multiple rounds. Results of Network Exchange Experiment Figures 2. Path of length (a) 2, (b) 3, (c) 4 and (d) 5 form intstructive examples of different phenomena in exchange networks. In two nodes path the equal split is more reasonable.

15 In three nodes path B receives the overwhelming majority of the money in her exchange (roughly 5/6). If the one exchange rule will be modified to allow B to take part on two exchanges in each round. B will get roughly equal footing with A and C. In four-node path B should have some amount of power over A, but it is weaker kind of power than in the three-nodes path. Experiments showed that B gets roughly between 7/12 and 2/3 of the money, but not more. In five-node path node C which occupies central position, is in fact weak if one exchange rule is used. Experiments showed that C does slightly better than A and E. For example, if we allowed A,C and E to take part in one exchange each, but allowed B and D to take part in two exchange each, C become powerful node.

16 Modeling Two-Person Interaction: The Nash Bargaining Solution. Two people, A and B, are negotiating over how to split $1 between them. But A also has an outside option of x, and B has an outside option of y. Figures 3. Two nodes bargaining with outside options. Notice that if x+y>1, then no agreement between A and B is possible, since they cannot divide dollar so that one gets at least x and the other gets at least y. Consequently, we will assume that x+y 1. A requires at least x from the negotiation, and B requires at least y. Consequently, the negotiation is really over how to split the surplus s=1-x-y. The natural prediction is that A and B will split surplus equally, so A gets x+s/2 and B gets y+s/2. Nash Bargaining Solution is x+s/2 = (x+1-y)/2 to A, and y+s/2 = (y+1-x)/2 to B. Status effects. Benefits about differential status can lead to deviations from theoretical prediction in bargaining.

17 Modeling Two-Person Interaction: Ultimatum Game. (i) Person A is given a dollar and told to propose a division of it to person B. That is, A should propose how much he keeps for himself, and how much he gives to B. (ii) Person B is then given the option of approving or rejecting the proposed division. (iii) If B approves, each person keeps the proposed amount. If B rejects, then each person gets nothing. Prediction: B s choice is between getting any positive amount of money and getting nothing. Hence, B should accept any positive amount of money. Experimental results: when a player B evaluates an outcome in which she walks away with only 10% of the total, there is a significant negative emotional payoff to being treated unfairly, and hence when we consider B s complete evaluation of the options, B finds a a greater overall benefit to rejecting the low offer and feeling good about it than accepting the low offer and feeling cheated. Moreover, since people playing the role of A understand that this is the likely evaluation that their partner B will bring to the situation, they tend to offer relatively balanced divisions to avoid rejection, because rejection results in A getting nothing as well. Modeling Network Exchange: Stable Outcomes Figure 4. Some examples of stable and unstable outcomes of network exchange on the three-node path and the four-node path: (a) unstable and (b) stable outcomes on the three-node path, (c) unstable and (d, e) stable outcomes on the four-node path. The darkened edges constitute matchings showing who exchanges with whom, and the numbers above the nodes represent the values. Instability: Given an outcome consisting of a matching and values for the nodes, an instability in this outcome is an edge not in the matching, joining two nodes X and Y, such that the sum of X s value and Y s value is less than 1. Stability: An outcome of network exchange is stable if and only if it contains no instabilities.

18 Modeling Network Exchange: Balanced Outcomes Figure 5. The difference between balanced and unbalanced outcomes: (a) not balanced, (b) balanced, and (c) not balanced. Balanced Outcome: An outcome (consisting of a matching and node values) is balanced if, for each edge in the matching, the split of the money represents the Nash bargaining outcome for the two nodes involved, given the best outside options for each node provided by the values in the rest of the network. To read: 12.2 experimental studies of power and exchange 12.3 results of network exchange experiments Advanced material. Exercises:

Bargaining and Power in Networks

Bargaining and Power in Networks From the book Networks, Crowds, and Markets: Reasoning about a Highly Connected World. By David Easley and Jon Kleinberg. Cambridge University Press, 2010. Complete preprint on-line at http://www.cs.cornell.edu/home/kleinber/networks-book/

More information

Chapter 11. Network Models of Markets with Intermediaries Price-Setting in Markets

Chapter 11. Network Models of Markets with Intermediaries Price-Setting in Markets From the book Networks, Crowds, and Markets: Reasoning about a Highly Connected World. By David Easley and Jon Kleinberg. Cambridge University Press, 2. Complete preprint on-line at http://www.cs.cornell.edu/home/kleinber/networks-book/

More information

Chapter 10. Matching Markets. 10.1 Bipartite Graphs and Perfect Matchings

Chapter 10. Matching Markets. 10.1 Bipartite Graphs and Perfect Matchings From the book Networks, Crowds, and Markets: Reasoning about a Highly Connected World. By David Easley and Jon Kleinberg. Cambridge University Press, 2010. Complete preprint on-line at http://www.cs.cornell.edu/home/kleinber/networks-book/

More information

Midsummer Examinations 2013

Midsummer Examinations 2013 [EC 7089] Midsummer Examinations 2013 No. of Pages: 5 No. of Questions: 5 Subject [Economics] Title of Paper [EC 7089: GAME THEORY] Time Allowed [2 hours (TWO HOURS)] Instructions to candidates Answer

More information

Chapter 7. Sealed-bid Auctions

Chapter 7. Sealed-bid Auctions Chapter 7 Sealed-bid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)

More information

Midterm Exam. Economics Spring 2014 Professor Jeff Ely

Midterm Exam. Economics Spring 2014 Professor Jeff Ely Midterm Exam Economics 410-3 Spring 014 Professor Jeff Ely Instructions: There are three questions with equal weight. You have until 10:50 to complete the exam. This is a closed-book and closed-notebook

More information

Bayesian Nash Equilibrium

Bayesian Nash Equilibrium . Bayesian Nash Equilibrium . In the final two weeks: Goals Understand what a game of incomplete information (Bayesian game) is Understand how to model static Bayesian games Be able to apply Bayes Nash

More information

Cournot s model of oligopoly

Cournot s model of oligopoly Cournot s model of oligopoly Single good produced by n firms Cost to firm i of producing q i units: C i (q i ), where C i is nonnegative and increasing If firms total output is Q then market price is P(Q),

More information

ECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015

ECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015 ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2015 These notes have been used before. If you can still spot any errors or have any suggestions for improvement, please let me know. 1

More information

Economics 200C, Spring 2012 Practice Midterm Answer Notes

Economics 200C, Spring 2012 Practice Midterm Answer Notes Economics 200C, Spring 2012 Practice Midterm Answer Notes I tried to write questions that are in the same form, the same length, and the same difficulty as the actual exam questions I failed I think that

More information

The Shapley Value and the Core

The Shapley Value and the Core The Shapley Value and the Core Lecture 23 The Shapley Value and the Core Lecture 23, Slide 1 Lecture Overview 1 Recap 2 Analyzing Coalitional Games 3 The Shapley Value 4 The Core The Shapley Value and

More information

If V = 4, there are only four times at which trade can happen before the pie vanishes.

If V = 4, there are only four times at which trade can happen before the pie vanishes. Problem Set 4 1. Imagine an alternating-offer bargain game, but instead of discounting future payoffs by δ and keeping the size of the pie fixed at 1, assume that in each period the value of the pie is

More information

21. Unverifiable Investment, Hold Up, Options and Ownership

21. Unverifiable Investment, Hold Up, Options and Ownership 21. Unverifiable Investment, Hold Up, Options and Ownership This chapter applies the tools of games with joint decisions and negotiation equilibrium to study the hold up problem in economics. We first

More information

Mechanism Design without Money I: House Allocation, Kidney Exchange, Stable Matching

Mechanism Design without Money I: House Allocation, Kidney Exchange, Stable Matching Algorithmic Game Theory Summer 2015, Week 12 Mechanism Design without Money I: House Allocation, Kidney Exchange, Stable Matching ETH Zürich Peter Widmayer, Paul Dütting Looking at the past few lectures

More information

Chapter 11: Game Theory & Competitive Strategy

Chapter 11: Game Theory & Competitive Strategy Chapter 11: Game Theory & Competitive Strategy Game theory helps to understand situations in which decision-makers strategically interact. A game in the everyday sense is: A competitive activity... according

More information

Best Response Function: gives each player's payoff-maximizing strategy as a function of the other players' strategies.

Best Response Function: gives each player's payoff-maximizing strategy as a function of the other players' strategies. OLIGOPOLY Readings Ch. 15 sections 1-4, Ch. 16 sections 1-10 1. A Little Bit of Game Theory Readings: Ch. 15 sections 1-4 A game consists of (at a minimum) players, strategies, and payoffs. The players

More information

Lesson 5: Competition and Market Structure

Lesson 5: Competition and Market Structure Lesson 5: Competition and Market Structure Overview: The entrepreneur will operate in a market that may, or may not, have others selling the same or similar products. If the entrepreneur is the only seller

More information

Lecture 11: Sponsored search

Lecture 11: Sponsored search Computational Learning Theory Spring Semester, 2009/10 Lecture 11: Sponsored search Lecturer: Yishay Mansour Scribe: Ben Pere, Jonathan Heimann, Alon Levin 11.1 Sponsored Search 11.1.1 Introduction Search

More information

Market Equilibrium and Applications

Market Equilibrium and Applications Market Equilibrium and Applications I. Market Equilibrium In the previous chapter, we discussed demand and supply, both for individual consumers and firms and for markets. In this chapter, we will combine

More information

Week 7 - Game Theory and Industrial Organisation

Week 7 - Game Theory and Industrial Organisation Week 7 - Game Theory and Industrial Organisation The Cournot and Bertrand models are the two basic templates for models of oligopoly; industry structures with a small number of firms. There are a number

More information

Universidad Carlos III de Madrid Game Theory Problem set on Repeated Games and Bayesian Games

Universidad Carlos III de Madrid Game Theory Problem set on Repeated Games and Bayesian Games Session 1: 1, 2, 3, 4 Session 2: 5, 6, 8, 9 Universidad Carlos III de Madrid Game Theory Problem set on Repeated Games and Bayesian Games 1. Consider the following game in the normal form: Player 1 Player

More information

It is totally forbidden to talk at any stage during the examination.

It is totally forbidden to talk at any stage during the examination. DO NOT LOOK AT THE EXAMINATION PAPER UNTIL YOU ARE TOLD TO DO SO READ CAREFULLY THE RULES OF THIS EXAM The first thing that you should do is fill in the oval corresponding to the number of your traccia.

More information

8 Modeling network traffic using game theory

8 Modeling network traffic using game theory 8 Modeling network traffic using game theory Network represented as a weighted graph; each edge has a designated travel time that may depend on the amount of traffic it contains (some edges sensitive to

More information

Lobbying contests with endogenous policy proposals

Lobbying contests with endogenous policy proposals Lobbying contests with endogenous policy proposals Johannes Münster March 31, 2004 Abstract Lobbyists choose what to lobby for. This endogeneity of policy proposals has recently been modelled by Epstein

More information

Industry profit in an oligopoly (sum of all firms profits) < monopoly profit.

Industry profit in an oligopoly (sum of all firms profits) < monopoly profit. Collusion. Industry profit in an oligopoly (sum of all firms profits) < monopoly profit. Price lower and industry output higher than in a monopoly. Firms lose because of non-cooperative behavior : Each

More information

Monopolistic Competition, Oligopoly, and maybe some Game Theory

Monopolistic Competition, Oligopoly, and maybe some Game Theory Monopolistic Competition, Oligopoly, and maybe some Game Theory Now that we have considered the extremes in market structure in the form of perfect competition and monopoly, we turn to market structures

More information

Market Game in Oil. 1. Understand how the interaction of buyers and sellers set the price in markets.

Market Game in Oil. 1. Understand how the interaction of buyers and sellers set the price in markets. Market Game in Oil This lesson has been modified, for use in Foundation for Teaching Economics materials by Kathy Ratté and Kenneth Leonard, from The Big Apple published in In The Marketplace, 1978. INTRODUCTION

More information

Oligopoly: How do firms behave when there are only a few competitors? These firms produce all or most of their industry s output.

Oligopoly: How do firms behave when there are only a few competitors? These firms produce all or most of their industry s output. Topic 8 Chapter 13 Oligopoly and Monopolistic Competition Econ 203 Topic 8 page 1 Oligopoly: How do firms behave when there are only a few competitors? These firms produce all or most of their industry

More information

Competitive markets as two-player games

Competitive markets as two-player games Competitive markets as two-player games Antonio Quesada Departament d Economia, Universitat Rovira i Virgili, Avinguda de la Universitat 1, 43204 Reus, Spain 23rd December 2008 126.14 Abstract With the

More information

1. Supply and demand are the most important concepts in economics.

1. Supply and demand are the most important concepts in economics. Page 1 1. Supply and demand are the most important concepts in economics. 2. Markets and Competition a. Market is a group of buyers and sellers of a particular good or service. P. 66. b. These individuals

More information

Chapter 29: Natural Monopoly and Discrimination

Chapter 29: Natural Monopoly and Discrimination Chapter 29: Natural Monopoly and Discrimination 29.1: Introduction This chapter discusses two things, both related to the fact that, in the presence of a monopoly, there is less surplus generated in the

More information

ECON 312: Oligopolisitic Competition 1. Industrial Organization Oligopolistic Competition

ECON 312: Oligopolisitic Competition 1. Industrial Organization Oligopolistic Competition ECON 312: Oligopolisitic Competition 1 Industrial Organization Oligopolistic Competition Both the monopoly and the perfectly competitive market structure has in common is that neither has to concern itself

More information

Intermediate Microeconomics. Chapter 13 Monopoly

Intermediate Microeconomics. Chapter 13 Monopoly Intermediate Microeconomics Chapter 13 Monopoly Non-competitive market Price maker = economic decision maker that recognizes that its quantity choice has an influence on the price at which it buys or sells

More information

MA300.2 Game Theory II, LSE

MA300.2 Game Theory II, LSE MA300.2 Game Theory II, LSE Lecture 10: Sequential Games with Imperfect Information 1. The Spence Signaling Model Or: a model of education in which you don t really learn anything... [But that s not why

More information

Oligopoly and Strategic Pricing

Oligopoly and Strategic Pricing R.E.Marks 1998 Oligopoly 1 R.E.Marks 1998 Oligopoly Oligopoly and Strategic Pricing In this section we consider how firms compete when there are few sellers an oligopolistic market (from the Greek). Small

More information

Game Theory and Algorithms Lecture 10: Extensive Games: Critiques and Extensions

Game Theory and Algorithms Lecture 10: Extensive Games: Critiques and Extensions Game Theory and Algorithms Lecture 0: Extensive Games: Critiques and Extensions March 3, 0 Summary: We discuss a game called the centipede game, a simple extensive game where the prediction made by backwards

More information

Econ 101A Final exam Th 14 December. Do not turn the page until instructed to.

Econ 101A Final exam Th 14 December. Do not turn the page until instructed to. Econ 101A Final exam Th 14 December. Do not turn the page until instructed to. 1 Econ 101A Final Exam Th 14 December. Please solve Problem 1, 2, and 3 in the first blue book and Problem 4 in the second

More information

Consumer and Producer Surplus. Consumer and Producer Surplus. Consumer Surplus. Consumer Surplus. Consumer Surplus Individual consumer surplus

Consumer and Producer Surplus. Consumer and Producer Surplus. Consumer Surplus. Consumer Surplus. Consumer Surplus Individual consumer surplus Consumer and Consumer and February 6, 2007 Reading: Chapter 6 Introduction Consumer surplus Producer surplus Efficiency and the gains from trade s 2 Introduction Connections to: Opportunity costs to consumers

More information

Economics 431 Homework 4 Due Wed July 25

Economics 431 Homework 4 Due Wed July 25 Economics 431 Homework 4 Due Wed July 5 Part I A variant of Stackelberg game There are two firms. Demand is linear, p = A BQ, and marginal cost is constant and equal to c. Suppose that firm 1 moves first

More information

Optimal Auctions Continued

Optimal Auctions Continued Lecture 6 Optimal Auctions Continued 1 Recap Last week, we... Set up the Myerson auction environment: n risk-neutral bidders independent types t i F i with support [, b i ] residual valuation of t 0 for

More information

Imperfect information Up to now, consider only firms and consumers who are perfectly informed about market conditions: 1. prices, range of products

Imperfect information Up to now, consider only firms and consumers who are perfectly informed about market conditions: 1. prices, range of products Imperfect information Up to now, consider only firms and consumers who are perfectly informed about market conditions: 1. prices, range of products available 2. characteristics or relative qualities of

More information

SECTION A (Multiple choice)

SECTION A (Multiple choice) MIDSUMMER EXAMINATIONS 2009 Subject Title of Paper Time Allowed ECONOMICS : MARKET POWER AND MARKET FAILURE Two Hours (2 hours) Instructions to candidates Candidates should attempt ALL of Section A, ONE

More information

EconS Bargaining Games and an Introduction to Repeated Games

EconS Bargaining Games and an Introduction to Repeated Games EconS 424 - Bargaining Games and an Introduction to Repeated Games Félix Muñoz-García Washington State University fmunoz@wsu.edu March 25, 2014 Félix Muñoz-García (WSU) EconS 424 - Recitation 6 March 25,

More information

Universidad Carlos III de Madrid Game Theory Problem Set - Dynamic Games

Universidad Carlos III de Madrid Game Theory Problem Set - Dynamic Games Universidad Carlos III de Madrid Game Theory Problem Set - Dynamic Games Session Problems 1, 2, 3, 4, 5, 6 1 (no SPNE) 2 7, 8, 9, 10, 11 3 12, 13, 14, 15, 16 4 17, 18, 19 5 Test 1. The next figure shows

More information

Simon Fraser University Spring 2015. Econ 302 D200 Final Exam Solution Instructor: Songzi Du Tuesday April 21, 2015, 12 3 PM

Simon Fraser University Spring 2015. Econ 302 D200 Final Exam Solution Instructor: Songzi Du Tuesday April 21, 2015, 12 3 PM Simon Fraser University Spring 2015 Econ 302 D200 Final Exam Solution Instructor: Songzi Du Tuesday April 21, 2015, 12 3 PM The brief solutions suggested here may not have the complete explanations necessary

More information

CHAPTER 3. Arbitrage and Financial Decision Making. Chapter Synopsis

CHAPTER 3. Arbitrage and Financial Decision Making. Chapter Synopsis CHAPTER 3 Arbitrage and Financial Decision Making Chapter Synopsis 3.1 Valuing Decisions When considering an investment opportunity, a financial manager must systematically compare the costs and benefits

More information

What are the conditions that lead to a perfectly competitive market?

What are the conditions that lead to a perfectly competitive market? Review: Lecture 1. Idea of constrained optimization. Definitions of economics. Role of marginal analysis. Economics as a way to explain. Also used to predict. Chapter 1 and 2. What is a market? What are

More information

18 INTERNATIONAL FINANCE* Chapter. Key Concepts

18 INTERNATIONAL FINANCE* Chapter. Key Concepts Chapter 18 INTERNATIONAL FINANCE* Key Concepts Financing International Trade The balance of payments accounts measure international transactions. Current account records exports, imports, net interest,

More information

Basics of Game Theory

Basics of Game Theory Basics of Game Theory Giacomo Bacci and Luca Sanguinetti Department of Information Engineering University of Pisa, Pisa, Italy {giacomo.bacci,luca.sanguinetti}@iet.unipi.it April - May, 2010 G. Bacci and

More information

Online Appendix for Anglo-Dutch Premium Auctions in Eighteenth-Century Amsterdam

Online Appendix for Anglo-Dutch Premium Auctions in Eighteenth-Century Amsterdam Online Appendi for Anglo-Dutch Premium Auctions in Eighteenth-Century Amsterdam Christiaan van Bochove, Lars Boerner, Daniel Quint January, 06 The first part of Theorem states that for any, ρ > 0, and

More information

Principles of Microeconomics Review D22-D29 Xingze Wang, Ying Hsuan Lin, and Frederick Jao (2007)

Principles of Microeconomics Review D22-D29 Xingze Wang, Ying Hsuan Lin, and Frederick Jao (2007) Principles of Microeconomics Review D-D Xingze Wang, Ying Hsuan Lin, and Frederick Jao (). Principles of Microeconomics, Fall Chia-Hui Chen November, Lecture Monopoly Outline. Chap : Monopoly. Chap : Shift

More information

THIS IS NOT A REQUIRED READING. Chapter 11B. Nonlinear Pricing. 11B.2 Perfect price discrimination, revisited

THIS IS NOT A REQUIRED READING. Chapter 11B. Nonlinear Pricing. 11B.2 Perfect price discrimination, revisited THIS IS NOT A REQUIRED READING. Chapter 11B Nonlinear Pricing 11B.1 Motivation and objectives In Chapter 11 we looked at sophisticated pricing strategies that implicitly differentiate among customers.

More information

Solutions to Exercises 8

Solutions to Exercises 8 Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.

More information

1. Write the number of the left-hand item next to the item on the right that corresponds to it.

1. Write the number of the left-hand item next to the item on the right that corresponds to it. 1. Write the number of the left-hand item next to the item on the right that corresponds to it. 1. Stanford prison experiment 2. Friendster 3. neuron 4. router 5. tipping 6. small worlds 7. job-hunting

More information

LECTURE 10: GAMES IN EXTENSIVE FORM

LECTURE 10: GAMES IN EXTENSIVE FORM LECTURE 10: GAMES IN EXTENSIVE FORM Sequential Move Games 2 so far, we have only dealt with simultaneous games (players make the decisions at the same time, or simply without knowing what the action of

More information

Price competition with homogenous products: The Bertrand duopoly model [Simultaneous move price setting duopoly]

Price competition with homogenous products: The Bertrand duopoly model [Simultaneous move price setting duopoly] ECON9 (Spring 0) & 350 (Tutorial ) Chapter Monopolistic Competition and Oligopoly (Part ) Price competition with homogenous products: The Bertrand duopoly model [Simultaneous move price setting duopoly]

More information

Internet Advertising and the Generalized Second Price Auction:

Internet Advertising and the Generalized Second Price Auction: Internet Advertising and the Generalized Second Price Auction: Selling Billions of Dollars Worth of Keywords Ben Edelman, Harvard Michael Ostrovsky, Stanford GSB Michael Schwarz, Yahoo! Research A Few

More information

Next Tuesday: Amit Gandhi guest lecture on empirical work on auctions Next Wednesday: first problem set due

Next Tuesday: Amit Gandhi guest lecture on empirical work on auctions Next Wednesday: first problem set due Econ 805 Advanced Micro Theory I Dan Quint Fall 2007 Lecture 6 Sept 25 2007 Next Tuesday: Amit Gandhi guest lecture on empirical work on auctions Next Wednesday: first problem set due Today: the price-discriminating

More information

CHAPTER 13 GAME THEORY AND COMPETITIVE STRATEGY

CHAPTER 13 GAME THEORY AND COMPETITIVE STRATEGY CHAPTER 13 GAME THEORY AND COMPETITIVE STRATEGY EXERCISES 3. Two computer firms, A and B, are planning to market network systems for office information management. Each firm can develop either a fast,

More information

Equilibrium: Illustrations

Equilibrium: Illustrations Draft chapter from An introduction to game theory by Martin J. Osborne. Version: 2002/7/23. Martin.Osborne@utoronto.ca http://www.economics.utoronto.ca/osborne Copyright 1995 2002 by Martin J. Osborne.

More information

How to measure the total economic well-being of a society?

How to measure the total economic well-being of a society? Welfare Economics continued Market efficiency How to measure the total economic well-being of a society? Suppose there is a all-powerful, well-intentioned dictator called a social planner who can allocate

More information

Chapter 15. Sponsored Search Markets. 15.1 Advertising Tied to Search Behavior

Chapter 15. Sponsored Search Markets. 15.1 Advertising Tied to Search Behavior From the book Networks, Crowds, and Markets: Reasoning about a Highly Connected World. By David Easley and Jon Kleinberg. Cambridge University Press, 2010. Complete preprint on-line at http://www.cs.cornell.edu/home/kleinber/networks-book/

More information

Terminology and Scripts: what you say will make a difference in your success

Terminology and Scripts: what you say will make a difference in your success Terminology and Scripts: what you say will make a difference in your success Terminology Matters! Here are just three simple terminology suggestions which can help you enhance your ability to make your

More information

Econ 201, Microeconomics Principles, Final Exam Version 1

Econ 201, Microeconomics Principles, Final Exam Version 1 Econ 201, Microeconomics Principles, Final Exam Version 1 Instructions: Please complete your answer sheet by filling in your name, student ID number, and identifying the version of your test (1 or 2).

More information

Department of Economics The Ohio State University Second Midterm Exam Answers Econ 805

Department of Economics The Ohio State University Second Midterm Exam Answers Econ 805 Prof. Peck Winter 010 Department of Economics The Ohio State University Second Midterm Exam Answers Econ 805 1. (30 points) a: Consider the following class of strategic form games, based on the parameter,

More information

6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games

6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games 6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games Asu Ozdaglar MIT February 4, 2009 1 Introduction Outline Decisions, utility maximization Strategic form games Best responses

More information

The Limits of Price Discrimination

The Limits of Price Discrimination The Limits of Price Discrimination Dirk Bergemann, Ben Brooks and Stephen Morris University of Zurich May 204 Introduction: A classic economic issue... a classic issue in the analysis of monpoly is the

More information

Chapter 9. Auctions. 9.1 Types of Auctions

Chapter 9. Auctions. 9.1 Types of Auctions From the book Networks, Crowds, and Markets: Reasoning about a Highly Connected World. By David Easley and Jon Kleinberg. Cambridge University Press, 2010. Complete preprint on-line at http://www.cs.cornell.edu/home/kleinber/networks-book/

More information

Econ 002 Exam 3 McLeod FORM A. Name ID#

Econ 002 Exam 3 McLeod FORM A. Name ID# Econ 002 Exam 3 McLeod FORM A Name ID# Firm Total Sales 1 $50 million 2 $50 million 3 $40 million 4 $30 million 5 $20 million 6 $10 million 1. Refer to the table above. Assuming there are only 6 firms

More information

Microeconomics Topic 3: Understand how various factors shift supply or demand and understand the consequences for equilibrium price and quantity.

Microeconomics Topic 3: Understand how various factors shift supply or demand and understand the consequences for equilibrium price and quantity. Microeconomics Topic 3: Understand how various factors shift supply or demand and understand the consequences for equilibrium price and quantity. Reference: Gregory Mankiw s rinciples of Microeconomics,

More information

11 PERFECT COMPETITION. Chapter. Key Concepts. Perfectly Competitive Firm s Demand Curve

11 PERFECT COMPETITION. Chapter. Key Concepts. Perfectly Competitive Firm s Demand Curve Chapter 11 PERFECT COMPETITION Key Concepts FIGURE 11.1 Perfectly Competitive Firm s Demand Curve Competition Perfect competition is an industry with many firms, each selling an identical good; many buyers;

More information

Competition and Regulation. Lecture 2: Background on imperfect competition

Competition and Regulation. Lecture 2: Background on imperfect competition Competition and Regulation Lecture 2: Background on imperfect competition Monopoly A monopolist maximizes its profits, choosing simultaneously quantity and prices, taking the Demand as a contraint; The

More information

chapter >> Consumer and Producer Surplus Section 3: Consumer Surplus, Producer Surplus, and the Gains from Trade

chapter >> Consumer and Producer Surplus Section 3: Consumer Surplus, Producer Surplus, and the Gains from Trade chapter 6 >> Consumer and Producer Surplus Section 3: Consumer Surplus, Producer Surplus, and the Gains from Trade One of the nine core principles of economics we introduced in Chapter 1 is that markets

More information

Forward exchange rates

Forward exchange rates Forward exchange rates The forex market consists of two distinct markets - the spot foreign exchange market (in which currencies are bought and sold for delivery within two working days) and the forward

More information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 4. Stable Marriage - An Application of Proof Techniques to Algorithmic

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 4. Stable Marriage - An Application of Proof Techniques to Algorithmic CS 70-2 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 4 Stable Marriage - An Application of Proof Techniques to Algorithmic Analysis Consider a dating agency that must

More information

1. Suppose demand for a monopolist s product is given by P = 300 6Q

1. Suppose demand for a monopolist s product is given by P = 300 6Q Solution for June, Micro Part A Each of the following questions is worth 5 marks. 1. Suppose demand for a monopolist s product is given by P = 300 6Q while the monopolist s marginal cost is given by MC

More information

Worldwide Casino Consulting Inc.

Worldwide Casino Consulting Inc. Card Count Exercises George Joseph The first step in the study of card counting is the recognition of those groups of cards known as Plus, Minus & Zero. It is important to understand that the House has

More information

Competition among Sellers Who Offer Auctions Instead of Prices

Competition among Sellers Who Offer Auctions Instead of Prices journal of economic theory 75, 141179 (1997) article no. ET972278 Competition among Sellers Who Offer Auctions Instead of Prices Michael Peters* and Sergei Severinov - Department of Economics, University

More information

Lecture 6 Part I. Markets without market power: Perfect competition

Lecture 6 Part I. Markets without market power: Perfect competition Lecture 6 Part I Markets without market power: Perfect competition Market power Market power: Ability to control, or at least affect, the terms and conditions of the exchanges in which one participates

More information

THE UNIVERSITY OF MELBOURNE MELBOURNE BUSINESS SCHOOL. MANAGERIAL ECONOMICS Term 1 1999 First Mid-Term Solutions DR.

THE UNIVERSITY OF MELBOURNE MELBOURNE BUSINESS SCHOOL. MANAGERIAL ECONOMICS Term 1 1999 First Mid-Term Solutions DR. THE UNIVERSITY OF MELBOURNE MELBOURNE BUSINESS SCHOOL MANAGERIAL ECONOMICS Term 1 1999 First Mid-Term Solutions DR. VIVEK CHAUDHRI Part A: Multiple Choice Questions Answer all of the following 10 questions

More information

Financial Intermediation

Financial Intermediation Financial Intermediation The last time you bought an apple at the grocery store did wonder at all who grew the apple? Probably not. You deal with the grocery store, a fruit intermediary among other things,

More information

Market Power and Efficiency in Card Payment Systems: A Comment on Rochet and Tirole

Market Power and Efficiency in Card Payment Systems: A Comment on Rochet and Tirole Market Power and Efficiency in Card Payment Systems: A Comment on Rochet and Tirole Luís M. B. Cabral New York University and CEPR November 2005 1 Introduction Beginning with their seminal 2002 paper,

More information

Game Theory: Supermodular Games 1

Game Theory: Supermodular Games 1 Game Theory: Supermodular Games 1 Christoph Schottmüller 1 License: CC Attribution ShareAlike 4.0 1 / 22 Outline 1 Introduction 2 Model 3 Revision questions and exercises 2 / 22 Motivation I several solution

More information

Chapter 9 examines firms under severe competition while chapter 11 illustrates monopoly firms that face no competition.

Chapter 9 examines firms under severe competition while chapter 11 illustrates monopoly firms that face no competition. The Firm and the Industry under erfect Competition The decisions of firms depend on consumer demand and production costs. Yet, they also depend on the behavior, the number, and the size of other firms

More information

Games of Incomplete Information

Games of Incomplete Information Games of Incomplete Information Jonathan Levin February 00 Introduction We now start to explore models of incomplete information. Informally, a game of incomplete information is a game where the players

More information

The Market-Clearing Model

The Market-Clearing Model Chapter 5 The Market-Clearing Model Most of the models that we use in this book build on two common assumptions. First, we assume that there exist markets for all goods present in the economy, and that

More information

Financial Markets. Itay Goldstein. Wharton School, University of Pennsylvania

Financial Markets. Itay Goldstein. Wharton School, University of Pennsylvania Financial Markets Itay Goldstein Wharton School, University of Pennsylvania 1 Trading and Price Formation This line of the literature analyzes the formation of prices in financial markets in a setting

More information

An Introduction to Sponsored Search Advertising

An Introduction to Sponsored Search Advertising An Introduction to Sponsored Search Advertising Susan Athey Market Design Prepared in collaboration with Jonathan Levin (Stanford) Sponsored Search Auctions Google revenue in 2008: $21,795,550,000. Hal

More information

CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

More information

Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets

Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that

More information

Answer Key: Problem Set 2

Answer Key: Problem Set 2 Answer Key: Problem Set February 8, 016 Problem 1 See also chapter in the textbook. a. Under leasing, the monopolist chooses monopoly pricing each period. The profit of the monopolist in each period is:

More information

Nash Equilibrium. Ichiro Obara. January 11, 2012 UCLA. Obara (UCLA) Nash Equilibrium January 11, 2012 1 / 31

Nash Equilibrium. Ichiro Obara. January 11, 2012 UCLA. Obara (UCLA) Nash Equilibrium January 11, 2012 1 / 31 Nash Equilibrium Ichiro Obara UCLA January 11, 2012 Obara (UCLA) Nash Equilibrium January 11, 2012 1 / 31 Best Response and Nash Equilibrium In many games, there is no obvious choice (i.e. dominant action).

More information

4 Eliminating Dominated Strategies

4 Eliminating Dominated Strategies 4 Eliminating Dominated Strategies Um so schlimmer für die Tatsache (So much the worse for the facts) Georg Wilhelm Friedrich Hegel 4.1 Dominated Strategies Suppose S i is a finite set of pure strategies

More information

Sequential lmove Games. Using Backward Induction (Rollback) to Find Equilibrium

Sequential lmove Games. Using Backward Induction (Rollback) to Find Equilibrium Sequential lmove Games Using Backward Induction (Rollback) to Find Equilibrium Sequential Move Class Game: Century Mark Played by fixed pairs of players taking turns. At each turn, each player chooses

More information

MATH MODULE 11. Maximizing Total Net Benefit. 1. Discussion M11-1

MATH MODULE 11. Maximizing Total Net Benefit. 1. Discussion M11-1 MATH MODULE 11 Maximizing Total Net Benefit 1. Discussion In one sense, this Module is the culminating module of this Basic Mathematics Review. In another sense, it is the starting point for all of the

More information

TAPPING ON THE BRAKES: ARE LESS ACTIVE MARKETS SAFER AND BETTER? Joseph E. Stiglitz Atlanta April 2014

TAPPING ON THE BRAKES: ARE LESS ACTIVE MARKETS SAFER AND BETTER? Joseph E. Stiglitz Atlanta April 2014 TAPPING ON THE BRAKES: ARE LESS ACTIVE MARKETS SAFER AND BETTER? Joseph E. Stiglitz Atlanta April 2014 MORE ACTIVE CAN TAKE MANY FORMS High frequency trading Liberalization: More kinds of trades and transactions

More information

UCLA. Department of Economics Ph. D. Preliminary Exam Micro-Economic Theory

UCLA. Department of Economics Ph. D. Preliminary Exam Micro-Economic Theory UCLA Department of Economics Ph. D. Preliminary Exam Micro-Economic Theory (SPRING 2011) Instructions: You have 4 hours for the exam Answer any 5 out of the 6 questions. All questions are weighted equally.

More information

Perfect Competition. We will use the second concept in here and your text, chapter 11.

Perfect Competition. We will use the second concept in here and your text, chapter 11. Perfect Competition There are two concepts of competition normally used in Economics: 1. The manner or process in which firms compete with one another for market share. 2. A description of a particular

More information

Microeconomic Theory Jamison / Kohlberg / Avery Problem Set 4 Solutions Spring 2012. (a) LEFT CENTER RIGHT TOP 8, 5 0, 0 6, 3 BOTTOM 0, 0 7, 6 6, 3

Microeconomic Theory Jamison / Kohlberg / Avery Problem Set 4 Solutions Spring 2012. (a) LEFT CENTER RIGHT TOP 8, 5 0, 0 6, 3 BOTTOM 0, 0 7, 6 6, 3 Microeconomic Theory Jamison / Kohlberg / Avery Problem Set 4 Solutions Spring 2012 1. Subgame Perfect Equilibrium and Dominance (a) LEFT CENTER RIGHT TOP 8, 5 0, 0 6, 3 BOTTOM 0, 0 7, 6 6, 3 Highlighting

More information

Economics of Insurance

Economics of Insurance Economics of Insurance In this last lecture, we cover most topics of Economics of Information within a single application. Through this, you will see how the differential informational assumptions allow

More information