For each set P below find a sequential equilibrium and verify that it is indeed a sequential equilibrium. • Note that sequential rationality cannot be determined from the normal‐form of a game. We know in equilibrium marginal utility per dollar spent will be equal for all goods. 1. If two bidders 17.3 Sequential Games. • Suppose we assume sequentially rational behavior for the players. An explanation for this is that any strategy (q;1 q) is optimal for information set 2.a, so any strategy pro le is a Nash equilibrium for the subgame rooted at 2.a. Since, as shown in [Halpern and Pass 2015], even Nash equilibria may not exist in Bayesian machine games, we clearly cannot expect a sequential equilibrium to exist in general. to fries must be in equilibrium (I asked you to do something similar in problem #7 of problem set 4). Rather than having a social planner, set up a market structure with -rms (who maximize pro-ts) and households (who maximize utility). Prop. Furthermore, suppose sequential rationality is common knowledge. Exercise 1 (similar to an exam question Winter 2002) Three consumers have identical Cobb-Douglas utility functions over two goods in the con-sumption set R2 + given by u(x 1i,x 2i) = x 1/2 1i x 1/2 2i. Exercise 331.1 in the book. (g) ConsideranyArrow-Debreuequilibrium. Properties of payo§s: 1 Player 1 is happy if player 2 accepts the gift: 1 In the case of a Friendly type, he is just happy because of altruism. (b) Show that if >2 3, then in all sequential equilibria there is no hiring on the equilibrium path. (h) Consider any Sequential Markets Equilibrium. 0 and as the set of conditioning events F on which sequential rationality is imposed expands to include all -nite subsets of a neighborhood basis for all players™open observable events. Both players randomize over their two strategy choices with probabilities .5 and .5. In general, buyers are restricted to purchase only one unit of the . Problem 1 below presents some games where PBEdoes not perform well, and introduces sequential equilibrium (SE) as a remedy to the exposed problems. We use backward induction again for this problem. (a) The supply curve shifts left. Problem Set #1. Equivalence of Arrow-Debreu and Sequential Markets Equilibria A full set of one-period Arrow securities is su¢ cient to make markets fisequentially complete.fl Any (nonnegative) consumption allocation is attainable with an appropriate sequence of Arrow security holdings fa t+1(st,s t+1)g satisfying all sequential markets budget constraints. Handout: Introduction to RBC Models. In game theory, trembling hand perfect equilibrium is a refinement of Nash equilibrium due to Reinhard Selten. On the Agenda 1 Formalizing the Game librium travel time of 63.3; the equilibrium flows are 2153 vehicles on link A and 5847 vehicles on link B. Question: Consider the Sequential Matching Pennies Game. Problem 1 (Sequential equilibrium) a) In the game of Figure1, Nature chooses Lwith probability 3 4. 2 * 1 * 2 1 1 1 games Pfries games P games fries . A strong plaintiff will win the case if it goes to trial, and a weak . This problem can be formulated as a two-person sequential game of two stages and with perfect information. Player 1 first chooses which game to play on the Nintendo Switch Player 2 observes the choice and chooses to either also play on the Switch or to instead use the IPAD. [Sections 15.5 and 15.7] More generally, it develops a dynamic programming method for a class of pol-icy games between the government and a continuum of households. Consider the following strategy profile, in which 1 plays a, and 2 plays L. This is a Nash equilibrium. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): The convergence of the sequential joint maximization method (Rutherford [10]) for searching economic equilibria is studied in the case of Cobb-Douglas utility functions. Player 2's information set will not be reached at the equilibrium, because player 1 will play L with probability 1. One type of problem solved using MPEC is finding market equilibrium in the energy sector. Obara (UCLA) Bayesian Nash Equilibrium February 1, 2012 6 / 28 5 Figure 2. Its only input is the finished good, which it buys from a competitive market with an upward sloping supply curve. This graphical solution may be stated mathematically as follows: x 2.0 2. is a sequential equilibrium. . Econ 504 (2008) Problem Set #1 Warm up: 1. The term information set is most usually applied when the game has a sequential component. A subgame is any part of a game that remains to be played after a given set of moves. Answers to Problem Set #1: GNPn.txt, PNFIC1n.txt, ps1n.m. We need to study the extensive form to determine if a strategy has this property. 2 In the case of an Enemy type, he enjoys seeing how player 2 unwraps a box with a frog inside! Consider the following signaling game. Engelbrecht-Wiggans and Weber (1979) find the equilibrium allo-cation of a fixed budget acros s a set of independent auctions. McCall Model McCall Sequential Search Model McCall Partial Equilibrium Search Model The simplest model of search frictions. Auctions Suggested Solutions by: Tibor Heumann 4/8/14 This problem set is due on Tuesday, 4/15/14. (Sequential Equilibrium) 226.1. This problem addressed by sequential equilibrium, which explicitly requires that the players play a best reply at every information set (sequential rationality) and that the players' beliefs are "consistent" with the other players' strategies. 5. β, where β is the set of indices of those m independent columns. 3. (c)Does this game have a Nash equilibrium (possibly in mixed strategies) that is not sub-game perfect? The model is augmented with an additional constraint that . Jean-Baptiste Lespiau, Samitha Samaranayake, Alexandre M. Bayen 1 1 ABSTRACT 2 This article considers the dynamic user equilibrium (DUE) problem for parallel networks. They are based on the following model of public goods and the solution concepts of competitive equilibrium, Lindahl equilibrium, and core: Consider an economy of a thousand (1000) identical households i ∈ H, a finite set of (a) Find all separating and pooling equilibria of the game above. The firm sells the same good to competitive buyers in the output market. We show that if the set of TMs that the agents can choose from is nite, then an . Now assume player 1 is allowed to quit the game in the initial node, hence she has three actions. We meet on Tuesdays (8-10am, UB4.228) and Wednesdays (8-10am, UB2.147). The above argument determines Player 2's beliefs at his information set. 0, the problem of agent iis to choose ci t;b i +1 1 0 that maximizes X1 t=0 tu ci t; 2(0;1) subject to (1a) for all t. Let us assume that bi 0 = 0 for all iis given. issues in definitions of sequential equilibrium in this setting, Once these 2. Questions 1, 2, and 3 are not as scary as they look. DEFINITION: A (sequential market) competitive equilibrium is a sequence fq tgand ci t;b i t+1 for i= 1;2 such that ŒGiven fq tg, ci t;b i t+1 solves i™s maximization problem . 500 cars. But it is a silly equilibrium, because if 2 ever found herself in a situation where . For the present belief, which puts probability 1 to the weak type, consider a . Do the following exercises from the rst homework of John Nachbar's Econ 504 (Spring 2007): 2(a), 4, and 7. Game Theory Problem Set 8 Solution 1. The action (strategy) set of player 1 is given by S 1 = f0;:::;100g;with choice . For general multistage games, we establish that an outcome is implementable in sequential equilibrium if and only if it is implementable sequential equilibrium (Kreps and Wilson 1982) ensure that the rationality test is applied to all information sets in an extensive-form game, because these concepts are defined relative to convergent sequences of fully mixed behavior strategies. this problem: Problem: Some information sets are reached with zero probability and so we can™t use Bayes rule to pin down beliefs Solution: Use completely mixed strategies to ensure that every information set is reached with positive probability This is the notion of sequential equilibrium So far Sequential Rationality and Weak Perfect Bayesian Equilibrium Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu June 16th, 2016 C. Hurtado (UIUC - Economics) Game Theory. There are two players, a plaintiff and a defendant in a civil suit. Because there are no subgames, this is also a subgame-perfect Nash equilibrium. Extensive Games Subgame Perfect Equilibrium Backward Induction Illustrations Extensions and Controversies NE not good enough for extensive games • There is something unsatisfactory about the Nash equilibrium concept in extensive games. The equilibrium constant K p = 4.1×10-13 at 600K. I bidder i's payo is u i(b;v) = 1(b i max j6=i b j)(v i b i). Player 1 Port Knights Animal Crossing IPAD IPAD 0.5) 0.09 What is the Nash Equilibrium of this . Subgame Perfect Nash Equilibrium Subgame Perfect Nash Equilibrium We are now going to define a refinement of Nash equilibrium that captures the notion of backward induction. (b) If the volume of the equilibrium mixture is decreased, will the number of AB molecules increase or decrease? Consequently, the above equilibrium can't be sequential. We ended the last problem set with the game which exposed the draw- . in order to be considered a subgame perfect equilibrium, a set of strategies must - be a nash equilibrium for each subgame . Walrasian Equilibrium (a.k.a. mation sets), a behavioral strategy profile is a subgame perfect equilibrium if and only if it is a sequential equilibrium (with the only feasible belief system that assigns prob 1 to the single history in each info set). . Economics questions and answers. In this study, a novel sequential optimality condition for general continuous optimization problems is established. In Section7, we consider when sequential equilibrium exists. The probabilities .9 and .1 are indeed computed through Bayes' rule. message and a separating equilibrium if he assigns to each type a different message. of sequential equilibrium. Given these beliefs and equations (3) and (4), it is very easy to see that no type will want to deviate. All Pay Auction. An Equilibrium Problem with Equilibrium Constraints (EPEC) is a mathematical program . Problem Set #2. This is a fun problem! . Menon Business Economics 4 PROBLEM SET Question 3 (Sequential moves; Nash and Subgame perfect Nash equilibrium) Exercise background: Consider the situation of Problem 2, but assume now that player 1 makes her demand before player 2 makes his demand. Let H i be the set of information sets at which player i moves. If the price is accepted, they trade the car at price p; no trade occurs otherwise. Perfect Bayesian equilibrium Perfect Bayesian equilibrium (PBE) strengthens subgame perfection by requiring two elements: - a complete strategy for each player i (mapping from info. The equilibrium price rises ("price for raw tea . We need to study the extensive form to determine if a strategy has this property. This implies that the same set of actions must be feasible at every node in an information set. For example, any of the game parts to the right of any box in the Pay-raise Voting Game is a subgame. Trembling-hand perfect equilibrium and sequential equilibrium aren't always the best Sequential Equilibrium •Information possessed by the players in an extensive-form game is represented in terms of information sets. 0, the problem of agent iis to choose ci t;b i +1 1 0 that maximizes X1 t=0 tu ci t; 2(0;1) subject to (1a) for all t. Let us assume that bi 0 = 0 for all iis given. There is also a family of pooling equilibria in which only one signal is received in equilibrium. It is shown that convergence is closely related to the behavior of certain inhomogeneous Markov chains. In a sequential equilibrium, 2 must play r and 1 must play y, with probability .5 assigned to each node in each information set. Bayesian Games Yiling Chen September 12, 2012. This paper studies algorithms for equilibrium problems with equilibrium constraints (EPECs). In such games, an agent can rationally choose to forget, so issues of imperfect recall arise. The aim of this work is to analyze lexicographic equilibrium problems on a topological Hausdorff vector space X, and their relationship with some other vector equilibrium problems. A represents the "normal" allele and is also represented by p in the H-W equilibrium equations. Existence results for the tangled lexicographic problem are proved via the study of a related sequential problem. They demonstrate that adding the restriction of sequential rationality partially alleviates the multiple equilibrium problem and thereby substantially expands the set of welfare criteria which can be implemented by carefully time-0 market structure sequential market structure The Two Welfare Theorems 1st Welfare Theorem: circumstances under which a competitive equilibrium is Pareto optimal (i.e. A two-link example. Problem 1 (Sequential equilibrium) a) In the game of Figure1, Nature chooses Lwith probability 3 4. less than or equal to 2. The network 3 dynamics are modeled using a Godunov discretization of the Lighthill-Williams-Richards partial differential 4 equation with a trapezoidal flux function. There are pure strategy Nash equilibrium which are not subgame perfect equilibrium OC . Finally, Weber (1983) analyzes sequential and simultaneous multigood auctions in which the bidders are not budget-con-strained. Jobs sampled sequentially. [l + .lS(vA/1000) 4) A QJ ;:: E ~ ~ I-SA 70 60 50 40 30 20 lO b. Graphical Solution to the Equilibrium Assignment Problem ~ SB sets to mixed actions) - beliefs for each player i (P i(v | h) for all information sets h of player i) Among other things, this eliminates the equilibrium se-lection problem, which occurs in some games where there are multiple equilibria. Consider the following reaction: 2H2O(g) + O2(g) ⇌ 2H2O2(g), starting with 1.0 bar each of H2O and O2, it was found at equilibrium that the pressure of H2O2 was 0.05 bar . We first find the output q 2 ∗ of firm 2 that maximizes firm 2's profit given the output q 1 of firm 1. An open sequential equilibrium is de-ned as a limit of (";F)-sequential equilibrium con-ditional distributions on outcomes as " ! Some matlab files to solve the simple RBC Model in the previous handout: ss.m, RBC.m. Proof sketch: Suppose were not a NE. The set of pure strategy Nash equilibria and the set of subgame perfect equilibria are identical B. the assumption that bargaining ends as soon as the second player rejects or accepts an offer is a criticism of sequential-move bargaining game. Sequential Games GamesinExtensiveForm,BackwardInduction, SubgamePerfectEquilibrium,Commitment June2016 Games in Extensive Form, Backward Induction, Subgame Perfect Equilibrium, Commitment ()Part 4: Game Theory IISequential Games June 2016 1 / 17 . . L3-sequential equilibrium. he makes different moves at the two nodes in the informa-tion set X ={x3,x4}.1 As pointed out by Halpern (1997), if the agent knows what strategy he is using at all times, and . If we assume that there is only one strategic player in an energy market (leader) and there are several smaller fringe firms we can setup an MPEC with the constraints bounded by a fixed demand in the network and firms focused on maximizing profits. . We propose a sequential nonlinear complementarity (SNCP) algorithm to . What Once you find your document (s), you can . refinements of Nash equilibrium permit the implementation of some additional SCCs. node. SA= 15. Furthermore, suppose sequential rationality is common knowledge. p + q = 1 thus, p = 1 - q = 1 - 0.01 = 0.99 (c) Calculate the percentage of carriers of the trait within the population. Complete Information.The rules of the all pay auction are: (i) the highest bid receives the object, (ii) each bidder pays his bid, independent of whether he wins or loses the object. rium. First, player 1 must pay $1 or pass. From the problem, we can write this as. (a) Show that for large enough there is a sequential equilibrium with no hiring on the equilibrium path. 2 Preliminaries on MPECs (c) Find a sequential equilibrium in which the probability of player 2 getting to move is strictly between 0 and 1. (a) Assuming the vol- ume is 2 L, calculate the equilibrium constant Kc for the reaction. The guaranteed security . Note that the belief ,u = g(m, p) may be credible because there exists a best response x which makes is self-fulfilling. Problem Set 3 is available here. (a) P = {2000, 7000} (b) P = [0, ∞). Equilibrium : The point in a game where both players have made their decisions and an outcome is reached . However, it holds in important classes of games, including single-agent games, games with pure adverse selection, games with pure moral hazard, and a class of social learning games. 1.2 Perfect Bayesian Equilibrium Let G be an extensiev form game. It's OK to work together on problem sets. Bayesian Nash, Trembling Hand and Sequential equilibrium. Games in Extensive Form, Backward Induction, Subgame Perfect Equilibrium, Commitment ()Part 4 . The strategy set (1 votes no; 2 votes yes; 3 votes yes) is a Nash equilibrium: each player's strategy is the best possible against those of the other two. Player 2 observes player 1 [s demand before making his choice. Competitive Equilibrium), for an Edgeworth Box: De nition: Price vector p and an allocation x = (x 1;x 2) in the Edgeworth box such that for i= 1;2 x i % ix ifor all x0i2B i(p) At equilibrium, the o er curves of the two consumers intersect Any intersection of the o er curves outside of !corresponds to a WE Only . than to reject, so in a subgame perfect equilibrium, player 2 must accept whenever i 99:The only degree of freedom left for player 2 . A trembling hand perfect equilibrium is an equilibrium that takes the possibility of off-the-equilibrium play into account by assuming that the players, through a "slip of the hand" or tremble, may choose unintended strategies, albeit with negligible probability. By selectively incor- Pooling Equilibrium. For instance, in a pooling equilibrium of our example, β1(A) = β1(B), whereas in a separating equilibrium, β1(A) 6= β1(B). at a singleton information set (and there are other restrictions as well). Sequential equilibrium is a refinement of Nash Equilibrium for extensive form games due to David M. Kreps and Robert Wilson.A sequential equilibrium specifies not only a strategy for each of the players but also a belief for each of the players. The problem here is that if the agent starts out using strat- . (a) Find a separating sequential equilibrium. . It ignores the sequential structure of the game and treats strategies as choices made once and for all. Learning Objective 17.3: Describe sequential move games and explain how they are solved. The information set after the quiche is o ffthe equilibrium path. 2. Consider the following sequential game. 15.10 The diagram shown here represents the equilibrium state for the reaction A2(g) + 2 B(g) 2 AB(g). L2-game theory. Some of the worksheets below are Equilibrium Physics Problems and Solutions Worksheets, Definition of equilibrium, Static and Dynamic Equilibrium, Equilibrium Equations, Equilibrium and Torque : Equilibrium and Torque, definition of static and dynamic equilibrium, Linear vs. Nash Equilibrium Lecture notes (PDF) 7 Application: Imperfect Competition Lecture notes (PDF) 8 Further Applications Lecture notes (PDF) 9 Backward Induction Lecture notes (PDF) 10 Application: Negotiation Lecture notes (PDF) 11 Subgame-Perfect Nash Equilibrium Lecture notes (PDF) 12 Repeated Games Lecture notes (PDF) 13 Application: Implicit . In this setting, we consider two notions of sequential equilibrium. We present a generalization of Scholtes's regularization scheme for MPECs and extend his convergence results to this new relaxation method. That is, fries fries games games P MU P MU = . . 2 Both types of player 1 prefer not to make a gift (obtaining a payo§ of 0), rather than making a gift that is rejected (with a Players 1 and 2 are in a sequential all-pay-own-bid auction for a prize worth $3. Economics. (Find the weak sequential equilibrium (equilibria) directly—no need to find the Nash equilibria.) In this game, any belief o ffthe equilibrium path is consistent. Do the following exercises from the Osborne-Rubinstein book. it corresponds to the solution to a PROBLEM SET 2 — SOLUTIONS Section A 1. (c) Show that in a sequential equilibrium, if the kth principal hires with positive probability then pT k 1 3 for T= A;B. Problem Set 7 Answer Key 1) A firm is a monopoly in its output market and a monopsony in its input market. Constructprices, interestrates, andborrowing/lending amounts so that the allocation of the Arrow-Debreu equilibrium is part of the allocation of a Sequential Markets Equilibrium at those prices. There are exceptions that are indicated in the calendar below. Solution: No, the equilibria found in part (b) include all Nash equilibria. If ( ) is a sequential equilibrium, then is a Nash equilibrium. DEFINITION: A (sequential market) competitive equilibrium is a sequence fq tgand ci t;b i t+1 for i= 1;2 such that ŒGiven fq tg, ci t;b i t+1 solves i™s maximization problem . This approach was already followed by the same authors in the case of variational inequalities. • Suppose we assume sequentially rational behavior for the players. •An information set h for player i is a set of i's decision nodes among which i cannot distinguish. because sequential equilibrium in this model imposes no restrictions on off-the-equilibrium beliefs. But assume that player 1 plays acompletely mixed strategy, playing L, M, and R with probabilities 1 , 3 . Consider the signaling game in the last page. I the set of players (bidders) is N I the set of states is V 1:::; V n I the set of actions for bidder i is A i = < + I the set of types for bidder i is V i I bidder i's interim belief is p i(v ijv i). In the context of mathematical programs with equilibrium constraints, the condition is proved to ensure Clarke stationarity. a. Formulate a balanced transportation problem that can be . Meetings and Organization. We are done. Problem Set 8 Solve before the classes April 22-24. (b) Which of these equilibria are sequential equilibria? Consumer 1 has initial endowment ω 1 = (2,8), while consumers 2 and 3 both have initial . a Nash equilibrium (where x is player 1's mixed strategy and y is player 2's mixed strategy) and (x0,y0) is a Nash equilibrium, then (x,y0) and (x0,y) are also Nash equilib-ria. That is, q 2 ∗ = q 2 ∗ (q 1).The profit of firm 2 can be formulated as sequential rationality Due: Thursday, March 28 at beginning of class Reading: Fudendberg and Triole, Sections 3.1-3.4, & 8.3 . We examine sequential equilibrium in the context of computational games (Halpern and Pass 2015), where agents are charged for computation. jumped as much as 30% from pre-lockdown levels") and the equilibrium quantity falls (tea output has fallen in China, . ante sequential equilibrium. What Problem 1 below presents some games where PBEdoes not perform well, and introduces sequential equilibrium (SE) as a remedy to the exposed problems. The definition of PBE is based on two notions (besides also propose a sequential nonlinear complementarity (SNCP) approach for solving EPECs . Alternative: Stigler, -xed sample search (choose a sample of n jobs Here, I will define sequential equilibrium and apply it to some important games. Answers to Problem Set #2. 1 CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Abstract. (b) Find a sequential equilibrium in which the probability of player 2 getting to move is 0. Recall that: De nition 1 A ebhaviaolr sattrgey for player i is a function i: H i ( A i) such that for any h i H i, the suporpt of i ( h i) is ontacined in the set of actions available at h i. eW now augment a plyear s strategy to explicitly account for his beliefs. called perfect Bayesian equilibrium (PBE), and it was shown that (i) PBE is a strict refinement of subgame-perfect equilibrium ((21)) and (ii) in general, the set of sequential equilibria ((15)) is a proper subset of the set of perfect Bayesian equilibria. A belief gives, for each information set of the game belonging to the player, a probability distribution on the nodes in the information set. Problem of an individual getting draws from a given wage distribution Decision: which jobs to accept and when to start work. M, Ch 8, 235-289. 2. What are the equilibrium pressures of CO, Cl2, and COCl if we start with 0.124 atm pressure of COCl at this temperature? Determine its profit-maximizing output. Problem Set 9. Rotational Velocity, …. Perfect Sequential Equilibrium An updating rule g(m, p) and metastrategies u,, a` form a perfect sequential equilibrium iff (i) 6; and az are each sequentially perfect for all teT; (ii) g is credible with respect to 61, 92. View Problem Set 2 - Solutions.pdf from EC 1101E at National University of Singapore. Recall that an extensive form game, E, specifies a host of objects, including a set of nodes, χ, an immediate predecessor mapping p(x) that induces a successor Handout: A Primer on Solving a Real Business Cycle Model. (f) Set up the Social Planner's problem for this economy. We ended the last problem set with the game which exposed the draw- . • Note that sequential rationality cannot be determined from the normal‐form of a game. So this game, that did not have a Nash equilibrium in pure strategies, does have a Nash equilibrium in mixed strategies. 7. SEQUENTIAL EQUILIBRIA IN A RAMSEY TAX MODEL BY CHRISTOPHER PHELAN AND ENNIO STACCHETTI1 This paper presents a full characterization of the equilibrium value set of a Ramsey tax model. F, Ch 3, 207-241, Ch 4, 319-364. Originally devised for constrained nonsmooth optimization, the proposed sequential optimality condition addresses the domain of the constraints instead of . The heterozygotes are the carriers of the trait and are represented by the 2pq term in the H-W equilibrium equations. of equilibrium; hence we need to use the Bayes' rule.
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