DOCSLIB.ORG

Game Theory in Electricity Markets

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Game Theory in Electricity Markets Game Theory in Electricity Markets Lecture 8: Extensive games with perfect information Athanasios Papakonstantinou mail: [email protected] web: athpap.net Centre for Electric Power and Energy Lecture key points The matching Subgame The extensive Backward pennies: now perfect game Induction and and then Nash equilibrium DTU Electrical Engineering Centre for Electric Power and Energy Lecture Agenda Non cooperative game theory Strategic (static, Extensive (multi- Repeated (many- one-shot) games stage) games shots) games Stackelberg games (leader-follower) a specific case of extensive games DTU Electrical Engineering Centre for Electric Power and Energy The matching pennies (ver 1) • Two players choose side of a coin simultaneously • What type of game and what is the equilibrium? P1 P2 w : Heads (q) w : Tails (1-q) 1 2 t : Heads (p) 1,-1 -1,1 1 t2: Tails (1-p) -1,1 1,-1 • If pennies match (HH, TT) P1 receives P2’s penny (+1 for P1 and -1 for P2) • If pennies do not match (HT, TH) P1 gives his penny to P2 (-1 for P1 and +1 for P1) DTU Electrical Engineering Centre for Electric Power and Energy The matching pennies (ver 1) • Two players choose side of a coin simultaneously • A zero sum game with mixed strategy Nash equilibria (p=0.5) P1 P2 w : Heads (q) w : Tails (1-q) 1 2 t : Heads (p) 1,-1 -1,1 1 t2: Tails (1-p) -1,1 1,-1 • If pennies match (HH, TT) P1 receives P2’s penny (+1 for P1 and -1 for P2) • If pennies do not match (HT, TH) P1 gives his penny to P2 (-1 for P1 and +1 for P1) DTU Electrical Engineering Centre for Electric Power and Energy The matching pennies (ver 2) • The two players do not choose simultaneously!! • Player 1 reveals (truthfully), then player 2 reveals after he has observed player 1: now there is a timeline and a tree: P1 P2 H T H 1, -1 -1, 1 T -1, 1 1, -1 Histories for P1 and P2: H1={H,T}: one action information sets H2={HH,HT,TH,TT}: two action info sets H2={HH}: P2’s reaction (H) to P1’s H DTU Electrical Engineering Centre for Electric Power and Energy The matching pennies (ver 2) • The two players do not choose simultaneously!! • Player 1 reveals (truthfully), then player 2 reveals without having observed player 1: forgetfulness (imperfect recall) P1 P2 H T H 1, -1 -1, 1 T -1, 1 1, -1 Histories for P1 and P2: H1={H,T} H2={H,T} Same matrix, different (hi)story History: a player’s sequence of actions DTU Electrical Engineering Centre for Electric Power and Energy Extensive games • Strategic games suppress the sequential nature of decision- making: agents choose their plans irrespective of what happens • Extensive games allow agents to change/ adapt as events unfold and can be represented as game trees • The extensive game: – Players – Actions – Payoff DTU Electrical Engineering Centre for Electric Power and Energy Extensive games • Strategic games suppress the sequential nature of decision- making: agents choose their plans irrespective of what happens • Extensive games allow agents to change/ adapt as events unfold • The extensive game: – Players – Actions – Terminal histories: a set of sequences of actions (if-then-else scenarios / scenario trees) – Payoff DTU Electrical Engineering Centre for Electric Power and Energy Extensive games • Extensive games with perfect information: tic-tac-toe, chess * From wikipedia, under creative commons • Games with finite horizon (unofficial chess rules: draw if a position is repeated tree times) • Strictly competitive games: there can be either a clear winner (and a loser) or a draw for both DTU Electrical Engineering Centre for Electric Power and Energy Extensive games • Extensive games with perfect information: tic-tac-toe, chess • Games with finite horizon (unofficial chess rules: draw if a position is repeated tree times) • Strictly competitive games: there can be either a clear winner (and a loser) or a draw for both DTU Electrical Engineering Centre for Electric Power and Energy Extensive games • For strictly competitive games there can be two outcomes: a) One player has a strategy that guarantees he wins b) Both players have a strategy that guarantees at leas a draw • What is the case for tic-tac-toe*? a) I play first: I play center, you play corners, I wedge else I play corner, you play center, I play corner (diagonal) b) You play first: my best outcome is to keep blocking you until we reach a draw (or you make a mistake) • What about chess? * https://www.quora.com/Is-there-a-way-to-never-lose-at-Tic-Tac-Toe DTU Electrical Engineering Centre for Electric Power and Energy Extensive games • Besides tic-tac-toe and chess there are several situations that can be modelled as extensive games • Incentives: Employer signs workers, workers decide on their behaviour, benefits and promotions are given accordingly • RnD: Firms decide on which technologies to invest • Monetary policy: Central banks decide on the monetary supply to create / counter inflation and individuals act accordingly • Market entry: Firms decide on whether they enter DTU Electrical Engineering Centre for Electric Power and Energy Market entry • The race for influence (exercise 1): The “Let it burn” association pushes for a policy change in Windville and Solartown, pioneers in adopting renewables. • From allocating limited resources to modelling the incumbent- challenger relation: break (or not) a monopoly • A challenger chooses to enter or not (in, out) in a race and the incumbent chooses to acquiesce (accept reluctantly) or fight if the challenger enters Players: challenger, incumbent Actions: in, out, acquiesce, fight Terminal histories: (in, acquiesce), (in, fight), out Payoffs: uc(in,acq.)=2, uc(out)=1, uc(in,fight)=0, ui(out)=2, ui(in,acq.)=1, ui(in,fight)=0 DTU Electrical Engineering Centre for Electric Power and Energy Market entry • The diagram illustration highlights the game’s sequential nature • What is the solution of this game? (for both the challenger and the incumbent?) DTU Electrical Engineering Centre for Electric Power and Energy The entry game Representing the extensive form as a strategic game: P1 the Challenger P2 the Incumbent (monopolist) P1 P2 Acquiesce Fight In 2,1 0,0 Out 1,2 1,2 Two pure strategy Nash equilibria: (in, acq.), (out, fight) Note that fight in (out, fight) does not appear if P1 chooses out (out, fight) is sustained by a non-credible threat of P2 DTU Electrical Engineering Centre for Electric Power and Energy The entry game Representing the extensive form as a strategic game: P1 the Challenger P2 the Incumbent (monopolist) Two pure strategy Nash equilibria: (in, acq.), (out, fight) Note that fight in (out, fight) does not appear if P1 chooses out (out, fight) is sustained by a non-credible threat of P2 Non-credible: P1 stays out because it expects that P2 will fight DTU Electrical Engineering Centre for Electric Power and Energy Extensive form equilibria • The non-credible NE addresses a complication in the equilibrium concept, visible when deriving the payoff matrix • In the strategic form all actions are observed simultaneously i.e. even if challenger chooses out, it observes the incumbent’s actions: same when using the payoff matrix for the extensive • The (out, fight) of the strategic form does not exist in extensive • In the extensive form there is a time-line, hence a challenger cannot observe the incumbent if it chooses out (because the action will never happen) • There has to be a process that rules out the non-credible equilibria: backwards induction followed by the definition of subgame equilibria DTU Electrical Engineering Centre for Electric Power and Energy Backward induction • For finite horizon games (like those we discussed) we start from the bottom of the tree and go backwards node by node • During this process we eliminate actions that are not sequentially rational e.g. fight in (out, fight) • The challenger reasons that if he enters, the incumbent will accommodate, because it is better than fighting. Given the incumbent’s behaviour, the challenger chooses to enter • Backward induction: a player in motion deduces for each of his avail. actions, the actions others (inc. himself) will take (remember tic-tac-toe and chess) and yields the most prefered terminal history (in, acq.) REMEMBER: challenger moves first DTU Electrical Engineering Centre for Electric Power and Energy The entry game (ver 2) • Backward induction not always sufficient for determining the solution of a game • If the challenger enters, the incumbent is indifferent between acquiescing or fighting • If C knows (or assumes) that I will acquiesce he will choose In (else out) or mixed strategy equilibrium (chance moves) • Equilibrium concept similar between strategic and extensive games: model a player’s behaviour in a steady state DTU Electrical Engineering Centre for Electric Power and Energy Subgame perfect equilibrium • A game tree consists of a set of nodes and a set of edges • Each edge connects two nodes (these nodes are adjacent) • For any pair of nodes there is a unique path that connects two nodes (??) • The length of a path is given by the number of edges contained • The beginning of the tree is the root and the adjacent nodes the successors DTU Electrical Engineering Centre for Electric Power and Energy Subgame perfect equilibrium • A subgame of a game tree begins at a nonterminal node and it includes all nodes and edges following (it concludes at payoffs) • A subgame beginning at node x is obtained by removing the edge connecting x and its predecessor s.t. the connected part containing x is the subgame DTU Electrical Engineering Centre for Electric Power and Energy Subgame perfect equilibrium • A Nash equilibrium of an extensive game is subgame perfect if the NE strategy can be applied at every stage (and remain eq.) • Subgame perfect Nash equilibrium is a Nash equilibrium DTU Electrical Engineering Centre for Electric Power and Energy Subgame perfect equilibrium • A Nash equilibrium of an extensive game is subgame perfect if the NE strategy can be applied at every stage (and remain eq.) • Subgame perfect Nash equilibrium is a Nash equilibrium The backward induction process: Subgame of length 1 P1 reasons that had he followed C and P2 followed E the best he can do is G, which for him yields less than other scenarios (hence G, E, C) not a subgame perfect eq.
Load more

Recommended publications
  • Uniqueness and Symmetry in Bargaining Theories of Justice
    Philos Stud DOI 10.1007/s11098-013-0121-y Uniqueness and symmetry in bargaining theories of justice John Thrasher Ó Springer Science+Business Media Dordrecht 2013 Abstract For contractarians, justice is the result of a rational bargain. The goal is to show that the rules of justice are consistent with rationality. The two most important bargaining theories of justice are David Gauthier’s and those that use the Nash’s bargaining solution. I argue that both of these approaches are fatally undermined by their reliance on a symmetry condition. Symmetry is a substantive constraint, not an implication of rationality. I argue that using symmetry to generate uniqueness undermines the goal of bargaining theories of justice. Keywords David Gauthier Á John Nash Á John Harsanyi Á Thomas Schelling Á Bargaining Á Symmetry Throughout the last century and into this one, many philosophers modeled justice as a bargaining problem between rational agents. Even those who did not explicitly use a bargaining problem as their model, most notably Rawls, incorporated many of the concepts and techniques from bargaining theories into their understanding of what a theory of justice should look like. This allowed them to use the powerful tools of game theory to justify their various theories of distributive justice. The debates between partisans of different theories of distributive justice has tended to be over the respective benefits of each particular bargaining solution and whether or not the solution to the bargaining problem matches our pre-theoretical intuitions about justice. There is, however, a more serious problem that has effectively been ignored since economists originally J.
    [Show full text]
  • Lecture Notes
    GRADUATE GAME THEORY LECTURE NOTES BY OMER TAMUZ California Institute of Technology 2018 Acknowledgments These lecture notes are partially adapted from Osborne and Rubinstein [29], Maschler, Solan and Zamir [23], lecture notes by Federico Echenique, and slides by Daron Acemoglu and Asu Ozdaglar. I am indebted to Seo Young (Silvia) Kim and Zhuofang Li for their help in finding and correcting many errors. Any comments or suggestions are welcome. 2 Contents 1 Extensive form games with perfect information 7 1.1 Tic-Tac-Toe ........................................ 7 1.2 The Sweet Fifteen Game ................................ 7 1.3 Chess ............................................ 7 1.4 Definition of extensive form games with perfect information ........... 10 1.5 The ultimatum game .................................. 10 1.6 Equilibria ......................................... 11 1.7 The centipede game ................................... 11 1.8 Subgames and subgame perfect equilibria ...................... 13 1.9 The dollar auction .................................... 14 1.10 Backward induction, Kuhn’s Theorem and a proof of Zermelo’s Theorem ... 15 2 Strategic form games 17 2.1 Definition ......................................... 17 2.2 Nash equilibria ...................................... 17 2.3 Classical examples .................................... 17 2.4 Dominated strategies .................................. 22 2.5 Repeated elimination of dominated strategies ................... 22 2.6 Dominant strategies ..................................
    [Show full text]
  • The Role of Self-Interest in Elite Bargaining
    The role of self-interest in elite bargaining Brad L. LeVecka,b, D. Alex Hughesa,c, James H. Fowlera,c,d, Emilie Hafner-Burtona,c, and David G. Victora,1 aLaboratory on International Law and Regulation, School of International Relations and Pacific Studies, cDepartment of Political Science, and dDivision of Medical Genetics, University of California, San Diego, La Jolla, CA 92093; and bDepartment of Political Science, University of California, Merced, CA 95343 Edited by David G. Rand, Yale University, New Haven, CT, and accepted by the Editorial Board November 11, 2014 (received for review May 28, 2014) One of the best-known and most replicated laboratory results in evident in less experienced populations (16, 17). Some research behavioral economics is that bargainers frequently reject low has examined whether leaders are psychologically or physiolog- offers, even when it harms their material self-interest. This finding ically different from nonelite populations—for example, research could have important implications for international negotiations on anxiety levels in leaders (18) and work on how leaders may be on many problems facing humanity today, because models of selected for skills in coordination, leadership, and followership international bargaining assume exactly the opposite: that policy (19). The few existing studies that have empirically examined makers are rational and self-interested. However, it is unknown actual decision making by elites in highly particular roles—such whether elites who engage in diplomatic bargaining will similarly as professional tennis players, soccer players, and traders—sug- reject low offers because past research has been based almost gest that, through a combination of learning and attrition, elites exclusively on convenience samples of undergraduates, members are more likely to exhibit self-interested, profit-maximizing be- of the general public, or small-scale societies rather than highly havior than their less experienced counterparts (20–23).
    [Show full text]
  • Nash Equilibrium
    Lecture 3: Nash equilibrium Nash equilibrium: The mathematician John Nash introduced the concept of an equi- librium for a game, and equilibrium is often called a Nash equilibrium. They provide a way to identify reasonable outcomes when an easy argument based on domination (like in the prisoner's dilemma, see lecture 2) is not available. We formulate the concept of an equilibrium for a two player game with respective 0 payoff matrices PR and PC . We write PR(s; s ) for the payoff for player R when R plays 0 s and C plays s, this is simply the (s; s ) entry the matrix PR. Definition 1. A pair of strategies (^sR; s^C ) is an Nash equilbrium for a two player game if no player can improve his payoff by changing his strategy from his equilibrium strategy to another strategy provided his opponent keeps his equilibrium strategy. In terms of the payoffs matrices this means that PR(sR; s^C ) ≤ P (^sR; s^C ) for all sR ; and PC (^sR; sC ) ≤ P (^sR; s^C ) for all sc : The idea at work in the definition of Nash equilibrium deserves a name: Definition 2. A strategy s^R is a best-response to a strategy sc if PR(sR; sC ) ≤ P (^sR; sC ) for all sR ; i.e. s^R is such that max PR(sR; sC ) = P (^sR; sC ) sR We can now reformulate the idea of a Nash equilibrium as The pair (^sR; s^C ) is a Nash equilibrium if and only ifs ^R is a best-response tos ^C and s^C is a best-response tos ^R.
    [Show full text]
  • Bounded Rationality in Bargaining Games: Do Proposers Believe That Responders Reject an Equal Split?
    No. 11 UNIVERSITY OF COLOGNE WORKING PAPER SERIES IN ECONOMICS BOUNDED RATIONALITY IN BARGAINING GAMES: DO PROPOSERS BELIEVE THAT RESPONDERS REJECT AN EQUAL SPLIT? BEN GREINER Department of Economics University of Cologne Albertus-Magnus-Platz D-50923 Köln Germany http://www.wiso.uni-koeln.de Bounded Rationality in Bargaining Games: Do Proposers Believe That Responders Reject an Equal Split?∗ Ben Greiner† June 19, 2004 Abstract Puzzled by the experimental results of the ’impunity game’ by Bolton & Zwick (1995) we replicate the game and alter it in a systematic manner. We find that although almost nobody actually rejects an offered equal split in a bargaining game, proposers behave as if there would be a considerably large rejection rate for equal splits. This result is inconsistent with existing models of economic decision making. This includes models of selfish play- ers as well as models of social utility and reciprocity, even when combined with erroneous decision making. Our data suggests that subjects fail to foresee their opponent’s decision even for one step in our simple bargaining games. We consider models of bounded rational decision making such as rules of thumb as explanations for the observed behavioral pattern. Keywords: ultimatum game, dictator game, impunity game, social utility, bounded rationality JEL Classification: C72, C92, D3 ∗I thank Bettina Bartels, Hakan Fink and Ralica Gospodinova for research assistance, the Strategic Interaction Group at MPI Jena for comments and serving as pilot participants, especially Susanne B¨uchner, Sven Fischer, Katinka Pantz and Carsten Schmidt for support in the conduction of the experiment, and Werner G¨uth, Axel Ockenfels, Ro’i Zultan, participants at the European ESA Meeting in Erfurt and at the Brown Bag seminars in Jena and Cologne for discussions and valuable comments.
    [Show full text]
  • Minimally Acceptable Altruism and the Ultimatum Game
    A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Rotemberg, Julio J. Working Paper Minimally acceptable altruism and the ultimatum game Working Papers, No. 06-12 Provided in Cooperation with: Federal Reserve Bank of Boston Suggested Citation: Rotemberg, Julio J. (2006) : Minimally acceptable altruism and the ultimatum game, Working Papers, No. 06-12, Federal Reserve Bank of Boston, Boston, MA This Version is available at: http://hdl.handle.net/10419/55606 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative Commons Licences), you genannten Lizenz gewährten Nutzungsrechte. may exercise further usage rights as specified in the indicated licence. www.econstor.eu No. 06‐12 Minimally Acceptable Altruism and the Ultimatum Game Julio J. Rotemberg Abstract: I suppose that people react with anger when others show themselves not to be minimally altruistic. With heterogeneous agents, this can account for the experimental results of ultimatum and dictator games.
    [Show full text]
  • Econs 424 % Bargaining Games and an Introduction to Repeated Games
    EconS 424 - Bargaining Games and an Introduction to Repeated Games Félix Muñoz-García Washington State University [email protected] March 25, 2014 Félix Muñoz-García (WSU) EconS 424 - Recitation 6 March25,2014 1/44 Watson, Ch. 19 # 8 In experimental tests of the ultimatum bargaining game, subjects who propose the split rarely o¤er a tiny share of the surplus to the other party. Furthermore, sometimes subjects reject positive o¤ers. These …ndings seem to contradict our standard analysis of the ultimatum bargaining game. Many scholars conclude that the payo¤s speci…ed in the basic model do not represent the actual preferences of the people who participate in the experiments. In reality, people care about more than their own monetary rewards. Félix Muñoz-García (WSU) EconS 424 - Recitation 6 March25,2014 2/44 Watson, Ch. 19 # 8 For example, people also act on feelings of spite and the ideal of fairness. Suppose that, in the ultimatum game, the responder’spayo¤ is y + a(y z), where y is the responder’smonetary reward, z is the proposer’smonetary take, and a is a positive constant. That is, the responder cares about how much money he gets and he cares about relative monetary amounts (the di¤erence between the money he gets and the money the other player gets). Assume that the proposer’spayo¤ is as in the basic model. Represent this game in the extensive form, writing the payo¤s in terms of m, the monetary o¤er of the proposer, and the parameter a. Félix Muñoz-García (WSU) EconS 424 - Recitation 6 March25,2014 3/44 Watson, Ch.
    [Show full text]
  • Bargaining Under a Deadline: Evidence from the Reverse Ultimatum Game ✩
    Games and Economic Behavior 45 (2003) 347–368 www.elsevier.com/locate/geb Bargaining under a deadline: evidence from the reverse ultimatum game ✩ Uri Gneezy,a,b Ernan Haruvy,c and Alvin E. Roth d,e,∗ a University of Chicago Graduate School of Business, Chicago, IL, USA b Technion, Israel Institute of Technology, Haifa, Israel c School of Management, University of Texas at Dallas, Richardson, TX, USA d Department of Economics, Harvard University, Cambridge, MA, USA e Harvard Business School, Boston, MA, USA Received 8 January 2003 Abstract We study a “reverse” ultimatum game, in which proposers have multiple chances to offer responders a division of some fixed pie. The game ends if the responder accepts an offer, or if, following a rejection, the proposer decides not to make a better offer. The unique subgame perfect equilibrium gives the proposer the minimum possible payoff. Nevertheless, the experimental results are not too different from those of the standard ultimatum game, although proposers generally receive slightly less than half of the surplus. We use the reverse ultimatum game to study deadlines experimentally. With a deadline, the subgame perfect equilibrium prediction is that the proposer gets the entire surplus. Deadlines are used strategically to influence the outcome, and agreements are reached near the deadline. Strategic considerations are evident in the differences in observed behavior between the deadline and no deadline conditions, even though agreements are substantially less extreme than predicted by perfect equilibrium. 2003 Elsevier Inc. All rights reserved. JEL classification: C7; C9 ✩ Accompanying materials to be found at http://www.utdallas.edu/~eharuvy/deadline/.
    [Show full text]
  • Reciprocity and Overoptimism in Public Goods Games
    Biased Beliefs and Reciprocal Behavior in Social Dilemmas Von der Fakultät für Wirtschaftswissenschaften der Rheinisch-Westfälischen Technischen Hochschule Aachen zur Erlangung des akademischen Grades eines Doktors der Wirtschafts- und Sozialwissenschaften genehmigte Dissertation vorgelegt von Dipl.-Ing. Anselm Hüwe Berichter: Univ.-Prof. Dr. rer. pol. Wolfgang Breuer Univ.-Prof. Dr. rer. pol. Christine Harbring Tag der mündlichen Prüfung: 21.04.2015 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar. Table of Contents 1 Introduction ............................................................................................................................. 1 2 Economic Experiments ........................................................................................................... 3 3 Game Theory and (Common) Belief in Rationality ................................................................ 8 3.1 Boundedly Rational Behavior ........................................................................................ 11 3.2 Biased Beliefs ................................................................................................................. 14 4 Social Preferences ................................................................................................................. 15 4.1 When Social Preferences Should Be Assumed, and When Not ..................................... 16 4.2 Evolutionary Analysis ...................................................................................................
    [Show full text]
  • Theory and Evidence on Bargaining University of Oxford, Michaelmas
    Behavioural Economics: Theory and Evidence on Bargaining University of Oxford, Michaelmas Term 2013 and Asia Summer Institute in Behavioral Economics National University of Singapore, 22 July-2 August 2013 Vincent P. Crawford, University of Oxford, All Souls College, and University of California, San Diego (with thanks to Sergei Izmalkov, Paul Milgrom, Al Roth and Muhamet Yildiz) Revised 3 July 2013 1 Bargaining Bargaining is at the heart of game-theoretic microeconomics. There are two prominent strands of theory, and some important experiments that raise questions about the theory, still unresolved. The theory includes the unstructured/cooperative theory of Nash (1950 Eonometrica), the structured/cooperative theory of Nash (1953 Econometrica) and the structured/noncooperative theory of Rubinstein (1982 Econometrica). The experiments include the unstructured bargaining experiments of Roth, Murnighan, and collaborators, culminating in Roth and Murnighan (1982 Econometrica), and structured bargaining experiments. In these slides I briefly consider both theory and experiments. 2 Rubinstein’s structured/noncooperative theory of bargaining Rubinstein’s theory postulates that bargaining proceeds via a given noncooperative game, in which two bargainers make alternating offers to each other on a fixed schedule until they reach an agreement. In Rubinstein’s version of the model there is no time limit. (Ingolf Stahl earlier studied a finite-horizon version, of which an extreme case is the one-period ultimatum game.) Delay is assumed to be costly (except in the ultimatum game, where delayed agreements are impossible anyway). The model allows the agreement to be about anything or things that are continuously variable, but I assume for simplicity that the bargainers are bargaining over the division of a dollar.
    [Show full text]
  • Backward Induction, Or SPNE’S
    LECTURE 10: GAMES IN EXTENSIVE FORM Jan Zouhar Games and Decisions 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 their opponent is) in most games played for amusement (cards, chess, other board games), players make moves in turns → sequential move games many economic problems are in the sequential form: English auctions price competition: firms repeatedly charge prices executive compensation (contract signed; then executive works) monetary authority and firms (continually observe and learn actions) … formally, we describe these as games in extensive form (representation using a game tree) Games and Decisions Jan Zouhar Class Game: Century Mark 3 rules: played by fixed pairs of players taking turns at each turn, each player chooses a number (integer) between 1 and 10 inclusive this choice is added to sum of all previous choices (initial sum is 0) the first player to take the cumulative sum above 100 (century) loses the game prize: 5 extra points for the final test (!!!) Volunteers? Games and Decisions Jan Zouhar Class Game: Century Mark (cont’d) 4 Analysis of the game what’s the winning strategy? broadly speaking, bring the sum to 89; then your opponent can’t possibly win actually, it’s enough to bring it to 78; then you can make sure to make it 89 later reasoning like this, the winning positions are: 100, 89, 78, 67, 56, 45, 34, 23, 12, 1 → the first mover can guarantee a win! winning strategy: in
    [Show full text]
  • Section Note 3
    Section Note 3 Ariella Kahn-Lang and Guthrie Gray-Lobe∗ February 13th 2013 Agenda 1. Game Trees 2. Writing out Strategies and the Strategy Space 3. Backwards Induction 1 Game Trees Let's review the basic elements of game trees (extensive form games). Remember, we can write any game in either its normal (matrix) form or its extensive (tree) form - which representation we use is mainly a question of which solution concept we want to implement. Game trees are particularly useful for representing dynamic (sequential) games, primarily be- cause they easily allow us to implement the simple solution concept of backwards induction (see later). 1.1 Dynamic Games vs. Simultaneous Games Many important economic applications in game theory cannot be described by a simultaneous move game. Recall that a simultaneous game is one where all players were choosing their actions without any information about actions chosen by other players. A dynamic game, in contrast, is one where at least one player at some point of the game is choosing his action with some information about actions chosen previously. While dynamic games are often sequential over time, what matters is not the timing, but the information a player holds about previous moves in the game, because this is what enables them to condition their strategy on previous moves. For this reason, the `order' in which we draw a game tree matters only in getting the flow of information correctly. In simultaneous games, we can draw the game tree either way round and it represents the same game. ∗Thanks to previous years' TF's for making their section materials available for adaptation.
    [Show full text]