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Labsi Working Papers
UNIVERSITY OF SIENA S.N. O’ H iggins Arturo Palomba Patrizia Sbriglia Second Mover Advantage and Bertrand Dynamic Competition: An Experiment May 2010 LABSI WORKING PAPERS N. 28/2010 SECOND MOVER ADVANTAGE AND BERTRAND DYNAMIC COMPETITION: AN EXPERIMENT § S.N. O’Higgins University of Salerno [email protected] Arturo Palomba University of Naples II [email protected] Patrizia Sbriglia §§ University of Naples II [email protected] Abstract In this paper we provide an experimental test of a dynamic Bertrand duopolistic model, where firms move sequentially and their informational setting varies across different designs. Our experiment is composed of three treatments. In the first treatment, subjects receive information only on the costs and demand parameters and on the price’ choices of their opponent in the market in which they are positioned (matching is fixed); in the second and third treatments, subjects are also informed on the behaviour of players who are not directly operating in their market. Our aim is to study whether the individual behaviour and the process of equilibrium convergence are affected by the specific informational setting adopted. In all treatments we selected students who had previously studied market games and industrial organization, conjecturing that the specific participants’ expertise decreased the chances of imitation in treatment II and III. However, our results prove the opposite: the extra information provided in treatment II and III strongly affects the long run convergence to the market equilibrium. In fact, whilst in the first session, a high proportion of markets converge to the Nash-Bertrand symmetric solution, we observe that a high proportion of markets converge to more collusive outcomes in treatment II and more competitive outcomes in treatment III. -
1 Sequential Games
1 Sequential Games We call games where players take turns moving “sequential games”. Sequential games consist of the same elements as normal form games –there are players, rules, outcomes, and payo¤s. However, sequential games have the added element that history of play is now important as players can make decisions conditional on what other players have done. Thus, if two people are playing a game of Chess the second mover is able to observe the …rst mover’s initial move prior to making his initial move. While it is possible to represent sequential games using the strategic (or matrix) form representation of the game it is more instructive at …rst to represent sequential games using a game tree. In addition to the players, actions, outcomes, and payo¤s, the game tree will provide a history of play or a path of play. A very basic example of a sequential game is the Entrant-Incumbent game. The game is described as follows: Consider a game where there is an entrant and an incumbent. The entrant moves …rst and the incumbent observes the entrant’sdecision. The entrant can choose to either enter the market or remain out of the market. If the entrant remains out of the market then the game ends and the entrant receives a payo¤ of 0 while the incumbent receives a payo¤ of 2. If the entrant chooses to enter the market then the incumbent gets to make a choice. The incumbent chooses between …ghting entry or accommodating entry. If the incumbent …ghts the entrant receives a payo¤ of 3 while the incumbent receives a payo¤ of 1. -
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. -
Notes on Sequential and Repeated Games
Notes on sequential and repeated games 1 Sequential Move Games Thus far we have examined games in which players make moves simultaneously (or without observing what the other player has done). Using the normal (strategic) form representation of a game we can identify sets of strategies that are best responses to each other (Nash Equilibria). We now focus on sequential games of complete information. We can still use the normal form representation to identify NE but sequential games are richer than that because some players observe other players’decisions before they take action. The fact that some actions are observable may cause some NE of the normal form representation to be inconsistent with what one might think a player would do. Here’sa simple game between an Entrant and an Incumbent. The Entrant moves …rst and the Incumbent observes the Entrant’s action and then gets to make a choice. The Entrant has to decide whether or not he will enter a market or not. Thus, the Entrant’s two strategies are “Enter” or “Stay Out”. If the Entrant chooses “Stay Out” then the game ends. The payo¤s for the Entrant and Incumbent will be 0 and 2 respectively. If the Entrant chooses “Enter” then the Incumbent gets to choose whether or not he will “Fight”or “Accommodate”entry. If the Incumbent chooses “Fight”then the Entrant receives 3 and the Incumbent receives 1. If the Incumbent chooses “Accommodate”then the Entrant receives 2 and the Incumbent receives 1. This game in normal form is Incumbent Fight if Enter Accommodate if Enter . -
Evolutionary Game Theory: ESS, Convergence Stability, and NIS
Evolutionary Ecology Research, 2009, 11: 489–515 Evolutionary game theory: ESS, convergence stability, and NIS Joseph Apaloo1, Joel S. Brown2 and Thomas L. Vincent3 1Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, Nova Scotia, Canada, 2Department of Biological Sciences, University of Illinois, Chicago, Illinois, USA and 3Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, Arizona, USA ABSTRACT Question: How are the three main stability concepts from evolutionary game theory – evolutionarily stable strategy (ESS), convergence stability, and neighbourhood invader strategy (NIS) – related to each other? Do they form a basis for the many other definitions proposed in the literature? Mathematical methods: Ecological and evolutionary dynamics of population sizes and heritable strategies respectively, and adaptive and NIS landscapes. Results: Only six of the eight combinations of ESS, convergence stability, and NIS are possible. An ESS that is NIS must also be convergence stable; and a non-ESS, non-NIS cannot be convergence stable. A simple example shows how a single model can easily generate solutions with all six combinations of stability properties and explains in part the proliferation of jargon, terminology, and apparent complexity that has appeared in the literature. A tabulation of most of the evolutionary stability acronyms, definitions, and terminologies is provided for comparison. Key conclusions: The tabulated list of definitions related to evolutionary stability are variants or combinations of the three main stability concepts. Keywords: adaptive landscape, convergence stability, Darwinian dynamics, evolutionary game stabilities, evolutionarily stable strategy, neighbourhood invader strategy, strategy dynamics. INTRODUCTION Evolutionary game theory has and continues to make great strides. -
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 .................................. -
SEQUENTIAL GAMES with PERFECT INFORMATION Example
SEQUENTIAL GAMES WITH PERFECT INFORMATION Example 4.9 (page 105) Consider the sequential game given in Figure 4.9. We want to apply backward induction to the tree. 0 Vertex B is owned by player two, P2. The payoffs for P2 are 1 and 3, with 3 > 1, so the player picks R . Thus, the payoffs at B become (0, 3). 00 Next, vertex C is also owned by P2 with payoffs 1 and 0. Since 1 > 0, P2 picks L , and the payoffs are (4, 1). Player one, P1, owns A; the choice of L gives a payoff of 0 and R gives a payoff of 4; 4 > 0, so P1 chooses R. The final payoffs are (4, 1). 0 00 We claim that this strategy profile, { R } for P1 and { R ,L } is a Nash equilibrium. Notice that the 0 00 strategy profile gives a choice at each vertex. For the strategy { R ,L } fixed for P2, P1 has a maximal payoff by choosing { R }, ( 0 00 0 00 π1(R, { R ,L }) = 4 π1(R, { R ,L }) = 4 ≥ 0 00 π1(L, { R ,L }) = 0. 0 00 In the same way, for the strategy { R } fixed for P1, P2 has a maximal payoff by choosing { R ,L }, ( 00 0 00 π2(R, {∗,L }) = 1 π2(R, { R ,L }) = 1 ≥ 00 π2(R, {∗,R }) = 0, where ∗ means choose either L0 or R0. Since no change of choice by a player can increase that players own payoff, the strategy profile is called a Nash equilibrium. Notice that the above strategy profile is also a Nash equilibrium on each branch of the game tree, mainly starting at either B or starting at C. -
1 Bertrand Model
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 with the strategic choices of its competition. In the former, this is trivially true since there isn't any competition. While the latter is so insignificant that the single firm has no effect. In an oligopoly where there is more than one firm, and yet because the number of firms are small, they each have to consider what the other does. Consider the product launch decision, and pricing decision of Apple in relation to the IPOD models. If the features of the models it has in the line up is similar to Creative Technology's, it would have to be concerned with the pricing decision, and the timing of its announcement in relation to that of the other firm. We will now begin the exposition of Oligopolistic Competition. 1 Bertrand Model Firms can compete on several variables, and levels, for example, they can compete based on their choices of prices, quantity, and quality. The most basic and funda- mental competition pertains to pricing choices. The Bertrand Model is examines the interdependence between rivals' decisions in terms of pricing decisions. The assumptions of the model are: 1. 2 firms in the market, i 2 f1; 2g. 2. Goods produced are homogenous, ) products are perfect substitutes. 3. Firms set prices simultaneously. 4. Each firm has the same constant marginal cost of c. What is the equilibrium, or best strategy of each firm? The answer is that both firms will set the same prices, p1 = p2 = p, and that it will be equal to the marginal ECON 312: Oligopolisitic Competition 2 cost, in other words, the perfectly competitive outcome. -
Evolutionary Stable Strategy Application of Nash Equilibrium in Biology
GENERAL ARTICLE Evolutionary Stable Strategy Application of Nash Equilibrium in Biology Jayanti Ray-Mukherjee and Shomen Mukherjee Every behaviourally responsive animal (including us) make decisions. These can be simple behavioural decisions such as where to feed, what to feed, how long to feed, decisions related to finding, choosing and competing for mates, or simply maintaining ones territory. All these are conflict situations between competing individuals, hence can be best understood Jayanti Ray-Mukherjee is using a game theory approach. Using some examples of clas- a faculty member in the School of Liberal Studies sical games, we show how evolutionary game theory can help at Azim Premji University, understand behavioural decisions of animals. Game theory Bengaluru. Jayanti is an (along with its cousin, optimality theory) continues to provide experimental ecologist who a strong conceptual and theoretical framework to ecologists studies mechanisms of species coexistence among for understanding the mechanisms by which species coexist. plants. Her research interests also inlcude plant Most of you, at some point, might have seen two cats fighting. It invasion ecology and is often accompanied with the cats facing each other with puffed habitat restoration. up fur, arched back, ears back, tail twitching, with snarls, growls, Shomen Mukherjee is a and howls aimed at each other. But, if you notice closely, they faculty member in the often try to avoid physical contact, and spend most of their time School of Liberal Studies in the above-mentioned behavioural displays. Biologists refer to at Azim Premji University, this as a ‘limited war’ or conventional (ritualistic) strategy (not Bengaluru. He uses field experiments to study causing serious injury), as opposed to dangerous (escalated) animal behaviour and strategy (Figure 1) [1]. -
Finitely Repeated Games
Repeated games 1: Finite repetition Universidad Carlos III de Madrid 1 Finitely repeated games • A finitely repeated game is a dynamic game in which a simultaneous game (the stage game) is played finitely many times, and the result of each stage is observed before the next one is played. • Example: Play the prisoners’ dilemma several times. The stage game is the simultaneous prisoners’ dilemma game. 2 Results • If the stage game (the simultaneous game) has only one NE the repeated game has only one SPNE: In the SPNE players’ play the strategies in the NE in each stage. • If the stage game has 2 or more NE, one can find a SPNE where, at some stage, players play a strategy that is not part of a NE of the stage game. 3 The prisoners’ dilemma repeated twice • Two players play the same simultaneous game twice, at ! = 1 and at ! = 2. • After the first time the game is played (after ! = 1) the result is observed before playing the second time. • The payoff in the repeated game is the sum of the payoffs in each stage (! = 1, ! = 2) • Which is the SPNE? Player 2 D C D 1 , 1 5 , 0 Player 1 C 0 , 5 4 , 4 4 The prisoners’ dilemma repeated twice Information sets? Strategies? 1 .1 5 for each player 2" for each player D C E.g.: (C, D, D, C, C) Subgames? 2.1 5 D C D C .2 1.3 1.5 1 1.4 D C D C D C D C 2.2 2.3 2 .4 2.5 D C D C D C D C D C D C D C D C 1+1 1+5 1+0 1+4 5+1 5+5 5+0 5+4 0+1 0+5 0+0 0+4 4+1 4+5 4+0 4+4 1+1 1+0 1+5 1+4 0+1 0+0 0+5 0+4 5+1 5+0 5+5 5+4 4+1 4+0 4+5 4+4 The prisoners’ dilemma repeated twice Let’s find the NE in the subgames. -
Game Theory]: Basics of Game Theory
Artificial Intelligence Methods for Social Good M2-1 [Game Theory]: Basics of Game Theory 08-537 (9-unit) and 08-737 (12-unit) Instructor: Fei Fang [email protected] Wean Hall 4126 1 5/8/2018 Quiz 1: Recap: Optimization Problem Given coordinates of 푛 residential areas in a city (assuming 2-D plane), denoted as 푥1, … , 푥푛, the government wants to find a location that minimizes the sum of (Euclidean) distances to all residential areas to build a hospital. The optimization problem can be written as A: min |푥푖 − 푥| 푥 푖 B: min 푥푖 − 푥 푥 푖 2 2 C: min 푥푖 − 푥 푥 푖 D: none of above 2 Fei Fang 5/8/2018 From Games to Game Theory The study of mathematical models of conflict and cooperation between intelligent decision makers Used in economics, political science etc 3/72 Fei Fang 5/8/2018 Outline Basic Concepts in Games Basic Solution Concepts Compute Nash Equilibrium Compute Strong Stackelberg Equilibrium 4 Fei Fang 5/8/2018 Learning Objectives Understand the concept of Game, Player, Action, Strategy, Payoff, Expected utility, Best response Dominant Strategy, Maxmin Strategy, Minmax Strategy Nash Equilibrium Stackelberg Equilibrium, Strong Stackelberg Equilibrium Describe Minimax Theory Formulate the following problem as an optimization problem Find NE in zero-sum games (LP) Find SSE in two-player general-sum games (multiple LP and MILP) Know how to find the method/algorithm/solver/package you can use for solving the games Compute NE/SSE by hand or by calling a solver for small games 5 Fei Fang 5/8/2018 Let’s Play! Classical Games -
8. Maxmin and Minmax Strategies
CMSC 474, Introduction to Game Theory 8. Maxmin and Minmax Strategies Mohammad T. Hajiaghayi University of Maryland Outline Chapter 2 discussed two solution concepts: Pareto optimality and Nash equilibrium Chapter 3 discusses several more: Maxmin and Minmax Dominant strategies Correlated equilibrium Trembling-hand perfect equilibrium e-Nash equilibrium Evolutionarily stable strategies Worst-Case Expected Utility For agent i, the worst-case expected utility of a strategy si is the minimum over all possible Husband Opera Football combinations of strategies for the other agents: Wife min u s ,s Opera 2, 1 0, 0 s-i i ( i -i ) Football 0, 0 1, 2 Example: Battle of the Sexes Wife’s strategy sw = {(p, Opera), (1 – p, Football)} Husband’s strategy sh = {(q, Opera), (1 – q, Football)} uw(p,q) = 2pq + (1 – p)(1 – q) = 3pq – p – q + 1 We can write uw(p,q) For any fixed p, uw(p,q) is linear in q instead of uw(sw , sh ) • e.g., if p = ½, then uw(½,q) = ½ q + ½ 0 ≤ q ≤ 1, so the min must be at q = 0 or q = 1 • e.g., minq (½ q + ½) is at q = 0 minq uw(p,q) = min (uw(p,0), uw(p,1)) = min (1 – p, 2p) Maxmin Strategies Also called maximin A maxmin strategy for agent i A strategy s1 that makes i’s worst-case expected utility as high as possible: argmaxmin ui (si,s-i ) si s-i This isn’t necessarily unique Often it is mixed Agent i’s maxmin value, or security level, is the maxmin strategy’s worst-case expected utility: maxmin ui (si,s-i ) si s-i For 2 players it simplifies to max min u1s1, s2 s1 s2 Example Wife’s and husband’s strategies