CS440/ECE448 Fall 2016 Midterm Review You Need to Be Able To
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The Determinants of Efficient Behavior in Coordination Games
The Determinants of Efficient Behavior in Coordination Games Pedro Dal B´o Guillaume R. Fr´echette Jeongbin Kim* Brown University & NBER New York University Caltech May 20, 2020 Abstract We study the determinants of efficient behavior in stag hunt games (2x2 symmetric coordination games with Pareto ranked equilibria) using both data from eight previous experiments on stag hunt games and data from a new experiment which allows for a more systematic variation of parameters. In line with the previous experimental literature, we find that subjects do not necessarily play the efficient action (stag). While the rate of stag is greater when stag is risk dominant, we find that the equilibrium selection criterion of risk dominance is neither a necessary nor sufficient condition for a majority of subjects to choose the efficient action. We do find that an increase in the size of the basin of attraction of stag results in an increase in efficient behavior. We show that the results are robust to controlling for risk preferences. JEL codes: C9, D7. Keywords: Coordination games, strategic uncertainty, equilibrium selection. *We thank all the authors who made available the data that we use from previous articles, Hui Wen Ng for her assistance running experiments and Anna Aizer for useful comments. 1 1 Introduction The study of coordination games has a long history as many situations of interest present a coordination component: for example, the choice of technologies that require a threshold number of users to be sustainable, currency attacks, bank runs, asset price bubbles, cooper- ation in repeated games, etc. In such examples, agents may face strategic uncertainty; that is, they may be uncertain about how the other agents will respond to the multiplicity of equilibria, even when they have complete information about the environment. -
Lecture 4 Rationalizability & Nash Equilibrium Road
Lecture 4 Rationalizability & Nash Equilibrium 14.12 Game Theory Muhamet Yildiz Road Map 1. Strategies – completed 2. Quiz 3. Dominance 4. Dominant-strategy equilibrium 5. Rationalizability 6. Nash Equilibrium 1 Strategy A strategy of a player is a complete contingent-plan, determining which action he will take at each information set he is to move (including the information sets that will not be reached according to this strategy). Matching pennies with perfect information 2’s Strategies: HH = Head if 1 plays Head, 1 Head if 1 plays Tail; HT = Head if 1 plays Head, Head Tail Tail if 1 plays Tail; 2 TH = Tail if 1 plays Head, 2 Head if 1 plays Tail; head tail head tail TT = Tail if 1 plays Head, Tail if 1 plays Tail. (-1,1) (1,-1) (1,-1) (-1,1) 2 Matching pennies with perfect information 2 1 HH HT TH TT Head Tail Matching pennies with Imperfect information 1 2 1 Head Tail Head Tail 2 Head (-1,1) (1,-1) head tail head tail Tail (1,-1) (-1,1) (-1,1) (1,-1) (1,-1) (-1,1) 3 A game with nature Left (5, 0) 1 Head 1/2 Right (2, 2) Nature (3, 3) 1/2 Left Tail 2 Right (0, -5) Mixed Strategy Definition: A mixed strategy of a player is a probability distribution over the set of his strategies. Pure strategies: Si = {si1,si2,…,sik} σ → A mixed strategy: i: S [0,1] s.t. σ σ σ i(si1) + i(si2) + … + i(sik) = 1. If the other players play s-i =(s1,…, si-1,si+1,…,sn), then σ the expected utility of playing i is σ σ σ i(si1)ui(si1,s-i) + i(si2)ui(si2,s-i) + … + i(sik)ui(sik,s-i). -
Lecture 4 Dynamic Programming
1/17 Lecture 4 Dynamic Programming Last update: Jan 19, 2021 References: Algorithms, Jeff Erickson, Chapter 3. Algorithms, Gopal Pandurangan, Chapter 6. Dynamic Programming 2/17 Backtracking is incredible powerful in solving all kinds of hard prob- lems, but it can often be very slow; usually exponential. Example: Fibonacci numbers is defined as recurrence: 0 if n = 0 Fn =8 1 if n = 1 > Fn 1 + Fn 2 otherwise < ¡ ¡ > A direct translation in:to recursive program to compute Fibonacci number is RecFib(n): if n=0 return 0 if n=1 return 1 return RecFib(n-1) + RecFib(n-2) Fibonacci Number 3/17 The recursive program has horrible time complexity. How bad? Let's try to compute. Denote T(n) as the time complexity of computing RecFib(n). Based on the recursion, we have the recurrence: T(n) = T(n 1) + T(n 2) + 1; T(0) = T(1) = 1 ¡ ¡ Solving this recurrence, we get p5 + 1 T(n) = O(n); = 1.618 2 So the RecFib(n) program runs at exponential time complexity. RecFib Recursion Tree 4/17 Intuitively, why RecFib() runs exponentially slow. Problem: redun- dant computation! How about memorize the intermediate computa- tion result to avoid recomputation? Fib: Memoization 5/17 To optimize the performance of RecFib, we can memorize the inter- mediate Fn into some kind of cache, and look it up when we need it again. MemFib(n): if n = 0 n = 1 retujrjn n if F[n] is undefined F[n] MemFib(n-1)+MemFib(n-2) retur n F[n] How much does it improve upon RecFib()? Assuming accessing F[n] takes constant time, then at most n additions will be performed (we never recompute). -
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 .................................. -
Exhaustive Recursion and Backtracking
CS106B Handout #19 J Zelenski Feb 1, 2008 Exhaustive recursion and backtracking In some recursive functions, such as binary search or reversing a file, each recursive call makes just one recursive call. The "tree" of calls forms a linear line from the initial call down to the base case. In such cases, the performance of the overall algorithm is dependent on how deep the function stack gets, which is determined by how quickly we progress to the base case. For reverse file, the stack depth is equal to the size of the input file, since we move one closer to the empty file base case at each level. For binary search, it more quickly bottoms out by dividing the remaining input in half at each level of the recursion. Both of these can be done relatively efficiently. Now consider a recursive function such as subsets or permutation that makes not just one recursive call, but several. The tree of function calls has multiple branches at each level, which in turn have further branches, and so on down to the base case. Because of the multiplicative factors being carried down the tree, the number of calls can grow dramatically as the recursion goes deeper. Thus, these exhaustive recursion algorithms have the potential to be very expensive. Often the different recursive calls made at each level represent a decision point, where we have choices such as what letter to choose next or what turn to make when reading a map. Might there be situations where we can save some time by focusing on the most promising options, without committing to exploring them all? In some contexts, we have no choice but to exhaustively examine all possibilities, such as when trying to find some globally optimal result, But what if we are interested in finding any solution, whichever one that works out first? At each decision point, we can choose one of the available options, and sally forth, hoping it works out. -
Backtrack Parsing Context-Free Grammar Context-Free Grammar
Context-free Grammar Problems with Regular Context-free Grammar Language and Is English a regular language? Bad question! We do not even know what English is! Two eggs and bacon make(s) a big breakfast Backtrack Parsing Can you slide me the salt? He didn't ought to do that But—No! Martin Kay I put the wine you brought in the fridge I put the wine you brought for Sandy in the fridge Should we bring the wine you put in the fridge out Stanford University now? and University of the Saarland You said you thought nobody had the right to claim that they were above the law Martin Kay Context-free Grammar 1 Martin Kay Context-free Grammar 2 Problems with Regular Problems with Regular Language Language You said you thought nobody had the right to claim [You said you thought [nobody had the right [to claim that they were above the law that [they were above the law]]]] Martin Kay Context-free Grammar 3 Martin Kay Context-free Grammar 4 Problems with Regular Context-free Grammar Language Nonterminal symbols ~ grammatical categories Is English mophology a regular language? Bad question! We do not even know what English Terminal Symbols ~ words morphology is! They sell collectables of all sorts Productions ~ (unordered) (rewriting) rules This concerns unredecontaminatability Distinguished Symbol This really is an untiable knot. But—Probably! (Not sure about Swahili, though) Not all that important • Terminals and nonterminals are disjoint • Distinguished symbol Martin Kay Context-free Grammar 5 Martin Kay Context-free Grammar 6 Context-free Grammar Context-free -
Adversarial Search
Artificial Intelligence CS 165A Oct 29, 2020 Instructor: Prof. Yu-Xiang Wang ® ExamplEs of heuristics in A*-search ® GamEs and Adversarial SEarch 1 Recap: Search algorithms • StatE-spacE diagram vs SEarch TrEE • UninformEd SEarch algorithms – BFS / DFS – Depth Limited Search – Iterative Deepening Search. – Uniform cost search. • InformEd SEarch (with an heuristic function h): – Greedy Best-First-Search. (not complete / optimal) – A* Search (complete / optimal if h is admissible) 2 Recap: Summary table of uninformed search (Section 3.4.7 in the AIMA book.) 3 Recap: A* Search (Pronounced “A-Star”) • Uniform-cost sEarch minimizEs g(n) (“past” cost) • GrEEdy sEarch minimizEs h(n) (“expected” or “future” cost) • “A* SEarch” combines the two: – Minimize f(n) = g(n) + h(n) – Accounts for the “past” and the “future” – Estimates the cheapest solution (complete path) through node n function A*-SEARCH(problem, h) returns a solution or failure return BEST-FIRST-SEARCH(problem, f ) 4 Recap: Avoiding Repeated States using A* Search • Is GRAPH-SEARCH optimal with A*? Try with TREE-SEARCH and B GRAPH-SEARCH 1 h = 5 1 A C D 4 4 h = 5 h = 1 h = 0 7 E h = 0 Graph Search Step 1: Among B, C, E, Choose C Step 2: Among B, E, D, Choose B Step 3: Among D, E, Choose E. (you are not going to select C again) 5 Recap: Consistency (Monotonicity) of heuristic h • A heuristic is consistEnt (or monotonic) provided – for any node n, for any successor n’ generated by action a with cost c(n,a,n’) c(n,a,n’) • h(n) ≤ c(n,a,n’) + h(n’) n n’ – akin to triangle inequality. -
Backtracking / Branch-And-Bound
Backtracking / Branch-and-Bound Optimisation problems are problems that have several valid solutions; the challenge is to find an optimal solution. How optimal is defined, depends on the particular problem. Examples of optimisation problems are: Traveling Salesman Problem (TSP). We are given a set of n cities, with the distances between all cities. A traveling salesman, who is currently staying in one of the cities, wants to visit all other cities and then return to his starting point, and he is wondering how to do this. Any tour of all cities would be a valid solution to his problem, but our traveling salesman does not want to waste time: he wants to find a tour that visits all cities and has the smallest possible length of all such tours. So in this case, optimal means: having the smallest possible length. 1-Dimensional Clustering. We are given a sorted list x1; : : : ; xn of n numbers, and an integer k between 1 and n. The problem is to divide the numbers into k subsets of consecutive numbers (clusters) in the best possible way. A valid solution is now a division into k clusters, and an optimal solution is one that has the nicest clusters. We will define this problem more precisely later. Set Partition. We are given a set V of n objects, each having a certain cost, and we want to divide these objects among two people in the fairest possible way. In other words, we are looking for a subdivision of V into two subsets V1 and V2 such that X X cost(v) − cost(v) v2V1 v2V2 is as small as possible. -
Introduction to Game Theory Lecture 1: Strategic Game and Nash Equilibrium
Game Theory and Rational Choice Strategic-form Games Nash Equilibrium Examples (Non-)Strict Nash Equilibrium . Introduction to Game Theory Lecture 1: Strategic Game and Nash Equilibrium . Haifeng Huang University of California, Merced Shanghai, Summer 2011 . • Rational choice: the action chosen by a decision maker is better or at least as good as every other available action, according to her preferences and given her constraints (e.g., resources, information, etc.). • Preferences (偏好) are rational if they satisfy . Completeness (完备性): between any x and y in a set, x ≻ y (x is preferred to y), y ≻ x, or x ∼ y (indifferent) . Transitivity(传递性): x ≽ y and y ≽ z )x ≽ z (≽ means ≻ or ∼) ) Say apple ≻ banana, and banana ≻ orange, then apple ≻ orange Game Theory and Rational Choice Strategic-form Games Nash Equilibrium Examples (Non-)Strict Nash Equilibrium Rational Choice and Preference Relations • Game theory studies rational players’ behavior when they engage in strategic interactions. • Preferences (偏好) are rational if they satisfy . Completeness (完备性): between any x and y in a set, x ≻ y (x is preferred to y), y ≻ x, or x ∼ y (indifferent) . Transitivity(传递性): x ≽ y and y ≽ z )x ≽ z (≽ means ≻ or ∼) ) Say apple ≻ banana, and banana ≻ orange, then apple ≻ orange Game Theory and Rational Choice Strategic-form Games Nash Equilibrium Examples (Non-)Strict Nash Equilibrium Rational Choice and Preference Relations • Game theory studies rational players’ behavior when they engage in strategic interactions. • Rational choice: the action chosen by a decision maker is better or at least as good as every other available action, according to her preferences and given her constraints (e.g., resources, information, etc.). -
Module 5: Backtracking
Module-5 : Backtracking Contents 1. Backtracking: 3. 0/1Knapsack problem 1.1. General method 3.1. LC Branch and Bound solution 1.2. N-Queens problem 3.2. FIFO Branch and Bound solution 1.3. Sum of subsets problem 4. NP-Complete and NP-Hard problems 1.4. Graph coloring 4.1. Basic concepts 1.5. Hamiltonian cycles 4.2. Non-deterministic algorithms 2. Branch and Bound: 4.3. P, NP, NP-Complete, and NP-Hard 2.1. Assignment Problem, classes 2.2. Travelling Sales Person problem Module 5: Backtracking 1. Backtracking Some problems can be solved, by exhaustive search. The exhaustive-search technique suggests generating all candidate solutions and then identifying the one (or the ones) with a desired property. Backtracking is a more intelligent variation of this approach. The principal idea is to construct solutions one component at a time and evaluate such partially constructed candidates as follows. If a partially constructed solution can be developed further without violating the problem’s constraints, it is done by taking the first remaining legitimate option for the next component. If there is no legitimate option for the next component, no alternatives for any remaining component need to be considered. In this case, the algorithm backtracks to replace the last component of the partially constructed solution with its next option. It is convenient to implement this kind of processing by constructing a tree of choices being made, called the state-space tree. Its root represents an initial state before the search for a solution begins. The nodes of the first level in the tree represent the choices made for the first component of a solution; the nodes of the second level represent the choices for the second component, and soon. -
Guru Nanak Dev University Amritsar
FACULTY OF ENGINEERING & TECHNOLOGY SYLLABUS FOR B.TECH. COMPUTER ENGINEERING (Credit Based Evaluation and Grading System) (SEMESTER: I to VIII) Session: 2020-21 Batch from Year 2020 To Year 2024 GURU NANAK DEV UNIVERSITY AMRITSAR Note: (i) Copy rights are reserved. Nobody is allowed to print it in any form. Defaulters will be prosecuted. (ii) Subject to change in the syllabi at any time. Please visit the University website time to time. 1 CSA1: B.TECH. (COMPUTER ENGINEERING) SEMESTER SYSTEM (Credit Based Evaluation and Grading System) Batch from Year 2020 to Year 2024 SCHEME SEMESTER –I S. Course Credit Course Title L T P No. Code s 1. CEL120 Engineering Mechanics 3 1 0 4 Engineering Graphics & Drafting 2. MEL121 3 0 1 4 Using AutoCAD 3 MTL101 Mathematics-I 3 1 0 4 4. PHL183 Physics 4 0 1 5 5. MEL110 Introduction to Engg. Materials 3 0 0 3 6. Elective-I 2 0 0 2 SOA- 101 Drug Abuse: Problem, Management 2 0 0 2 7. and Prevention -Interdisciplinary Course (Compulsory) List of Electives–I: 1. PBL121 Punjabi (Compulsory)OR 2 0 0 2. PBL122* w[ZYbh gzikph OR 2 0 0 2 Punjab History & Culture 3. HSL101* 2 0 0 (1450-1716) Total Credits: 20 2 2 24 SEMESTER –II S. Course Code Course Title L T P Credits No. 1. CYL197 Engineering Chemistry 3 0 1 4 2. MTL102 Mathematics-II 3 1 0 4 Basic Electrical &Electronics 3. ECL119 4 0 1 5 Engineering Fundamentals of IT & Programming 4. CSL126 2 1 1 4 using Python 5. -
The Conservation Game ⇑ Mark Colyvan A, , James Justus A,B, Helen M
Biological Conservation 144 (2011) 1246–1253 Contents lists available at ScienceDirect Biological Conservation journal homepage: www.elsevier.com/locate/biocon Special Issue Article: Adaptive management for biodiversity conservation in an uncertain world The conservation game ⇑ Mark Colyvan a, , James Justus a,b, Helen M. Regan c a Sydney Centre for the Foundations of Science, University of Sydney, Sydney, NSW 2006, Australia b Department of Philosophy, Florida State University, Tallahassee, FL 32306, USA c Department of Biology, University of California Riverside, Riverside, CA 92521, USA article info abstract Article history: Conservation problems typically involve groups with competing objectives and strategies. Taking effec- Received 16 December 2009 tive conservation action requires identifying dependencies between competing strategies and determin- Received in revised form 20 September 2010 ing which action optimally achieves the appropriate conservation goals given those dependencies. We Accepted 7 October 2010 show how several real-world conservation problems can be modeled game-theoretically. Three types Available online 26 February 2011 of problems drive our analysis: multi-national conservation cooperation, management of common-pool resources, and games against nature. By revealing the underlying structure of these and other problems, Keywords: game-theoretic models suggest potential solutions that are often invisible to the usual management pro- Game theory tocol: decision followed by monitoring, feedback and revised decisions. The kind of adaptive manage- Adaptive management Stag hunt ment provided by the game-theoretic approach therefore complements existing adaptive management Tragedy of the commons methodologies. Games against nature Ó 2010 Elsevier Ltd. All rights reserved. Common-pool resource management Multi-national cooperation Conservation management Resource management Prisoners’ dilemma 1.