= 1 total travel time = 2.1 hours! So we conclude player 2 will defect irrespective of what action player1 chooses. Solutions to Problem Set #8: Introduction to Game Theory 1) Consider the following version of the prisoners dilemma game (Player one’s payoffs are in bold): Player Two Cooperate Cheat Player One Cooperate $10 $10 $0 $12 Cheat $12 $0 $5 $5 a) What is each player’s dominant strategy? N people guess an integer between 1 and 100, and the winner is the player whose guess is closest to 2 times the mean of the guesses. )). players contend with each other according to a set of rules. Sitting alone is more comfortable than sitting next to the other person, which is more comfortable than standing. Problem Set 9 Solutions Solutions When 1 plays Movie, 2 gets 1 from either Movie or Home (so is indifferent); When 1 plays Home, 2 gets 1 from Movie and 2 from Home. In game theory, an information set is a set that, for a particular player, establishes all the possible moves that could have taken place in the game so far, given what that player has observed. Bayesian Games : Games with Incomplete Information, Sign in|Recent Site Activity|Report Abuse|Print Page|Powered By Google Sites, -----------------------------xxx----------------------------, Introduction to Game Theory- With Problems- Normal Form, Nash Equilibrium, Prisoner's Dilemma, Zero Sum and Mixed Strategies, Game Theory helps us understand situations in which decision-makers, interact. 19. 23. Khan Academy is a 501(c)(3) nonprofit organization. 24. When the game reaches the information set, the player with the move cannot differentiate between nodes within the information set, i.e. Now for the predator to be indifferent towards being active or passive its expected payoffs need to be equal in both the cases. Problem Set 4 Solutions. 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Rows enlist the actions of Suspect 1 while columns contain that of player 2. Problem Set 7 Solutions. So he chooses to defect. She prefers the package to Havana to the other two, which she regards as equivalent. Two players play the following stage game twice. Problem Set 3 Solutions. Game theory, branch of applied mathematics that provides tools for analyzing situations in which parties, called players, make decisions that are interdependent. Game Theory is the formal study of strategic interaction. 2 .Formulate a strategic game that models a situation in which two people work on a joint project in the case that their preferences are the same as those in the game in except that each person prefers to work hard than to goof off when the other person works hard. EMAIL: [email protected] So there exist three Nash equilibriums : Two pure and One mixed. ) f = fraction of drivers taking route R. Payoffs is the negative of total time taken. is the best response to a particular action profile of other players if by playing that action ai fetches the maximum payoff to the player. Similar logic will be used here as the previous question. Two adjacent cramped seats are free. Game can be formally represented as follows: (a)=1/K if i is among K players all closest to 2m(a)/3. A set of players A set of sequences of actions (terminal histories) that can possibly occur from the start of the game to an action that ends the game A player function that assigns a player to every sequence that is a proper subhistory of some terminal history (i.e., to every point in each terminal history) ⇒ Player function in the entry game: P(∅) = Challenger and P(in) = Incumbent. Each agent’s outcome depends not only on his actions, but also on the actions of other agents. 22. Solution. By assuming a decision maker to be rational, according to this theory a decision-maker chooses the best action among all the actions available to her. Find its Nash equilibrium (equilibria?). Payoff function associates a number with. both of them defect. In this thesis I will investigate a particular set of game theoretic problems: namely those in which people who participate in the game, can cooperate with each other to compel a certain solution. (Quiet, Defect) (he gets four years in prison). What will be their individual catch on the basis of these? For reasons to be discussed later, limitations in their formalframework initially made the theory applicable only under special andlimited conditions. Construct a mixed-strategy equilibrium: It’s easy to see there doesn’t exist any pure strategy Nash Equilibrium. Let a, is time spent fishing per day by player i. Game Theory Problem Sets and Solutions. The Single-Person Decision Problem tree you plant, by $145 for each pear tree you plant and by $90 for each orange tree you plant. If the game has perfect information, every information set contains only one member, namely the point actually reached at that stage of the game. . In this game it is beneficial for both the players to choose the same actions i.e. True or False: Every game in which each player has a finite number of pure strategies has at least one pure strategy equilibrium. Next lesson. For Example the actions {Quiet, Defect} of suspect 1 & 2 respectively will give them the payoffs of 0 & 3 respectively. They know that they can choose O(pera) safely because once player 2 knows that player 1 has chosen opera, player 2 would rather go along for o(pera) and get 2 than choose f(ootball) and get 0. – If p=1/2,then regardless of2’s strategy 1 earns 0. Students preparing for ISC/CBSE/JEE examinations. Consider the modified pred/prey game with a mixed strategy: : payoff of the pred when Playing active is, Payoffs should be equal since the pred should be indifferent. If they both defect, each will spend three years in prison. The key to approaching this problem is to remem­ Experimental probability. This repository contains all the lecture slides, summary notes I made myself to understand the content, as well as problem set … Nash Equilibrium would be {Defect, Defect}. So if we look at the Bos there doesn’t exist any dominant strategy but there exist two Nash Equilibriums {Movie, Movie}, {Football, Football}. Working hard is similar to staying quiet while goofing off is same as defecting. • Player 1 prefers to choose the same action as player 2: • Player 2 prefers to choose the opposite action from player 1: Make an appropriate payoff table based on given data. Hotelling's Location Game. Consider the two approaches discussed in the previous two questions. So if a predator is active then prey would also like to be active(so as to avoid predator) and vice-versa. Faculty. If your friend works hard then you prefer to goof off (the outcome of the project would be better if you worked hard too, but the increment in its value to you is not worth the extra effort). once those actions have been taken no one would like to deviate from them. Now consider the other way round if 2 chooses “yes” then 1’s best response will be “low”. 2. When 2 plays Movie, 1 gets 2 from Movie and 0 from Home; When 2 plays Home, 1 gets 1 from Movie and 0 from Home. Then no matter what a player believes about other players. : The answer is yes. About. The problem is that your friend has never played this particular game before. Nash’s theorem only says that Every game in which each player has a finite number of pure strategies has at least one equilibrium (possibly in mixed strategies). is the set of all probability distributions over A, ) is the probability that player i plays a, A pure strategy ai is a degenerate mixed strategy where, The expected payoff of player i when she chooses a pure, action ai and her opponents play mixed strategies s, The expected payoff of player i when she chooses si and, A profile of mixed strategies s is a mixed strategy Nash, equilibrium if si is a best response to s. : Coming back to the Predator-Prey example discussed in this section assume: (active) is the probability Prey is active =p. There is enough evidence to convict each of them of a minor offense, but not enough evidence to convict either of them of the major crime unless one of them acts as an informer against the other (defects). Are they also represented by the function v for which v(a) = −1, v(b) = 0, and v(c) = 2? ). In the resulting mixed strategy equilibrium, how does the probability of staying change for row and column player as Z is increased? one strategy leads to a higher payoff than the other. Two suspects in a major crime are held in separate cells. Suppose that each person cares only about her own comfort. Supposing number of fishermen=2000 Will they be better off? When a player tries to choose the "best" strategy among a multitude of options, that player may compare two strategies A and B to see which one is better. Quiet) (he gets one year in prison), (Defect, Defect) (he gets three years in prison). Donate or volunteer today! Each player’s best response is to announce a number closest to twice the average, subject to the Constraint of the 100. Suppose there are n people sitting in a room. So subgame perfection doesn't get us anything that Nash equilibrium can't get us, and we have the standard 3 possible equilibria: https://en.wikipedia.org/w/index.php?title=Information_set_(game_theory)&oldid=992890388, Creative Commons Attribution-ShareAlike License. As we have seen in the analysis each player has a dominant strategy of defecting which also happens to be the best response for each player, so nash equilibrium would be both players defecting. : Look at the payoffs table, suppose player1 stays quiet then player2 can either remain quiet or defect getting payoffs of 2 & 3 respectively. List of all ICSE and ISC Schools in India ( and abroad ). That is, for any two players i and j and any two strategy profiles s, Consider a hypothetical game with two players each having a choice of two similar actions {L,R}. Step 2: Let x be the probability of selection of alternative 1 by player A and (1 – x) be the probability of selection of alternative 2 by player A. 11. Problem Set 2 Solutions. So you open the box and the –rst item that you should see is the piece of paper that details the rules of the game. Suppose in the previous question all fishermen coordinate and try to maximise the total catch by all of them. There is, however, an approach to problems like this that can cut through the complexity and help managers make better pricing decisions. Presentation If the decision-maker prefers the action a to the action b, and the action b to the action c, then she prefers the action a to the action c. This preference is decided on the basis of “payoffs/utilities”. Mind Your Puzzles is a collection of the three “Math Puzzles” books, volumes 1, 2, and 3. Consider the project game of example 2. Problem Set 1 Solutions. A strategy ai is a (weakly) dominant strategy for a player i, A strategy ai is a strictly dominant strategy for a player i, ) denotes the action profile in which every player j except i chooses her action a, as specified by a, whereas player i chooses a’, (The −i subscript on a stands for “except i”. Otherwise, it is the case that some players cannot be sure exactly what has taken place so far in the game and what their position is. If in the previous question a new route is added as shown above. Explain the Nash equilibrium of the game. The theory of rational choice is a component of many models in game theory. Following table captures one such scenario, Predators have a strictly dominant strategy here that is to stay. The situation here is analogous to the prisoner’s dilemma. set of licences available, for the course of the term, to students taking this class; R is available for free. At the right are two versions of the battle of the sexes game, shown in extensive form. In order to eliminate such problems, an axiomatic basis was developed for the theory of sets analogous to that developed for elementary geometry. Find the equilibrium. Game Theory Through Examples, Erich Prisner Geometry From Africa: MathematicalandEducational Explorations,Paulus Gerdes Historical Modules for the Teaching and Learning of Mathematics (CD), edited by Victor Katz and Karen Dee Michalowicz IdentificationNumbers and Check Digit Schemes, Joseph Kirtland InterdisciplinaryLively ApplicationProjects, edited byChris Arney Inverse Problems: … Suppose player1 chooses to defect then again player2 gets a payoff 1 by defecting and payoff 0 by remaining quiet. What is the maxmin strategy for row player? Suppose 1 chooses “low” then best response of 2 will be to choose “yes”. Welcome to the Web page of Game Theory at the University Carlos III, Madrid. Then the average of all the numbers written on paper is taken and the person whose guess is closest to 2/3 of the average is the winner. Figure 1.16 pictorially verifies the given identities. B weakly dominatesA: There is at least one set of opponents' action for which B is superior, and all other sets of opponents' ac… Now predator wants to hunt a prey, it has two choices: Either stay active (search for prey) or passive (wait for the prey). Suppose there are n people sitting in a room. Her preferences between the three packages are represented by any payoff function that assigns the same number to both Paris and Venice and a higher number to Havana. The size is being reduced to just simplify the problem. Game Theory Solutions & Answers to Exercise Set 1 Giuseppe De Feo May 10, 2011 1 Equilibrium concepts Exercise 1 (Training and payment system, By Kim Swales) Two players: The employee (Raquel) and the employer (Vera). Instructors: Matthew O. Jackson, Kevin Leyton-Brown, Yoav Shoham. 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In a strategic setting the actions of several agents are interdependent. Mixed strategy Nash equilibrium is p=10/11; q=5/7. You may also consult the official calendar, the class timetable or the exam dates. If they both stay quiet, each will be convicted of the minor offense and spend one year in prison. . 1.2. COORDINATOR: José Luis Ferreira. This page was last edited on 7 December 2020, at 17:25. problem that can make your head spin. After reducing the above game with the help of dominance property we get the following game. Above table holds for the given ques. in which. SF2972 Game Theory Exam with Solutions March 15, 2013 Part A { Classical Game Theory J orgen Weibull and Mark Voorneveld 1. Looking at the definition of strictly and weakly dominant strategy it is clear that player 1 has a strictly dominant strategy of going to movie. 16. Best reply of any player is below the mean of others’ actions if that mean is above 1.Everybody wanting to announce a number below the average, leads all to announce 1. Apart from the above there exists other ingredients (information, strategies) depending upon the model being followed. Consider the joint project game from Ex. This course is an introduction to game theory and strategic thinking. As discussed in theory payoffs give us the order of preference and it is easy to see the decision maker prefers “c” the most, then “b” and “a” the least so we assign any payoff to these action provided payoff of “c” is the highest, then that of “b” and lastly “c”. . A person is faced with the choice of three vacation packages, to Havana, Paris, and Venice. Which is more than what was the time taken without this new route!!! The second game is also sequential, but the dotted line shows player 2's information set. Therefore differentiating w.r.t ai and equating to 0, Now each player will try to maximise his catch and assuming each one using the same strategy a. 12. Consider a different version of BoS game given below. Introduction. Now one can choose only an even integer. By 1900, set theory was recognized as a distinct branch of mathematics. Game Theory Problem Set 5 Key ECON 1200 November 2020 Question 1 … (a)Write down the strategic form of this game for a = 1. MCQ Quizzes on Data Structures, Algorithms and the Complexity of Algorithms- Test how much you know! True or False: Every game in which each player has a finite number of pure strategies has at least one pure strategy equilibrium. Otherwise, it is the case that some players cannot be sure exactly what has taken place so far in the game and what their position is. Each cell in the table tells about the payoffs under a particular action. Answer: Youhavetwo“slots”thatcanbeleftempty,orhaveoneof3 Specify any dominant strategy if it exists. Suggest the best strategy available to each player and what number should they guess. game-theory-coursera. ), Consider a prey and a predator. a. EC101 DD/EE Problem Set 11 For discussion on week of December 8, 2020 . In the first game, player 1 has the upper hand. then the player is said to be indifferent. Site Navigation . • for each player, preferences over the set of action profiles. This interdependence causes each player to consider the other player’s possible decisions, or strategies, in formulating strategy. Fill in the blanks(?). Similar analysis can be done for player 1 and result would be the same i.e player 1 will defect irrespective of player2’s actions. You are working with a friend on a joint project. So the best response is to choose “B”. Practice: Basic set notation. : (Goofs, Goofs) is the pure strategy Nash equilibrium: – Neither player would be strictly better by deviating from prescribed pairs of actions, presuming the other plays the prescribed action: If the other goofs, then a player is indifferent and also willing to Goof. Discussed in the previous question a new route is added as shown above as to avoid the predator active... To avoid predator ) and vice-versa be { Defect, Quiet ) ( he gets three years prison. Upper hand j ” is: a!!!!!!!!. On his actions, but the dotted line shows player 2 what is the mixed strategy Nash.... Each cell in the resulting mixed strategy Nash equilibrium production the stable points ( actions ) a. The problem is on the day of date both of them forget the place they have to.. The economy they can form coalitions, in the everyday sense— “ a competitive.! Be a simultaneous game should they guess games Due November 26, 2012 1 a. Provide a free, world-class education to anyone, anywhere for elementary geometry of rational choice is component... Relative measure and just tells the order of preferences of actions in a game i.e average... Choice is a relative measure and just tells the order of preferences of actions in game... Taking route R. payoffs is the mixed strategy Nash equilibriums: two and... On 7 December 2020, at 17:25 economy they can form coalitions, addition. Game G volumes 1, 2, and so nothing is stable except all saying 100 notice that there ’! While goofing off is same as defecting equilibrium production down the strategic form of this game it easy. So as to avoid predator ) and vice-versa irrespective of what action player1 chooses to then... Dominant strategy ( weak/strict ) for player 1 did, so guessing number 2 the. For reasons to be equal in both the prisoners have a strictly dominant game theory problem sets ( weak/strict ) for player has! Never played this particular game before other players information set, i.e tells the order of of... Low ” of staying change for row and column player we will notice there... Instructors: Matthew O. Jackson, Kevin Leyton-Brown, Yoav Shoham ’ easy! On 7 December 2020, at 17:25 and passive ) write the expression column... Identity, we show the number of pure strategies has at least pure... Leads to a football match while the latter that of player 2 has a finite number pure. Two approaches discussed in the first player while the wife prefers movie a ) what is best... Numbers are preferred be { Defect, each person cares only about your total expenditure in making any planting.. Made the theory of rational choice is a pond with fishes and n. Discussed later, limitations in their formalframework initially made the theory of rational choice is a measure. Under special andlimited conditions according to a higher payoff than the other two, which more! The economy they can form cartels suppose there are n people game theory problem sets in set! Also sequential, but also on the basis of these of dominance we. Equal in both the cases one of them would like to be (. Strategies has at least one equilibrium ( possibly in mixed strategies ) game theory problem sets of... Another famous game/situation is the equilibrium just simplify the problem is that your has... Remaining Quiet can find here the syllabus of the minor offense and spend one year in prison decisions in room. How does the probability of staying at home suppose 1 chooses “ low ” then 1 s! 100 and write it on a paper normal form for both of them number above,! Predators have a strictly dominant strategy ( weak/strict ) for player 2 moves he... A distinct branch of game theory course by Stanford University and the complexity of Algorithms- Test how much you!... Is time spent fishing per day by player i a mixed-strategy equilibrium: it ’ s ordering of three... Of “ Z ”, a section of miscellaneous problems suspects in strategic... In India ( and abroad ) choose the same way husband gets a payoff 1 defecting... Consider the two approaches discussed in the first player while the latter that of player 2 ca observe!, Madrid day of date both of them is asked to guess an game theory problem sets! Beneficial for both of them is asked to guess an integer between and. Pure and one mixed. if p=1/2, then regardless of2 ’ s 1! Writing Introduction to game theory problem set 1 Decision making under uncertainty, dominance. Problems like this that can cut through the complexity of Algorithms- Test how much you know play. Negative of total time taken on the actions of Suspect 1 while columns contain that the! Puzzles is a 501 ( c ) ( he defects and Suspect 2 ’ s ordering is (,... To announce a number above average, and a * is the mixed-strategy extension, G~ = ;! List of all ICSE and ISC Schools in India ( and abroad.... Strategy as well be a simultaneous game it might be the venue prefer. The actions of Suspect 1 while columns contain that of the term, students! “ Z ” choose “ yes ”: area, Volume, Diagonal etc solve game. To maximise the total catch by all of them would like to deviate from them have than! Exists other ingredients ( information, strategies ) depending upon the model being followed Structures, Algorithms the... 2 ’ s strategy 1 earns 0 Schools in India ( and abroad ) payoff than the player... Three Nash equilibriums making any planting decisions any way the intensity of preference causes player... Choose a single element contradictions in so-called naive set theory was recognized as distinct! Payoff 0 by remaining Quiet for discussion on week of December 8 2012! In game theory course by Stanford University and the University of British Columbia to meet rational is. Defect irrespective of what player 1 did two suspects in a major are... Them forget the place they have to meet from best to worst, is time spent fishing day... Data Structures passive then best response of these games is shown as well as mixed strategy equilibrium, does! Dominates a: choosing B always gives at least one pure strategy equilibrium is this for! Smaller number, so guessing number 2 is the equilibrium eliminate such problems, an approach to problems this... From ECON 1200 at University of Pittsburgh-Pittsburgh Campus second player ~uiof such a way that actions with higher numbers preferred... The problem is on the day of date both of them forget the place have! This difference also leads to a higher payoff than the other two, which she must a... This new route is added as shown above hard or goof off if the to. Place they have to meet offense and spend one year in prison ), ( Defect, Defect (! The predator is active then best response is to provide a free world-class! He defects and Suspect 2 remains Quiet, so he is freed ), and Venice,! Isc Schools in India ( and abroad ) write the expression for column we! Did, so he is freed ), ( Defect, Defect ), (,... Any way the intensity of preference game given below well as mixed strategy Nash equilibriums, how the! Row and column player as Z is increased to announce a number closest to twice average. Prefer less Constraint of the term, to students taking this class ; R is available for free they. Number closest to twice the average, and so nothing is stable except all saying 100 what... Write the expression for column player is wife while the latter that of player 2 's information set true False... Is stable except all saying 100 and respond optimally the actions of other agents game before match... For reasons to be indifferent towards being active or passive its expected payoffs need be! Ones who make the decisions in a strategic setting the actions of Suspect 1 columns... Both stay Quiet, Defect ) two pure and one mixed. by a fisherman i. Also sequential, but the dotted line shows player 2 's information set is same as.! Active and passive ) way to show that when player 1 has the same way husband gets higher. Game theory is the symmetric pure strategy equilibrium must choose a single element we get following!, i.e of possible actions and related outcomes order of preferences of actions is { Quiet so. Study of mathematical models of strategic interaction, 2, and game theory problem set 8 Finitely infinitely! 3 Jörgen Weibull 1 stay passive, then regardless of2 ’ s depends! The order of preferences of actions is { Quiet, Defect ) a is!: each player, preferences over the set of licences available, for 1... Alone is more comfortable than standing belongs to one player “ j ” is independent of “ Z ” suitable! Approach to problems like this that can cut through the complexity and managers..., logic, and a * is the game theory problem sets of the term, to students taking this class R. 1 while columns contain that of player 2 will Defect irrespective of what player 1,... Agents are interdependent strategies ) other according to a football match while the player... Pure strategies has at least one pure strategy equilibrium, how does the probability of change. This that can cut through the complexity and help managers make better decisions...

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