What is Game theory: Definition and 83 Discussions

Game theory is the study of mathematical models of strategic interaction among rational decision-makers. It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed zero-sum games, in which each participant's gains or losses are exactly balanced by those of the other participants. In the 21st century, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers.
Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty.
Game theory was developed extensively in the 1950s by many scholars. It was explicitly applied to evolution in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. As of 2014, with the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole, eleven game theorists have won the economics Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of evolutionary game theory.

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  1. Fra

    A Game theory, quantum physics and the Bell inequality (paper)

    The reason for highlighting this paper is to highlight (and exemplify by how different researcher think about this) a new "perspective" to understanding the different between "ignorance of the physicists" which is reallly what Bell assumptions imples. And "ignorance of they players", which is...
  2. M

    Game theory: competitive auction for the money in a chest

    Hi, I am back yet again with another problem I was reading the following question and attempting it. It was an interview problem, so it isn't technically homework, but I don't know where else to post it. I think there are elements of game theory involved, but I have no academic background in...
  3. M

    Game theory: dice game to aim for sum no greater than 9

    Hi, I was attempting the following question and would appreciate any insight on how others would approach this game theory/probability-type question. Question: You have been chosen to play a game involving a 6-sided die. You get to roll the die once, see the result, and then may choose to...
  4. Leo Liu

    Discussing Insurance, Risk Tolerance and Game Theory

    The title is my question, and this post is meant to be a place for discussions. I think it is somewhat related to game theory and microeconomics, but I am not sure how to accurately categorize it. Let me give you an example. Let us suppose that Mr. Goose, an ideal person with a national average...
  5. Frabjous

    Prob/Stats Two-Person Game Theory by Rapoport

    From a retiree’s stack of left behind books, I picked up a copy of Two-Person Game Theory (1966) by Rapoport. Is this an acceptable first (and possibly only) reading on the topic?
  6. S

    I Who would win a perfect game of chess?

    While chess hasn't been solved yet, other games have. For example, I know that in in some games, like connect four, if both players play perfectly, the player who goes first will always win. On the other hand, some games, like tic tac toe, a perfect game will result in a draw; in fact, I...
  7. Posty McPostface

    Can external factors enforce ethical outcomes in the prisoner's dilemma?

    What can 'game theory' tell us about life? The prisoner's dilemma is an issue of Pareto optimality, wherein the best possible outcome is one where both parties cooperate with each other to derive the highest Pareto optimality. But, the issue is that the highest Pareto optimality for the...
  8. Posty McPostface

    Game theory and "Mutually Assured Destruction"

    In Stanley Kubrick's Dr. Strangelove or : How I Learned to Stop Worrying and Love the Bomb, is depicted, rather comically, a mad scientist who understands that the doomsday weapon is actually an absolute deterrent against nuclear war. The only misfortune is that nobody knew about it in time as...
  9. E

    About the strategy of reducing the total suffering in a queue

    This is my funny theory (may be I have found already known things...).Let us assume the following abstract situation. We have a special place where people can get some kind of service (for instance any bureaucratic office). There is only one service clerk who spend a fixed time (we will call it...
  10. P

    Can human traits be formalized?

    I've been wondering about modeling human traits that seem to have a universal aspect of nature. Though this is worthy of another topic, on a more fundamental level or serving as a good example, can 'masculinity' have a more objective measure of measuring it and formalizing it. In other words, is...
  11. N

    I Stochastic Minimax partition problem card game

    Hey all, ran into a game theory problem I can't solve. A and B have a set of 10 random numbers from 1-10, players can make so called "piles", a pile has a goal number from 1 to 10, if 6,3 are on the table, a pile of nine may be started, the pile is added to by adding sets of numbers that sum to...
  12. V

    A Game Theory: Are the payoff functions πi continuous?

    How do I show that the payoff function πi isn´t continuous? Why do best replies not always exist?
  13. S

    Cohesion in legislatures (game theory)

    Homework Statement Homework EquationsThe Attempt at a Solution I've got a feeling that the problem is either incomplete or unclear, because so far I've come up with nothing but vague ideas. If it is clear, then please let me know so this thread can be closed/deleted.
  14. 1

    Where do physics and game theory meet?

    This is going to sound really strange, but I don't want to have this question inside of me and never ask it. I love business/economics/game theory. However, the subject I've studied the most in my life at this point is physics. The question of what I want to write about for my PhD thesis is...
  15. W

    I Game Theory: Strategy for game with non-square payoff matrix

    Hi, suppose two players are a playing a game with a non-square payoff matrix, like for example this one: ...a...b... A: (1,3) (1,0 B: (0,0) (2,1) C: (3,1) (0,3) How would one go about finding an optimal mixed strategy for something like this? I mean, if this was a 3x3 matrix then one could find...
  16. B

    Is Game Theory a Self-Contained Subject?

    I'm very intrigued by game theory after watching a little puzzle on YouTube that was part of a game show. The solution or best strategy turned out to be solved through game theory. Anyhow, I further checked out an introduction to game theory online and there didn't seem to be any "regular'...
  17. C

    Mathematical Game Theory (Von Neumann Morganstern Utility)

    1: If u: omega---> reals is a Von Neumann Morganstern Utiliy function and L is a lottery, prove that expectation E is "linear" ie: E(Au(L)+B)=AEu(L)+B2. Given none:The Attempt at a Solution : My attempt at a solution has gone nowhere. I found a stanford and princeton game theory notes that went...
  18. nomadreid

    Game theory -- Nash equilibrium

    Suppose we have a game, played in which Alice and Bob play mixed strategies: (sorry for the dots, but I don't know how to put a table or tab spacing in this text box) ..............Bob ..........Dove, prob q......Hawk, prob (1-q) ...Dove , prob p......(2,3).........(4,5) Alice ...Hawk, prob...
  19. D

    Game Theory Q: Proving Player Win w/Axiom of Choice

    Hey PF! Can you help me with something: Players alternately choose 0's or 1's. A play of this infinite game is thus a sequence of 0's and 1's. Such a sequence can be considered as the binary expansion of a real number between 0 and 1. Given a set ##E## of real numbers satisfying ##0 < x < 1...
  20. M

    MHB Reaction functions, differentiation question _ Game theory economics

    Consider a scenario where two individuals—Robert and Stuart—are under- taking a joint project, where the value generated from the project depends on the effort expended by both individuals. Let x be Robert’s level of effort and y the level of effort of Stuart; the value of the project for each...
  21. C

    Mathematical Game Theory (Drawing gametree for Tic-Tac-Toe

    Homework Statement Draw out the game tree for tic-tac-toe Homework EquationsThe Attempt at a Solution I'm just thinking about it and I don't know how to start. To draw out the entire gametree, we'd have thousands and thousands of branches. So based on symmetry there is only 3 starting points...
  22. C

    Microeconomics with Game Theory

    Hello. I'm currently looking for a textbook which emphasizes microeconomic theory on a more advanced level compared to the mass-marketed books, (e.g. Mankiw, McConell, ...) and a textbook that contains more game theory. Does anyone have suggestions?
  23. R

    Help with shared constraints (game theory)

    I've been struggling with shared constraints problems for a while now. I have a game between two players with a shared constraint. For example, player 1 is trying to maximize f(x,y) by choosing x, and player 2 is trying to maximize g(x,y) by choosing y. The players are competing in a...
  24. D

    [Game Theory] A pedestrian is hit by a car. How many people will help?

    Homework Statement Consider the following social problem. A pedestrian is hit by a car and lies injured on the road. There are n people in the vicinity of the accident. The injured pedestrian requires immediate medical attention, which will be forthcoming if at least one of the n people call...
  25. A

    MATLAB Critical points of system of ODE in MATLAB - Game theory and poker

    Hello there, I hope I'm posting in the right section. I have been doing some work on evolutionary game theory and poker. I will give a brief description of how I got here. I have eight strategies i = 1, 2, \ldots, 8 and the eight proportions of the population playing each strategy is...
  26. B

    Which Strategies Survive Iterative Deletion in Game Theory?

    I would like to know which strategies survive iterative deletion of strictly dominated strategies D E F A 0, 1 0, 0 10, 4/5 B 3, 1 1, 2 0, 1 C 1, 5/2 2, 3/2 0, 2 Since there are only weakly dominated strategies, the answer is A, B, C and D, E, F. Is this correct?*
  27. W

    Looking for a good intro text for game theory

    Hey all, so my university's game theory course never runs due to lack of students signing up for it, so I'm taking it upon myself to learn what I can about this fascinating subject. I'm specifically looking for a game theory text which is: a) introductory b) for upper-level math undergraduates...
  28. I

    Game Theory: Nim - Winning Strategy for 5 & 1000 Stones

    Homework Statement Two players: A and B. Players can take 1 or 2 stones per turn. The player who takes the last stone wins. The question is: is there any winning strategy with 5 stones?, does it change with 1000 stones?. Homework Equations None. The Attempt at a Solution If...
  29. A

    Prerequisites for Learning Game Theory

    Hey, I am looking to teaching myself a new subject along with statistical mechanics. I am very interested in some of the implications of game theory and the notion that we can approximately quantify and predict decisions made by large groups of people. Unfortunatly, as an engineering graduate my...
  30. Einj

    References for Nash's game theory

    Hi all. I recently watched a documentary about John Nash's game theory and I'd like to study more about that. My present mathematical knowledge is that of a particle physics student at his fourth year. Is my preparation enough? If so, can someone advise some references where I could study?
  31. M

    Game theory: Optimal Strategy. Need help with a system of equations.

    This problem is not schoolwork, it is from "The Mathematics of Poker". I have three equations based off a 3x3 Rock, Paper, Scissors payout table where winning with scissors is given a +2 payout instead of +1. Homework Statement Find the solution to the system of equations Homework...
  32. K

    Two-person sequential game, game theory

    Hello! Here is my solution. Is it right? Thank you! Homework Statement An investor, whom we will refer to as player I, is considering bidding for a company called Fortune F. Currently, the shares of F are valued at $100. If I can take over the company it is known that F would be worth $110...
  33. W

    What Are the Prerequisites for Understanding Game Theory?

    any prerequisites to cover the following Course content: The courses examines the theory of games and its applications in economics: Game trees with perfect information, NIM, combinatorial games. Backward induction. Extensive and strategic (normal) form of a game. Nash equilibrium...
  34. R

    Game Theory ( 2 player zero-sum game)

    Hi I have just started a game theory module and have been given the following exercise. Can anyone help me with the following question. I have a feeling I can use a minimax argument to answer this but I am not sure how to go about this. Suppose a two-player zero-sum game has the following...
  35. M

    Game Theory 2x2 Matrix: Finding Optimal Strategy for (-3,-4), (-7,2)

    Homework Statement {(-3,-4),(-7,2)} Find the value of the game (basically saying find x*) Homework Equations E((x,1-x), 1)=E((x,1-x),2)) The Attempt at a Solution Basically I used dot product of (x,1-x) and column 1 (-3,-7). I got 4x-7 for that, which is my E((x,1-x). Then I found the dot...
  36. P

    How to Solve Bayesian Game Equations?

    Homework Statement My Textbook gives the following example problem: Cournot Duopoly with incomplete information. The profit functions are given by: u_i = q_i(θ_¡ - q_i - q_j) Firm 1 has one type θ_1 = 1, but firm 2 has private information about its type θ_2. Firm 1 believes that...
  37. C

    Simple summation, difficult inference. (game theory)

    hi guys, new user, long time lurker. the following simple proof is proposed with the highlighted summations. you do not need to know what the proof is of to answer my question (it is that the payoff for a skew symmetric game, rock paper scissors, is zero). i need help understanding how you...
  38. R

    Game Theory Help: Solving Task Before WWII Invasion

    Hi! I really need a help in solving certain task. Before the Allied invasion of France during WWII [bonus point for the month and the year of this invasion ], the Germans had to decide where to place their defenses. They had three choices: They could concentrate their defenses...
  39. C

    Please help me question on the game theory, thanks

    A person has a utility function defined over her wealth given by u(w)=ln(w). Her initial wealth is $2 and she faces a risky prospect in which she will lose $1 with probability 0.4 and gain $1 with probability 0.6. (i) Show that this person is risk averse by demonstrating that her utility...
  40. M

    Game Theory: Winning Moves for (20, 21, 28)

    (1 pt) We shall denote a position in three-pile Nim by (a,b,c), so that there are a chips in the first pile, b in the second, and c in the third. Given the following position in Nim, list all winning moves. As an example, if the piles are (2,2,2) then we can list all winning moves as...
  41. R

    Game Theory a problem which is a bit similar to the Impossible Puzzle

    Game Theory a problem which is a bit similar to the "Impossible Puzzle" From numbers 1 to 10, two integers X, and Y (not necessarily distinct) are chosen by a referee . The referee informs secretly to Joe the integer U where U = X + Y . The referee informs secretly to Bob the integer V where V...
  42. C

    Fun Game Theory, Guessing a Number With a Twist

    Fun Game Theory, Guessing a Number With a "Twist" You and I are playing a game. I begin by picking an integer from 1 to 2011 (inclusive). On each turn you try to guess my number. I then tells you whether your guess is too high, too low, or correct. If your guess is not correct, I add or...
  43. A

    A few basic questions about combinatorial game theory

    I am trying to teach myself the basic theory behind combinatorial game theory regarding impartial games, the solution to Nim, and the Sprague-Grundy theorem. I understand most of it, but I have a few questions about parts that are still unclear to me. 1. Why are Nim-sums computed using binary...
  44. R

    Solving Chop Games with Nim Heap Equivalent: Problem 1

    1) Problem 1 For every i; j with 1<=i; j<=8 fi nd a nim heap which is equivalent to the i x j game of Chop. CHOP Game Start with an i x j array of boxes viewed as a plank which is secured only at the lower left hand corner. Each player is a pirate, and they alternate turns. On each turn a...
  45. B

    Game theory teaches robots how to deceive

    I came across an article in the "Electronics Weekly" titled Game theory teaches robots how to deceive It asks, "Are there ethical issues in teaching robots to lie? Yes and no. Why can't robots learn how to lie? We do it all the time. Since robots will have to deal with humans and work...
  46. R

    Winning Strategy in a Diamond Game: Proving Player 1's Advantage

    So I'm taking a course on game theory and as an intro he left us with this question. I'd like to have it solved fornext class as it is for bonus marks.I'm not sure how to add an attachment here so I will describe the game and board and hopefully someone can tell me how to upload a photo in the...
  47. D

    Win Free Stuff at Work: Game Theory Strategies

    In the office I work, there is a popular game when someone wants to give something away. When a person has something to give away, they send out an email to 'n' people. The email directs the recipients to try and guess the lowest positive integer they can that is unique amoung all the responses...
  48. H

    Game Theory - A graphical solution to all 2-person n x n strategy games?

    I was not sure where to post this. It didn't seem to fit in the homework/self-study category, and I believe it is thought-provoking enough to warrant posting in the general forums. I am sorry if I am wrong. In game theory, 2-person non-zero-sum games with 2n strategies can be graphically...
  49. S

    How to introduce Game Theory to HS students without boring them

    I am to give a short presentation to high school students about game theory. How can I make it as to not make it boring for them? I don't think I'm going to use much mathematics because I doubt at their level that they're going to understand it that much if I delve too deep into the mathematics.
  50. thrill3rnit3

    Is Game Theory the Perfect Summer Learning Opportunity for High Schoolers?

    Hello, I will be a rising high school senior next year, and I'm looking for something worthwhile to learn over this summer. I'm really interested in learning mathematical game theory. I was wondering on the prerequisites before learning game theory (I've done maths up to linear algebra and real...