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The document provides an overview of game theory, including definitions of key concepts. It discusses: 1) Game theory as the mathematical analysis of conflict situations where players make rational decisions. It aims to find optimal strategies. 2) Key concepts in game theory including games, moves, strategies, information, payoffs, extensive and normal forms, and equilibria such as Nash equilibrium. 3) Examples of games and equilibrium concepts including prisoners' dilemma, mixed strategies, and maximin strategies. Game theory has applications in economics, politics, and military strategy.
GT Presentation
- 1. GAME THEORY The Complete Strategyst By J. D. William
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- 4. 4 Why We SelectedThe Book “The complete Strategyst by J.D. William” Evolutionary Not as a set of theorems but a box of tools Stunning Interplay Cooperation and Conflict strengths and weaknesses of market economy
- 5. 5 Definition, History andSignificance of Game Theory Arif Hussain Types of Games Basic Concepts of Game Theory Kinds of Strategies Nash Equilibrium and Prisoners Dilemma Examples of Application of Game Theory In real World Sequence of Presentation
- 6. 6 What is GameTheory? Mathematical analysis of a conflict of interest To find optimal choices will lead to a desired outcome under given conditions That
- 7. 7 Game Theory Actually becoming afield of major interest economics sociology political military
- 8. 8 A study ofways to win in a situation under the given the conditions of the situation
- 9. 9 Game Theory isa set of tools and techniques for decisions in which each opponent aspires to optimize his own decision. involving two or more intelligent opponents under uncertainty at the expense of the other opponents Formal Definition
- 10. Ideas underlying gametheory Apparent in Bible Charles Darwin Talmud The works of Descrates Sun Tzu History Of Game Theory 10
- 11. Game Theory Outgrowth ofThree Seminal Works Augustin Cournot Emile Borel, Francis Ysidro Edgeworth History Of Game Theory 11
- 12. Systematic and ModernAnalysis 1944 “Theory of Games and Economic Behavior” 1950 Modern methodological framework by John Nash World War -II British Naval Officers first applied game theory History Of Game Theory 12
- 13. 1994 Johns Nash J.C.Harsanyi R. Selton NoblePrizes Shared 1996 William Vickery James Mirrlees History Of Game Theory 13
- 14. 14 Flexible Decision Structure Correct solutionsno matter what happens Constant reassessment of the planning process Analyze through other players' eyes Don't just play the game, change the rules look beyond the traditional roles Significance of Game Theory
- 15. Elements of Game Theessential elements of a game are: Players. The individuals who make decisions. Rules of The Game. Who moves when? What can they do? Outcomes. What do the various combinations of actions produce? Payoffs. What are the players’ preferences over the outcomes? Information. What do players know when they make decisions? Chance. Probability distribution over chance events, if any 15
- 16. Basic Concepts ofGame Theory Game Move Information Strategy Payoffs Extensive Form Normal Form Equilibria 16
- 17. Game A conflictin interest among players (individuals or groups) There exists a set of rules that define the terms of exchange of information and pieces The conditions under which the game begins, and the possible legal exchanges in particular conditions The entirety of the game is defined by all the moves to that point, leading to an outcome. 17
- 18. Move The wayin which the game progresses between states through exchange of information and pieces Moves are defined by the rules of the game Moves can be made in either alternating fashion, occur simultaneously for all players, or continuously for a single player until he reaches a certain state or declines to move further Moves may be choice or by chance. For example, choosing a card from a deck or rolling a die is a chance move with known probabilities On the other hand, asking for cards in blackjack is a choice move. 18
- 19. Information A stateof perfect information is when all moves are known to all players in a game Games without chance elements like chess are games of perfect information While games with chance involved like blackjack are games of imperfect information. 19
- 20. Strategy A strategyis the set of best choices for a player for an entire game It is an overlying plan that cannot be upset by occurrences in the game itself. 20
- 21. Difference Between AMove is a single step a player can take during the game. A strategy is a complete set of actions, which a player takes into account while playing the game throughout Move Strategy 21
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- 23. Pure strategy Mixed Strategy Totally mixed strategy. 23 Kinds of Strategies
- 24. Pure Strategy Apure strategy provides a complete definition of how a player will play a game. In particular, it determines the move a player will make for any situation he or she could face. A player‘s strategy set is the set of pure strategies available to that player. select a single action and play it Each row or column of a payoff matrix represents both an action and a pure strategy 24
- 25. Mixed Strategy Astrategy consisting of possible moves and a probability distribution (collection of weights) which corresponds to how frequently each move is to be played. A player would only use a mixed strategy when she is indifferent between several pure strategies, and when keeping the opponent guessing is desirable - that is, when the opponent can benefit from knowing the next move. 25
- 26. Totally Mixed Strategy. A mixed strategy in which the player assigns strictly positive probability to every pure strategy In a non-cooperative game, a totally mixed strategy of a player is a mixed strategy giving positive probability weight to every pure strategy available to the player. 26
- 27. Payoff The payoffor outcome is the state of the game at it's conclusion In games such as chess, payoff is defined as win or a loss In other situations the payoff may be material (i.e. money) or a ranking as in a game with many players. 27
- 28. Extensive and NormalForm Extensive Form The extensive form of a game is a complete description of: - The set of players - Who moves when and what their choices are - What players know when they move - The players’ payoffs as a function of the choices that are made. - In simple words we also say it is a graphical representation (tree form) of a sequential game. 28
- 29. The Normal Form The normal form is a matrix representation of a simultaneous game For two players, one is the "row" player, and the other, the "column" player Each rows or column represents a strategy and each box represents the payoffs to each player for every combination of strategies Generally, such games are solved using the concept of a Nash equilibrium. . 29
- 30. Equilibrium Equilibrium isfundamentally very complex and subtle The goal to is to derive the outcome when the agents described in a model complete their process of maximizing behaviour Determining when that process is complete, in the short run and in the long run, is an elusive goal as successive generations of economists rethink the strategies that agents might pursue. 30
- 31. Nash Equilibrium Definedas a set of strategies such that none of the participants in the game can improve their payoff, given the strategies of the other participants. Identify equilibrium conditions where the rates of output allowed the firms to maximize profits and hence no need to change. No price change is an equilibrium because neither firm can benefit by increasing its prices if the other firm does not 31
- 32. Limitations of NashEquilibrium For some games, there may be no Nash equilibrium; continuously switch from one strategy to another There can be more than one equilibrium 10, 10 100, -30 -20, 30 140, 25 Firm 2 Firm 1 No Price Change No Price Change Price Increase Price Increase Both firms increasing their price is also a Nash equilibrium 32
- 33. Dominant Strategies Onefirm’s best strategy may not depend on the choice made by the other participants in the game Leads to Nash equilibrium because the player will use the dominant strategy and the other will respond with its best alternative Firm 2’s dominant strategy is not to change price regardless of what Firm 1 does 33
- 34. Dominated Strategies Analternative that yields a lower payoff than some other strategies a strategy is dominated if it is always better to play some other strategy, regardless of what opponents may do It simplifies the game because they are options available to players which may be safely discarded as a result of being strictly inferior to other options. 34
- 35. Continued…. A strategys¡ in set S is strictly dominated for player i if there exists another strategy, s¡’ in S such that, Πi(s¡’) > Πi(s¡) In this case, we say that s¡’ strictly dominates s¡ In the previous example for Firm 2 no price change is a dominant strategy and price change is a dominated strategy 35
- 36. Maxim in Strategies Highly competitive situations (oligopoly) Risk-averse strategy – worst possible outcome is as beneficial as possible, regardless of other players Select option that maximizes the minimum possible profit 36
- 37. Each firmfirst determines the minimum profit that could result from each strategy Second, selects the maximum of the minimums Hence, neither firm should introduce a new product because guaranteed a profit of at least $3 million Maxim in outcome not Nash equilibrium- loss avoidance rather than profit maximization 4, 4 3, 6 6, 3 2, 2 Firm 1 Firm 2 Firm 2 Minimum Firm 1 Minimum New Product No New Product No New Product New Product 3 2 23 37
- 38. Mixed Strategies Purestrategy – Each participant selects one course of action Mixed strategy requires randomly mixing different alternatives Every finite game will have at least one equilibrium 38
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- 40. Example Battle of theSexes Two agents need to coordinate their actions, but they have different preferences Original scenario: • husband prefers football • wife prefers opera Another scenario: • Two nations must act together to deal with an international crisis • They prefer different solutions This game has two pure-strategy Nash equilibrium and one mixed-strategy Nash equilibrium How to find the mixed-strategy Nash equilibrium? 40
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- 42. Types of Games One-Person Games Zero-Sum Games Non zero sum game Two-Person Games Repeated Games 42
- 43. One-Person Games Aone-person games has no real conflict of interest Only the interest of the player in achieving a particular state of the game exists Single-person games are not interesting from a game- theory perspective because there is no adversary making conscious choices that the player must deal with
- 44. Zero-Sum Games Azero-sum game is one in which no wealth is created or destroyed Whatever one player wins, the other loses Therefore, the player share no common interests There are two general types of zero-sum games: with and without perfect information If the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero Example a. Rock, Paper, Scissors b. Poker game
- 45. Example of pokergame Let’s there are three players, Aslam, Akram and Saleem each starting with Rs 100, in a total of Rs 300. They meet at Akram’s house and play for a couple of hours. At the end of the evening Aslam has Rs 200, Akram has Rs 60 and Saleem has Rs 40. The total amount of money between them is still Rs 300. Aslam has a increase of Rs 100, Akram is down Rs 40 and Saleem is down Rs 60. The total of these three numbers is zero (100-40-60), so it is a zero-sum game.
- 46. Example In non-zero-sumgames, one player's gain needn't be bad news for the other(s) Indeed, in highly non-zero-sum games the players' interests overlap entirely In 1970, when the three Apollo 13 astronauts were trying to figure out how to get their stranded spaceship back to earth, they were playing an utterly non-zero- sum game, because the outcome would be either equally good for all of them or equally bad.
- 47. Non Zero SumGame In game theory, situation where one decision maker's gain (or loss) does not necessarily result in the other decision makers' loss (or gain). In other words, where the winnings and losses of all players do not add up to zero and everyone can gain: a win-win game. Example Prisoner's dilemma
- 48. Two-Person Games Two-persongames are the largest category of familiar games A more complicated game derived from 2-person games is the n-person game These games are extensively analyzed by game theorists However, in extending these theories to n-person games a difficulty arises in predicting the interaction possible among players since opportunities arise for cooperation and collusion.
- 49. Repeated Games Inrepeated games, some game G is played multiple times by the same set of agents G is called the stage game Each occurrence of G is called an iteration or a round Usually each agent knows what all the agents did in the previous iterations, but not what they’re doing in the current iteration Usually each agent’s payoff function is additive Examples 1. Iterated Prisoner’s Dilemma 2. Repeated Ultimatum Game 3. Repeated Matching Pennies 4. Repeated Stag Hunt
- 50. Sequential Games Agame where one player chooses his action before the others choose theirs Importantly, the later players must have some information of the first's choice, otherwise the difference in time would have no strategic effect Extensive form representations are usually used for sequential games, since they explicitly illustrate the sequential aspects of a game. Combinatorial games are usually sequential games. Sequential games are often solved by backward induction.
- 51. Simultaneous Games Agame where each player chooses his action without knowledge of the actions chosen by other players Normal form representations are usually used for simultaneous games. Example Prisoner dilemma .
- 52. Application of GameTheory Philosophy Resource Allocation and Networking Biology Artificial Intelligence Economics Politics Warfare
- 53. Non Cooperative Games Becausethe two participants are interrogated separately, they have no idea whether the other person will confess or not 53
- 54. Co-operative Games Possibilityof negotiations between participants for a particular strategy If prisoners jointly decide on not confessing, they would avoid spending any time in jail Such games are a way to avoid prisoner’s dilemma 54
- 55. Sequential Games Oneplayer acts first & then the other responds 2 firms contemplating the introduction of an identical product in the market 1st firm- develop brand loyalties, associate product with the firm in minds of consumers Thus, first mover advantage 55
- 56. An example forsequential games Firm 2 No new product Introduce new product Firm 1 No new product $2, $2 $-5, $10 Introduce new product $10, $-5 $-7, $-7 • Assume firms use maximum criterion, so neither should introduce a new product and earn $2 mn each • Firm 1 introduces a new product, firm 2 will still decide to stay out because right now it is losing $5 mn, opposed to $7 mn otherwise. 56
- 57. Conclusion By usingsimple methods of game theory, we can solve for what would be a confusing array of outcomes in a real-world situation Using game theory as a tool for financial analysis can be very helpful in sorting out potentially messy real-world situations, from mergers to product releases.