Game theory


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Game theory

  2. 2. History 1928 –Game theory did not really exist as a unique field until John von Neumann & Morgenstern published a paper. 1950 -John Nash developed a criterion for mutual consistency of players strategies, known as Nash equilibrium, applicable to a wider variety of games than the criterion proposed by von Neumann and Morgenstern. 1965 –Reinhard Selten introduced his solution concept of subgame perfect equilibria, which further refined the Nash equilibrium. 1970 –Game theory was extensively applied in biology, largely as a result of the work of John Maynard Smith and his evolutionarily stable strategy. 2007 –Leonid Hurwicz, together with Eric Maskin and Roger Myerson, was awarded the Nobel Prize in Economics "for having laid the foundations of mechanism design theory 2
  3. 3. Whats IT??Game theory is a method ofstudying strategic decisionmaking. More formally, it is "thestudy of mathematical models ofconflict and cooperationbetween intelligent rationaldecision-makers. Strategic- It is a setting wherethe outcomes that affect youdepends on actions not just onyour own actions but on actionsof others. 3
  4. 4. Why game theory ???Because the press tells us to … ―Managers have much to learn from game theory —provided they use it to clarify their thinking, not as a substitute for business experience.‖ The Economist, 15 June 2007 ―Game Theory, long an intellectual pastime, came into its own as a business tool.‖ Forbes, 3 July 2009 ―Game theory is hot.‖ The Wall Street Journal, 13 February 2011Because recruiters tell us to … ―Game theory forces you to see a business situation over many periods from two perspectives: yours and your competitor’s.‖ Judy Lewent – CFO, Merck ―Game theory can explain why oligopolies tend to be unprofitable, the cycle of over capacity and overbuilding, and the tendency to execute real options earlier than optimal.‖ Tom Copeland – Director of Corporate Finance, McKinsey 4
  5. 5. Terminology Players Strategies ◦ Choices available to each of the players  Might be conditioned on history Payoffs ◦ Some numerical representation of the objectives of each player  Could take account fairness/reputation, etc.  Does not mean players are narrowly selfish 5
  6. 6. Standard Assumptions Rationality ◦ Players are perfect calculators and implementers of their desired strategy Common knowledge of rules ◦ All players know the game being played Equilibrium ◦ Players play strategies that are mutual best responses 6
  7. 7. Strategies for Studying Games ofStrategy  Two general approaches ◦ Case-based  Pro: Relevance, connection of theory to application  Con: Generality ◦ Theory  Pro: General principle is clear  Con: Applying it may not be Types of firms : 1)Perfect competition 2)Monopolistic or Imperfect competition. ex- Motor car industry , mobile markets etcetera. 7
  8. 8. Where It Exists?An easy strategy A complex strategyAsking your friend out for dinner? Geopolitical talks? 8
  9. 9. Representation of Games The games studied in game theory are well-defined mathematical objects. A game consists of a set of players, a set of moves (or strategies) available to those players, and a specification of payoffs for each combination of strategies. Categories :Extensive formNormal formCharacteristic function formPartition function form 9
  10. 10. Extensive form Games here are played on trees (as pictured to the left). Here each vertex (or node) represents a point of choice for a player. The player is specified by a number listed by the vertex. The lines out of the vertex represent a possible action for that player. The payoffs are specified at the bottom of the tree. 10
  11. 11. Normal/Strategic Form The normal (or strategic form) game is usually represented by a matrix which shows the players, strategies, and payoffs. More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions. 11
  12. 12. Types of GamesCooperative : A game is cooperative if the players are able to form binding commitments. For instance the legal system requires them to adhere to their promises. Often it is assumed that communication among players is allowed in cooperative games. Cooperative games focus on the game at large.Non-cooperative : In game theory, a non-cooperative game is one in which players make decisions independently. Non-Cooperative games are able to model situations to the finest details, producing accurate results.Hybrid games : contain cooperative and non-cooperative elements. 12
  13. 13. Symmetric : A symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. Prisoners dilemma, and the stag hunt are all symmetric games.Asymmetric : Most commonly studied asymmetric games are games where there are not identical strategy sets for both players.Zero-sum games : In zero-sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero (more informally, a player benefits only at the equal expense of others).Non Zero-sum games :These type of games have outcomes whose net results are greater or less than zero 13
  14. 14. Simultaneous games : Are games where both players move simultaneously, or if they do not move simultaneously, the later players are unaware of the earlier players actions (making them effectively simultaneous).Sequential games (or dynamic games) : They are games where later players have some knowledge about earlier actions. This need not be perfect information about every action of earlier players; it might be very little knowledge. For instance, a player may know that an earlier player did not perform one particular action.Perfect information and Imperfect information : A game is one of perfect information if all players know the moves previously made by all other players. Thus, only sequential games can be games of perfect information, since in simultaneous games not every player knows the actions of the others. Many card games are games of imperfect information, for instance poker. 14
  15. 15.  Combinatorial games : Games in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are called combinatorial games. Examples include chess and go. Games that involve imperfect or incomplete information may also have a strong combinatorial character. A related field of study, drawing from computational complexity theory, is game complexity, which is concerned with estimating the computational difficulty of finding optimal strategies. Discrete and continuous games : Continuous games includes changing strategy set. Much of game theory is concerned with finite, discrete games, that have a finite number of players, moves, events, outcomes, etc. 15
  16. 16.  Infinitely long games : Pure mathematicians are not so constrained, and set theorists in particular study games that last for infinitely many moves, with the winner (or other payoff) not known until after all those moves are completed. The focus of attention is usually not so much on what is the best way to play such a game, but simply on whether one or the other player has a winning strategy. Meta-games : These are games the play of which is the development of the rules for another game, the target or subject game. Metagames seek to maximize the utility value of the rule set developed. 16
  17. 17. Lesson 1 Dominance Principle : Do Not Play A Strictly Dominated Strategy. Player 2 α β α 0,0 3,-1 Player 1 β -1,3 1,1 We say that my strategy α strictly dominates my strategy β if my payoff from α is strictly greater than that from β regardless of what others do. Here playing strategy β is a strictly dominated strategy. 17
  18. 18. GAME 1Put in the box below either the letter α (alpha) or β (beta) .Think of this as agreat bid. I will randomly pair your form with another form and neither yourpair or you would know with whom you were paired. Here is how grades maybe assigned :- you pair α, β – A,C α, α – B minus , B minus β, α – C , A β, β –B plus , B plus 18
  19. 19. α, β – A, C α, α – B minus , B minus β, α – C , A β, β –B plus , B plus MY PAIR ME (X`) (X) α β α β MYME α B A PAIR α C(X) minus B (X`) minus β C B plus β A B plus 19
  20. 20. MY PAIR (X`) α β Me (X) α B minus , A, B minus C β C, B plus , A B plusOutcome matrix We cared about our own grade1)2) We cared about other grades } 2 Different payoffs 20
  21. 21. MY PAIR (X`) α β If the pair chooses alpha and I choose alpha then I get zero and if player ME chooses alpha and I (X) α 0,0 3, -1 choose beta I get -1. 0 > -1 β -1, 3 1,1 If the player chooses beta and IWe try to put numbers into perspective and these are called choose alpha I get 3 and if player# utilities chooses beta and Ijust represent that we optimise our own gain and others loss. choose beta I get 1(A,C) – 3 3>1(B minus , B minus )- 0 Hence in both cases she receives a higher payoff if she chooses alpha.
  22. 22. MY PAIR (X`) α βME(X) α 0,0 -1, -3 β -3,-1 1,1 Replacing (A,C) = -1,(C,A) = -3 22
  23. 23. Penalty kick game Goal keeper l rShooter L 4,-4 9,-9 M 6,-6 6,-6 R 9,-9 4,-4 23
  24. 24. FINALS OF UEFA 2008MAN Vs CHELSEA It seemed that Chelsea’s strategy ofgoing to Van der Sar’s left had beenhatched by someone on the Utdbench. As Anelka prepared to takeChelsea’s 7th penalty Van der Sarpointed to the left corner. NowAnelka had a terrible dilemma. Thiswas game theory in its rawest form.So Anelka knew that Van der Sarknew that Anelka knew that Van derSar tended to dive right against rightfooters. Instead Anelka kicked rightbut it was at mid-height whichIgnacio warned against. 24
  25. 25. Lesson 2 Put yourself in others shoes and try to figure out what they will do. Destroyer Easy Hard Easy 1,1 1,1Savior Hard 0,2 2,0 Here savior should look upon the destroyers payoff before making a move. 25
  26. 26. Prisoners Dilemma Two Prisoners are on trial for a crime and each one faces a choice of confessing to the crime or remaining silent. The payoffs are given below: Prisoner 2 Confess Silent Confess 4,4 1,5Prisoner 1 Silent 5,1 2,2 26
  27. 27. Solution to Prisoners Dilemma Prisoner 2 Confess Silent Confess 4,4 1,5Prisoner 1 Silent 5,1 2,2 The only stable solution in this game is that both prisoners confess; in each of the other 3 cases, at least one of the players can switch from ―silent‖ to ―confess‖ and improve his payoff. 27
  28. 28. Odds to Prisoners Dilemma What would the Prisoner 1 and Prisoner 2 decide if they could negotiate? They could both become better off if they reached the cooperative solution…. which is why police interrogate suspects in separate rooms. Equilibrium need not be efficient. Non-cooperative equilibrium in the Prisoner’s dilemma results in a solution that is not the best possible outcome for the parties. Alternative: Implied contract if there were a long relationship between the parties— (partners in crime) are more likely to back each other 28
  29. 29. CPU Competition Intel and AMD compete fiercely to develop innovations in CPUs for PCs As a result of this, CPU speeds have increased dramatically, but there are few differences between the products of the two companies Even though both companies expended huge amounts of money to gain a competitive advantage, their relative competitive position ends up unchanged. Both companies are worse off than if they had each slackened the pace of innovation This is an example of a prisoner’s dilemma 29
  30. 30. Sale Price Guarantees Globus and many other stores like Pantaloons offer sale price guarantees ◦ If an item comes on sale in the time period after you bought it, they will match the difference in prices ◦ Thus, consumers wishing to buy now are ―protected‖ against regrets from future price reductions 30
  31. 31. Walking Down the Demand Curve  Globus would like to sell goods at high prices to those with high willingness to pay for them and then lower prices to capture those with lower willingness to pay.  They might do this by running sales after new items have been out for a while  But high value consumers will anticipate this and wait for the sale to occur. 31
  32. 32. The Power of Commitment By offering to rebate back the difference in prices, Globus makes the sale strategy less profitable for its ―future‖ self. This commits it to less discounting in the future In fact It enables it to charge higher prices today…and tomorrow. 32
  33. 33. Coordination Game Coordination game involves multiple outcomes that can be stable. A simple Coordination game involves two players choosing between two options, wanting to choose the same. Battle of the Sexes : 33
  34. 34. Solution to Battle Of the Sexes Solution: Both should play the same game. Everyone benefits from being in cooperative group, but each can do better by exploiting cooperative efforts of others. 34
  35. 35. Anti Coordination Game Routing Congestion Game : Solution: The players must coordinate to send traffic on different connection points. 35
  36. 36. Rules of the Game The strategic environment ◦ Players – I , j ◦ Strategies- S(i) – strategy of player I ◦ S(i) – set of possible strategies of player I ◦ S – strategy profile ◦ Payoffs- µ (i) ◦ µi (s) – profile utilities 36
  37. 37.  The rules ◦ Timing of moves ◦ Nature of conflict and interaction ◦ Informational conditions ◦ Enforceability of agreements or contracts The assumptions ◦ Rationality ◦ Common knowledge 37
  38. 38. Prisoners Dilemma (communication is not legal)communication wont help 38
  39. 39. Investor game(where both can communicate legally.)A communication can help L R U 1,1 0,0 D 0,0 1,1 39
  40. 40. So far we have used the dominant strategy solution and iterativeelimination of dominated strategy (IEDS) solution concepts to solvestrategic form games. A third approach is to use the Nash equilibriumconcept. 40
  41. 41. Nash Equilibrium- Necessary requirementsfor a Nash equilibrium are that each playerplay a best response against a conjecture,and the conjecture must be correct. 41
  42. 42. Going to movies He Person 1 Person 2 Person 3She Movie1 2,1 0,0 0,-1 Movie 2 0,0 1,2 0,-1 Movie 3 -1,0 -1,0 -2,-2Either both people go for movie 1 orBoth go for Movie 2You call this each persons ―equilibrium‖ or ―Nash equilibrium ― 42
  43. 43. How do we identify the Nash equilibria in a game?Look for the dominant strategy.Eliminate the dominated strategies.Play a minimax strategy. In a zero sum game you choose that strategy inwhich your opponent can do you the least harm from among all of thebad outcomes.Cell-by-cell inspection or trial and error 43
  44. 44.  1- no regrets as we choosing the best move 2- Self fulfilling believes 44
  45. 45. Coordination game –1) Your farewell party2) LED technology3) Tv shows4) Software platforms coordination 45
  46. 46.  selfish routing in large networks like the Internet theoretical basis to the field of multi-agent systems. Use of NLP for optimizing results. 46
  47. 47. Diverse applications1)Economics and business2) Political science -game-theoretic models in which the players are often voters, states, special interest groups, and politicians.3) Biology -Evolutionary game theory (EGT) is the application of game theory to evolving populations of lifeforms in biology.4) Computer science and logic-Algorithmic game theory and within it algorithmic mechanism design combine computational algorithm design and analysis of complex systems with economic theory.5) Philosophy-common beliefs or knowledge6) Legal problems and behavioral economics. 47