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Game Theory

Game Theory. Jacob Foley. http://www.youtube.com/watch?v=HCinK2PUfyk http://www.youtube.com/watch?v=l0ywiYboCLk. Overview. Introduction and history Total-conflict games Partial-conflict games Three-person voting game. What is Game Theory .

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Game Theory

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  1. Game Theory Jacob Foley

  2. http://www.youtube.com/watch?v=HCinK2PUfyk • http://www.youtube.com/watch?v=l0ywiYboCLk

  3. Overview • Introduction and history • Total-conflict games • Partial-conflict games • Three-person voting game

  4. What is Game Theory • Game- two or more individuals compete to try to control the course of events • Uses mathematical tools to study situations involving both conflict and cooperation

  5. History • The first known discussion of game theory occurred in a letter written by James Waldegrave in 1713 • Theory of Games and Economic Behavior by John von Neumann in 1944 • Eight game theorists have won Nobel prizes in economics

  6. Definitions • Player-maybe be people, organizations or countries • Strategies- course of action they may take based on the options available to them • Outcomes- the consequences of the strategies chosen by the players • Preferences- each player has a perfered outcome

  7. Game theory analyzes the rational choice of strategies • Areas Applied • Bargaining tactics in labor-management disputes • Resource allocation decisions • Military Choices in international crises

  8. What makes it different • Analyzes situations in which there are at least two players • The outcome depends on the choices of all the players • Players can cooperate but it is not necessary

  9. Why is it important? • Provided theoretical foundations in economics • Applied in political science (study of voting, elections, and international relations) • Given insight into understanding the evolution of species and conditions under which animals fight each other for territory

  10. Two-Person Total-Conflict • Location Game • Two Young Entrepreneurs with a new restaurant in the mountains • Lisa likes low elevations • Henry likes higher elevations • Routes A, B, and C run east-west • Highways 1, 2 and 3 run north-south • Henry selects one of the routes • Lisa selects one of the Highways • Selection is made simultaneously

  11. Heights of the intersections

  12. How do they choose • Maximin- the maximum value of the minimum numbers in the row of a table • Minimax- the minimum value of the maximum numbers in the columns of a table • Saddlepoint- the outcome when the row minimum and the column maximum are the same

  13. Solution • In total-conflict games, the value is the best outcome that both players can guarantee • In our example the value is 5 • The value is given by each player choosing their maximin and minimax strategies

  14. Example 2: Restricted-Location • Use the same information from previous problem • However, the county officials outlaw restaurants on Route B and Highway 2

  15. Results • There are no saddlepoints • If both choose their minimax and maximin strategy, we will result in 7 • However, they could try to out think the other which could result in 10 or 2

  16. Duel Game

  17. Flawed Approach • Pitcher- If I choose F I hold the batter down to .300 or less but the batter is likely to guess F which gives him at least .200 and actually .300 • Batter- Because the pitcher will try to surprise me with C, I should guess C. I would then average .500. • Pitcher- But if batter guess C, I should really throw F. Thus leading to an average of .100 for the batter

  18. As we see…we can keep going over and over… • Pure Strategy- Each of the definite courses of action that a player can choose • Mixed Strategy- Course of action is randomly chosen from one of the pure strategies by: • Each pure strategy is assigned some probability, indicating the relative frequency with which that pure strategy will be played • The specific strategy used in any given play of the game can be selected at using some appropriate random device

  19. Expected Value of E • In each of the n payoffs, s1, s2, ……, sn, will occur with the probability p1, p2, ………pn, respectively. • The expected Value E • E=p1s1 +p2s2+………..+ pn*sn • And we assume p1+p2+……+pn=1

  20. Matching Pennies • Two players • Each has a penny • They both show either heads or tails at the same time • If the match, player 1 gets the pennies • If they are not a match, player 2 gets the pennies

  21. Payoff Matrix

  22. Results • H & T are pure strategies for both players • There is no way one player can outguess the other • Each player should use a mixed strategy choosing H half the time and T half the time • For player 1: • E(h)= ½(1) + ½(-1) = 0 • E(t)= ½(-1) + ½(1) =0

  23. Cont. • The expected value for player 2 is the same • This means the game is fair, which means the expected value = 0 and therefore favors neither player when at least one player uses an optimal mixed strategy • If one player does not use the 50-50 strategy the player that does gains an advantage

  24. Another example

  25. Results • Player 1 • E(H) = 5*(p) + (-3)(1-p) = 8p-3 • E(T) = (-3)(p) +(1-p)=-4p +1 • 8p-3=-4p+1 • 12p = 4 • P=1/3 • Therefore, • E(H) = 8(1/3) -3 = E(T) = -4(1/3) + 1 =-1/3 => p=1/3 • So their optimal mixed straigy is (1/3, 2/3) with expected value of 1/3

  26. Cont. • Using same calculations for player 2 we get the same optimal mixed stratigy of (1/3, 2/3) • However, the expected value for player 2 is 1/3 • Therefore, we have a zero-sum game.

  27. Lets go back to the baseball game

  28. What should the pitcher do? • E(f)= (0.3)p + (0.2)(1-p) = 0.1p + 0.2 • E(c)= (0.1)p + 0.5(1-p) = -0.4p + 0.5 • Solution is at the intersection of these two lines • -0.4p + 0.5 = 0.1p + 0.2 • p = 0.6 • Giving E(f)=E(c)=E=0.26 • Thus, the Pitcher should pitch F with p = 3/5 and C with p=2/5 so the batter will not be better than .260

  29. What should the batter do? • E(f)= (0.3)q + (0.1)(1-q) = 0.2q + 0.1 • E(c)= (0.2)q + (0.5)(1-q) = -0.3q + 0.5 • 0.2q + 0.1 = -0.3q + 0.5 • q=0.8 • E(f) = E(c) = E = 0.260 • Therefore, he should guess F with p=4/5 and C with p=1/5 which gives him a batting average of 0.260 • So this gives us an outcome of 0.260

  30. Partial-Conflict Games • These are games in which the sum of payoffs to the players at different outcomes varies • There can be gains by both players if the cooperate but this could be difficult

  31. Prisoners’ Dilemma • Two-person variable-sum game • Shows the workings behind arms races, price wars, and some population problems • In these games, each player benefits from cooperating • There is no reason for them to cooperate without a credible threat of retaliation for not cooperating • Albert Tucker, Princeton mathematician, named the game the Prisoners’ Dilemma in 1950

  32. So the actual game • Two people are accused of a crime • Each person has a choice: • Claim their innocence • Sign a confession accusing the partner of committing the crime • It is in their interest to confess and implicate their partner to receive reduce sentence • However, if both confess, both will be found guilty • As a team, their best interest is to deny having committed the crime

  33. Apply it to the real world army race • Two nations, Red and Blue • A: Arm in preparation for war • D: Disarm or negotiate an arms-control agreement • Rank from best to worse (41)

  34. What should they do? • Red • If Blue selects A- Red receives a payoff of 2 for A and 1 for D, so choose A • If Blue select D- Red receives a payoff of 4 for A and 3 for D, so choose A • Red has a dominate strategy of A • So a rational Red nation will choose A • Similarly, Blue will choose A

  35. Results • If the nations work independently, we get an outcome of (A,A) with payoff of (2,2) • This is a Nash Equilibrium- where no player can benefit by departing by itself from its strategy associated with an outcome • So, each player can corporate, play independent, or defect • Defect dominates cooperate and playing independent for both players • However, defect by both players results in a worse outcome than the mutual-cooperation outcome

  36. Another Example “Chicken” • Two Drivers coming at each other at high speeds

  37. Results • Neither player has a dominate strategy • The Nash Equilibrium are (4,2) and (2,4) • This means that getting the result of (3,3) will be unlikely because each players has an incentive to deviate to get a high payoff

  38. Larger Games • Lets look for a 3x3x3 game • We find the optimal solution by looking at individuals dominant strategy • Reducing it to a 3x3 game and we solve like a 2 person games we have been doing

  39. Example: Truel • A duel with 3 people • Each player has a gun and can either fire or not fire at either of the other players • Goal is to survive 1st and survive with as few other players as possible • http://www.youtube.com/watch?v=rExm2FbY-BE&feature=related

  40. Game Tree

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