An Introduction to Game Theory
Order ID 53563633773 Type Essay Writer Level Masters Style APA Sources/References 4 Perfect Number of Pages to Order 5-10 Pages
Dominant Strategies Teaching Example 1. In this 2-by-2 game Rose has 2 pure strategies: Top or Bottom. Colin can choose Left or Right. Can an economist predict which of the 4 outcomes will occur? Rose, Colin Left Right Top 1, 2 0, 1 Bottom 2, 1 1, 0 When is it okay to lie? 2. Getting Caught. Anita, Barbara and Carly share a townhouse. On Tuesday the girls promised each other to stop eating those oversized muffins from Tim Hortons. Yet, Wednesday morning Anita and Barbara catch each other in the line-up at Tim Hortons. Both girls have broken their promise. Should they tell Carly? UA UB Confess Lie Confess -10, -10 -5, -20 Lie -20, -5 -15, -15 Prisoners Dilemma 3. You and Him are taken into police custody and put in separate interrogation rooms. You can either Deny the crime or Deal (plea bargain). He has the same options. The payoffs give the number of years you will spend in jail (less is better). Can we predict the outcome? What if this was a Repeated Game? YearsYOU, YearsHIM Deny Deal Deny 1, 1 6, 0 Deal 0, 6 3, 3 Credible Threat 4. Big Pig, Little Pig (Bergstrom) Two pigs are in a Skinner Box. There are 5 kilograms of tasty pig food in a trough. If one of the pigs crosses the room and pushes the button an additional 5 kilograms of food will drop into the trough. Each pig has 2 pure strategies: Push or Wait. One of the pigs called Big Pig is faster at running across the room and is faster at eating. Little Pig is at a disadvantage. Is there a Dominant Strategy equilibrium? Could Little Pig issue a Credible Threat that would result in an equilibrium? FoodBP, FoodLP Wait Push Wait 3, 2 10, 0 Push 7, 3 9, 1 Group Projects 5. Two students are forced to by their professor to work on a group project. Their grade, G depends on how much effort each student puts into the project G = e1 + e2. Each student can either Work Hard e = 5 or Slack Off e = 1. Working has costs C1 = 2e1 and C2 = 2e2. Rewards in the payoff matrix should be computed as R1 = (e1 + e2) 2e1 and R2 = (e1 + e2) 2e2. Complete the table. Is there a Dominant Strategies equilibrium? Discuss Repeated Games in this context. Discuss Reputation in this context. R1, R2 e2 = 5 e2 = 1 e1 = 5 e1 = 1 Nash Equilibrium Teaching Example 6. Is there a Dominant Strategies Equilibrium is this game? Discuss the Nash Equilibrium (Stability) RYOU, RHIM Left Right Top 2, 1 0, 0 Bottom 0, 0 1, 2 Battle of the Sexes 7. A loving couple must decide what to watch on the living room television. There is another TV in the basement. The payoffs in the table below are often called a Game of Complements. Explain. Is there a Dominant Strategies Equilibrium? Identify Nash Equilibria. Discuss the First Mover Advantage UHIM, UHER Kenny vs Spenny Gilmore Girls Kenny vs Spenny 50, 30 20, 20 Gilmore Girls 20, 20 30, 50 Saying I Love You 8. First Mover Advantage Is there a Dominant Strategy when it comes to saying, I love you? Explain why there is no First Mover Advantage in this game. UHE, USHE I Love You Talk about Economics I Love You 15, 15 -10, 0 Talk about Economics 0, -10 5, 5 Chicken 9. Two high-school boys take their fathers cars to opposite ends of a parking lot and speed towards a head-on collision. Any boy who turns is labelled the Chicken and will undergo peer torment for the next week. Going straight is regarded as heroic. Is there a Dominant Strategy? Why is this often called a Game of Substitutes. Identify Nash Equilibria. Comment. Do we care that there are two Nash? Is there a First Mover Advantage? Discuss Reputation. UYOU, UHIM Turn Straight Turn 0, 0 5, 10 Straight 10, 5 -5, -5 Games of Substitutes 10. Avoiding your Ex (Another Game of Substitutes a.k.a Dis-coordination) RYOU, RYOUR EX Montanas Kelseys Montanas -200, -200 200, 200 Kelseys 200, 200 -200, -200 Mixed Strategies Equilibrium Teaching Example 11. Player One, who chooses Rows calculates the variable p = Probability (Top) in an attempt to keep Player Two indifferent between a pure strategy of Left and a pure strategy of Right. Player Two calculates the variable q = Probability (Left) in an effort to keep his opponent indifferent between Top and Bottom. We arrive at a Stalemate. Since both players are indifferent, there is an equilibrium. The probability of top is equal to p = __________ The probability of Left is equal to q = __________ Assume independence. What is the most likely outcome in this game? RYOU, RHIM P(Left) = q P(Right) = 1 q P(Top) = p 1, 3 2, 1 P(Bottom) = 1 p 2, 1 1, 2 Baseball is Boring! 12. A speedy runner has just reached First Base and Second Base is unoccupied. The runner has pure strategies to Steal second base or Wait. The pitcher has pure strategies to Pitch (Throw to Home) or Pickoff (Throw to First). If the runner attempts to Steal while the pitcher implements a Pickoff, the runner gets a very low reward and the pitcher is happy. If the runner attempts to Steal while the pitcher throws home, the runner is happy, and the pitcher is unhappy. Calculate the mixed strategies equilibrium. Assume independence. What is the most likely outcome? Explain why baseball is boring to watch. RPITCHER, RRUNNER Steal q Wait 1?q Pitch: p 0, 4 2, 0 Pickoff: 1?p 3, 0 0, 1 Asymmetric Payoffs 13. Adam and Bruce are looking for dates. The only available women are Carly and Diane. If both boys ask the same girl she will turn them both down (substitutes). Payoffs in this game are asymmetric. Discuss. Calculate the mixed strategies equilibrium. Interpret the probabilities calculated in Part b. UADAM, UBRUCE Bruce asks Carly Bruce asks Diane Adam asks Carly 0, 0 1, 1 Adam asks Diane 9, 1 0, 0 Rock, Paper, Scissors 14. Player A (Rows) calculates p1 = Probability (Rock) and p2 = Probability (Paper) and obviously p3 = 1 p1 p2 in an effort to leave her opponent indifferent between pure strategies. Player B (Columns) calculates q1 = Probability (Rock) and q2 = Probability (Paper) in an effort to leave his opponent indifferent between Rock, Paper and Scissors. Discuss the resulting equilibrium. What would be the Mixed Strategy when Sheldon and Leonard play Rock, Paper, Scissors, Lizard, Spock? $YOU, $HIM Rock Paper Scissors Rock 0, 0 1, 1 1, 1 Paper 1, 1 0, 0 1, 1 Scissors 1, 1 1, 1 0, 0 Closing Comments 15. Discuss why All Dominant Strategy Equilibria must also be Nash Equilibria. Mixed Strategy Equilibria do not exist in a game where there is a Dominant Strategies Equilibrium. Old Test Questions 16. Adam and Andrea Adams have been married for two years. Bob and Bernice Loblaws have been married for three. Every Tuesday night Adam and Bob go bowling. One Tuesday night Andrea accidentally runs into Bernice at some questionable Mississauga bar. Trouble is that Andrea is wrapped around some guy named Thor and Bernice is flirting with Fabio. Each lady faces two possible Pure Strategies. She can either Talk and tell her spouse about his friends wife or Lie; saying nothing. Her payoff depends rather critically on the strategy followed by the other woman. Payoffs below are in terms of utility. Based on the matrix above identify the equilibrium outcome. Briefly discuss whether it would be worthwhile for the two women to form a Pact prior to returning home. If so, how would that Pact work? R1, R2 Player Two Talk Lie Player One Talk -5, -5 5, -20 Lie -20, 5 0, 0 17. A market is supplied by two firms. Each firms revenue R depends on whether they advertise or not. Each firms revenue also depends on whether their competitor advertises. Payoffs in revenue are shown in the matrix below. Advertising costs $2. Profits are revenues minus advertising costs. Assuming that firms act to maximize profits, what is the equilibrium outcome of this game? Would it be worthwhile for these two firms to form a Pact? If so, what would be the terms of this agreement? R1, R2 Player 2 Ad No Ad Player 1 Ad 3, 3 5, 0 No Ad 0, 5 2, 2 18. Cathy and Karen have exclusive rights to provide babysitting services in their neighborhood. Each babysitter can either charge $5 or $10 per hour. Their weekly revenues depend critically on what price the other sitter charges. A payoff matrix shows their potential revenues. Briefly discuss the equilibrium outcome of this game. Re-write and solve this payoff matrix as a tree-diagram. Discuss whether it would be worthwhile for the girls to form a pact in this situation. R1, R2 Karen 5 10 Cathy 5 200, 200 400, 100 10 100, 400 250, 250 19. The payoff matrix to a game is provided below. Payoffs depend on the unknown parameters of ?, ?, ? and ?. (a) Solve for the Mixed Strategy Equilibrium in terms of these unknown parameters. Solve for restrictions which must be placed on the values of these parameters which ensure that [Top | Right] is a Dominant Strategy Equilibrium. Solve for restrictions which must be placed on the values of these parameters to
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