Game Theory

Prisoner's Dilemma A crime has been committed and 2 suspects have been arrested and put in separate rooms. The police make them the following offer: If you each confess, I will send both of you to prison for 5 years. If none of you confess, I will send both of you to prison for 2 years. If your friend confess and you do not, I will send you to prison for 10 years and will send your friend to prison for 1 year. If you confess and your friend does not, I will send you to prison for 1 year and I will send your friend to prison for 10 years.

Games in Strategic Form Players Strategies Payoffs (costs or benefits)

(Payoff of player A, Payoff of player B) Prisoner's Dilemma Prisoner A Confess Not Confess Confess Not Confess (Payoff of player A, Payoff of player B) 5 (years in prison),5 10,1 1,10 2,2 Prisoner B Two individuals are arrested for allegedly being involved in a serious crime. They are held in separate cells. The DA privatively tells each of them that if he is the only one to confess, then he will be rewarded with a sentence of only one year and the other prisoner will go to prison for 10 years. But, if both confess they will both stay in prison for 5 years. Finally, if neither confess there will not be enough proof to convict them for the serious crime but they will be convicted for a lesser crime; they will both stay in prison for 2 years. We need to check first if there are strictly dominated strategies. If prisoner A believes that prisoner B will confess, he will be better off confessing because he will go to prison for 5 years instead of 10. If prisoner A believes that prisoner B will not confess, he will be better off confessing because he will go to prison for 1 year instead of 2. Therefore, no matter how prisoner A believes prisoner B will act he will confess. Confess is thus a strictly dominant strategy for player A. This game is symmetric and player B will act in the same way. The solution of the game is that both player will confess.

War Game Two countries are deciding whether to spend their budget on health or defense. 1- If both countries spend their budget on health they both have an utility of $100. If they both spend their budget on defense they get an utility of $0. If one country spends the budget on health and the other one spends on defense, the country spending the money on defense gets an strategic advantage over the other one. Therefore, the country spending the money on health gets a payoff of -100 and the country spending the money on defense gets a payoff of 200. A)- Who are the players. B)- What are their strategies. C)- Is there an equilibrium in strictly dominated strategies. D)- Is there a Nash Equilibrium.

(country I) 100, (country II)100 War Game Country I Health Defense Health Defense (country I) 100, (country II)100 200,-100 -100,200 0,0 Country II

Equilibrium in Strictly Dominated Strategies A strategy is strictly dominant for a player if regardless of what the other player does he or she will play it. In an equilibrium in dominant strategies both players play strictly dominant strategies. Pareto Optimality versus Equilibrium

Pigs in a Box A strong and a weak pig are kept in a box. The box has a lever on one side and a food dispenser on the other. When the lever is pushed, food appears at the dispenser. If the weak pig pushes the lever, the strong pig waits at the dispenser and eats all the food. If the strong pig pushes the lever, the weak pig waits at the dispenser and eat part of the food. When the strong pig arrives at the dispenser, it pushes away the weak pig and eat the leftovers. Pushing the lever burns 10 calories and there are 100 calories worth of food at the dispenser.

Pigs in a Box (strong) 90, (weak)-10 100,-10 15,75 0,0 Strong Pig Does the strong pig have a Strictly Dominant Strategy? Strong Pig Push Lever Wait by Dispenser Push Lever Wait by Dispenser (strong) 90, (weak)-10 100,-10 15,75 0,0 Weak Pig

Battle of the Sexes A husband prefers to go to a boxing match and his wife prefers to go to the opera, but they like doing things together.

Battle of the Sexes (husband) 5, (wife) 3 0,0 1,1 3,5 Husband Boxing Opera Boxing Opera (husband) 5, (wife) 3 0,0 1,1 3,5 Wife

Nash Equilibrium An outcome is a Nash Equilibrium if neither player want to deviate from it “taking the opponent behavior as given”. If the husband thinks that the wife is going to opera, he will go to the opera. If the wife thinks that the husband is going to the opera, she will go to the opera. Both players going to the opera is a NE. Both players going to the boxing game is also a NE. Dominant strategy equilibrium is a much stronger concept (“regardless of what the other player does”). An equilibrium in dominant strategies is always a NE but not vice versa. If a game has a DE equilibrium this is the only equilibrium of the game.

Bank Runs: Suppose that a bank has only two customers Bank Runs: Suppose that a bank has only two customers. Each customer has deposited $100 in the bank. The bank has invested the combined deposits ($200) in a long term investment. The bank can liquidate the investment now for $120, but if it waits another period, it can liquidate the investment for $250. Each customer must choose (simultaneously) whether to leave her deposit in the bank. If both leave their money in the bank, then they will each receive $125 next period. If both withdraw now, then they get back only the $60 each that the bank can recover now. If one withdraws and the other stays in, then again the bank must liquidate the investment early, but the withdrawer gets $100 and the other gets only the remaining $20. Suppose that the customers do not care when they get their money, but rather only care about the amount of money that they will receive.

27- Study domination a)- To withdraw is a dominant strategy for both of the customers b)- To leave the money in the bank is a dominant strategy for both of the customers. c)- To leave the money in the bank is a dominant strategy for only one of the customers.. d)- To withdraw the money is a dominant strategy for only one of the customers. e)- There are no dominant or dominated strategies. 28- Equilibrium a)- This game does have an equilibrium in dominated strategies. b)- This game does not have a Nash equilibrium. c)- The only Nash equilibrium is for both customers to withdraw the money. d)- The only Nash equilibrium is for both customers to leave the money in the bank. e)- There is more than one Nash equilibria.

Ice-cream Sellers Two Ice-cream sellers have to decide where to locate on a linear beach. Both sellers charge the same price. Consumers will buy from the nearest seller.