1. Stag hunt. Two guys go hunting for food, and each can choose to hunt stag or hunt hares. Only if both players hunt stag will they succeed; if only one player hunts stag, he will fail. On the other hand, the players are able to successfully hunt hares on their own-but hares are much smaller, and so have a smaller payoff than catching (and sharing) a stag. The payoff matrix therefore looks like the following: if both go for Stag - the payoff of each is 8; if Player 1 goes for Stag, Player 2 for Hare - payoffs are 0 and 2, if Player 1 goes for Hare, Player 2 for Stag - payoffs are 2 and 0, if both go for Hares - 2 and 2.
Draw the game as a table. What are the Nash equilibria of this game? How does this game differ from a prisoner's dilemma, and how can participants achieve the optimal (both hunt Stag) outcome?
2. Chicken. In the movies about the 60th in USA the following scene sometimes occurs. Two young lads position their cars on the narrow road facing each other and remove the brakes. Then they start racing towards each other. The crowd of the equally young friends cheers. At some point the players simultaneously either have to keep on going or swerve into the ditch. Model and draw this is a game of simultaneous actions. The payoffs that you choose should reflect the following rankings of outcomes. If both lads go straight they both die which is the worst for each. If one of the players swerves (no matter what the other does) the outcome for him is public humiliation which is bad but not as bad as death. If player 1 goes straight and his opponent swerves this is the best outcome for player 1. If the roles of the players are reversed the payoffs are reversed. What are the Nash equilibria of this game?
3. Another Chicken. Suppose before racing player 1 can publicly remove his steering wheel and throw it away. If he does not remove the wheel the game continues as the simultaneous actions games of chicken above. You may use the same payoffs as you designed for that game. If player 1 removes the wheel he can only go straight after that. If his opponent swerves player 1 gets even higher payoff than on the other branch. Model this game as a combination of the game tree and the table. What is the equilibrium of this game? Should player 1 remove the wheel?
4. Entry game. Exxon currently owns the only petrol station operating in Wagga-Dubba and collects the payoff of 10 if it remains like that. Mobil may stay out of this lucrative market and collect the payoff of 0. Or it may enter the market in which case Exxon may accommodate the entrant or start a price war. If Mobil enters and Exxon accommodates the payoffs are 5 for each. If Mobil enters and Exxon starts a price war the payoffs are negative 1 for each. Suppose the decisions whether to enter or not and whether to rtart the war or not are made simultaneously.
a. Draw the game as a table, fill in the payoffs. What are the Nash equilibria of the game?
b. Now suppose Mobil makes their move first and Exxon may respond with the war after observing Mobil's move. Draw this game as a tree and write the payoffs. What are the backward induction equilibria of the game?
5. R&D race. Two pharmaceutical firms are considering a research program for a new drug. The cost of the program is C to each firm; the potential profits from selling the drug on the market are Π. If each firm pays the cost and engages in R&D, each stands a 50% chance of winning R&D race and obtaining the profits Π. (In this case each firm has Π/2 - C as a payoff.) If only one firm pays the cost, it will get all the profits for sure, while the other firm loses nothing.
a) Draw the game tree, assuming that firm 1 moves first.
b) What is the backward induction Equilibrium of this game when Π=30, C=10? When Π=30, C=20?