Tuesday, 11/27/01
5. A and B are playing a game where A can play either Top or Bottom and B can play either Left or Right. They will choose their strategies simultaneously. The payoff matrix in this game is:
| 7,7 | 12,10 |
| 10,12 | 11,11 |
where A's payoffs are shown first. (a) Does this game have a dominant strategy equilibrium? Defend your answer carefully. (b) How many Nash equilibria does this game have? Defend your answer carefully.
The first thing you need to do is proceed systematically like Dr. Schwab did when solving these simultaneous move games:
First suppose B picks Left. What is A's best move?After doing your underlining, you should be able to identify any dominant strategies very easily. If both players have a dominant strategy, you'll have a dominant strategy equilibrium.
How do we look for Nash equilibria? First, the only candidates for NE are boxes with both underlined because those are the only situations where everyone is happy with what they are doing. So what's the difference between dominant strategy and Nash equilibria? Both have double underlines... dominant strategy equilibria are special cases of Nash equilibria.
If you're at a dominant strategy equilibrium, do you want to change your mind? Of course not, your strategy is dominant! But if you are at a Nash equilibrium, must you be playing a dominant strategy? Nope. All dominant strategy equilibria are Nash, but not all Nash are dominant strategy equilibria.
6. The Environmental Protection Agency (EPA) has established a regulation for Firm. Firm must decide whether or not to comply with this regulation; EPA must decide whether or not to inspect and thus find out whether or not Firm has complied. If Firm complies, it always earns 4; if it does not comply it earns 0 if EPA inspects and 10 if EPA does not inspect. If EPA inspects it earns -4 if Firm does not comply and 6 if Firm does comply; if EPA does not inspect it earns -10 if Firm does not comply and 10 if Firm does comply. Show that this game does not have a Nash equilibrium.
Here's what you might want to do - try setting this one up as a table like a Prisoner's Dilemma game or the game in Problem 5. You can do this one in exactly the same way as Problem 5.
7. Consider two firms that can either advertise or not advertise. If they both advertise they each earn a profit of 5. If one advertises and the other does not, then the firm that advertises earns 10 and the other firm earns 3. If neither advertises, then each earns an amount 8. Firms make their decisions on advertising simultaneously. Show that both firms will advertise.
Again, it will help to set up the table and work out the best moves for each player by doing the underlining. Once you do that, everything is very obvious.
8. Rich McDuck is the owner of the only cable TV company in Disneyland. He earns an annual profit of 100. His nephew Donald is planning to start his own company in order to provide more options to the population of Disneyland. This game is played sequentially. First, Donald must decide whether or not to enter the market; then Rich must decide whether or not to drop the price to 0. If Donald enters the market and Rich does not drop the price to 0, both competitors earn an annual profit of 50; if Donald does not enter the market and Rich drops the price to 0, both competitors earn an annual profit of 0; if Donald enters the market and Rich drops the price to 0, Rich earns an annual profit of 0 and Donald has an annual loss of 50; if Donald does not enter the market and Rich does not drop the price to 0, Rich continues to earn an annual profit of 100 and Donald earn 0. Use backward induction to find each player’s best strategy.
This one is best set up as a tree. How can you tell when to set something up like a tree and when to set it up as a table with cells? A good rule is that simultaneous move games are best done as tables and sequential move games are best done as trees. Here, we are told that Donald moves before Rich. Sequential move, so tree is better.