Game study is the study of strategic interaction where one player’s decision depends on what the other player does. What the opponent does also depends upon what he thinks the first player will do.
Examples of Game Theory
- Both players have a dominant strategy.
A DOMINANT strategy occurs when there is an optimal choice of strategy for each player no matter what the other does.
UP 8,3 5,4
P1 DOWN 7,5 2,6
- If P2 chooses left P!1will choose UP
- If p2 chooses right P1 will choose UP
- Therefore UP is a dominant strategy for P1
P2 will always choose right no matter what P1 does
The unique equilibrium is (up, right). This is despite the fact that (down, left) is pareto superior.
- One player has a dominant strategy
Push lever wait for swill
Push lever 8,-2 1,7
Wait for swill 10,-2 0,0
- piglet will always wait
- Pig will have to push
There are many games which don’t have a dominant strategy.
Definition: A Nash equilibrium occurs when the payoff to player one is the best given the other’s choice.
And player’s 2 choice is the best given the other’s choice.
UP 5,4 3,10
DOWN 9,2 0,1
If P1 goes UP, P2 prefers right since 10>4. But if P1 goes down then P2 prefers left since 2>1. If P2 goes left then P1 goes down since 9>5. If P2 goes right then P1 goes UP since 3>0
Nash Equilibrium and the Prisoners Dilemma
There are 2 outcomes which are stable (UP,RIGHT) and (DOWN, LEFT) which are “stable”: neither player would wish to change his action given the action of the other player. This is a NASH equilibrium
Confess -3,-3 0,-6
Deny -6,0 -1,-1
Repeated Games and Game Theory
If games are repeated then there is the possibility of punishing people for cheating, this will provide an incentive for sticking to the pareto optimal approach.
However if they are repeated a finite number of times then there will be an incentive to cheat. If the game is played 10 times then the player will defect on the 10th round so why cooperate. So therefore you may as well defect on round 9 and so round 8 as well
If it is played an infinite number of times then it will be different. The best strategy then is to play tit for tat. If a player defects in one round you retaliate in the next round. In other words you do what ever your opponent does and this is an incentive to enforce the cartel.
Game Theory: A game of entry deterrence
If a new firm enters the market then the payoff will depend on whether the incumbent fights or accepts. If the incumbent fights they both get 0. If it does not fight then the incumbent gets 1 and the entrant gets 2. Therefore the equilibrium is for the new firm to enter and the incumbent to accept.
However, if the incumbent can give a credible threat that he will fight then he may be able to persuade the entrant to stay out. He could do this by investing in extra capacity, which would give him a bigger payoff in a price war. This would deter entry. So although the monopolist would never use this he would prevent entry.