econ game theory questions
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Lecture9.pptxMixed Strategy Nash Equilibrium
Chapter 11
Yuhta Ishii
Economics 402
Introduction
So far, we have focused on Nash equilibria in pure strategies.
Each player plays a strategy with probability one.
Remember that some normal form games don’t have any Nash equilibria in pure strategies.
E.g. matching pennies.
In contrast, there will always be a Nash equilibrium in mixed strategies.
Matching Pennies
There is no NE in pure strategies
Game is a zero-sum game where whatever player 1 wins, player 2 loses.
The sum of payoffs in each cell is 0.
In many such games, typically there is no NE in pure strategies.
Once I know what the other player will play (which is the case in NE), I never have an incentive to stay the ``loser’’.
Matching Pennies
In such games, randomization overcomes this problem.
There will always exist some NE in mixed strategies (could be in pure strategies) in every finite normal form game.
In zero-sum games, this is natural.
For example, in matching pennies, you wouldn’t want to be predictable in your play.
So you randomize.
Matching Pennies
Matching Pennies
Matching Pennies
Definition
Property 1: Relationship to (pure strategy) NE
As you might guess, any pure strategy NE (from previous lectures) is still a mixed strategy NE.
A pure strategy NE is just a mixed strategy NE in which each player plays a mixed strategy that assigns probability one to exactly one strategy.
So mixed strategy NE are more general.
Property 2: Best Response Condition
Property 2: Best Response Condition
Property 3: Rationalizability
General Procedure for finding mixed strategy NE
Step 1: Find the set of rationalizable pure strategies by performing iterated elimination of strictly dominated strategies.
Step 2: In the reduced game where each player only plays rationalizable strategies, write equations for each players’ indifference conditions.
Step 3: Solve these equations to determine equilibrium randomization probabilities.
Nash’s Theorem
The important discovery of Nash was the following theorem.
Nash’s theorem: Every finite normal form game (finite number of players and finite number of strategies for each player) has at least one Nash equilibrium in either pure strategies or mixed strategies.
Example: Lobbying Game
Two firms simultaneously and independently decide whether to lobby (L) or not (N) the government in hopes of trying to generate favorable legislation.
Example 2: Tennis serves
Example 2: Tennis Serves
Example 3:
Compute all of the mixed strategy Nash equilibria of the above game.
