econ game theory questions
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Lecture6.pptxRationalizability and Iterated Dominance
Economics 402 (Spring 2020)
Yuhta Ishii
Pennsylvania State University
Game
Guess a whole number between 1 and 100.
The person whose guess is closest to 2/3 of the average in the classroom wins 10 dollars.
Write your name and your guess on a sheet and hand it in.
We will discuss this later in the class.
Introduction
We previously talked about the idea of strict dominance.
Rational behavior involves at least avoiding strictly dominated strategies.
But sophisticated players reason beyond this.
Reading: Ch. 7
Example 1:
Suppose player 1 is rational.
Can we say which strategy he will choose?
No, for each of his strategies there is a belief that makes it optimal.
Example 1:
What about player 2?
Can we say which strategy he will choose?
We don’t know for sure, but now we know that he will choose only between and .
Example 1:
We as an analyst came to the conclusion that 2 will not play or .
But since player 1 also knows that 2 is rational, and 2 has the payoffs in the above normal form, he should also conclude that will never play .
Example 1:
We as an analyst came to the conclusion that 2 will not play .
But since player 1 also knows that 2 is rational, and 2 has the payoffs in the above normal form, he should also conclude that will never play .
Example 1:
We as an analyst came to the conclusion that 2 will not play .
But since player 1 also knows that 2 is rational, and 2 has the payoffs in the above normal form, he should also conclude that will never play .
Example 1:
But now if 1 knows that 2 will never play , then player 1 should never play .
is strictly dominated in this smaller game.
So if player 1 knows that player 2 is rational, and player 1 is rational, then player 1 will never play .
Example 1:
Player 2 will then reason similarly: Player 2 knows
that player 1 knows that player 2 is rational.
that player 1 is rational.
So player 2 will realize that player 1 will never play .
Example 1:
Given this, player 2 will realize that he should play .
The only surviving strategy for player 1 is and the surviving strategy for player 2 is .
This general procedure is what is called the iterated elimination/deletion of strictly dominated strategies.
Iterated Elimination/Deletion of Strictly Dominated Strategies
Step 1: Delete all strictly dominated strategies for each player. Call the set of strategies that remain for player .
Step 2: Consider the reduced game, where each player only plays strategies in . Eliminate all strictly dominated strategies for each player in this reduced game. Call the remaining strategies for player , .
Repeat this process until no more strategies can be eliminated.
Call the surviving set of strategies for player , .
This set will be called the set of rationalizable strategies for player i.
More on this name in the following slide.
Rationalizability
An alternative way to describe this iteration procedure is what is called rationalizability.
Remember last time, we saw that:
a strategy is in if and only if it is also in .
In words, a strategy is not strictly dominated if and only if it is a best response for some belief.
So the iterated elimination of strictly dominated strategies is the same as the following procedure:
Rationalizability Procedure
Step 1: Keep all strategies that are best responses for some belief. Call the set of strategies that are kept for player .
Step 2: Consider the reduced game, where each player only plays strategies in . Keep for each player the set of strategies that are best responses for some belief in the reduced game. Call the remaining strategies for player , .
Repeat this process until no more strategies can be eliminated.
Call the surviving set of strategies for player , .
This set will be called the set of rationalizable strategies for player i.
These are called rationalizable, because these are the strategies that can be rationalized as best responses with some belief over opponents’ rationalizable strategies.
Rationalizability and Iterated Elimination of Strictly Dominated Strategies
As I said before, these two are exactly the same procedure, because of the fact that .
So in order to find the rationalizable strategies, you can just perform the iterated elimination of strictly dominated strategies.
Rationalizable Strategy Profiles
A strategy profile is rationalizable if each player plays a rationalizable strategy.
We denote this set as .
For example, the rationalizable strategy profile in the above game was: (B,Z).
Some Examples
Example 1
What are the rationalizable strategies for each player:
Example
What are the rationalizable strategies for each player:
Example 2
What are the rationalizable strategies for each player:
Example 2
What are the rationalizable strategies for each player:
Example 2
What are the rationalizable strategy profiles?
Example 2
What are the rationalizable strategy profiles?
Example 3:
What are the rationalizable strategy profiles of this game?
All strategy profiles
{(A,X), (A,Y), (B,X),(B,Y)}
{(A,X), (A,Y)}
{(A,X), (B,Y)}
Example 3:
What are the rationalizable strategy profiles of this game?
All strategy profiles
{(A,X), (A,Y), (B,X),(B,Y)}
{(A,X), (A,Y)}
{(A,X), (B,Y)}
What are the rationalizable strategies for each player?
Example: Eisner-Katzenberg game
Battle of the Sexes
What are the rationalizable strategies for each player?
,
,
Battle of the Sexes
What are the rationalizable strategies for each player?
,
,
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Common Knowledge
Introduction
The idea of common knowledge is at the heart of iterated elimination/deletion of strictly dominated strategies (and thus also rationalizability).
Some fact is common knowledge if
each player knows that fact;
each player knows that all players know the fact;
each player knows that all players know that all players know the fact;
Etc.
Common Knowledge in Iterated Elimination
What we are assuming is that both payoffs and rationality are common knowledge:
For example, we were able to delete in this game because we assumed:
Player 2 knows both
that player 1 knows that player 2 is rational.
that player 1 is rational.
Common Knowledge Assumptions in Game Theory
Common knowledge assumption allow us to sharpen our predictions:
We implicitly use it in iterated elimination of strictly dominated strategies/rationalizability.
However, this is oftentimes a simplifying assumption:
Many economic contexts don’t feature common knowledge of rationality or of payoffs.
We assume it for simplicity for now.
Much of modern game theory explores how to relax this unrealistic assumption.
Hat Game
To illustrate the importance of common knowledge assumptions, let’s play a game.
3 players.
Each player is given a hat to wear, which is either red or blue.
Each player cannot see the color of his own hat, but can see the color of the other players’ hats.
Hat game: Analysis
Hat Game
There were three red hats: I denote this as (R,R,R)
I first asked each of you, ``Do you know the color of your own hat?’’
Each one answered no.
Now when I announced that ``at least one of you is wearing a red hat,’’ player 3 is able to conclude the color of his/her hat to be red.
Why?
Hat Game
What’s going on? Consider the situation after I made the announcement.
What is player 3 thinking?
He sees two red hats and so he knows that either the color configuration is (R,R,R) or (R,R,B).
Hat Game
Suppose hypothetically that the true color configuration were (R,R,B).
Let’s replay this game.
Player 1 would still be uncertain about his/her hat.
Claim: Player 2 would be able to tell that his/her hat is red.
Player 2 is uncertain in this situation between (R,R,B) and (R,B,B)
Player 2 knows that because player 1 was uncertain, his hat must be red.
Here player 2 knows that player 1 knows that there is at least one red hat.
Hat Game
Given this, player 3 once (s)he sees that player 2 is uncertain, should conclude that the true configuration is (R,R,R)!
What is going on?
Common Knowledge
The idea of common knowledge is much stronger than just the idea of knowledge.
Notice that when I revealed to the players that ``there is at least one person with a red hat,’’ I revealed some thing that everyone already knew.
Yet, it had an effect on what player 3 was able to conclude!
Before the announcement, every player was uncertain about the color of his/her hat.
Common Knowledge
What happened?
The revelation that ``at least one person is wearing a red hat’’ made this fact common knowledge.
Everyone already knew this fact.
But now everyone knows that everyone knows it.
Everyone knows that everyone knows that everyone knows it, etc.
Common Knowledge
More precisely, consider the following statement:
“Player 2 knows that player 1 knows that there is at least one red hat.”
This fact is uncertain to player 3 before the announcement.
The announcement made this statement certain to player 3.
This example illustrates the importance of not just knowledge but common knowledge.
Back to the ``Choose 1-100 game.”
Why didn’t the prediction of the game coincide with what actually happens in the classroom?
One reason why iterated deletion of strictly dominated strategies may not be a good prediction is that typically in many real world scenarios, rationality and the normal form are not common knowledge.
Common Knowledge
