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Q: game theory in microeconomics ( Answered 5 out of 5 stars,   0 Comments )
Subject: game theory in microeconomics
Category: Business and Money > Economics
Asked by: vitaminc-ga
List Price: $10.00
Posted: 08 Oct 2002 19:42 PDT
Expires: 07 Nov 2002 18:42 PST
Question ID: 74262
1)Consider the following two-person game:
                (a1,a2   b1,c2)
                (c1,b2   d1,d2)......actually all braket
where for i=1,2, ai>ci and bi>di. Compute the best response functions
for the two players. What are the Nash equilibria of this game?
2)There are 10 students in a game theory class. The teacher proposes
the following to the students. Each student has to write
simultaneously either "yes" or "no" on a slip of paper. For every
student who write "yes", the teacher will pay $1 to all other
students; and, in addition, if there are at least two "yes"-sayers her
will pay one additional dollar to all "yes"-sayers.
 a)What are the pure strategy Nash equilibria of the above game?
 b)What strategy would you adopt if you were playing this game?
Explain your    choice.

Request for Question Clarification by rbnn-ga on 08 Oct 2002 20:26 PDT
what does "actually all braket" mean?

Clarification of Question by vitaminc-ga on 08 Oct 2002 21:40 PDT
Left    Right
Up    a1,a2   b1,c2
Down  c1,b2   d1,d2

i mean there is a big bracket for all these four items.
well, you can ignore it.
just check the arrangement above(that's what i mean)

Clarification of Question by vitaminc-ga on 08 Oct 2002 21:42 PDT
it's payoff for the game theory
Subject: Re: game theory in microeconomics
Answered By: rbnn-ga on 09 Oct 2002 23:38 PDT
Rated:5 out of 5 stars
1. If player 1 chooses UP, then player 2s best response is LEFT since
   If player 1 chooses DOWN, then player 2s best response if LEFT
since b2>d2

   If player 2 chooses LEFT then player 1s best response is UP since
   If player 2 chooses RIGHT then player 1's best response is UP since
b1>d1 .

 The only Nash equilibrium for the game is therefore (UP,LEFT).

2. a)
We examine the number of students who choose yes, from 0 through 10,

First, suppose no students choose YES. Can any student improve his
payoff by changing his vote to YES?
The answer is no; even though, if the one student does this, all the
other students' payoffs increase, the
definition of Nash equilibrium is that if a player can improve his OWN
value by changing only his OWN strategy then
there is no equilibrium.

Hence, the state (also called strategy profile) in which all students
choose NO is a Nash equilibrium.

In general, suppose there are k students choosing YES and 10-k
students choosing NO. If k>=2 then each YES student gets k dollars
(k-1 from the other YES students and 1 because there are two YES
students); the NO students each get k dollars as well. Hence, the
higher k is the more money the students get, so that the only Nash
equilibrium for k>=2 is when all the students choose YES and they each
get $10.00.

The state with one YES, however, is not a NASH equilibrium, because a
NO student gets only $1, but could increase his payoff to $2 by
switching his vote.

So, the only Nash equilibria are those states in which all students
make the same vote.

(Note that the phrase in the problem "for every 
student who write "yes", the teacher will pay $1 to all other 
students" has to be read carefully - I misread it at first as implying
only the NO voters get the extra cash.

2b) This is quite complex because my *utility* function might be
different from my *game payoff*. I might want my fellow students
to get money; or I might not to. I might want the teacher to lose
money; or I might not to. I also might want to SIGNAL to the other
students, or to the teacher, what my utility function is for future

I could also use a mixed strategy (here I am only consider pure
strategy equilibria, where there is no randomization).

Still, I think the answer sought, and what I would do is: choose YES.

In no case do I ever LOSE money by choosing YES rather than NO; no
matter what the
other students do I either stay the same or gain money by choosing

Search Strategy:
For this problem I used a rather enjoyable book I'd read not long ago
called Game Theory Evolving, by Herbert Gintis, Princeton, 2000.

Request for Answer Clarification by vitaminc-ga on 11 Oct 2002 14:32 PDT
well,i figure it out that i got question wrong.(for Q1)
it should be bi<di.

and i think there is no dominant stragety here, so we have to use
mixed strategy to solve it, not the pure one.


Clarification of Answer by rbnn-ga on 11 Oct 2002 19:29 PDT
I believe that if question 1 is modified so that bi<di, instead of the
original bi>di, then (UP,LEFT) and DOWN,RIGHT are both pure-strategy
Nash equilibrium.
vitaminc-ga rated this answer:5 out of 5 stars
now i got it
thanks a lot ^^

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