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
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
The only Nash equilibrium for the game is therefore (UP,LEFT).
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
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
For this problem I used a rather enjoyable book I'd read not long ago
called Game Theory Evolving, by Herbert Gintis, Princeton, 2000.