2. Nash Equilibrium
The Nash Equilibrium is the solution to a game in which two or
more players have a strategy, and with each participant
considering an opponent’s choice, he has no incentive, nothing
to gain, by switching his strategy.
In the Nash Equilibrium, each player's strategy is optimal when
considering the decisions of other players. Every player wins
because everyone gets the outcome they desire. To quickly test
if the Nash equilibrium exists, reveal each player's strategy to
the other players. If no one changes his strategy, then the Nash
Equilibrium is proven.
3. Example
Sam
Tom
A B
A 1, 1 1, -1
B -1, 1 0, 0
Imagine a game between Tom and Sam. In this simple game, both players can
choose strategy A, to receive $1, or strategy B, to lose $1. Logically, both
players choose strategy A and receive a payoff of $1. If you revealed Sam's
strategy to Tom and vice versa, you see that no player deviates from the
original choice. Knowing the other player's move means little and doesn't
change either player's behavior. The result A, A represents a Nash
Equilibrium.
4. Mixed Strategies
Idea: use a prob. distribution that cannot be exploited by other player
– payoff should be equal independent of the choice of strategy of
other player
– guarantees minimum gain (maximum loss)
C D
A 2 0
B -5 3
Player 1
Payoff to P1 when playing A = x(2) + (1-x)(0) = 2x
Payoff to P1 when playing B = x(-5) + (1-
x)(3) = 3 – 8x
2x = 3 – 8x, thus x = 3/10
How should Player 2 play?
x (1-x)
Player 2
5. Mixed Strategies
C D
A 2 0
B -5 3
Player 1
Player 2
Player 2 mixed strategy
– 3/10 C , 7/10 D
– maximizes its loss independent of P1 choices
Player 1 has same reasoning
Payoff to P2 when playing C = x(-2) + (1-x)(5) = 5 - 7x
Payoff to P2 when playing D = x(0)
+ (1-x)(-3) = -3 + 3x
5 – 7x = -3 + 3x, thus x = 8/10
Payoff to P1 = 6/10
x
(1-x)
6. In a matrix game 𝐴, each entry is an ordered pair of real numbers
(𝑥 , 𝑦) indicating payoff s of 𝑥 to Rose and 𝑦 to Colin.
Considering only the first coordinates of these entries (i.e. just
the payoffs to Rose) gives a matrix 𝑅 that we call Rose’s payoff
matrix . Similarly, the second coordinates form a matrix 𝐶, called
Colin’s payoff matrix
Example
8. Best Respond
Suppose that p is a mixed strategy for Rose and the 𝑖 th entry of p is 0.
In this case, Rose will never select row 𝑖when she plays p.
If the 𝑖 th entry of p is positive, then she will select row 𝑖
some of the time. We say that p calls on row 𝑖 if 𝑝 𝑖 > 0
we say that Colin’s mixed strategy q calls on column 𝑗 if q 𝑗 > 0
Example
We will suppose throughout this example that Colin will always play
the mixed strategy q
Rose’s
expected payoff for each possible pure strategy she could select:
9. Best Respond
Now suppose that Rose is going to play the mixed strategy
p = [ p1 , p2 , p3 ]
Rose’s expected payoff when she plays p is
We must have 𝑝 1 , 𝑝 2 , 𝑝 3 ≥ 0 and 𝑝 1 + 𝑝 2 + 𝑝 3 = 1. It follows that
12. They give her an expected payoff of 80 / 31 . More generally, her best
responses to q will be all mixed strategies that only call on rows two,
three, and four (i.e. mixed strategies of the form [0 𝑝2 𝑝3 𝑝4 0]). In
particular, her strategy p is indeed a best response to q.
13. Theorem
In every 2 × 2 matrix game, one of the following holds:
Iterated removal of dominated strategies reduces the matrix to
1 × 1. This row and column form a pure Nash equilibrium.
Rose and Colin both have mixed strategies that equate the other
player’s results and these form a Nash equilibrium.
14. p = [𝑝 1 −𝑝 ] for Rose that equates Colin’s results. Equating Colin’s first
and second column payoff s gives us
𝑝 (−2) + (1 − 𝑝 )3 = 𝑝 (2) + (1 − 𝑝 )(−1).
Thus 8𝑝 =4 and 𝑝= 1 / 2 , so p = [ 1/2 1/2 ]. Now we will look for a
strategy q= or Colin that equates Rose’s results. Equating Rose’s first
and second row payoff s yields
q (1) + (1-q)0= q(-1)+ (1-q)3
This gives us 5𝑞 =3 so 𝑞= 3 / 5 and q = Since the strategies p and q
both equate the other player’s payoff s, they are best responses to one
an-other. Therefore, p = [ 1/2 1/2 ] and q = form a Nash equilibrium.
15. Hawk vs. Dove
-5 10
0 4
-5, -5 10, 0
0, 10 4, 4
The expected payoff when two hawks engage is −5 for each.
When a hawk and a dove compete, the hawk takes the resource for 10
and the dove gets 0.
One of the doves will eventually give up and the other will get the +10
resource, so the expected payoff for each dove will be 4.
The probability that 𝐼 will face a hawk is 𝑝 and the probability 𝐼 will
encounter a dove is 1 − 𝑝
p P-1
I
16. Theorem
Every symmetric matrix game has a symmetric Nash equilibrium.
It is natural to suspect that d = will be a symmetric Nash equilibrium.
To check this, let 𝑅 be the payoff matrix for the row player and
consider the vector 𝑅d = Every pure strategy is a best response to
d, so Proposition 8.4 implies that d ⊤ is a best response to d. So, d is a
symmetric Nash equilibrium.
17. Monopoly
Start by considering a monopoly just one company producing
the commodity. This company makes only one decision: the
quantity to produce. Given this quantity, the demand curve
sets the price.
Cost=cQ.
Revebue=PQ=(a-bQ)Q.
The profit 𝑈 is equal to revenue minus cost, so
U = ( a – bQ)Q – cQ=-bQ^2+(a-c)Q.
Find the quantity that maximizes profit
18. To maximize profit, the company should produce exactly
units. With this level of production, yearly profit will be
