2. GAME 1
When do we state that above payoff matrix is a
representation of a zero-sum game without equilibrium
point?
Please elaborate your reason.
Player II
One Two
Player
I
One a b
Two d c
3. GAME 2
What is the best strategy if you were the convict in this case? Give the
reasons.
What is the type of this game?
Sheriff
Highway Forest
Convic
t
Highw
ay 0 1
Forest 1 1-1/n
4. GAME 3
Players I and II simultaneously call out one of the numbers: “one” or
“two”.
Player I wins if the sum of the numbers is odd.
Player II wins if the sum of the numbers is even.
The amount paid to the winner by the loser is always the sum of the
numbers in dollars.
The payoff matrix is as follows.
5. GAME 3
Problems:
Is there any equilibrium point? Give your reason.
Suppose Player I play to call “one” 60% and to call “two”
40% of the time, what is the average value?
Is that the minimax value? If it isn’t please give your
suggested value.
Player II’s call
One Two
Player I’s
call
One -2 3
Two 3 -4
6. GAME 4
Player 1 maximize its payoff, while Player 2 is the
opposite. Unfortunately some payoffs are missing.
Does this game have its equilibrium point (EP)?
Explain your reasoning to determine this EP.
Player 2
I II III IV
Player
1
A 2 ? ? ?
B 3 ? ? ?
C 4 ? ? ?
D 5 6 7 8
7. GAME 5
From a set of 3 cards, numbered as 1, 2, 3, Player 1 select a card at
will.
Player 2 tries to guess this selected card.
After each guess, Player 1 signals High, Low, of Correct depending on
Player 2’s guess.
The game is over when the card is successfully guessed by Player 2.
Player 2 pays Player 1 an amount equals to the number of trials Player
2 made.
Problems:
What payoff matrix should be constructed?
What type of game is it? Please give your reason
8. SOLUTION 1
Assume there is no EP and let’s start from p11.
If a ≥ b, then b < c, as otherwise b is an EP.
Since b < c, we must have c > d, as otherwise c is an EP. Continuing
thus, we see that d < a.
In other words, if a ≥ b, then a > b < c > d < a.
By symmetry, if a ≤ b, then a < b > c < d> a.
20 points
9. SOLUTION 2
As the convict, let’s say choose “highway” at p
(p)(0)+(1-p)(1)=(p)(1)+(1-p)(1-1/n)
p =1/(n+1) and
choose “forest” at (1-p)= n/(n+1)
The expected value to escape= n/(n+1)
Two-person zero-sum game without equilibrium point
15 points
5 points
10. SOLUTION 3
Maximin=-2; minimax=3 no EP
P1 calls “one” 60% & calls “two” 40%:
If P2 calls “one”: 60%(-2)+40%(3)=0
If P2 calls “two”: 60%(3)+40%(-4)=0.2
Hence the “at least value” of P1 in this case = 0
VN Minimax value P1 calls “one” at “p”
p(-2)+(1-p)(3)=p(3)+(1-p)(-4)
p=7/12 VN Minimax=1/12
8 points
8 points
4 points
11. SOLUTION 4
Player 1 viewpoint: no matter “?”
In each row of ? > p11,p21,p31,p41 player 1 will choose p11,p21,p31,p41 as
minima and p41 as maximin value
Player 2 viewpoint:
In each row of ? < p11,p21,p31,p41 player 2 will choose p41 and may choose ?s
or p42,p43,p44 as maxima but for sure will choose p41 as minimax value
Hence the EP is p41
2 points
9 points
9 points
12. SOLUTION 5
Maximin=1;Minimax=2Two-person zero-sum game
without equilibrium point
Player 2
123 132 21/2
3
312 321
Player
1
1 1 1 2 2 3
2 2 3 1 3 2
3 3 2 2 1 1
The payoffs represent the amount Player
1 gets
15 points
5 points
