MATH20270 Game Theory - Problem Sheet 1. Zero-Sum Pure ...

MATH20270 Game Theory - Problem Sheet 1.
(Zero-Sum Pure Strategy Games)
1. Here are some game tables. Identify any equilbrium strategies and values.
(a)
Ann \ Bill I II III IV
A 2 -15 -4 -100
B 6 6 5 9
C 5 -7 -1 20
D 2 -19 4 12
(c)
↓ Ann \ Bill→ I II III IV
A 4 -15 -4 -100
B 5 6 5 9
C 0 -7 -1 20
D -2 -19 4 12
(b)
Ann ↓ \ Bill → Heads Tails
Heads 1 -1
Tails -1 1
(d)
↓ Ann \ Bill→ I II III
A 10 2 1
B 19 8 6
C 19 7 -2
2. Show that if a game has more than one combination of equilibrium strategies then the
corresponding game values are equal. (In other words, prove the lemma stating that all
equilibrium values of a matrix are the same.)
3. In the game,
↓ Ann \ Bill→ I II III
A 1 2 1
B 6 3 8
C 4 1 9
if dominated strategies are successively removed from the game table an equilbrium point is
reached. Give the sequence of strategy removals that leads to this equilibrium point.
4. Prove that a 2 × 2 zero-sum game with equilibrium point must have a dominated strategy.
5. Give an example of a two player zero-sum game with equilibrium point but no dominated
strategies.
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6. Simplify the following games by removing any dominated strategies:
(a)
↓ Ann \ Bill→ I II III
A 1 -2 -5
B -1 1 -3
C 0 2 4
(b)
↓ Ann \ Bill→ I II III
A 4 -3 -1
B 0 -1 -1
C 0 -1 -2
(c)
↓ Ann \ Bill→ I II III IV V
A 1 -2 -4 -3 4
B -1 2 0 -3 -4
C 0 3 1 -1 0
7. For this exercise, you will need to go to the course web page mathsci.ucd.ie/modules/math20270
and click on the Random Game Generator. Enter your student number and generate a 3 × 3
zero-sum game. Write down the reference number for the game table produced when answering
the following:
(a)
i. Calculate the maximin and minimax of the game.
ii. Does the game have an equilibrium point?
iii. Simplify the game as much as possible by removing any dominated strategies.
(b) Repeat, this time with a generated 5 × 5 game.
8. Show that, in any two player zero-sum game, the maximin value is always less than or equal
to the minimax value.
9. Prove that a two player zero-sum game has an equilibrium point (i.e. an entry which is
smallest in its row and largest in its column) if, and only if, the maximin and minimax
values are equal.
10. Write down a game table representing the well-known game of “Rock-Paper-Scissors”. Calculate
the maximin and minimax values. Does this game have an equilibrium point?
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