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

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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