MATH4321 – Game Theory
Mid-term Test – Spring 2016
Time allowed: 90 minutes
Course instructor: Professor Yue Kuen KWOK
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1. Consider the zero-sum game with game matrix
(
).
Let X = (x, 1 x) be the probability vector of the mixed strategy of the Row player.
(a) Plot the expected payoff E(X, 1), E(X, 2), E(X, 3) and E(X, 4) as straight lines over
the range:
Find the highest point of the lower envelope.
[3 points]
(b) Find the value of the game and the optimal mixed strategy by the Row player.
Explain why the fourth column would not be used by the Column player.
[3 points]
(c) Find the general form of the continuum of optimal mixed strategies of the Column
player.
[4 points]
Hint: Column 3 =
(Column 1 + Column 2).
2. In a symmetric zero-sum game where the two players can switch roles, so the game
matrix satisfies:
.
(a) Show that the value of a symmetric game is always zero.
[3 points]
) is a saddle point, then (
) is a saddle point. In addition,
(b) Show that if (
(
) and (
) are also saddle points.
[4 points]
3. Consider the bimatrix game that models the game of chicken.
Two cars are headed toward each other at a high speed. Each player has two options: turn
off, or continue straight ahead. The payoff matrix is listed below:
Turn
Straight
Turn
(19, 19)
(-42, 68)
Straight (68, -42)
(-45, -45)
(a) Find the best response functions of the Row player and Column player.
[2 points]
(b) Find the two pure Nash equilibriums and the one mixed Nash equilibrium. [3 points]
(c) Verify by using the definition of mixed Nash equilibrium that the answer you found
in part (b) is indeed a Nash equilibrium. Find the expected payoff of each player.
[4 points]
(d) Describe the concept of the safety value of a player in a bimatrix game. Find the
safety value of the Row player.
[4 points]
4. A strategy profile X is said to be weakly Pareto-dominating another profile Y if all players
have higher or equal payoff under profile X, with strict inequality for at least one player.
Prove or disprove:
If a strategy profile weakly Pareto-dominates all other strategy profiles, then it must be a
Nash equilibrium.
[3 points]
5. An important theorem in mixed Nash equilibrium is the equality of payoff theorem. In a
two-player bimatrix game, Player I would obtain the same payoff from each of the pure
strategies in the mixed Nash equilibrium.
(a) Provide an intuitive argument for the above theorem.
[2 points]
(b) Prove the following mathematical result:
Let X* = (
) and Y* = (
) e a mixed Nash equilibrium. Suppose
(
)
(
)
for some k, then
[5 points]
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