Dr. Jens Maßberg
Simon J¨ger a Institut f¨r Optimierung und Operations Research u summer semester 2015

Mathematics of Games
Exercise Session 1
Exercise Session 1 due on 23.04.2015, by 8:15am, N24-226.
Total : 20 Points
Hand-in in groups of at most 3 persons!

1. Apply the iterated strict dominance elimination to the following 2-person game. Write down for each iteration, which strategies are strictly dominated and thus get deleted.
Hint: In one step a strategy is only strictly dominated by a mixed strategy, but not by a pure one!

A1
A2
A3
A4

B1 B2 B3 B4
5,7 11,2 4,7 4,8
6,6 10,1 3,2 5,0
3,0 8,4 5,6 2,2
4,7 5,7 6,8 3,10

[8 Points]
2. Each of n players gets a unique marker color. Now each player i (i ∈ {1, . . . , n}) simultaneously chooses his position: a real number xi ∈ [0, 1]. All points in [0, 1] are colored:
Each point y gets player i’s marker color for that i with xi closest to y.
If a point y ∈ [0, 1] has the same distance to more than one player’s position xi , its color is determined at random (uniformly). If one position xi is chosen by more than one player, the points with minimum distance to xi are colored randomly (uniformly) with one of those players’ marker colors. Each player wants to color a largest possible part of the interval with his marker color.
(i) If there are two players, does a pure-strategy Nash Equilibrium exist for? If so, is it unique? Either give a pure-strategy NE (with justification) or explain why one does not exist. [4 Points]
(ii) If there are two players, which pure-strategy for player i strictly dominate xi = 0?
Which pure-strategy for player i strictly dominate xi = 1? (The pure-strategy s ∈ Si strictly dominates s ∈ Si if ui (s, s−i ) > ui (s , s−i )∀s−i ∈ j=i Sj .) [4 Point]
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(iii) If there are three players, does a pure-strategy Nash Equilibrium exist for? If so, is it unique? Either give a pure-strategy NE (with justification) or explain why one does not exist. [4 Points]
[12 Points]
3. Suppose there are n ≥ 2 firms in the Cournot oligopoly model. Let qi denote the quantity produced by firm i, and let Q = q1 + . . . + qn denote the aggregate quantity on the market. Let P denote the market-clearing price and assume that inverse demand is given by P (Q) = a − Q (assuming Q < a, else P = 0). Assume that costs are assymetric: the total cost of firm i from producing quantity qi is Ci (qi ) = ci qi . Following Cournot, suppose that the firms choose their quantities simultaneously. What is the pure-strategy
Nash Equilibrium if 0 < ci < a/n for each firm? Consider n = 2 at first and then n ≥ 3.
[4 + 8 = 12 Points]

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