Exploring dominated strategies in HR

Problem 1 [16pt]
This problem is from 2020 exam.
Consider the following simultaneous-move game between players A and B
A B l r
U x, 1 −1, 0
D 0, y z, 1
where x ∈ R, y ∈ R, z ∈ R are some parameters.
(a) Find conditions on x, y and z so that the strategy profile (U, l) is a Nash equilibrium.
(b) Find conditions on x, y and z so that strategy U is strictly dominated.
(c) Find conditions on x, y and z so that, for some β ∈ (0, 1), a proper mixed strategy
(β, 1 − β) of player B (i.e., B plays l with probability β and r with probability (1 − β))
is the best response to some strategy of player A (formally: there exists β ∈ (0, 1) and a
mixed strategy of player A such that (β, 1 − β) is the best response to that strategy).
(d) [5pt] Give an example of values x, y and z and a mixed strategy (α, 1 − α) of player
1 such that player B’s best response is to play l and r with equal probabilities.
Problem 2 [30pt]
In this problem, we will continue to explore dominated strategies in Cournot problem
(problem 3 from tutorial 7).
Suppose that there are two firms, 1 and 2, with strategies qi ∈ [0, 1], i ∈ {1, 2}, and
payoff function πi = (1 − (q1 + q2))qi
. You may have seen in other subjects that the Nash
equilibrium of this game is (q1, q2) = (1/3, 1/3). In tutorials, we have done only three
steps of elimination of strictly dominated strategies and arrived to the set of strategies
[5/16, 3/8], but if we continue in the same fashion, the set will shrink to just 1/3 (after
infinitely many steps).
In the definition of strictly dominated strategy, strategy qi
is dominated if there is a
strategy ˆqi which gives a higher payoff against any strategy of the opponent.
Suppose ˆqi = 0.3 and you want to show that some strategy qi
is dominated using ˆqi
.
For example, we have shown that strategies qi > 1/2 are dominated by ˆqi = 1/2.
(a) Show that ˆqi = 0.3 cannot be used to show that qi = 0.6 is dominated. (The key
here is to understand what, mathematically, you need to obtain; the equation itself is
simple.)
(b) What strategies can be shown to be dominated using ˆq1 = 0.3
Consider now a game between three players, where

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