HW 2 1. Consider a Chicken game where two players are driving ...

HW 2
1. Consider a Chicken game where two players are driving toward each other. Each player
chooses between going straight, swerving to the left, and swerving to the right. The choices
are made simultaneously. If one goes straight while the other swerves right or left, the one
who goes straight gets a payoff of 3 while the other gets –1. If each swerves to his left, each
gets 0. If each swerves to his right, again each gets 0. If both go straight, or if one swerves to
his left while the other swerves to his right, then the cars crash and each gets the payoff –6.
a) Show the payoff matrix for this game.
b) Find all Nash equilibria in pure strategies.
2. Two start ups are competing for leadership in a software market. The leader wins, and the
other loses. Each firm can invest some x ∈ [0.001, 1] unit for research and development by
paying cost of
. If a firm invests x units and the other firm invests y units, the former wins
with probability
. Therefore, the payoff of the former start up will be − . All these are
common knowledge.
a) Compute all pure strategy Nash equilibria.
3. (A synergistic relationship) Two individuals are involved in a synergistic relationship. If both
individuals devote more effort to the relationship, they are both better off. For any given
effort of individual j, the return to individual i’s effort first increases, then decreases.
Specifically, an effort level is a nonnegative number, and individual i’s preferences (for i = 1,
2) are represented by the payoff function ai(c + aj − ai), where ai is i’s effort level, aj is the
other individual’s effort level, and c > 0 is a constant.
a) Model the above as a strategic game.
b) Write the best response correspondence of player i.
c) Solve for Nash Equilibrium.
4. (A joint project) Two people are engaged in a joint project. If each person i puts in the effort
xi, a nonnegative number equal to at most 1, which costs her c(xi), the outcome of the project
is worth f (x1, x2). The worth of the project is split equally between the two people, regardless
of their effort levels. Formulate this situation as a strategic game. Find the Nash equilibria of
the game when (a) f (x1, x2) = 3x1x2 and c(xi) = for i = 1, 2, and (b) f (x1, x2) = 4x1x2 and
c(xi) = xi for i = 1, 2. In each case, is there a pair of effort levels that yields both players
higher payoffs than the Nash equilibrium effort levels?
5. (Cournot’s duopoly game with linear inverse demand and a fixed cost) Find the Nash
equilibria of Cournot’s game when there are two firms, the inverse demand function is given
by
− ≤
=
0 >
where Q = q1 + q2
and the cost function of each firm i ∈ {1, 2} is given by
= + > 0
0 = 0

where c ≥ 0, f > 0, and c < α. (Note that the fixed cost f affects only the firm’s decision of
whether or not to operate; it does not affect the output a firm wishes to produce if it wishes to
operate.)
6. Definition (Strict Nash equilibrium) An action profile a* is a strict Nash equilibrium if for
every player i we have ui(a*) > ui(ai, a-i*) for every action ai ≠ ai* of player i.
a) Prove that a strict Nash Equilibrium is a Nash Equilibrium.
b) Consider the following game:
L R
T 2, 1 1, 1
M 1, 1 2, 0
B 2, 0 1, 1
Find the Nash Equilibria of the game above. How many of them are strict?
c) Consider the following figure: Bi(aj) stands for the best response function of player i, the
shaded area of player 1’s best response function indicates that for a2 between and ,
player 1 has a range of best responses. For example, all actions of player 1 from ∗∗ to
∗∗∗ are best responses to the action ∗∗∗ of player 2.
Find all the Nash Equilibria and the strict Nash equilibria of the game whose best
response functions are given in the figure.