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.  The Game of Product Choice

27. Three automobile firms, A, B, and C, intend to produce one
new model each. In each case the firm has choose an economy (E),
medium priced (M), or luxury (L) model. The profit that each firm
expects to make will depend on the choices made by all three. All
annual profit in millions of dollars ae shown in the tables below.
Since each firm has its own established reputation and dealer
network, their profit expectations are not identical, but each
knows the profit expectations of the others for every possible
combination of models. Secrecy is not possible and each will know
the plans of the others early enough for each to change its mind.
Each adopts a game theoretic approach to the problem of deciding
which model it should produce.

A ”decision set” will refer to one set of possible actions or
“strategies” by each player. For example A=E, B=M, C=L, or written
as a triplet (E,M,L) in order of A, then B, then C.
An “outcome” refers to a possible set of 3 payoffs, one for each
player in turn. For example, (16, 17, 27) is the extreme resulting
from (M, M, M).

1.   How many possible decision sets are there?
2.   What is C’s preferred outcome?
3.   Does any firm in this game have a dominant strategy?
If not, explain why not, if yes, identify and explain why it is
dominant.
4.   If one player’s strategy can be identified, we can
present a familiar 2-player payoff matrix for the remaining players
showing a set of “reduced outcomes” identifying only the payoffs of
these 2 players. Show such a payoff matrix with reduced outcomes in
interior cells.
5.   Identify any Nash equilibria. If none exist explain
why. If any does exist, identify each such and explain why it is a
Nash equilibrium.
6.   Players act simultaneously and choose only once, but
do so knowing their rivals possible choices. What decision set will
they arrived at? Explain. (Which models would you expect each of
the firms to produce and why?)
7.   Suppose instead players act sequentially, and
(because they have industrial spies in their rival firms and can
revise their strategies) can make a different choice after their
rivals act.
If A chooses E, what decision set do they all arrive at, and what
is the full outcome (i.e. payoffs for each of the 3)
8.   With the same situation as in (7) but B chooses 1st
and chooses E, what decision set and full payoff results?

Show transcribed image text Firm As expected profit If B produces If C produces E M L E M L E M L If A produces E 15 18 20 18 22 23 20 23 25 M 26 17 23 17 16 19 22 19 23 L 23 21 16 21 20 18 15 18 19 Firm B's expected profit If A produces E M L E M L E M L If C produces If B produces E 13 17 19 18 21 22 20 22 24 M 26 18 24 18 17 19 24 19 24 L 24 26 17 25 23 21 16 21 19 Firm Cs expected profit If A produces M L E M L E M L If B produces If C produces E 24 27 29 26 31 32 28 33 35 M 23 24 28 24 27 28 27 29 30 L 23 26 25 25 25 23 20 28 24

Firm As expected profit If B produces If C produces E M L E M L E M L If A produces E 15 18 20 18 22 23 20 23 25 M 26 17 23 17 16 19 22 19 23 L 23 21 16 21 20 18 15 18 19 Firm B's expected profit If A produces E M L E M L E M L If C produces If B produces E 13 17 19 18 21 22 20 22 24 M 26 18 24 18 17 19 24 19 24 L 24 26 17 25 23 21 16 21 19 Firm Cs expected profit If A produces M L E M L E M L If B produces If C produces E 24 27 29 26 31 32 28 33 35 M 23 24 28 24 27 28 27 29 30 L 23 26 25 25 25 23 20 28 24