# 博弈论代写 | GAME THEORY IN THE SOCIAL SCIENCES Fall 2020 Problem Set 4

GAME THEORY IN THE SOCIAL SCIENCES
Fall 2020
Problem Set 4

1. Equilibria: Consider a two player game in which player 1 can choose
 or . The game ends if he chooses  while it continues to player 2 if
he chooses . Player 2 can then choose  or , with the game ending
after  and continuing again with player 1 after . Player 1 then can
choose  or , and the game ends after each of these choices. Imagine
that the payoffs following choice  by player 1 are (2 0), following 
by player 2 are (3 1), following  by player 1 are (0 0), and following
 by player 1 are (1 2).
2. Veto Power: Two players must choose between three alternatives: ,
 and . Player 1’s preferences are given by  Â1  Â1  while player
2’s preferences are given by  Â2  Â2 . The rules are that player 1
moves first and can veto one of the three alternatives. Then, player 2
chooses which of the remaining two alternatives will be chosen.
3. Entering an Industry: A firm (player 1) is considering entering
an established industry with one incumbent firm (player 2). Player 1
must choose whether to enter or to not enter the industry. If player 1
enters the industry then player 2 can either accommodate the entry,
or fight the entry with a price war. Player 1’s most preferred outcome
is entering with player 2 not fighting, and his least preferred outcome
is entering with player 2 fighting. Player 2’s most preferred outcome
is player 1 not entering, and his least preferred outcome is player 1
entering with player 2 fighting.
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4. Entry Deterrence: NSG is considering entry into the local phone
market in the Bay Area. The incumbent S&P, predicts that a price
war will result if NSG enters. If NSG stays out, S&P earns monopoly
profits valued at \$10 million (net present value, or NPV of profits),
while NSG earns zero. If NSG enters, it must incur irreversible entry costs of \$2 million. If there is a price war, each firm earns \$1
million (NPV). S&P always has the option of accommodating entry
(not starting a price war). In such a case, both firms earn \$4 million
(NPV). Suppose that the timing is such that NSG first has to choose
whether or not to enter the market. Then S&P decides whether to
“accommodate entry” or “engage in a price war.”
5. The Stackelberg Leader: Suppose there are two firms (the industry
is a “duopoly”)  = 1 2. Each firm’s cost function is given by () =
 for all  (“unit cost” is constant, equal to  for both firms). The
inverse demand function is linear where it is positive, given by
() = ½  −  if  ≤ 
0 if 
where  = 1 + 2 and 1  2. Each firm’s strategic variable
is output, as in Cournot’s model, but the firms make their decisions
sequentially, rather than simultaneously — firm 1 chooses its output,
then the firm 2 does so, knowing the output chosen by the first firm.
What is the subgame perfect equilibrium of this game?
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