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1. 1. Consider the game below, and answer the following questions:
(a) Find all the Nash Equilibria for the subgame starting with player 1s second move.
(b) Can player 2 play U as part of a subgame perfect Nash Equilibrium? Can player 2 play
U as part of a Nash Equilibrium?
(c) Can player 2 achieve a payo® of 3 as part of a subgame perfect Nash Equilibrium? Can
player 2 achieve a payo® of 3 as part of a Nash Equilibrium?
(d) Find a pure strategy Nash Equilibrium for the whole game that is not subgame perfect.
(3/4,3)
1
2
U D
(0,1/2)
2 2
1
L
C R
L C R
U D
L R
(0,0) (1,1) (1,2) (2,1) (0,1) (0,1)
DECISIONS, GAMES, AND RATIONAL CHOICE
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2. 2. Consider the following simpli¯ed version of poker" with 2 players. Each player i is given
a number ti. The numbers are drawn independently and uniformly from the interval [0; 1].
Each player must decide whether to Bid or Fold. If both players Fold then both players
receive -1. If one player Bids and the other player Folds then the player that Bids receives
+1 and the player that Folds receives -1. If both players Bid then the player with the higher
number receives +k and the other player receives ¡k. (Note, the probability of a tie is zero
and can be ignored.)
(a) Set k = 2. Suppose the players play simultaneously. Find a Bayesian Nash equilibrium.
(Hint: look for an equilibrium in which both players use cuto® strategies of the form Fold if
ti · ®i, and Bid if ti > ®i.") Is this equilibrium unique? Be sure to describe the equilibrium
completely and rigorously.
(b) Suppose k increases. What happens to the cuto® levels? In words, what is the intuition
for the change?
(c) Again set k = 2. What if the players bid sequentially? That is, suppose player 1 must
decide whether to Bid or Fold ¯rst, and player 2 observes his choice before deciding whether
to Bid or Fold. Do the players have the same cuto® levels, or are they di®erent? If di®erent,
which player has the higher cuto® level? In words, what is the intuition for this result?
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3. 3. MegaByteTechAssociates (MBTA) has a potential investment project. This project would
develop an application for MacroStuft a prestigious cwidget manufacturer. The game pro­
ceeds in 3 time periods. In period 1, MBTA must make an irreversible decision about how
much to invest in development. An investment level of I costs I2
=2, and creates a product
that can be manufactured at cost cI and has value to MacroStuft of VM I, where VM > c
(the product is a software template so only one unit is needed by MacroStuft). In period 2,
MacroStuft observes I, chooses a price p ¸ 0, and makes a take-it-or-leave-it o®er to buy
the product at price p. In period 3 MBTA accepts or rejects the o®er. If MBTA accepts the
o®er, then it receives a payo® of p¡cI ¡I2
=2 and MacroStuft receives a payo® of VM I¡p.
If MBTA rejects the o®er, then they receive a payo® of ¡I2
=2 and MacroStuft receives a
payo® of 0.
(a) Write out an extensive form representation of the game. Find the pure strategy, subgame­
perfect Nash equilibrium of the game. What is the optimal investment level (the level
that maximizes the total surplus generated by the product)? How does it compare to the
equilibrium investment level? Intuitively, why does MBTA choose the level of investment it
does, even though the project creates more value than it costs?
Now modify the game in the following way. Suppose there are 5 time periods. The
¯rst three are as given above. Now, however, if MBTA refuses the MacroStuft o®er, then
it can o®er the product to another company, Solar. Speci¯cally, if MBTA refuses to sell
to MacroStuft, then the game moves to period 4. In period 4, MBTA chooses a price q,
and makes a take-it-or-leave-it o®er to Solar to buy the product at price q. Solar values
the product at VSI, and the cost of production is the same as above. In period 5, Solar
either rejects or accepts the o®er and the game ends. If Solar rejects, then both it and
MacroStuft receive payo®s of 0, and MBTA receives a payo® of ¡I2
=2. If Solar accepts the
o±ce, then it receives a payo® VSI¡q, MBTA receives q¡cI¡I2=2, and MacroStuft receives
¡r (MacroStuft is a jealous company, and ¡r represents its distress from seeing a competitor
obtain the project).
(b) Find the Pure strategy subgame perfect Nash equilibrium of this game. Does your answer
depend on the relative magnitudes of VM and VS? If equilibrium investment by MBTA is
di®erent than in part (a), give an intuitive explanation for the di®erence.
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4. Return to the three-period game of part (a), but now allow MacroStuft to make the
following o®er. In period 0 (before MBTA makes its investment decision), MacroStuft o®ers
a legally enforceable and unbreakable option contract, that committs it to buy the project
for VM per unit of I at period 3 in return for a ¯xed payment of F now. MBTA may then
accept or reject the option contract, and in either case play proceeds as before. Now of
course, if MBTA buys the option the payo®s are di®erent.
(c) Write out an extensive form for this game paying special attention to the payo®s on the
branch where MBTA accepts the option contract. Find the pure strategy subgame perfect
nash equilibrium in this game. What fee F will be charged for the option? What does this
example tell us about the value of enforceable contracts?
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