MIC: Ex 2, Oct 2007
1. Show that the game depicted below has a unique mixed strategy Nash
equilibrium.
L M R
U 1,-2 -2,1 0,0
M -2,1 1,-2 0,0
D 0,0 0,0 1,1
Hint: show that it has a unique pure strategy equilibrium. Then show that
player 1 (for example) cannot put positive weight on both U and M. Then show
that she cannot put positive weight on both U and D, without any weight on
M, and so on.)
Solution: Let player 2 play his three strategies with probabilities p; q; r re-
spectively. If 1 mixes between U and M, then p 2q = q 2p ) p = q )
_
V1 (U ) 0:5: Then if x1 a; are standard.
b) Show that this game has a pure strategy Nash equilibrium if and only if
a = 0:5:
Only if: Assume a y > a:
c) Suppose we modify consumer behavior so that if both vendors locate at
a; then fraction purchase from 1 and 1 purchase from vendor 2, where
1
can be chosen to be any number between 0 and 1. Show that for any value of a;
there exists a value such that the game has a pure strategy Nash equilibrium.
Choose = a; then x = y = a is an equilibrium.
3. Two players use the following procedure to divide a cake. Player 1 divides
the cake into two pieces, and player 2 chooses one of the two pieces, and 1 obtains
the remaining one. The cake is continuously divisible, and each player likes all
parts of it.
a) Suppose that the cake is perfectly homogeneous, so that each player only
cares about the size of the piece. Formulate this as an extensive game and solve
for a subgame perfect equilibrium.
b) Suppose that the cake is not homogeneous so that di¤erent players eval-
uate parts of it di¤erently. Let the set C represent the cake and let P C
represent a piece. If P 0 P and P 6= P 0 ; then each player strictly prefers P to
P 0 : Player’ preferences are continuous, i.e. if i strictly prefers P to P 0 ; then
s
there is a strict subset of P which i also prefers to P 0 : Consider a subgame
perfect equilibrium where player 1 partititons the cake fP1 ; P2 g and player 1
chooses P2 : Show that player 2 must be indi¤erent between P1 and P2 ; and that
player 1 weakly prefers P1 to P2 : Give an example where 1 strictly prefers P1
to P2:
4. The members of a hierarchical group of hungry lions face a piece of prey.
If lion 1 does not eat the prey, the prey escapes and the game ends. If it eats
the prey, it becomes fat and slow and lion 2 can eat it. If lion 2 does not eat lion
1, the game ends. If it eats lion 1, then it may be eaten by lion 3, and so on.
Each lion prefers to eat than be hungry, but prefers to be hungry than be eaten.
Formulate this as a game in extensive form. Find the backwards induction
outcomes for any number n of lions. (You need to prove this by induction).
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