14.12: Problem Set 4 Solution
Ruitian Lang
November 14, 2010
1
The stage game has three Nash equilibria:
(
S, S
)
,
(
M, M
), and
(
2
3
S
+
1
3
M,
1
3
S
+
2
3
M
)
, and the payoffs
associated with these equilibria are (2
,
1)
,
(1
,
2) and
(
2
3
,
2
3
)
. Therefore, the players can coordinate on different
Nash equilibria to punish anyone who deviates. Consider the following strategy profile for example. Player
1 plays
M
and Player 2 plays
S
in the first period. If the outcome of the first period is (
M, S
), then on odd
dates they play (
M, M
) and on even dates they play (
S, S
). If the outcome of the first period is not (
M, S
),
then they play
(
2
3
S
+
1
3
M,
1
3
S
+
2
3
M
)
thereafter. To check whether it is a subgame perfect equilibrium, it
suffices to check that neither player has incentive to deviate in the first period, since the outcome in the other
periods does not affect continuation strategies and they are playing Nash equilibria of the stage game from
period 2 on. In period 1, playing
S
gives Player 1 2 + 2
k
2
3
, and playing
M
gives her 0 +
k
+ 2
k
. Therefore,
she has no incentive to deviate when
2 + 2
k
2
3
≤
0 +
k
+ 2
k,
or
k
≥
6
5
. Similarly, Player 2 has no incentive to deviate when
k
≥
6
5
too. Therefore, the above strategy
profile is a subgame perfect equilibrium when
k
≥
2.
2
We conjecture that the strategy profile is the following: at each date
t
= 3
n
+
k
, the proposer proposes wage
w
k
, and the responder accepts a wage off
w
if and only if her payoff from wage
w
is greater than or equal
to her payoff from wage
w
k
. (i.e., the union accepts
w
if and only if
w
≥
w
k
for
k
= 2, and the employer
accepts
w
if and only if
w
≤
w
k
for
k
= 0
,
1.) At
t
= 3
n
, when the union offers wage
w
, the employer knows
that if agreement is not reached this period they will agree on
w
1
in the following period, so he accepts
w
when 1

w > δ
(1

w
1
) and reject
w
when 1

w < δ
(1

w
1
). He has no incentive to deviate from the plan
”accepting if and only if
w
≤
w
0
” if and only if 1

w
0
=
δ
(1

w
1
). The similar analysis for
t
= 3
n
+ 1 and
1
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t
= 3
n
+ 2 yields two similar conditions. Therefore, no responder has incentive to deviate if and only if
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 Spring '11
 LieWang
 Game Theory, Nash, Subgame perfect equilibrium

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