Val: Stall/Stall/ Do it
Earl: Do it/ Do it
Course of play: Val stalls, Earl does it. Game ends.

Suppose Earl’s strategy is Stall/Stall.
What is Val’s best response?
If Val does this, what are Earl’s best responses?
What if Val’s strategy is Stall/Stall/Stall?

Nick and Rachel divide 4 candy bars. They take turns choosing.
Nick goes first. What should Nick choose first?
Preferences are:
For Nick For Rachel
Snickers Milky Way
Milky Way
Kit Kat
Baby Ruth
Kit Kat
Baby Ruth
Snickers
Hint: No matter what happens,
Nick will get two bars. Rachel will never choose Snickers.

Rachel
Nick
MW
KK
BR
BR KK MW
Nick Nick
BR MW KK
Sn
6
11
8
10
9
9
6
11
9
9
8
10
Nick
KK
Rachel
9
9
Sn
MW
Rachel
BR
BR
KK
Sn
Nick Nick Nick
KK Sn BR
KK BR
Rachel
7
6
9
9
5
7
7
6
6
6

6
11
Rachel
Nick
MW
KK
BR
BR KK MW
Nick Nick
BR MW KK
Sn
8
10
9
9
6
11
9
9
8
10
Nick
KK
Rachel
9
9
Sn
MW
Rachel
BR
BR
KK
Sn
Nick Nick Nick
KK Sn BR
KK BR
Rachel
7
6
9
9
5
7
7
6
6
6

6
11
Rachel
Nick
MW
KK
BR
BR KK MW
Nick Nick
BR MW KK
8
10
9
9
6
11
9
9
8
10
Sn
Nick
KK
Rachel
9
9
Sn
MW
Rachel
BR
BR
KK
Sn
Nick Nick Nick
KK Sn BR
KK BR
Rachel
7
6
9
9
5
7
7
6
6
6
In the simplified game, where Nick’s first move must be Sn or MW,
Subgame perfect equilibria:
Nick plays MW/KK/MW/MW/Sn/Sn/KK
Rachel plays MW/BR or MW/KK

Go to Movies
Bob
Bob
Go shoot pool
Alice
Go to A Go to B
Go to A
Alice
Go to B Go to A
Alice
Go to B
Go to A Go to B
2
3
0
0
1
1
3
2
2.5
1
2.5
0
Alice knows whether Bob is shooting pool before she decides which movie to go to.

1) Identify the proper subgames.
2) Find their SPNE’s
3) SPNE for full game is a strategy profile whose substrategy profiles are SPNE

• Two proper subgames.
• One starts where Alice discovers that Bob is shooting pool.
• One starts where Alice discovers that Bob is going to the movies.
• These have no proper subgames, so any NE for these subgames is subgame perfect
• So our next step is to find Nash equilibria for each.

Go to Movies
Bob
Bob
Go shoot pool
Alice
Go to A Go to B
Go to A
Alice
Go to B Go to A
Alice
Go to B
Go to A Go to B
2
3
0
0
1
1
3
2
2.5
1
2.5
0
Alice knows whether Bob is shooting pool before she decides which movie to go to.

In the subgame where Alice decides what to do when Bob shoots pool, the NE is for Alice to go to Movie A.

The other subgame in strategic form
Bob
Movie A
Movie B
A
Movie A l i
2,3 c
1,1 e
Movie B
0,0
3,2
Pure strategy equilibria where both go to A and where both go to B.
Mixed strategy equilibrium where Alice goes to movie A with probability q
Such that Bob is indifferent between the two movies.
That is, 2q+0(1-q)=1q+3(1-q), which implies q=2/3.
Similar calculation shows that Alice is indifferent between the two movies when
Bob goes to Movie B with probability 2/3.
Expected payoff to Bob in mixed strategy N.E. Is 2q=4/3.

Subgame perfect NE strategy profiles
1) For Bob, Movies, Movie B if movies. For Alice,
A if Bob shoots pool, B if Bob goes to movies
2) For Bob, Shoot pool, Movie A if movies. For
Alice, A if Bob shoots pool, A if Bob goes to movies.
3) For Bob, Shoot pool, Randomize with probability 2/3 of going to B if movies.
For Alice, Movie A if Bob shoots pool, Randomize with probability 2/3 of going to A if movies

• Suppose Bob chooses to go to the movies rather than to shoot pool.
• What can Alice conclude that he expects her to do?
• If he expects this what should she do?
• So doesn’t that leave us with only one plausible equilibrium?
• Notion of forward induction…

• Backward induction. Draw conclusions about what other player(s) will do by working back from the end and assuming they are rational.
• Forward induction. Drawing conclusions about what other player(s) believe by looking at what they have done and assuming they are rational.

How many pure strategy Nash equilibria does this game have?
Player 2
Player 1
Top
Bottom
Left
1, 2
1,0
Right
0,2
3,4

How many Nash equilibria does it have, including all mixed strategies?
Player 2
Player 1 Top
Bottom
Left
1, 2
1,0
Right
0,2
3,4
A) 0 B) 1 C) 2 D) 3 E) 4

This game has no mixed Nash equilibria that are not pure
Player 2
Player 1 Top
Bottom
Left
1, 2
1,0
Right
0,2
3,4
Note: Bottom weakly dominates top for Player 1.
Right weakly dominates Left for Player 2
Player 1 will mix only if Player 2 plays Left for sure.
But if Player 1 mixes, Player 2’s best response is Right.
So we can’t have Player 1 mixing.
Player 2 will mix only if Player 1 plays Top for sure.
But if Player 2 mixes, Player 1’s best response is Bottom.
So we can’t have Player 2 mixing.

How does number of pure and mixed strategy equilibria vary with X?
Player 1
Top
Bottom
Player 2
Left
1, 2
1,0
Right
0,X
3,4
How many Nash equilibria are there if X>2?
A) 0 B) 1 C) 2 D)3 E) 4

Player 2
Player 1 Top
Bottom
Left
1, 2
1,0
Right
0,X
3,4
There are 2 pure strategy N.E.
Bottom weakly dominates top for Player 1, but there is no weak domination for Player 2.
Player 1 will mix only if she is sure that Player 2 will go left.

Player 2
Left Right
Player 1
Top
Bottom
1, 2
1,0
0,X
3,4
Player 1 will mix only if she is sure that Player 2 will go left.
Let p be the probability Player 1 goes Top
Player 2 will go left only if 2p≥pX+4(1-p)
Equivalently (6-X)p≥4 or p≥4/(6-X)

Player 2 will go left only if 2p≥pX+4(1-p)
Equivalently (6-X)p≥4 or p≥4/(6-X)
When X<2, there are mixed strategy Nash equilibrium in which Player 1 plays Top with any probability p≥4/(6-x) and Player 2 Plays Left for sure.

• For X<2, there are many mixed strategy Nash equilibria and 2 pure strategy Nash equilibria.
• For X=2, there are 2 pure strategy Nash equilibria and no mixed strategy equilibria that are not pure.
• For X>2 there is exactly one Nash equilibrium, a pure strategy.

Todd and Steven
Divide the Estate
Problem 8.10

• Round 1: Todd makes offer of Division.
Steven accepts or rejects.
Round 2: If Steven rejects, estate is reduced to
100d pounds. Steven makes a new offer and
Todd accepts or rejects.
Round 3: If Todd rejects, estate is reduced to
100d 2 pounds. Todd makes new offer and
Steven accepts or rejects. If Steven rejects, both get zero.

• In last subgame, Steven must either accept or reject Todd’s offer. If he rejects, both get 0. If he accepts, he gets what Todd offered him.
• If Todd offers any small positive amount ε,
Steven’s best reply is to accept.
• So in next to last subgame, Todd would offer
Steven ε and take 100d 2 -ε for himself.

Todd
Propose
Steven
Accept
Reject
Steven
Propose
Todd
Accept Reject
Todd
Propose
Steven
Reject
Accept

• At node where Steven has offered Todd a division, there are 100d units to divide. Todd would accept 100d 2 or more, would reject less.
• So at previous node Steven would offer Todd
100d 2 and would have 100(d-d 2 ) for himself.

• Now consider the subgame where Todd makes his first proposal.
• At this point there are 100 pounds of gold to divide.
• Todd sees that Steven would accept anything greater than 100(d-d 2 ).
• So Todd would offer Steven 100(d-d 2 )+ε and keep 100(1-d+d 2 )-ε for himself.

First node: Offer Steven 100(d-d 2 )+ε
Second node: If Steven rejects Todd’s offer and makes a counteroffer to Todd: Accept 100d 2 or more, reject less.
Third node: If Todd rejects Steven’s counter offer, make a new offer to Steven of a small ε.

First node: Accept any offer greater than
100(d-d 2 ), reject smaller offers.
Second node: If Steven rejects Todd’s first offer, then offer Todd 100d 2
Third node: Accept any positive offer.

Suppose d =.9, then Todd offers Steven
100(d-d 2 )=100(.9-.81)= 9 and keeps 91 for himself.
If d=.5, Todd offers Steven 100(d-d 2 )= 25 and keeps 100-25=75 for himself.
Notice that Steven’s share 100(d-d 2 ) is largest when the decay rate is d=.5.
What happens with more rounds of bargaining?