CHAPTER 29
Game Theory
TRUE/FALSE
1.
A situation where everyone is playing a dominant strategy must be a
Nash equilibrium.
2.
In a Nash equilibrium, everyone must be playing a dominant strategy.
3.
In the prisoner’s dilemma game, if each prisoner believed that the
other prisoner would deny the crime, then both would deny the crime.
4.
A general has the two possible pure strategies, sending all of his
troops by land or sending all of his troops by sea. An example of a mixed strategy
is where he sends 1/4 of his troops by land and 3/4 of his troops by sea.
5.
While game theory predicts noncooperative behavior for a single play
of the prisoner’s dilemma, it would predict cooperative tit-for-tat behavior if the
same people play prisoner’s dilemma together for, say, 20 rounds.
6.
A two-person game in which each person has access to only two
possible strategies will have at most one Nash equilibrium.
7.
If a game does not have an equilibrium in pure strategies, then it will
not have an equilibrium in mixed strategies either.
8.
A game has two players, and each has two strategies.
The strategies are Be Nice and Be Mean. If both players play Be Nice, both
get a payoff of 5. If both players play Be Mean, both get a payoff of 23. If one
player plays Be Nice and the other plays Be Mean, the player who played Be Nice
gets 0 and the player who played Be Mean gets 10. Playing Be Mean is a dominant
strategy for both players.
T/F ANSWERS:
1. T. Dominant strategy is the best response to all rival strategies. If
both play their dominant strategy, they are simultaneously offering
best responses to rival’s choice and would have no incentive to
change. This is a Nash equilibrium.
2. F. Player’s does not need to have a dominant strategy for a Nash
equilibrium to exist.
3. F. Confess is the dominant strategy for each player even if they
believe rival would deny.
4. F. Mixed strategy would assign a probability of sending ALL troops
by land and sea. For example, .25 probability of sending by land and
.75 probability of sending by sea.
5. F. Noncooperative behavior would still be the dominant strategy if
players know how many rounds would be played.
6. F. there can be two Nash equilibria in two-person two-strategy game.
eg. Battle of Sexes game.
7. F. Example, Matching penny game.
8. T. Playing mean gives both players higher payoff no matter what
rival plays.
MULTIPLE CHOICE
1.
A game has two players. Each player has two possible strategies. One
strategy is Cooperate, the other is Defect. Each player writes on a piece of paper
either a C for cooperate or a D for defect. If both players write C, they each get a
payoff of $100. If both players write D, they each get a payoff of 0. If one player
writes C and the other player writes D, the cooperating player gets a payoff of S
and the defecting player gets a payoff of T. To defect will be a dominant strategy
for both players if
a.
S + T > 100.
b.
T > 2S.
c.
S < 0 and T > 100.
d.
S < T and T > 100.
e.
S and T are any positive numbers.
2.
In the game matrix below, the first payoff in each pair goes to player
A who chooses the row, and the second payoff goes to player B, who chooses the
column. Let a, b, c, and d be positive constants.
Player B
Left
Right
a,1
b,1
Player A
Top
1,c
1,d
Bottom
If player A chooses Bottom and player B chooses Right in a Nash
equilibrium, then we know that
a.
b > 1 and d < 1.
b.
c < 1 and b < 1.
c.
b < 1 and c < d.
d.
b < c and d < 1.
e.
a < 1 and b < d.
3.
In the town of Torrelodones, each of the N > 2 inhabitants has $100.
They are told that they can all voluntarily contribute to a fund that will be evenly
divided among all residents. If $F are contributed to the fund, the local K-Mart will
match the private contributions so that the total amount to be divided is $2F. That
is, each resident will get back a payment of $2F/N when the fund is divided. If the
people in town care only about their own net incomes, in Nash equilibrium, how
much will each person contribute to the fund?
a.
$0
b.
$10
c.
$20
d.
$50
e.
$100
4.
Big Pig and Little Pig have two possible strategies, Press the Button,
and Wait at the Trough. If both pigs choose Wait at the Trough, both get 2. If both
pigs choose Press the Button, then Big Pig gets 5 and Little Pig gets 5. If Little Pig
presses the button and Big Pig waits at the trough, then Big Pig gets 10 and Little
Pig gets 0. Finally, if Big Pig presses the button and Little Pig waits at the trough,
then Big Pig gets 6 and Little Pig gets 2. In Nash equilibrium,
a.
Little pig will get a payoff of zero.
b.
Little Pig will get a payoff of 5 and Big Pig will get a payoff of 5.
c.
both pigs will wait at the trough.
d.
e.
Little Pig will get a payoff of 2 and Big Pig will get a payoff of 6.
the pigs must be using mixed strategies.
5.
Suppose that in a Hawk-Dove game, the payoff to each player is − 6 if
both play Hawk. If both play Dove, the payoff to each player is 3, and if one plays
Hawk and the other plays Dove, the one that plays Hawk gets a payoff of 8 and the
one that plays Dove gets 0. In equilibrium, we would expect hawks and doves to
do equally well. This happens when the proportion of the total population that
plays Hawk is approximately
a.
.45.
b.
.23.
c.
.11.
d.
.73.
e.
1.
1. ANS: C. Payoffs are C, C = 100, 100; D, D = 0, 0; C, D = S, T and D,
C = T, S. If 2nd player plays C, the first gets 100 for C and T for D. If
the 2nd plays D, the first gets S for C and T for D. If T > 100 and S <
0, playing D gives higher payoff no matter what opponent plays.
2. ANS: C For Right to be B’s best response to A’s Bottom strategy, c
< d and for Bottom to be A’s best response to B’s Right strategy, b <
1.
3. ANS: A. Suppose there are 3 players. Any player who contributes 0
will receive 100 + 2 (2F)/3 = 100 + 4/3F if the other two contributes F
each. She will receive {100 – F + 2(3F)/3} = 100 + F, if she also
contributes F. If others contribute 0, she will receive 100 if she also
contributes 0 and she will receive 100 – F + 2F/3 = 100 - 1/3F if she
contributes F. Contributing 0 yields higher payoff for any player no
matter whether other players contribute F or 0..
4. ANS: E. There is no pure strategy Nash equilibrium. Therefore,
equilibrium must involve mixed strategy.
5. ANS: A. The population proportion of hawks and doves gives the
probabilities that a player will meet either a hawk or a dove. The
mixed strategy Nash equilibrium probability of encountering a hawk
is 5/11 = .454. Therefore, if this is the proportion oh Hawk’s in the
population, expected payoff of Hawk’s and doves will be the same.
PROBLEM
Consider the following game: Column and Row are the two players. Row’s
strategies are top and bottom and column’s strategies are left and right.
COLUMN
LEFT
TOP
-3, -3
RIGHT
2,0
ROW
BOTTOM 0, 2
1, 1
a. Find all the pure strategy equilibria when players move
simultaneously.
b. Find mixed strategy equilibrium.
c. If row moves first what is the sub game perfect equilibrium of the
sequential move game? Draw the game to illustrate.
ANSWER
a. Both TOP/RIGHT and BOTTOM/LEFT are pure strategy Nash
equilibrium
b. Let PL be the probability COLUMN plays left. Then:
Expected payoff to ROW from TOP is: PL(-3) + (1-PL)2 and from
BOTTOM: PL(0) + (1-PL)1
Equating the expected payoffs: -3PL + 2 -2P1 = 1 – PL OR PL = ¼
Let PT be the probability ROW plays TOP. Then:
Expected payoff to COLUMN from LEFT is: PT(-3) + (1-PL)2 and
from RIGHT: PT(0) + (1-PT)1
Equating payoffs: PT = ¼. So the mixed strategy equilibrium is
ROW playing TOP with probability ¼ and COLUMN playing
RIGHT with probability ¼.
c. ROW knows that if she plays TOP, COLUMN will respond with
RIGHT (0>-3) and ROW will then end up with 2. If ROW starts with
BOTTOM she knows again that COLUMN will respond with LEFT
(2>1) and ROW will end up with 0. Knowing the end result of both
options, clearly ROW will go with TOP and COLUMN will respond
with RIGHT, which is the sub-game perfect Nash equilibrium of the
sequential game.