Say that three people play a word game in which each person chooses a letter simultaneously. Unlike in question 4, they
Posted: Sat Mar 19, 2022 5:51 pm
Say that three people play a word game in which each person
chooses a letter simultaneously. Unlike in question 4, they do not
choose in sequence! Person 1 chooses the first letter, either B, C,
or S. Person 2 chooses the second letter, either A, I, or O. Person
3 chooses the third letter, either G, R, or T. For example, if
person 1 chooses C, person 2 chooses A, and person 3 chooses T,
then they form the word CAT. If person 1 chooses B, person 2
chooses I, and person 3 chooses R, then they form BIR, which is not
a word. Note that BAG, BAR, BAT, BIG, BIT, BOG, CAR, CAT, COG, COT,
SAG, SAT, SIR, and SIT are words. BIR, BOR, BOT, CAG, CIG, CIR,
COR, CIT, SIG, SOG, SAR, SOR, and SOT are not words. Say that
person 1 wants to form a word, but wants the word to be as early in
alphabetical order as possible. For example, person 1 would prefer
BIG over BOG. Person 2 wants to form a word, but wants the word to
be as late in alphabetical order as possible. For example, person 2
would prefer CAT over CAR. If a word is not formed, then persons 1
and 2 get payoff 0. Person 3, however, is a "spoiler" and does not
want a word to be formed.
a. Make a strategic form game. Please use the numbers 0, 1, 2, .
. . , 14 for person 1's and person 2's payoffs and the numbers 0, 1
for person 3's payoffs. Find all (pure strategy) Nash
equilibria.
b. Now say that person 1 is a "spoiler" and does not want a word
to be formed. Now say that person 2 wants to form a word, but wants
the word to be as early in alphabetical order as possible. Person 3
wants to form a word, but wants the word to be as late in
alphabetical order as possible. Make a strategic form game. Please
use the numbers 0, 1 for person 1's payoff and the numbers 0, 1, 2,
. . . , 14 for person 2's and person 3's payoffs. Find all (pure
strategy) Nash equilibria.
c. Now say that person 2 is a "spoiler" and does not want a word
to be formed. Now say that person 1 wants to form a word, but wants
the word to be as late in alphabetical order as possible. Person 3
wants to form a word, but wants the word to be as early in
alphabetical order as possible. Make a strategic form game. Please
use the numbers 0, 1 for person 2's payoff and the numbers 0, 1, 2,
. . . , 14 for person 1's and person 3's payoffs. Find all (pure
strategy) Nash equilibria.
chooses a letter simultaneously. Unlike in question 4, they do not
choose in sequence! Person 1 chooses the first letter, either B, C,
or S. Person 2 chooses the second letter, either A, I, or O. Person
3 chooses the third letter, either G, R, or T. For example, if
person 1 chooses C, person 2 chooses A, and person 3 chooses T,
then they form the word CAT. If person 1 chooses B, person 2
chooses I, and person 3 chooses R, then they form BIR, which is not
a word. Note that BAG, BAR, BAT, BIG, BIT, BOG, CAR, CAT, COG, COT,
SAG, SAT, SIR, and SIT are words. BIR, BOR, BOT, CAG, CIG, CIR,
COR, CIT, SIG, SOG, SAR, SOR, and SOT are not words. Say that
person 1 wants to form a word, but wants the word to be as early in
alphabetical order as possible. For example, person 1 would prefer
BIG over BOG. Person 2 wants to form a word, but wants the word to
be as late in alphabetical order as possible. For example, person 2
would prefer CAT over CAR. If a word is not formed, then persons 1
and 2 get payoff 0. Person 3, however, is a "spoiler" and does not
want a word to be formed.
a. Make a strategic form game. Please use the numbers 0, 1, 2, .
. . , 14 for person 1's and person 2's payoffs and the numbers 0, 1
for person 3's payoffs. Find all (pure strategy) Nash
equilibria.
b. Now say that person 1 is a "spoiler" and does not want a word
to be formed. Now say that person 2 wants to form a word, but wants
the word to be as early in alphabetical order as possible. Person 3
wants to form a word, but wants the word to be as late in
alphabetical order as possible. Make a strategic form game. Please
use the numbers 0, 1 for person 1's payoff and the numbers 0, 1, 2,
. . . , 14 for person 2's and person 3's payoffs. Find all (pure
strategy) Nash equilibria.
c. Now say that person 2 is a "spoiler" and does not want a word
to be formed. Now say that person 1 wants to form a word, but wants
the word to be as late in alphabetical order as possible. Person 3
wants to form a word, but wants the word to be as early in
alphabetical order as possible. Make a strategic form game. Please
use the numbers 0, 1 for person 2's payoff and the numbers 0, 1, 2,
. . . , 14 for person 1's and person 3's payoffs. Find all (pure
strategy) Nash equilibria.