Is there a game with exactly two SPE s.t. the outcome of one of them is for both players better than the...
I saw this task in a textbook. I would be sure that this is impossible (I could solve this if there could be more than two SPE or if it had be better for just one player), but they present it as if not.
My reasoning is that if there are have to be 2 SPE, one player must have a choice at some node (he can decide for any of two strategies because they yield the same outcome). But if it were to happen again, there would certainly be more than 2. So only one choice in the backward induction. But that gives both the players the same payoff.
game-theory nash-equilibrium
asked Dec 2 '18 at 18:18
SlowerPhoton
408111
add a comment |
I saw this task in a textbook. I would be sure that this is impossible (I could solve this if there could be more than two SPE or if it had be better for just one player), but they present it as if not.
My reasoning is that if there are have to be 2 SPE, one player must have a choice at some node (he can decide for any of two strategies because they yield the same outcome). But if it were to happen again, there would certainly be more than 2. So only one choice in the backward induction. But that gives both the players the same payoff.
game-theory nash-equilibrium
asked Dec 2 '18 at 18:18
SlowerPhoton
408111
1
What does SPE stand for?
– saulspatz
Dec 2 '18 at 19:20
Sub-game perfect equilibrium, sorry for not specifying that.
– SlowerPhoton
Dec 2 '18 at 19:20
add a comment |
I saw this task in a textbook. I would be sure that this is impossible (I could solve this if there could be more than two SPE or if it had be better for just one player), but they present it as if not.
My reasoning is that if there are have to be 2 SPE, one player must have a choice at some node (he can decide for any of two strategies because they yield the same outcome). But if it were to happen again, there would certainly be more than 2. So only one choice in the backward induction. But that gives both the players the same payoff.
game-theory nash-equilibrium
asked Dec 2 '18 at 18:18
SlowerPhoton
408111
I saw this task in a textbook. I would be sure that this is impossible (I could solve this if there could be more than two SPE or if it had be better for just one player), but they present it as if not.
My reasoning is that if there are have to be 2 SPE, one player must have a choice at some node (he can decide for any of two strategies because they yield the same outcome). But if it were to happen again, there would certainly be more than 2. So only one choice in the backward induction. But that gives both the players the same payoff.
game-theory nash-equilibrium
game-theory nash-equilibrium
asked Dec 2 '18 at 18:18
SlowerPhoton
408111
asked Dec 2 '18 at 18:18
SlowerPhoton
408111
asked Dec 2 '18 at 18:18
SlowerPhoton
408111
asked Dec 2 '18 at 18:18
SlowerPhoton
408111
asked Dec 2 '18 at 18:18
SlowerPhoton
408111
408111
1
What does SPE stand for?
– saulspatz
Dec 2 '18 at 19:20
Sub-game perfect equilibrium, sorry for not specifying that.
– SlowerPhoton
Dec 2 '18 at 19:20
add a comment |
1
What does SPE stand for?
– saulspatz
Dec 2 '18 at 19:20
Sub-game perfect equilibrium, sorry for not specifying that.
– SlowerPhoton
Dec 2 '18 at 19:20
1
1
What does SPE stand for?
– saulspatz
Dec 2 '18 at 19:20
What does SPE stand for?
– saulspatz
Dec 2 '18 at 19:20
Sub-game perfect equilibrium, sorry for not specifying that.
– SlowerPhoton
Dec 2 '18 at 19:20
Sub-game perfect equilibrium, sorry for not specifying that.
– SlowerPhoton
Dec 2 '18 at 19:20
add a comment |
1 Answer
1
active
oldest
votes
Note that subgame perfection is more general than
backward induction, as it applies also to games with imperfect information.
Consider a coordination game. For instance,
$$begin{bmatrix}
2,2 & 0,0 \
0,0 & 1,1 \
end{bmatrix}$$
Write it in extensive form and it will have two equilibria in pure strategies and one in mixed strategies. The two equilibria in pure strategies satisfy the condition that one is better than the other.
For a game with exactly two equilibria, use the special coordination game
$$begin{bmatrix}
1,1 & 0,0 \
0,0 & 0,0 \
end{bmatrix}$$
answered Dec 2 '18 at 21:19
mlc
4,87931332
I feel like using imperfect information is cheating. Do you know of an example using perfect information only? We can consider pure strategies only.
– SlowerPhoton
Dec 14 '18 at 0:36
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Note that subgame perfection is more general than
backward induction, as it applies also to games with imperfect information.
Consider a coordination game. For instance,
$$begin{bmatrix}
2,2 & 0,0 \
0,0 & 1,1 \
end{bmatrix}$$
Write it in extensive form and it will have two equilibria in pure strategies and one in mixed strategies. The two equilibria in pure strategies satisfy the condition that one is better than the other.
For a game with exactly two equilibria, use the special coordination game
$$begin{bmatrix}
1,1 & 0,0 \
0,0 & 0,0 \
end{bmatrix}$$
answered Dec 2 '18 at 21:19
mlc
4,87931332
I feel like using imperfect information is cheating. Do you know of an example using perfect information only? We can consider pure strategies only.
– SlowerPhoton
Dec 14 '18 at 0:36
add a comment |
Note that subgame perfection is more general than
backward induction, as it applies also to games with imperfect information.
Consider a coordination game. For instance,
$$begin{bmatrix}
2,2 & 0,0 \
0,0 & 1,1 \
end{bmatrix}$$
Write it in extensive form and it will have two equilibria in pure strategies and one in mixed strategies. The two equilibria in pure strategies satisfy the condition that one is better than the other.
For a game with exactly two equilibria, use the special coordination game
$$begin{bmatrix}
1,1 & 0,0 \
0,0 & 0,0 \
end{bmatrix}$$
answered Dec 2 '18 at 21:19
mlc
4,87931332
I feel like using imperfect information is cheating. Do you know of an example using perfect information only? We can consider pure strategies only.
– SlowerPhoton
Dec 14 '18 at 0:36
add a comment |
Note that subgame perfection is more general than
backward induction, as it applies also to games with imperfect information.
Consider a coordination game. For instance,
$$begin{bmatrix}
2,2 & 0,0 \
0,0 & 1,1 \
end{bmatrix}$$
Write it in extensive form and it will have two equilibria in pure strategies and one in mixed strategies. The two equilibria in pure strategies satisfy the condition that one is better than the other.
For a game with exactly two equilibria, use the special coordination game
$$begin{bmatrix}
1,1 & 0,0 \
0,0 & 0,0 \
end{bmatrix}$$
answered Dec 2 '18 at 21:19
mlc
4,87931332
Note that subgame perfection is more general than
backward induction, as it applies also to games with imperfect information.
Consider a coordination game. For instance,
$$begin{bmatrix}
2,2 & 0,0 \
0,0 & 1,1 \
end{bmatrix}$$
Write it in extensive form and it will have two equilibria in pure strategies and one in mixed strategies. The two equilibria in pure strategies satisfy the condition that one is better than the other.
For a game with exactly two equilibria, use the special coordination game
$$begin{bmatrix}
1,1 & 0,0 \
0,0 & 0,0 \
end{bmatrix}$$
answered Dec 2 '18 at 21:19
mlc
4,87931332
answered Dec 2 '18 at 21:19
mlc
4,87931332
answered Dec 2 '18 at 21:19
mlc
4,87931332
answered Dec 2 '18 at 21:19
mlc
4,87931332
4,87931332
I feel like using imperfect information is cheating. Do you know of an example using perfect information only? We can consider pure strategies only.
– SlowerPhoton
Dec 14 '18 at 0:36
add a comment |
I feel like using imperfect information is cheating. Do you know of an example using perfect information only? We can consider pure strategies only.
– SlowerPhoton
Dec 14 '18 at 0:36
I feel like using imperfect information is cheating. Do you know of an example using perfect information only? We can consider pure strategies only.
– SlowerPhoton
Dec 14 '18 at 0:36
I feel like using imperfect information is cheating. Do you know of an example using perfect information only? We can consider pure strategies only.
– SlowerPhoton
Dec 14 '18 at 0:36
add a comment |
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1
What does SPE stand for?
– saulspatz
Dec 2 '18 at 19:20
Sub-game perfect equilibrium, sorry for not specifying that.
– SlowerPhoton
Dec 2 '18 at 19:20