Mixed Nash Equilibria and Negative Probability The 2019 Stack Overflow Developer Survey Results Are InAre all Nash equilibrium pure strategies also Nash equilibrium mixed strategies.Find the Nash equilibria in pure strategies in the following games.Finding Nash Equilibria of a finte game of 2 players.Finding Nash Equilibria for this Bimatrix GameMixed Nash equilibrium for non-square matrix gameMixed strategy nash equilbriumMixed Nash equilibria for zero matrixNash Bargaining Mechanism Security LevelGame Theory - Mixed strategy Nash equilibriaNash Equilibrium and Mixed NE Problem
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Mixed Nash Equilibria and Negative Probability
The 2019 Stack Overflow Developer Survey Results Are InAre all Nash equilibrium pure strategies also Nash equilibrium mixed strategies.Find the Nash equilibria in pure strategies in the following games.Finding Nash Equilibria of a finte game of 2 players.Finding Nash Equilibria for this Bimatrix GameMixed Nash equilibrium for non-square matrix gameMixed strategy nash equilbriumMixed Nash equilibria for zero matrixNash Bargaining Mechanism Security LevelGame Theory - Mixed strategy Nash equilibriaNash Equilibrium and Mixed NE Problem
$begingroup$
I am currently attempting to look for mixed equilibria in this payoff matrix
$beginbmatrix(boldsymbol1,1) & (boldsymbol5,2) & (boldsymbol0,0) \(boldsymbol2,5) & (boldsymbol3,3) & (boldsymbol1,0) \ (boldsymbol0,0) & (boldsymbol0,1) & (boldsymbol2,2) endbmatrix$
(You can call the row player Player 1 for simplicity and Player 2 the column player. I put the payoffs for player 1 in bold).
Now, I did find three pure NE's (that is to say (2,5), (5,2) and (2,2)), as well as a few mixed equilibria for the various subgames. I can't, however, find a mixed equilibrium for the entire game (in particular, if I assign probability $p_1$ to the first column, it turns out to be $-frac19$ in a linear system of equations.)
How am I supposed to interpret that negative probability? I know this game is supposed to have an odd number of Nash equilibria, but the I obtain an even number.
Besides what I wrote in the original problem, I also found the following mixed equilibria in the subgames. I am labeling the column player's actions l, c, and r (from left to right) and the row player plays u, m, d from the top down. So we have:
-(u, m) played with probability 2/3 and 1/3; and (l,c) played with probability 2/3 and 1/3
-(m,d) played with probability 2/7 and 5/7 ; and (l,r) played with probability 1/3 and 2/3
-(m,d) played with probability 1/4 and 3/4 ; and (c,r) played with probability 1/4 and 3/4
-(u,d) played with probability 2/3 and 1/3 ; and (l,r) played with probability 2/3 and 1/3
-(u,d) played with probability 1/3 and 2/3; and (c,r) played with probability 2/7 and 5/7
probability game-theory nash-equilibrium
asked Mar 6 '18 at 17:39
AnonymousAnonymous
64
$endgroup$
add a comment |
$begingroup$
I am currently attempting to look for mixed equilibria in this payoff matrix
$beginbmatrix(boldsymbol1,1) & (boldsymbol5,2) & (boldsymbol0,0) \(boldsymbol2,5) & (boldsymbol3,3) & (boldsymbol1,0) \ (boldsymbol0,0) & (boldsymbol0,1) & (boldsymbol2,2) endbmatrix$
(You can call the row player Player 1 for simplicity and Player 2 the column player. I put the payoffs for player 1 in bold).
Now, I did find three pure NE's (that is to say (2,5), (5,2) and (2,2)), as well as a few mixed equilibria for the various subgames. I can't, however, find a mixed equilibrium for the entire game (in particular, if I assign probability $p_1$ to the first column, it turns out to be $-frac19$ in a linear system of equations.)
How am I supposed to interpret that negative probability? I know this game is supposed to have an odd number of Nash equilibria, but the I obtain an even number.
Besides what I wrote in the original problem, I also found the following mixed equilibria in the subgames. I am labeling the column player's actions l, c, and r (from left to right) and the row player plays u, m, d from the top down. So we have:
-(u, m) played with probability 2/3 and 1/3; and (l,c) played with probability 2/3 and 1/3
-(m,d) played with probability 2/7 and 5/7 ; and (l,r) played with probability 1/3 and 2/3
-(m,d) played with probability 1/4 and 3/4 ; and (c,r) played with probability 1/4 and 3/4
-(u,d) played with probability 2/3 and 1/3 ; and (l,r) played with probability 2/3 and 1/3
-(u,d) played with probability 1/3 and 2/3; and (c,r) played with probability 2/7 and 5/7
probability game-theory nash-equilibrium
asked Mar 6 '18 at 17:39
AnonymousAnonymous
64
$endgroup$
add a comment |
$begingroup$
I am currently attempting to look for mixed equilibria in this payoff matrix
$beginbmatrix(boldsymbol1,1) & (boldsymbol5,2) & (boldsymbol0,0) \(boldsymbol2,5) & (boldsymbol3,3) & (boldsymbol1,0) \ (boldsymbol0,0) & (boldsymbol0,1) & (boldsymbol2,2) endbmatrix$
(You can call the row player Player 1 for simplicity and Player 2 the column player. I put the payoffs for player 1 in bold).
Now, I did find three pure NE's (that is to say (2,5), (5,2) and (2,2)), as well as a few mixed equilibria for the various subgames. I can't, however, find a mixed equilibrium for the entire game (in particular, if I assign probability $p_1$ to the first column, it turns out to be $-frac19$ in a linear system of equations.)
How am I supposed to interpret that negative probability? I know this game is supposed to have an odd number of Nash equilibria, but the I obtain an even number.
Besides what I wrote in the original problem, I also found the following mixed equilibria in the subgames. I am labeling the column player's actions l, c, and r (from left to right) and the row player plays u, m, d from the top down. So we have:
-(u, m) played with probability 2/3 and 1/3; and (l,c) played with probability 2/3 and 1/3
-(m,d) played with probability 2/7 and 5/7 ; and (l,r) played with probability 1/3 and 2/3
-(m,d) played with probability 1/4 and 3/4 ; and (c,r) played with probability 1/4 and 3/4
-(u,d) played with probability 2/3 and 1/3 ; and (l,r) played with probability 2/3 and 1/3
-(u,d) played with probability 1/3 and 2/3; and (c,r) played with probability 2/7 and 5/7
probability game-theory nash-equilibrium
asked Mar 6 '18 at 17:39
AnonymousAnonymous
64
$endgroup$
I am currently attempting to look for mixed equilibria in this payoff matrix
$beginbmatrix(boldsymbol1,1) & (boldsymbol5,2) & (boldsymbol0,0) \(boldsymbol2,5) & (boldsymbol3,3) & (boldsymbol1,0) \ (boldsymbol0,0) & (boldsymbol0,1) & (boldsymbol2,2) endbmatrix$
(You can call the row player Player 1 for simplicity and Player 2 the column player. I put the payoffs for player 1 in bold).
Now, I did find three pure NE's (that is to say (2,5), (5,2) and (2,2)), as well as a few mixed equilibria for the various subgames. I can't, however, find a mixed equilibrium for the entire game (in particular, if I assign probability $p_1$ to the first column, it turns out to be $-frac19$ in a linear system of equations.)
How am I supposed to interpret that negative probability? I know this game is supposed to have an odd number of Nash equilibria, but the I obtain an even number.
Besides what I wrote in the original problem, I also found the following mixed equilibria in the subgames. I am labeling the column player's actions l, c, and r (from left to right) and the row player plays u, m, d from the top down. So we have:
-(u, m) played with probability 2/3 and 1/3; and (l,c) played with probability 2/3 and 1/3
-(m,d) played with probability 2/7 and 5/7 ; and (l,r) played with probability 1/3 and 2/3
-(m,d) played with probability 1/4 and 3/4 ; and (c,r) played with probability 1/4 and 3/4
-(u,d) played with probability 2/3 and 1/3 ; and (l,r) played with probability 2/3 and 1/3
-(u,d) played with probability 1/3 and 2/3; and (c,r) played with probability 2/7 and 5/7
probability game-theory nash-equilibrium
probability game-theory nash-equilibrium
asked Mar 6 '18 at 17:39
AnonymousAnonymous
64
asked Mar 6 '18 at 17:39
AnonymousAnonymous
64
edited Mar 7 '18 at 15:09
Anonymous
asked Mar 6 '18 at 17:39
AnonymousAnonymous
64
asked Mar 6 '18 at 17:39
AnonymousAnonymous
64
asked Mar 6 '18 at 17:39
AnonymousAnonymous
64
64
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Negative probability is meaningless. Getting a negative probability in your calculation simply means that there is no mixed strategy Nash equilibrium in which either player randomizes over all three of their pure strategies.
Of all the mixed strategy profiles you listed, only the first and third ones are MSNEs. Note that the indifference condition is only a necessary condition, not a sufficient one. You need to ensure that the mixed strategies leading to indifference are indeed best responses.
For example, in your second profile, when Player 1 plays $frac27m+frac57d$, it is true that Player 2 is indifferent between $l$ and $r$ (payoff is $frac107$), but choosing $c$ leads to a higher payoff ($frac117>frac107$). So the best response to $frac27m+frac57d$ is the pure strategy $c$ as opposed to $frac13l+frac23r$. The other non-MSNE profiles have the same problem.
answered Mar 6 '18 at 23:37
Herr K.Herr K.
6181717
$endgroup$
$begingroup$
Thank you for that. Could you please help me identify all the mixed equilibria then? Because if we include the pure ones I am left with an even number of those.
$endgroup$
– Anonymous
Mar 7 '18 at 9:17
$begingroup$
@Anonymous instead of me giving you the answer, why don't you show what you've done and ask for verification?
$endgroup$
– Herr K.
Mar 7 '18 at 11:10
$begingroup$
I added them to the main problem now.
$endgroup$
– Anonymous
Mar 7 '18 at 15:09
$begingroup$
@Anonymous: Please see my updated answer.
$endgroup$
– Herr K.
Mar 7 '18 at 17:47
$begingroup$
Thank you again for explaining this to me.
$endgroup$
– Anonymous
Mar 7 '18 at 18:32
add a comment |
Your Answer
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1 Answer
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1 Answer
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$begingroup$
Negative probability is meaningless. Getting a negative probability in your calculation simply means that there is no mixed strategy Nash equilibrium in which either player randomizes over all three of their pure strategies.
Of all the mixed strategy profiles you listed, only the first and third ones are MSNEs. Note that the indifference condition is only a necessary condition, not a sufficient one. You need to ensure that the mixed strategies leading to indifference are indeed best responses.
For example, in your second profile, when Player 1 plays $frac27m+frac57d$, it is true that Player 2 is indifferent between $l$ and $r$ (payoff is $frac107$), but choosing $c$ leads to a higher payoff ($frac117>frac107$). So the best response to $frac27m+frac57d$ is the pure strategy $c$ as opposed to $frac13l+frac23r$. The other non-MSNE profiles have the same problem.
answered Mar 6 '18 at 23:37
Herr K.Herr K.
6181717
$endgroup$
$begingroup$
Thank you for that. Could you please help me identify all the mixed equilibria then? Because if we include the pure ones I am left with an even number of those.
$endgroup$
– Anonymous
Mar 7 '18 at 9:17
$begingroup$
@Anonymous instead of me giving you the answer, why don't you show what you've done and ask for verification?
$endgroup$
– Herr K.
Mar 7 '18 at 11:10
$begingroup$
I added them to the main problem now.
$endgroup$
– Anonymous
Mar 7 '18 at 15:09
$begingroup$
@Anonymous: Please see my updated answer.
$endgroup$
– Herr K.
Mar 7 '18 at 17:47
$begingroup$
Thank you again for explaining this to me.
$endgroup$
– Anonymous
Mar 7 '18 at 18:32
add a comment |
$begingroup$
Negative probability is meaningless. Getting a negative probability in your calculation simply means that there is no mixed strategy Nash equilibrium in which either player randomizes over all three of their pure strategies.
Of all the mixed strategy profiles you listed, only the first and third ones are MSNEs. Note that the indifference condition is only a necessary condition, not a sufficient one. You need to ensure that the mixed strategies leading to indifference are indeed best responses.
For example, in your second profile, when Player 1 plays $frac27m+frac57d$, it is true that Player 2 is indifferent between $l$ and $r$ (payoff is $frac107$), but choosing $c$ leads to a higher payoff ($frac117>frac107$). So the best response to $frac27m+frac57d$ is the pure strategy $c$ as opposed to $frac13l+frac23r$. The other non-MSNE profiles have the same problem.
answered Mar 6 '18 at 23:37
Herr K.Herr K.
6181717
$endgroup$
$begingroup$
Thank you for that. Could you please help me identify all the mixed equilibria then? Because if we include the pure ones I am left with an even number of those.
$endgroup$
– Anonymous
Mar 7 '18 at 9:17
$begingroup$
@Anonymous instead of me giving you the answer, why don't you show what you've done and ask for verification?
$endgroup$
– Herr K.
Mar 7 '18 at 11:10
$begingroup$
I added them to the main problem now.
$endgroup$
– Anonymous
Mar 7 '18 at 15:09
$begingroup$
@Anonymous: Please see my updated answer.
$endgroup$
– Herr K.
Mar 7 '18 at 17:47
$begingroup$
Thank you again for explaining this to me.
$endgroup$
– Anonymous
Mar 7 '18 at 18:32
add a comment |
$begingroup$
Negative probability is meaningless. Getting a negative probability in your calculation simply means that there is no mixed strategy Nash equilibrium in which either player randomizes over all three of their pure strategies.
Of all the mixed strategy profiles you listed, only the first and third ones are MSNEs. Note that the indifference condition is only a necessary condition, not a sufficient one. You need to ensure that the mixed strategies leading to indifference are indeed best responses.
For example, in your second profile, when Player 1 plays $frac27m+frac57d$, it is true that Player 2 is indifferent between $l$ and $r$ (payoff is $frac107$), but choosing $c$ leads to a higher payoff ($frac117>frac107$). So the best response to $frac27m+frac57d$ is the pure strategy $c$ as opposed to $frac13l+frac23r$. The other non-MSNE profiles have the same problem.
answered Mar 6 '18 at 23:37
Herr K.Herr K.
6181717
$endgroup$
Negative probability is meaningless. Getting a negative probability in your calculation simply means that there is no mixed strategy Nash equilibrium in which either player randomizes over all three of their pure strategies.
Of all the mixed strategy profiles you listed, only the first and third ones are MSNEs. Note that the indifference condition is only a necessary condition, not a sufficient one. You need to ensure that the mixed strategies leading to indifference are indeed best responses.
For example, in your second profile, when Player 1 plays $frac27m+frac57d$, it is true that Player 2 is indifferent between $l$ and $r$ (payoff is $frac107$), but choosing $c$ leads to a higher payoff ($frac117>frac107$). So the best response to $frac27m+frac57d$ is the pure strategy $c$ as opposed to $frac13l+frac23r$. The other non-MSNE profiles have the same problem.
answered Mar 6 '18 at 23:37
Herr K.Herr K.
6181717
edited Mar 7 '18 at 17:46
answered Mar 6 '18 at 23:37
Herr K.Herr K.
6181717
answered Mar 6 '18 at 23:37
Herr K.Herr K.
6181717
answered Mar 6 '18 at 23:37
Herr K.Herr K.
6181717
6181717
$begingroup$
Thank you for that. Could you please help me identify all the mixed equilibria then? Because if we include the pure ones I am left with an even number of those.
$endgroup$
– Anonymous
Mar 7 '18 at 9:17
$begingroup$
@Anonymous instead of me giving you the answer, why don't you show what you've done and ask for verification?
$endgroup$
– Herr K.
Mar 7 '18 at 11:10
$begingroup$
I added them to the main problem now.
$endgroup$
– Anonymous
Mar 7 '18 at 15:09
$begingroup$
@Anonymous: Please see my updated answer.
$endgroup$
– Herr K.
Mar 7 '18 at 17:47
$begingroup$
Thank you again for explaining this to me.
$endgroup$
– Anonymous
Mar 7 '18 at 18:32
add a comment |
$begingroup$
Thank you for that. Could you please help me identify all the mixed equilibria then? Because if we include the pure ones I am left with an even number of those.
$endgroup$
– Anonymous
Mar 7 '18 at 9:17
$begingroup$
@Anonymous instead of me giving you the answer, why don't you show what you've done and ask for verification?
$endgroup$
– Herr K.
Mar 7 '18 at 11:10
$begingroup$
I added them to the main problem now.
$endgroup$
– Anonymous
Mar 7 '18 at 15:09
$begingroup$
@Anonymous: Please see my updated answer.
$endgroup$
– Herr K.
Mar 7 '18 at 17:47
$begingroup$
Thank you again for explaining this to me.
$endgroup$
– Anonymous
Mar 7 '18 at 18:32
$begingroup$
Thank you for that. Could you please help me identify all the mixed equilibria then? Because if we include the pure ones I am left with an even number of those.
$endgroup$
– Anonymous
Mar 7 '18 at 9:17
$begingroup$
Thank you for that. Could you please help me identify all the mixed equilibria then? Because if we include the pure ones I am left with an even number of those.
$endgroup$
– Anonymous
Mar 7 '18 at 9:17
$begingroup$
@Anonymous instead of me giving you the answer, why don't you show what you've done and ask for verification?
$endgroup$
– Herr K.
Mar 7 '18 at 11:10
$begingroup$
@Anonymous instead of me giving you the answer, why don't you show what you've done and ask for verification?
$endgroup$
– Herr K.
Mar 7 '18 at 11:10
$begingroup$
I added them to the main problem now.
$endgroup$
– Anonymous
Mar 7 '18 at 15:09
$begingroup$
I added them to the main problem now.
$endgroup$
– Anonymous
Mar 7 '18 at 15:09
$begingroup$
@Anonymous: Please see my updated answer.
$endgroup$
– Herr K.
Mar 7 '18 at 17:47
$begingroup$
@Anonymous: Please see my updated answer.
$endgroup$
– Herr K.
Mar 7 '18 at 17:47
$begingroup$
Thank you again for explaining this to me.
$endgroup$
– Anonymous
Mar 7 '18 at 18:32
$begingroup$
Thank you again for explaining this to me.
$endgroup$
– Anonymous
Mar 7 '18 at 18:32
add a comment |
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