Mixing
Find the threshold strategy of a game with incomplete information
$begingroup$
I have a question on how to find the equilibrium outcomes of the following game as a function of $(epsilon_1, epsilon_2)$.
The game
There are 2 players.
Let $Y_i$ denote the action of player $i$ for each $iin {1,2}$.
For each $iin {1,2}$, player $i$ chooses between action $1$ or $0$.
For each $iin {1,2}$, if player $i$ chooses action $1$, she gets $-frac{1}{2}Y_j+epsilon_i$ as payoff, with $jneq iin {1,2}$
For each $iin {1,2}$, if player $i$ chooses action $0$, she gets $0$ as payoff.
For each $iin {1,2}$, $epsilon_i$ is private information of player $i$.
Players play Bayesian Nash Equilibrium.
We assume that $(epsilon_1, epsilon_2)$ are i.i.d. uniformly distributed in $[-1,1]$.
Question
Show that, for each $iin {1,2}$, player $i$ chooses $1$ if and only if $epsilon_igeq frac{1}{5}$
My thoughts: I really don't know how to answer the question. I tried with solving the following fixed point problem:
$$
begin{cases}
alpha_1=PrBig[(epsilon_1-frac{1}{2})times alpha_2+epsilon_1times (1-alpha_2)geq 0Big]\
alpha_2=PrBig[(epsilon_2-frac{1}{2})times alpha_1+epsilon_2times (1-alpha_1)geq 0Big]\
end{cases}
$$
where $alpha_i$ is the probability that player $i$ playes $i$ and inside the square brackets we have the expected profit of each player.
The system gives $alpha_1=alpha_2=frac{2}{5}$ and it doesn't seem to give/suggest the threshold that is in the question.
game-theory nash-equilibrium
asked Jan 24 at 12:47
STFSTF
541422
$endgroup$
add a comment |
$begingroup$
I have a question on how to find the equilibrium outcomes of the following game as a function of $(epsilon_1, epsilon_2)$.
The game
There are 2 players.
Let $Y_i$ denote the action of player $i$ for each $iin {1,2}$.
For each $iin {1,2}$, player $i$ chooses between action $1$ or $0$.
For each $iin {1,2}$, if player $i$ chooses action $1$, she gets $-frac{1}{2}Y_j+epsilon_i$ as payoff, with $jneq iin {1,2}$
For each $iin {1,2}$, if player $i$ chooses action $0$, she gets $0$ as payoff.
For each $iin {1,2}$, $epsilon_i$ is private information of player $i$.
Players play Bayesian Nash Equilibrium.
We assume that $(epsilon_1, epsilon_2)$ are i.i.d. uniformly distributed in $[-1,1]$.
Question
Show that, for each $iin {1,2}$, player $i$ chooses $1$ if and only if $epsilon_igeq frac{1}{5}$
My thoughts: I really don't know how to answer the question. I tried with solving the following fixed point problem:
$$
begin{cases}
alpha_1=PrBig[(epsilon_1-frac{1}{2})times alpha_2+epsilon_1times (1-alpha_2)geq 0Big]\
alpha_2=PrBig[(epsilon_2-frac{1}{2})times alpha_1+epsilon_2times (1-alpha_1)geq 0Big]\
end{cases}
$$
where $alpha_i$ is the probability that player $i$ playes $i$ and inside the square brackets we have the expected profit of each player.
The system gives $alpha_1=alpha_2=frac{2}{5}$ and it doesn't seem to give/suggest the threshold that is in the question.
game-theory nash-equilibrium
asked Jan 24 at 12:47
STFSTF
541422
$endgroup$
add a comment |
$begingroup$
I have a question on how to find the equilibrium outcomes of the following game as a function of $(epsilon_1, epsilon_2)$.
The game
There are 2 players.
Let $Y_i$ denote the action of player $i$ for each $iin {1,2}$.
For each $iin {1,2}$, player $i$ chooses between action $1$ or $0$.
For each $iin {1,2}$, if player $i$ chooses action $1$, she gets $-frac{1}{2}Y_j+epsilon_i$ as payoff, with $jneq iin {1,2}$
For each $iin {1,2}$, if player $i$ chooses action $0$, she gets $0$ as payoff.
For each $iin {1,2}$, $epsilon_i$ is private information of player $i$.
Players play Bayesian Nash Equilibrium.
We assume that $(epsilon_1, epsilon_2)$ are i.i.d. uniformly distributed in $[-1,1]$.
Question
Show that, for each $iin {1,2}$, player $i$ chooses $1$ if and only if $epsilon_igeq frac{1}{5}$
My thoughts: I really don't know how to answer the question. I tried with solving the following fixed point problem:
$$
begin{cases}
alpha_1=PrBig[(epsilon_1-frac{1}{2})times alpha_2+epsilon_1times (1-alpha_2)geq 0Big]\
alpha_2=PrBig[(epsilon_2-frac{1}{2})times alpha_1+epsilon_2times (1-alpha_1)geq 0Big]\
end{cases}
$$
where $alpha_i$ is the probability that player $i$ playes $i$ and inside the square brackets we have the expected profit of each player.
The system gives $alpha_1=alpha_2=frac{2}{5}$ and it doesn't seem to give/suggest the threshold that is in the question.
game-theory nash-equilibrium
asked Jan 24 at 12:47
STFSTF
541422
$endgroup$
I have a question on how to find the equilibrium outcomes of the following game as a function of $(epsilon_1, epsilon_2)$.
The game
There are 2 players.
Let $Y_i$ denote the action of player $i$ for each $iin {1,2}$.
For each $iin {1,2}$, player $i$ chooses between action $1$ or $0$.
For each $iin {1,2}$, if player $i$ chooses action $1$, she gets $-frac{1}{2}Y_j+epsilon_i$ as payoff, with $jneq iin {1,2}$
For each $iin {1,2}$, if player $i$ chooses action $0$, she gets $0$ as payoff.
For each $iin {1,2}$, $epsilon_i$ is private information of player $i$.
Players play Bayesian Nash Equilibrium.
We assume that $(epsilon_1, epsilon_2)$ are i.i.d. uniformly distributed in $[-1,1]$.
Question
Show that, for each $iin {1,2}$, player $i$ chooses $1$ if and only if $epsilon_igeq frac{1}{5}$
My thoughts: I really don't know how to answer the question. I tried with solving the following fixed point problem:
$$
begin{cases}
alpha_1=PrBig[(epsilon_1-frac{1}{2})times alpha_2+epsilon_1times (1-alpha_2)geq 0Big]\
alpha_2=PrBig[(epsilon_2-frac{1}{2})times alpha_1+epsilon_2times (1-alpha_1)geq 0Big]\
end{cases}
$$
where $alpha_i$ is the probability that player $i$ playes $i$ and inside the square brackets we have the expected profit of each player.
The system gives $alpha_1=alpha_2=frac{2}{5}$ and it doesn't seem to give/suggest the threshold that is in the question.
game-theory nash-equilibrium
game-theory nash-equilibrium
asked Jan 24 at 12:47
STFSTF
541422
asked Jan 24 at 12:47
STFSTF
541422
asked Jan 24 at 12:47
STFSTF
541422
asked Jan 24 at 12:47
STFSTF
541422
asked Jan 24 at 12:47
STFSTF
541422
541422
add a comment |
add a comment |
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