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Maximin zero-sum continuous game
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In my first course on optimisation, we were solving zero-sum games with a payoff matrix involving mixed strategies solutions, by reducing it to a linear program. But, what if we have a payoff function instead?
Player I and II can choose from intervals now. After scaling, both players choose a number in $[0,1]$. The payoff function is $A : [0,1]^2 to [0,1]$, assuming that the range of $A$ is bounded. You can also assume that $A in C^infty([0,1]^2)$. You are very welcome to loosen assumptions.
$$beginaligned max_f min_y & int_0^1 A(t,y)f(t) dt \mbox such that &int_0^1 f(t) dt= 1\ & qquad f(t) geq 0endaligned$$
Perhaps it is too hard to solve this analytically. I suppose one could use an $N times N$ grid (for large $N$) instead of $ [ 0,1 ] ^2 $ and then form a payoff matrix. It seems better for computational purposes anyway.
optimization calculus-of-variations
edited Jul 22 at 14:29
Rodrigo de Azevedo
12.6k41751
asked Jul 22 at 14:12
George Aliatimis
2110
add a comment |Â
up vote
0
down vote
favorite
In my first course on optimisation, we were solving zero-sum games with a payoff matrix involving mixed strategies solutions, by reducing it to a linear program. But, what if we have a payoff function instead?
Player I and II can choose from intervals now. After scaling, both players choose a number in $[0,1]$. The payoff function is $A : [0,1]^2 to [0,1]$, assuming that the range of $A$ is bounded. You can also assume that $A in C^infty([0,1]^2)$. You are very welcome to loosen assumptions.
$$beginaligned max_f min_y & int_0^1 A(t,y)f(t) dt \mbox such that &int_0^1 f(t) dt= 1\ & qquad f(t) geq 0endaligned$$
Perhaps it is too hard to solve this analytically. I suppose one could use an $N times N$ grid (for large $N$) instead of $ [ 0,1 ] ^2 $ and then form a payoff matrix. It seems better for computational purposes anyway.
optimization calculus-of-variations
edited Jul 22 at 14:29
Rodrigo de Azevedo
12.6k41751
asked Jul 22 at 14:12
George Aliatimis
2110
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In my first course on optimisation, we were solving zero-sum games with a payoff matrix involving mixed strategies solutions, by reducing it to a linear program. But, what if we have a payoff function instead?
Player I and II can choose from intervals now. After scaling, both players choose a number in $[0,1]$. The payoff function is $A : [0,1]^2 to [0,1]$, assuming that the range of $A$ is bounded. You can also assume that $A in C^infty([0,1]^2)$. You are very welcome to loosen assumptions.
$$beginaligned max_f min_y & int_0^1 A(t,y)f(t) dt \mbox such that &int_0^1 f(t) dt= 1\ & qquad f(t) geq 0endaligned$$
Perhaps it is too hard to solve this analytically. I suppose one could use an $N times N$ grid (for large $N$) instead of $ [ 0,1 ] ^2 $ and then form a payoff matrix. It seems better for computational purposes anyway.
optimization calculus-of-variations
edited Jul 22 at 14:29
Rodrigo de Azevedo
12.6k41751
asked Jul 22 at 14:12
George Aliatimis
2110
In my first course on optimisation, we were solving zero-sum games with a payoff matrix involving mixed strategies solutions, by reducing it to a linear program. But, what if we have a payoff function instead?
Player I and II can choose from intervals now. After scaling, both players choose a number in $[0,1]$. The payoff function is $A : [0,1]^2 to [0,1]$, assuming that the range of $A$ is bounded. You can also assume that $A in C^infty([0,1]^2)$. You are very welcome to loosen assumptions.
$$beginaligned max_f min_y & int_0^1 A(t,y)f(t) dt \mbox such that &int_0^1 f(t) dt= 1\ & qquad f(t) geq 0endaligned$$
Perhaps it is too hard to solve this analytically. I suppose one could use an $N times N$ grid (for large $N$) instead of $ [ 0,1 ] ^2 $ and then form a payoff matrix. It seems better for computational purposes anyway.
optimization calculus-of-variations
edited Jul 22 at 14:29
Rodrigo de Azevedo
12.6k41751
asked Jul 22 at 14:12
George Aliatimis
2110
edited Jul 22 at 14:29
Rodrigo de Azevedo
12.6k41751
edited Jul 22 at 14:29
Rodrigo de Azevedo
12.6k41751
edited Jul 22 at 14:29
Rodrigo de Azevedo
12.6k41751
12.6k41751
asked Jul 22 at 14:12
George Aliatimis
2110
asked Jul 22 at 14:12
George Aliatimis
2110
asked Jul 22 at 14:12
George Aliatimis
2110
2110
add a comment |Â
add a comment |Â
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