Game theory - a shooter problem
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I just met an interesting question but did not know how to approach it...
Suppose two gunmen (A and B) are moving in a straight line towards each other in a fixed speed. Each of them only have one bullet in the gun. The distance between them is $1$ meter. They have the same probability of hitting each other when shooting. The probability of a shooter knocking the other out, is $1-x$, where $x$ is the distance between them.
The thing is, A has a radar, letting him check whether B has fired. So you can imagine that if he knows B already shot and missed, he will keep walking until he has probability $1$ of shooting his opponent. We assume that once someone decides to shoot and is successful in hitting his opponent (shooting takes no time), the victim will immediately die without any ability to shoot back.
According to the final situation, their utility is defined as follows: if $A$ is alive while $B$ is dead, $A$ gets $U_{1}$, $B$ gets $0$ (similar if $A$ dies and $B$ survives); if both die, they both get $U_{2}$; if they all survive, they both get $U_{3}$. We have, $U_1>U_3>U_2$>0.
What is their best strategy respectively? Assume that they want to maximize their utility first, and if two strategies provides same utility, they prefer the one that gives the opponent less utility.
Thank you for your help! I know the problem is really long.
game-theory conditional-probability
asked Nov 11 at 7:34
illusion
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up vote
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down vote
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I just met an interesting question but did not know how to approach it...
Suppose two gunmen (A and B) are moving in a straight line towards each other in a fixed speed. Each of them only have one bullet in the gun. The distance between them is $1$ meter. They have the same probability of hitting each other when shooting. The probability of a shooter knocking the other out, is $1-x$, where $x$ is the distance between them.
The thing is, A has a radar, letting him check whether B has fired. So you can imagine that if he knows B already shot and missed, he will keep walking until he has probability $1$ of shooting his opponent. We assume that once someone decides to shoot and is successful in hitting his opponent (shooting takes no time), the victim will immediately die without any ability to shoot back.
According to the final situation, their utility is defined as follows: if $A$ is alive while $B$ is dead, $A$ gets $U_{1}$, $B$ gets $0$ (similar if $A$ dies and $B$ survives); if both die, they both get $U_{2}$; if they all survive, they both get $U_{3}$. We have, $U_1>U_3>U_2$>0.
What is their best strategy respectively? Assume that they want to maximize their utility first, and if two strategies provides same utility, they prefer the one that gives the opponent less utility.
Thank you for your help! I know the problem is really long.
game-theory conditional-probability
asked Nov 11 at 7:34
illusion
11
New contributor
illusion is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
At 1 meter, the radar seems unnecessary. Is it option 1: that there is certainty of knowing the other shot? Or is it option 2: that only 1 can be certain if the other shot? In other words, option 2 is: if player 1 shoots, then player 2 knows player 1 shot, but if player 2 shoots then player 1 is uncertain is player 2 shot.
– Jay Schyler Raadt
23 hours ago
@JaySchylerRaadt Thanks for pointing out, it is option 2. Only A has the radar but B does not.
– illusion
15 hours ago
Is $U_2>0$? How big is a step = how fast does x change? Does it speed up or slow down as they get closer or is p(x) uniformly distributed across x?
– Jay Schyler Raadt
14 hours ago
@JaySchylerRaadt All 3 utilities are positive; they are walking to each other in a constant speed
– illusion
9 hours ago
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I just met an interesting question but did not know how to approach it...
Suppose two gunmen (A and B) are moving in a straight line towards each other in a fixed speed. Each of them only have one bullet in the gun. The distance between them is $1$ meter. They have the same probability of hitting each other when shooting. The probability of a shooter knocking the other out, is $1-x$, where $x$ is the distance between them.
The thing is, A has a radar, letting him check whether B has fired. So you can imagine that if he knows B already shot and missed, he will keep walking until he has probability $1$ of shooting his opponent. We assume that once someone decides to shoot and is successful in hitting his opponent (shooting takes no time), the victim will immediately die without any ability to shoot back.
According to the final situation, their utility is defined as follows: if $A$ is alive while $B$ is dead, $A$ gets $U_{1}$, $B$ gets $0$ (similar if $A$ dies and $B$ survives); if both die, they both get $U_{2}$; if they all survive, they both get $U_{3}$. We have, $U_1>U_3>U_2$>0.
What is their best strategy respectively? Assume that they want to maximize their utility first, and if two strategies provides same utility, they prefer the one that gives the opponent less utility.
Thank you for your help! I know the problem is really long.
game-theory conditional-probability
asked Nov 11 at 7:34
illusion
11
New contributor
illusion is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I just met an interesting question but did not know how to approach it...
Suppose two gunmen (A and B) are moving in a straight line towards each other in a fixed speed. Each of them only have one bullet in the gun. The distance between them is $1$ meter. They have the same probability of hitting each other when shooting. The probability of a shooter knocking the other out, is $1-x$, where $x$ is the distance between them.
The thing is, A has a radar, letting him check whether B has fired. So you can imagine that if he knows B already shot and missed, he will keep walking until he has probability $1$ of shooting his opponent. We assume that once someone decides to shoot and is successful in hitting his opponent (shooting takes no time), the victim will immediately die without any ability to shoot back.
According to the final situation, their utility is defined as follows: if $A$ is alive while $B$ is dead, $A$ gets $U_{1}$, $B$ gets $0$ (similar if $A$ dies and $B$ survives); if both die, they both get $U_{2}$; if they all survive, they both get $U_{3}$. We have, $U_1>U_3>U_2$>0.
What is their best strategy respectively? Assume that they want to maximize their utility first, and if two strategies provides same utility, they prefer the one that gives the opponent less utility.
Thank you for your help! I know the problem is really long.
game-theory conditional-probability
game-theory conditional-probability
asked Nov 11 at 7:34
illusion
11
New contributor
illusion is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 11 at 7:34
illusion
11
New contributor
illusion is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 9 hours ago
asked Nov 11 at 7:34
illusion
11
New contributor
illusion is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 11 at 7:34
illusion
11
asked Nov 11 at 7:34
illusion
11
11
New contributor
illusion is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
illusion is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
illusion is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
At 1 meter, the radar seems unnecessary. Is it option 1: that there is certainty of knowing the other shot? Or is it option 2: that only 1 can be certain if the other shot? In other words, option 2 is: if player 1 shoots, then player 2 knows player 1 shot, but if player 2 shoots then player 1 is uncertain is player 2 shot.
– Jay Schyler Raadt
23 hours ago
@JaySchylerRaadt Thanks for pointing out, it is option 2. Only A has the radar but B does not.
– illusion
15 hours ago
Is $U_2>0$? How big is a step = how fast does x change? Does it speed up or slow down as they get closer or is p(x) uniformly distributed across x?
– Jay Schyler Raadt
14 hours ago
@JaySchylerRaadt All 3 utilities are positive; they are walking to each other in a constant speed
– illusion
9 hours ago
add a comment |
At 1 meter, the radar seems unnecessary. Is it option 1: that there is certainty of knowing the other shot? Or is it option 2: that only 1 can be certain if the other shot? In other words, option 2 is: if player 1 shoots, then player 2 knows player 1 shot, but if player 2 shoots then player 1 is uncertain is player 2 shot.
– Jay Schyler Raadt
23 hours ago
@JaySchylerRaadt Thanks for pointing out, it is option 2. Only A has the radar but B does not.
– illusion
15 hours ago
Is $U_2>0$? How big is a step = how fast does x change? Does it speed up or slow down as they get closer or is p(x) uniformly distributed across x?
– Jay Schyler Raadt
14 hours ago
@JaySchylerRaadt All 3 utilities are positive; they are walking to each other in a constant speed
– illusion
9 hours ago
At 1 meter, the radar seems unnecessary. Is it option 1: that there is certainty of knowing the other shot? Or is it option 2: that only 1 can be certain if the other shot? In other words, option 2 is: if player 1 shoots, then player 2 knows player 1 shot, but if player 2 shoots then player 1 is uncertain is player 2 shot.
– Jay Schyler Raadt
23 hours ago
At 1 meter, the radar seems unnecessary. Is it option 1: that there is certainty of knowing the other shot? Or is it option 2: that only 1 can be certain if the other shot? In other words, option 2 is: if player 1 shoots, then player 2 knows player 1 shot, but if player 2 shoots then player 1 is uncertain is player 2 shot.
– Jay Schyler Raadt
23 hours ago
@JaySchylerRaadt Thanks for pointing out, it is option 2. Only A has the radar but B does not.
– illusion
15 hours ago
@JaySchylerRaadt Thanks for pointing out, it is option 2. Only A has the radar but B does not.
– illusion
15 hours ago
Is $U_2>0$? How big is a step = how fast does x change? Does it speed up or slow down as they get closer or is p(x) uniformly distributed across x?
– Jay Schyler Raadt
14 hours ago
Is $U_2>0$? How big is a step = how fast does x change? Does it speed up or slow down as they get closer or is p(x) uniformly distributed across x?
– Jay Schyler Raadt
14 hours ago
@JaySchylerRaadt All 3 utilities are positive; they are walking to each other in a constant speed
– illusion
9 hours ago
@JaySchylerRaadt All 3 utilities are positive; they are walking to each other in a constant speed
– illusion
9 hours ago
add a comment |
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illusion is a new contributor. Be nice, and check out our Code of Conduct.
illusion is a new contributor. Be nice, and check out our Code of Conduct.
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At 1 meter, the radar seems unnecessary. Is it option 1: that there is certainty of knowing the other shot? Or is it option 2: that only 1 can be certain if the other shot? In other words, option 2 is: if player 1 shoots, then player 2 knows player 1 shot, but if player 2 shoots then player 1 is uncertain is player 2 shot.
– Jay Schyler Raadt
23 hours ago
@JaySchylerRaadt Thanks for pointing out, it is option 2. Only A has the radar but B does not.
– illusion
15 hours ago
Is $U_2>0$? How big is a step = how fast does x change? Does it speed up or slow down as they get closer or is p(x) uniformly distributed across x?
– Jay Schyler Raadt
14 hours ago
@JaySchylerRaadt All 3 utilities are positive; they are walking to each other in a constant speed
– illusion
9 hours ago