#1
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Countries Invading
Country x is planning to attack country y, and country y is anticipating the attack. Country x can either attack by land or by sea. Country y can either prepare for a land defense or a sea defense. Both countries must choose either an all land or all sea strategy, they may not divide their forces. The following are the probabilities of a successful invasion according to both strategies used.
If x launches an attack by sea and y prepares a defense by sea the probability of a successful invasion is 80%. If x launches an attack by sea and y prepares a defense by land the probability of a successful invasion is 100%. If x launches an attack by land and y prepares a defense by land the probability of a successful invasion is 60%. If x launches an attack by land and y prepares a defense by sea the probability of a successful invasion is 100%. What should the strategy of country x be, assuming the goal is to maximize the probability of a successful invasion? Assume the goal of country y to be to minimize the probability of a successful invasion. What is the final probability of a successful invasion assuming both utilize an optimal strategy? [I will post the results after the answers are in] |
#2
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Re: Countries Invading
Game theory 101. First week. Dominant strategy available. Thus no need to randomize via min-max. Yawn.
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#3
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Re: Countries Invading
[ QUOTE ]
I'm smarter than you. [/ QUOTE ] Isn't that what you really wanted to say? |
#4
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Re: Countries Invading
No it wasn't. Just that this isn't an interesting problem.
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#5
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Re: Countries Invading
I never took formal game theory, but I did take micro econ (which the professor tried to teach us game theory) so here goes: Thoughts in white
<font color="white"> assigning probability to attack by land for X as q, and probability for defense of land for Y as p we have the EV of our attack set as 0.6*p*q+1*p*(1-q)+1*(1-p)*q+0.8*(1-p)*(1-q) which comes out to 0.6*p*q+p-p*q-p*q+q+0.8-0.8*p-0.8*q+0.8*p*q Which comes out to: -0.6*p*q+0.2*p+0.2*q+0.8 Now I'm treading in unknown waters. I was going to take a first derivative with respect to p but that doesn't seem to get me anywhere to solving for p in terms of q (rather it seems to give me q to minimize p!) So we take a guess and rearrange it as such: 0.2*p-0.6*q*p+0.2*q+0.8 = 0.2*p+q*(-0.6*p+0.2)+0.8 So thus I feel like I want to render Y's choice irrelevant which means setting -0.6*p+0.2 =0 or p=1/3 resulting in EV = 0.8+.2*1/3=0.866666666 To do the same for figuring out q: 0.2*q-0.6*q*p+0.2*p+0.8= 0.2*q+p(-0.6*q+0.2)+0.8 Setting -0.6*q+0.2 = 0 we get 1/3 for q as well resulting in EV strategy 0.8666666 Optimal strategy for both is 1/3 land 2/3 sea, resulting in a victory probability of 0.8666666666 (or a failure probability of .1333333333) </font> PS, cool graphic |
#6
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Re: Countries Invading
[ QUOTE ]
Dominant strategy available. [/ QUOTE ] There is no dominant strategy that I can see for either player. |
#7
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Re: Countries Invading
[ QUOTE ]
Game theory 101. First week. Dominant strategy available. Thus no need to randomize via min-max. Yawn. [/ QUOTE ] Just a suggestion: Try not to make incorrect observations while you are implying something is so easy that it's boring |
#8
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Re: Countries Invading
udontknow's solution looks right to me.
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#9
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Re: Countries Invading
"Both countries must choose either an all land or all sea strategy, they may not divide their forces."
Did udontknow's answer not violate this directive? |
#10
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Re: Countries Invading
The strategy isn't to send 1/3 across land and 2/3 across sea, but it is to roll a three sided die, and if it comes up 1, send it across land, and any other number, across sea.
(Yes, I know there's no such thing as a three sided die) |
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