Game Theory Problem! Inside...

Discussion in 'The Lounge' started by Rix, Dec 17, 2009.

?

Who?

  1. Fred

    0 vote(s)
    0.0%
  2. Bob

    2 vote(s)
    100.0%
  1. Rix

    36

    Nov 8, 2009
    189
    5
    36
    You are entering a 2v2 first person shooter tournament, first to 10 kills wins. You average 1 kill and 4 deaths per minute. You're choosing a team mate to enter the tournament with. Bob averages 1 kill and 0 deaths a minute. Fred averages 7 kills and 8 deaths a minute. Trying to maximize the probability of your team winning the tournament, who should you choose (and why)?

    So far, nobody has answered the question correctly. Obviously, many have guessed the correct answer but none have understand why it's correct.

    I created this problem entirely myself. It raises an important issue in game theory: is having a player with a higher K/D ratio and higher Kills minus Deaths always the better option?

    I posted this here in case anyone's interested in game theory.
     

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  2. milrtime83

    milrtime83 Well-Known Member
    78

    Nov 4, 2009
    668
    74
    78
    Fred seems like the right answer.

    With Bob you are averaging more deaths per kill. At a point when your team has 24 kills you would have 48 deaths if Bob were on your team but only 36 if Fred were on your team.

    Edit: I picked the wrong option in the poll, I meant to pick Fred.
     
  3. milrtime83

    milrtime83 Well-Known Member
    78

    Nov 4, 2009
    668
    74
    78
    It is 10 kills per team. So you need to look at your teams kill/death rate and can't just compare the ratios of potential partners. Sure Bob won't die, but you dieing 4 times per minute will lose the game long before you win because your team is getting so few kills. With Bob your teams ratio is 2kills/4deaths or 0.5. With Fred it is 8kills/12deaths or 0.667. In both situations you are likely to lose but your odds are slightly better with Fred.

    edit: How did this post get above KlaymenDK's? It was a response to his post when I originally posted it.
     
  4. KlaymenDK

    KlaymenDK Well-Known Member
    143

    May 29, 2009
    1,217
    130
    143
    Denmark
    Bob seems like the right answer. With Fred you are averaging more deaths per kill.

    I don't know if:
    -- if the 10 kill goal is per team or for a single player,
    -- we're supposed to deal with fractions here (or does the game last any number of whole minutes?),
    -- if the kill/death rate is linear (do I kill first, then die four times, or what?).

    It's not only about getting kills, it's also about denying your opponent, so you need to not die. Since you are the common factor in both scenarios, you are essentially irrelevant. So, comparing Bob and Fred it it evident that every minute, Bob is gaining one point (kills minus deaths) while Fred is actually losing one point. Based on that, I'd pick Bob.
     
  5. savajio

    savajio Member
    16

    Dec 7, 2009
    13
    8
    16
    You should realize you can't win and Bob and Fred should team up. :p
     
  6. Rix

    36

    Nov 8, 2009
    189
    5
    36
    Fred is the correct answer, here's why:

    With Fred, your team's average is 8 kills and 12 deaths a minute. With Bob its 2 kills and 4 deaths a minute. Note the ratios. Let's work out the exact probability that each hypothetical team would win.

    Using the binomial theorem, an event is a kill or a death. In the game is is guaranteed that there will be at least 10 events (could win with 10-0) and at most 19 events (because the score could go to 10-9). We calculate the probability of Fred's team winning by calculating the chance of getting 10 or more kills in the next 19 events. It doesn't matter that it may be all 19 kills and the game would have ended by then, it's just how it's worked out. We do the same for bobs team. Obviously we need the event success rate which is 0.4 for Bob's team and 0.333... for Fred's team. With Fred, your chances of winning are 18.6092.%, with Bob it's 6.47662%, a big difference. You're almost three times more likely to win with Fred than Bob.

    If anyone has mathematica and in interested in this problem, use this formula
    Manipulate[ Sum[(z^i)*(1 - z)^(y - i)*(y!/(i!*(y - i)!)), {i, x, y}], {x, 1, 20,
    1}, {y, 1, 40, 1}, {z, 0, 1}]
    x is the required kills to win (10 in this case), y is the maximum number of events that can occur (19 in this case). You can always calculate y by adding the total number of kills that each team needs to win then subtracting 1 (you don't have to have the scores even). z is the probability of event success. It then outputs the probability your team wins.

    milrtime83 is the winner of this challenge. To the victor, the spoils :D
     

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