Parachute game solved (?) -- re: Gamesapalooza

[EverettKings]

I was reading the gamesapalooza game thread here and think Ive figured out the parachute game, which turns out to be kind of lame once everybody knows how to play (if my thinking is correct). I've put the rules here for convenience:
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Game: Parachute Drop

Object of the game: Be the last player standing. At least one player is eliminated each round.

Rules: Each player starts with 100 HP. You are jumping from an airplane. Each round, you will choose a number between 0 and the number of HP you have to inflict upon yourself as damage. (Basically, when you pull your parachute cord.)

The less HP you take, the slower you float to the ground. The last player each round to hit the ground is shot. Then a new round is played and numbers you chose in previous rounds are subtracted from your available 100 HP.

If you exceed the maximum number of HP you have available, you SPLAT and die.

If two people tie for the slowest, they are both shot and share the rank for standings purposes.

Game play: Submit your HP number to the moderator for each round by the specified deadline. Failure to submit for the first round results in 0 championship points being awarded. Failure to submit in subsequent rounds results in a default pick of "0 HP."

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Also:

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Rules clarification: Someone will be shot, even if people are eliminated due to non-submission.

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and
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RULES CLARIFICATIONS

Someone will be shot after each jump, regardless of anyone splatting.

If there is a tie for the lowest submission, both players will be shot. They will share a place and split the corresponding championship points. (i.e. if two players tie for first place, they will each receive 22.5 championship points)

Splats happen before shootings. This means that that the person shot will rank above the person who splatted.

Splats happen in order that they hit the ground. A person who submits 5 and splats would be ranked above a person who submits 3 and splats. (They are ranked in the order they die.)

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So we're at the first jump. Obviously, if you bid 0, you will die that round and lose the game in last place or maybe tie some other moron for last. So nobody should bid 0.

If everyone understands this, then since nobody is bidding zero, anybody who bids 1 will die for sure and lose the game. So nobody should bid 1 either.

You see where this is going, and the logic holds until everyone is bidding 99 (because 98 is a sure death).

If everyone's bidding 99, then you will NOT bid 100 because you will 'splat' and die before everyone else and lose. You also don't bid less than 99 because you will die and not even tie. So optimal strategy is for everyone to bid 99 and split the win.


The only reason not to bid 99 is if several people do not understand this and you can count on them bidding less than 99, allowing you to get away with bidding 15 or whatever. That's why the game worked for gamesapalooza, because people either didn't think about it or counted on at least one other person to not think about it.


amirite?

[bravos1]

well in your scenario, no one wins either.. all bid 99 and there is a 10 way tie, hence all 10 get shot and no one wins.

[EverettKings]

which is optimal since its a fair game and nobody gains on anyone else. given that everyone else is playing optimally as I define it, your best option is to play along.

[LuckayLuck]

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The only reason not to bid 99 is if several people do not understand this and you can count on them bidding less than 99, allowing you to get away with bidding 15 or whatever. That's why the game worked for gamesapalooza, because people either didn't think about it or counted on at least one other person to not think about it.

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this is why I didn't vote for 99
I'm a genius indeed

[Neil S]

Everett: your 'solution' is unfortunately lacking because you don't take into account the possibility of mixed strategies.

[EverettKings]

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Everett: your 'solution' is unfortunately lacking because you don't take into account the possibility of mixed strategies.

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Actually my solution defines optimal. Do you see why?


I did consider mixed strategies, and there may be multiple optima, but can you find one? I feel like all mixed strategies will collapse to this pure one by sheer dominance.

[bravos1]

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Everett: your 'solution' is unfortunately lacking because you don't take into account the possibility of mixed strategies.

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Or idiots playing

[Neil S]

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Everett: your 'solution' is unfortunately lacking because you don't take into account the possibility of mixed strategies.

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Actually my solution defines optimal. Do you see why?


I did consider mixed strategies, and there may be multiple optima, but can you find one? I feel like all mixed strategies will collapse to this pure one by sheer dominance.

[/ QUOTE ]Well, it depends on how we define the payoffs. If everyone goes 99 then splats round 2, what's the payoff for that versus a single winner?

[EverettKings]

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Everett: your 'solution' is unfortunately lacking because you don't take into account the possibility of mixed strategies.

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Actually my solution defines optimal. Do you see why?


I did consider mixed strategies, and there may be multiple optima, but can you find one? I feel like all mixed strategies will collapse to this pure one by sheer dominance.

[/ QUOTE ]Well, it depends on how we define the payoffs. If everyone goes 99 then splats round 2, what's the payoff for that versus a single winner?

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It's irrelevant actually. The rules say that the points for first place are split evenly amongst the players that tie for that place. But since points only matter in a relative sense, if you can tie everyone else it doesn't matter how many points are awarded for it.

I'll assume that splitting first 10 ways is better than voluntarily taking 10th and getting all of the points for that.

99 has to be optimal. Assume that everyone is bidding 99. Your best strategy is to bid 99. Therefore it is an optimal solution.

[Spook]

Rules clarification.
Points will be awarded based on the round you are eliminated in.
#round*(#entrants-#live players at start of round)