Excellent work.
I have just invented a simple game for miners. I believe it clarifies the problem with the proportional reward system (at least for people who are more comfortable with words than with formulae). If you think that I've missed the point or made a subtle error then please correct me. Also, feel free to use/extend this as you see fit.
This is a 4-player game. There is a central pot into which each player must deposit a bitcoin to play. A fair coin is flipped repeatedly. If a tails comes up at any point then the game ends immediately and the 4 players split what is left in the pot evenly.
If the coin comes up heads on the first flip then player 1 takes a bitcoin from the pot.
After the second head, player 2 takes a bitcoin.
After the third head, player 3 takes a bitcoin.
After the fourth head, player 4 takes a bitcoin.This game comes with the question:
If you were invited to play this game, which player would you like to be?Of course, the idea is that:
- Honest 24/7 proportional miners will be happy as player 3 or 4.
- People who have an ethical problem with pool hopping will be disgusted with the game and refuse to have any part in it.
- Pool hoppers will ask to be player 1.
Thanks for posting your game teukon - it kept me busy for quite a while. I like the fact that it's easily understandable, and it's something that makes for a simple thought experiment - and it's clear to see that it's unfair. It's not a good analogy of proportional pool hopping vs full time mining since the sequential payout part = which I think you intend to mimic the skew toward smaller block sizes - has a much too large an effect. But generally it's a good introduction to proportional pool mining, moral judgements aside.
It got me to thinking for the last day or so - how could I change your game to be a better analogy of prop pool mining and at the same time keep it a simple thought experiment? The short answer is that I couldn't. But your game did inspire me, and I extended it into a dice game that mimics proportional pool mining quite well. It's no longer a good thought experiment, but I think it would be a good teaching tool.
DicecoinDicecoin is a dice game played at the Zero Sum Casino. It uses a 10 sided die, and there can be any number of players.
Rules:
1. A die is thrown until a '10' appears.
2. Each throw costs a total of $10, which is split evenly among all players present for the throw.
3. When the '10' appears, the prize of $100 is split proportionally amongst all players, per throw played.
4. Players can leave or join a game at any time.
5. There must be more than one player for a game to continue.
Assumptions:
1. Gamblers at the Zero Sum Casino are just as irrational about gambling as anyone else - they will stay in a game that lasts a long time and costing a lot because they want to recoup or at least offset their losses. They also jump into a game in progress because it has been going on for a while and "it's due for a '10' ".
2. More rational players also play to the end of a game because they know that at the Zero Sum Casino game rules are written so that on average their wins = losses.
3. There are never any shortage of players.
Example: A game lasts for 20 throws. There are ten players and all players stay until the end of the game and no one else joins. Each throw costs each player:
$10 divided by the number of players = $1, for a total cost of $20 per player.
Each player wins $100 divided amongst the players proportional to the number of throws they played:
$100/10/20 = $0.50 per player per throw played = $10 per player. The total profit = winnings - loss = -$10. So for this example, the players lose $10 each.
Question: is there a winning strategy for this gambling game? What is it?
Extra credit questions: What is the percentage improvement in winnings (winnings for your strategy/winnings for no strategy), and what is your expected profits in dollars per game? Frame your answer in mathematical terms as per
How to hop 6: Basic probability for bitcoin miners.
So here's a competition for everyone:
Post your strategy here, with your reasoning. If you can answer the first question, have the winning strategy and your reasoning is sound, you'll
win 1 btc from me. The winner for the extra credit questions will be the poster with the simplest, clearest and most informative explanation, will
win 4btc from me. Good chartporn will be helpful but not necessary.
Rules:
- The winner will be the first poster with a correct answer before midnight 19-10-2011 UTC.
- You can post multiple answers if you think your first answer might be incorrect, but only the most recent answer will be considered.
- Feel free to discuss each other's answers.
I'll post hints here from time to time.
A read of
How to hop 5: Back to basics and
How to hop 6: Basic probability for bitcoin miners might be helpful.
Thanks again for such a great idea, teukon!
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