CORDIAL MINUET ENSEMBLE

Love that Rounders GIF, Jere!

A good example here is a game where you have a 1/1M chance of winning a billion dollars, you can play a game once a day for $1.

The EV of a single play of this game is $999 (bet $1, and win $1000 on average).  But intuitively, no one would play this game.  The positive EV there is "off" somehow, intuitively.  And it's off in reality, because if you play the game every day for the rest of your life, you probably won't win, and you'll be out thousands of dollars.

Suppose you were allowed to play multiple times in one day, and you had $100K, your life savings.  You have an "edge" here... would you go all-in and play the game 100K times?  You'd expect to walk away with $99M by doing this.

This is a variation of this paradox:

http://en.wikipedia.org/wiki/St._Petersburg_paradox

Bernoulli solved this paradox by realizing you should maximize the geometric mean, not the EV.  The geometric mean is the same as the Kelly Criterion.

ALL IN means that there will be at least one 0 term in your geometric mean if you don't have 100% chance of winning, pulling the whole thing to 0.  Essentially, "do this long enough, and you will eventually be broke."

What you are saying is that you are playing for a fraction of your bankroll anyway, and that's true.  By viewing one table in isolation, because it will end after one all-in win (or loss) for you, and there are no re-buys at that table, the same "long term" reasoning doesn't apply.

But if you think you have an edge going INTO a table, and you don't want to eventually go broke, you should not put your whole bankroll into that one table.  See the Rounders GIF.

Going all-in in a situation where you have a known edge (like you have a 36 in turn 1, or AA in poker) is actually not optimal in some senses.

Yes, in terms of the isolated case, the more you bet in a positive edge situation, the more you will win on average in those situations.  Betting more increased your expected payoff.

BUT, if you take that to the limit, you arrive at "I will win the most, on average, if I bet everything I have."  Because an edge is not a guarantee, you WILL lose often enough in those positive edge situations.  That means if you play this way, you will experience ruin very quickly.  Yes, you "maximize your expectation," but that's not an optimal thing to do over a series of bets where you actually want to survive financially.

http://en.wikipedia.org/wiki/Kelly_criterion

This criterion is used by pro blackjack players, poker players, horse racing people, and even hedge funds.

There's an AMAZING book about the history of this criterion called "Fortune's Formula."  That book was one of the inspirations for Cordial Minuet.

(This ties into the "bankroll management" stuff that was discussed earlier.  Also, when we're talking about Kelly-optimal betting, we could be talking about fractions of the entire bankroll or fractions of this given table buy-in.)

Well, mzo, I don't want the solution here to be too specific.  Leaving after you get a bad first number is one problem, but what about leaving after two numbers?  This was also somewhat motivated by the tournament "search for co-conspirator" problem with leaving tables at will.

But really, mzo, if someone leaves a table after their first pick, you've won a coin from them for nothing.  Annoying, but not exactly good for them.  It's really the nature of this kind of game---they're in for the ante, but beyond that, they risk nothing.  Same in poker.  "Know when to walk away."  That means you can walk away at any time.

I now realized that just setting a table rake (players starting with 95 chips instead of 100) doesn't totally solve the tournament table-hopping-searching problem, because a player who leaves a table early punishes the other "innocent" player too.  Yeah, I'm down 5% by leaving, but I'm trying to accomplish something by leaving (search out my cooperating partner), where you're just down 5% for nothing.

So, Jere's solution is perfect here.

Still, I don't know if "last player at the table gets it" is right.  What if, on round 1, we both bet 20 chips, and I win.  Now our stacks are 75, 115.  I decide to leave.  Then you get the 10 table-pot chips, and leave with 85?

It's almost like it should go to the winner only if the loser leaves....

Or... automatically go to the winner of the first round.... what about that?  But that's just a forced-high-ante on the first round.  Essentially, it diminishes the skillful potential of round 1.  Folding is GOOD, but this would discourage folding.

So, instead of this being a reward for holding out, or a reward for winning, what if it's just anti-leave insurance that goes back to each player if they stay X rounds.

You start with 95 chips, and after each round, you get one more chip, until you get your original 100 back.  If you leave, you're remaining insurance chips could simply be lost, while your opponent gets theirs back (or your opponent could get both).

Still, this is complication that would need explaining to players.

I'm still leaning toward treating this as a non-problem for non-tournament games.  For tournament games, a fixed table-start rake handles it, with the unfortunate side-effect of punishing the innocent player too. 

There's probably a simple, server-side fix here for tournaments... like, if your opponent leaves and neither person is back to 100 chips, then the remaining opponent gets their portion of the table rake back.  This could happen invisibly and just show up in the post-game outcome screen (for the rare cases where it needs to happen in a tournament---the rest of the time in the tournament, players would just be starting with 95 chips and playing as usual).

Yeah, I was thinking about doing it on a separate server.  Then I could really control the structure, entry fee, and so on.

Otherwise, people would be starting with varying balances, etc.

BUT, I have been thinking of ways to run a tournament for "everyone" on the main server.  Not sure what the win criteria would be.... profit since the start of the tournament?

Oh, and #2 applies only to v11, which I'm about to release.

Yeah, I was hoping to make time-limited free-for-all work, but you're right that it is exploitable.

In your IPD example, the bots weren't even able to pick their partners, but still won through recognizing each other.  So, even with randomly-assigned opponents, collusions could still happen whenever two cooperators are randomly assigned to each other.  "If you ever see me, dump all your chips to me!"

An edge is an edge, unfortunately.

On the other hand, gaining that edge, however small we could make it through randomization, would still require multiple entry fees.  But even if you never meet / recognize each other, you've paid 2x for 2 seats at the tournament, doubling your chances of winning if you each have the same skill, so worst case even if you have no edge, you're breaking even.  Any edge you get, from the rare chance that you meet and inflate one bankroll, will push you beyond breaking even.

Thus, someone would be motivated to organize an unbounded number of collaborators to enter a tournament together.  The limit would be the size of the prize pool, but normally, the prize pool would grow based on the number of entrants.

I wonder if the rake could be useful here somehow.  If the rake is 40% of the pot (giving you only 20% of what your opponent pushed in, with 80% of what your opponent pushed in going to the house), wouldn't people prefer to play strangers over shared-purse collaborators?

If I have two accounts with X each, and I play myself all-in, then afterward, I have (1.2 X) of my 2x left, which is a 40% loss.  Will that larger bank roll of 1.2 X now give me enough of an edge in the rest of the tournament to recoup my loss?

On the other hand, if I play all-in against someone else's X, afterward, I have 1.2 X again, but this time it's a 20% gain.

It seems like any rake over 25% would make this work, losing more to the rake than you're transferring to yourself---BUT, the question of the edge still remains.  If the pot is 100X, and you gain a 1% edge by doing this, then you're ahead.

It seems like only a true bracket system, where bankrolls are not a factor, would stop this.  A $5 entry fee, where after 30 minutes, the bottom 50% is culled, and the remaining folks are reset back to $5.  Repeat until one person is left. 

Someone could stay in longer by employing stooges, but eventually, all the stooges would be weeded out.

This is a good point!  I'll work on fixing this.

Yeah, I've thought about this....

They're having a Casino Night type fundraiser there on Saturday evening, December 13.  See "Party on the 13th" here:

https://www.facebook.com/babycastles

They're working on setting up their own, local Cordial Minuet kiosks for in-person cash games.

Hmm... what if, before you make your pre-reveal bet, you could mouse over your squares and preview what a given reveal would do on the score graph?  (Using the same mouse-over-to-adjust-score-graph mechanism that is in place in other parts of the game.)