Toot Toot! Puzzle Boat!

This weekend I went on a bit of an adventure. I played board games at Sky's place on Saturday (fun, but not much of an adventure) but instead of going home I hitched a ride with Pounder and spent the night at his place. Then all day Sunday (where all day started at 2pm because Arizona sucks) we hung out in Snuggles' basement working on Puzzle Boat 2.

A Puzzle Boat is essentially a large bundle of puzzles that you solve by yourself or in a team of whatever size you want. You start off with a couple of presumably easy puzzles and each time you complete a puzzle you get a password to enter into the website which unlocks more puzzles! There are some 'meta' puzzles that use the answers to the base puzzles and then one big main puzzle that puts everything together. The ultimate goal is to solve that big puzzle first. There are over 100 individual puzzles and our team of 7 people solved 13 in 10 hours so it's not exactly a quick endeavour!

We have 13 more puzzles currently unlocked that are in various states of completion and like 80 or 90 more still to unlock after that. We have another day to puzzle away (next Saturday I think) but finishing it then seems pretty out there too. So we've decided we're allowed to work on the 13 we have now but not unlock any new ones until we meet up again which seems like a pretty fair way to do things when we're not trying to win. That way we won't meet back with all the really fun stuff done and a bunch of cryptic crosswords still to do. Not that there's anything wrong with cryptic crosswords... But the one that we unlocked so far had the cryptic crossword part completely ignored.

Winning seems like the sort of thing that could have been plausible in an alternate universe but it would have involved a larger team, with people who have done this before, and who didn't mostly have to work Monday morning. As far as I can tell no one is even close to being done yet so a team that could work continuously for a couple days in a row is a really important piece it seems. That isn't our team, and that's fine, because you don't have to play for first place every time. You can play for fun, and yesterday was tons of fun. Lots of little 'a-ha' moments that really make the person who saw the trick feel good without the people who were stuck feeling bad because we're all on the same team and there are SO MANY of those moments to go around. And while a bigger team would have a better chance of winning it also means each person gets to touch fewer puzzles and that's not necessarily more fun. One of the 13 we solved just sat there having people rotate in to look at it forever until Lino finally made the right leap of logic to get the ball rolling again so you do want a fair number of people, but not too many. One room of people seemed like a pretty good number, though I probably should have brought my laptop since I kept needing to borrow others.

What's also cool about this sort of thing is when you have 100+ puzzles there gets to be a little bit of everything. Some sudokus, a cryptic crossword, a diagramless crossword, anagrams, ciphers, some image lookups, maybe there'll be a book code at some point. Browser games! Song recognition! All kinds of logic puzzles! Woo!

It was a bit of an adventure getting back to Toronto this morning and I never sleep well away from home but it was all worthwhile. All aboard the puzzle boat! TOOT TOOT!

Perfectly Logical Beings

The pirate puzzle that's been discussed here the last few days came from a book about preparing for a programming interview. After explaining the solution to the puzzle the author then goes on a little rant about 'perfectly logical beings'. The puzzle didn't explicitly state that's what the pirates were (one of the tricks the interviewer might throw at you is leaving that information to you to derive) but in puzzle terms it was pretty clear regardless. The whole greedy, logical thing and all.

Perfectly logical beings don't make sense in the real world. Their inputs and outputs can all be precisely determined in a relatively easy manner once you figure out the trick to the puzzle. Real people are much more complicated, though I must admit I am quite intrigued to learn more about what Danielle was talking about in the Facebook thread with regards to actually working those sorts of things out in a non-puzzle sense.

Daniel and Sthenno both brought up what would happen if pirate E broke ranks and voted against the solution 1-0-1-0-98 even though we 'proved' that it's in his best interest to vote yes. In the real world maybe E can do that. He may have some input that we didn't think to account for (maybe he truly thinks one is the loneliest number and would have voted for any split that got him a non-one number) which causes him to behave what we perceive to be 'irrationally'. I'm sure he has a good explanation for what he did. But that would make him a real, complicated person and not a perfectly logical being.

So what happens in the puzzle when E breaks rank? He doesn't. The puzzle world simply doesn't work that way. The puzzle is a little logic problem designed to be solved in a reasonably short period of time with very limited information. Especially in an interview situation you may win bonus points with the interview by thinking 'outside the box' and giving the pirates backstories that cause weird votes. Maybe you'll just make the interviewer annoyed and lose the job. Who knows!

Sthenno also said he'd better hope I'm not actually dividing pirate plunder with him in this way since I'm going to die if I do. Now, I like my friends from University and all but there's no way I'm ever going to let them vote on if I get to live or die. (Can you imagine betting your life that you know how Bung is going to act in any given situation?) But I do play a lot of board games where this sort of decision can come up... Take El Grande, for example. You draft actions in that game. Some actions really impact board position. Other actions moderately impact board position and score points. Often the situation comes up where you can take the point scoring action or you can 'move the king' and set up the point scoring action to be really good for you. The trick is convincing the next guy to score points for you! (He can take the action but not activate it if he thinks it's going to be 'too good' for you.) Is a 10-2-0-0-0 split good enough? 10-4-0-0-0? 6-8-4-0-0? It depends on who you're playing with and the state of the board. Games like that (Modern Art, Dominant Species, and even Carcassonne) are all about putting yourself in a position where other people will score you points but you can't make it so they're only scoring you points or they won't do it.


But I digress. I've put a lot of thought into what happens when E isn't actually a perfectly logical being so I might as well say what I'd do in B's shoes. Here's what I believe to be true...

• D wants to get down to 2 people at this point and likely thinks she can capitalize on E's apparent randomness to get all the loot. She's probably voting against anything I propose and is definitely voting against anything C proposes after I die.
• I don't trust E. For all I know he just likes to see people die and would even vote against a split giving him all the loot. Who needs cash when you can have blood?
• Death is now a very real option for me. 
• Death is also a very real option for C. I probably need his vote to not die, but the trick is he probably needs my vote for him to not die as well. Unless he's willing to bargain with E he needs my split to pass.

I've gone back and forth in my head about what I'd actually do. My first guy feeling was actually to propose 0-0-0-100. That's right, all the loot to me. I'd be banking on the fact that C trusts E as much as I do and that he realizes voting against my split means he dies too.

Then I thought that the life of a pirate probably isn't what I want out of life and I just want to survive. The best way to do that is the 0-0-100-0 split. C can have all the loot and we both get to live. I'm guaranteed to survive this time! (I know C isn't a lunatic since he did actually vote for 1 coin when A offered it.)

But then I'm thinking of the ultimatum game... I don't really want to make C angry. It may well go against all that greed stands for, but E threw greed out the window when he killed off A. So rather than make C an offer he simply can't refuse (0-0-100-0) I'm going to make him one he shouldn't refuse. 0-0-50-50. And then we stop sailing around with E.

Ultimatum Game Results

Today I was planning on looking at the 'solution' to the ultimatum game and how it relates to what happens in the pirate puzzle if the last pirate breaks form. Unfortunately I'm not a good enough/fast enough writer to find a way to pull those two things together and this post was just looking like a big wall of unrelated text. So I'm going to split things into two and delay talking about crazy pirates for another day.

Yesterday I mentioned the Ultimatum Game where Alice is asked to divide $100 between herself and Bob. Bob then votes on the split and either the split carries or you both get nothing. If we all behaved like the puzzle pirates from two days ago then the correct split would be 99-1 and the stranger might be sad but would surely accept. One dollar is better than no dollars, after all.

It turns out in reality people aren't perfectly logical beings. I haven't been able to find actual data from most of the experiments which were run, only summaries from people who may or may not know what they're talking about. But it seems like in general Alice seems to offer much closer to a fair split, and with good reason. Bob frequently rejects splits which he views as being unfair. Apparently when you get into the $80-$20 split range more than 70% of splits are rejected.

I was talking about this with Andrew yesterday and he said that it's an interesting thought experiment while not much is on the line but if you were splitting up, say, $100k then all of a sudden the $20k seems like it should get taken more often, right? It turns out researchers thought of this too and ran the experiments with huge sums of money. To do so they went to a less developed country and started offering large sums of local currency. I found one study that took place in north-eastern India. The average yearly income in those villages was reported to be around 17k rupees and they were offering 20k in their highest stakes experiment. So they were giving these people more than a yearly salary in one game! They mucked with the experiment a little by straight up telling their Alice's that the optimal strategy is to offer as little as possible to Bob while having Bob still say yes. (They did this because they were trying to test how Bob played the game, not how Alice played the game, and all previous experiments were plagued with too many high offers. It turns out people just play nice too often!)

They succeeded in lowering the average offer significantly with those instructions. They also succeeded in showing that once you start offering a huge amount of money the behaviour changes.  When they were splitting up 200 rupees a full 50% of Bob's said no to offers under 10% of the pot. When splitting 20000 rupees only 5% did. It's hard to say no to a month's worth of money even when you know someone is 'screwing' you by taking 12 months himself. (Also of interest as the stakes got higher Alice would take a bigger chunk for herself. I guess the thinking is that if Bob is pretty much guaranteed to take a month's salary there's no point in giving him even more than that, right?)

A very interesting result came from an experiment done in New Guinea. Apparently the culture there is based on reciprocity. (As Sheldon would say, "You haven't given me a gift, you've given me an obligation!") There Alice would sometimes offer Bob more than half of the money. And Bob would say no!

So it turns out game theory doesn't really apply to a lot of these situations. There's something more at work than just maximizing incoming money. The concept of fairness was brought up a lot in the articles I was reading. Or maybe it's not a stranger and you realize you actually have to live with them going forward so you're better off giving them a 'fairer' shake so they're not bitter at you forever. It seems like the psychoeconomists are looking into it still. I find it interesting, anyway.

Monopoly, Ultimatums, and Logical Pirates

Yesterday's puzzle got a couple correct answers in the comments of both the blog and on the Facebook notes thread. (I should really search harder for a solution which would allow combining both sources of comments...) There was also some good discussion about what would happen if one of the pirates wasn't a "perfectly logical being" which is actually where I was planning on heading with tomorrow's post. So I'm going to essentially ignore that for now but will be returning soon. Today I'm going to provide a tangentially related anecdote, the game theory "game" that sparked my remembrance of that puzzle, and the solution for those who may not read comments.


Last month was Monopoly at McDonalds. I was out eating lunch there with Andrew and he pulled Park Place off of his drink. If you collect both Park Place and Boardwalk you win a cool million dollars. The conversation turned to what would happen if I had Boardwalk on my drink. Andrew seemed to assume it was a given that we'd just split it and take $500k each. But part of me is thinking that Boardwalk is more likely to be the rare one so my expected value in terms of sheer money would be to reject that offer and just buy some more drinks in the hopes of getting a Park Place of my own. (Or to haggle him down to a better split I guess.) But in reality the marginal utility of the extra $100k between $500k and $600k isn't all that much and certainly isn't worth any strife that might be caused by making such demands from a friend. So if I'd had Boardwalk I'm sure we would have just taken $500k each and been ecstatic.

I wanted to continue the thought experiment though, so I asked Andrew what he'd do if a random dude had walked up to him and offered to buy Park Place off of him. What values would he accept? (Clearly he'd sell it to me for $500k, so what about the random dude?) The conclusion Andrew came to was he simply wouldn't sell it to a random dude no matter what the price was. The thing is we don't know which piece is actually worth the million bucks. One of Boardwalk and Park Place is worth almost a million dollars and the other is practically worthless. (It has some value to the owner of the real piece but is worth nothing to everyone else.) So if some guy showed up and offered a large amount of money then the odds are he's trying to scam Andrew out of the real piece. $50k sounds great but there's no way a stranger with the real piece is paying that much for the common one was his argument. After all, the stranger could just go buy 30 drinks, get 60 pieces, and almost certainly get his million bucks. If the stranger was offering a low amount of money (say, $50) then Andrew still wouldn't take it. Even though the stranger is probably just trying to save himself from buying 30 drinks there's a small chance he'd be trying to pull a fast one and it wouldn't be worth the risk.

Personally I like the idea of getting $50 for my worthless piece of paper, though I think what I'd do is try to find a second person in the restaurant with a Park Place and try to haggle the stranger to buy them both for $100. I don't think the stranger is scamming me at all but I will admit it would be a huge blow to my sanity if I sold it to him and it turned out to be the real piece and I probably wouldn't be willing to take that risk either. (Mostly I don't think it would ever happen at all. I know if I thought I had the winner I wouldn't advertise it in a room full of strangers. I'd quietly go home, hide the winner, and then just eat at McDonalds for a while.)


Which brings us to the 'game' I only just found out about earlier this week in my reading... The Ultimatum Game. The basic idea of the game is a random dude (probably in a top hat with a handlebar moustache) is going around making the following proposal to two strangers (in this case referred to as Alice and Bob.):

Alice is given $100 dollars and is told to split that money up into two piles. One pile for herself and the other pile for Bob. Then Bob gets to vote on the split. He can either vote for the split in which case Alice and Bob walk away with some cash. Or he can vote against the split in which case our moustachioed hero takes his money back. Either way the game is over. Alice and Bob can't negotiate over the split or communicate in any way. Alice makes a split and then Bob decides whether to take it or not.

So the question is, if you're Alice what split do you propose? If you're Bob what splits do you accept?


Now, the solution to yesterday's puzzle.

The trick to this puzzle is to find the base case and figure out what happens. Then work your way backwards until you end up at the current case. Since you only need half of the votes to pass a proposal the base case is when there are two pirates left since whatever is proposed is guaranteed to pass. (Obviously you vote for your own split since otherwise you're dead.) I'm going to break form from how everyone else labeled the pirates and assert that the pirates are, in reverse order of seniority, E-D-C-B-A.

Two pirates alive (E & D): D knows that whatever he proposes will pass and she's greedy so she offers a 0-100 split. It passes and D is rich!

Three pirates alive (E, D, & C): C knows that if his proposal gets shot down that the split will be 0-100. He also knows that he just needs one other vote in order to have his proposal pass. He needs to bribe E or D to vote with him. Clearly he can't bribe D to vote with him since D is already going to get all the money if C loses. On the other hand E is going to get absolutely nothing if this vote fails. One coin is better than no coins, after all, so C proposes a 1-0-99 split. E is logical and greedy and knows it isn't getting any better so he has to accept.

Four pirates alive (E, D, C, & B): B knows that if his proposal gets shot down that the split will be 1-0-99. He also knows he needs just one vote in order to win. The big loser is the last split was D. She has visions of coming home with all the coins way back in the base case but she now knows that's a pipe dream. She's going to get nothing if B dies. B knows that D knows this and once again one coin is better than no coins. His proposed split is 0-1-0-99 and it passes.

Five pirates alive (E, D, C, B, & A): I hope the pattern is becoming clear. A knows she needs to get 2 more votes and she knows that if she dies the split will be 0-1-0-99. The two easiest people for her to bribe and going to be E and C since they'll be getting nothing if A dies. So A proposes a split of 1-0-1-0-98 and walks away with most of the loot.

As Snuggles might say, "It's good to be captain!"

Greedy Logical Pirates

Here's a little logic puzzle from a book my old roommate Blake gave me many years ago. I really liked this one when I read it (I believe I've told it to a few people over the years) and I've been doing some other reading recently that reminded me of it. Can you figure it out?

There are five pirates who discover some buried treasure on an island. There are 100 gold coins and they need to divide the coins up amongst themselves. The way their pirate code works is the lead pirate proposes a split of the coins and then all the pirates vote on that proposal. If at least half the pirates vote YEA then that split carries and they move on with their plundering lives with their new booty. Otherwise they execute the lead pirate and start over from the top with the next pirate in line making a proposal.

The pirates are all greedy, logical, and don't want to die. You're the lead pirate. What split do you propose? Why?

False Dichotomies

I read a backgammon strategy book a while ago where the author made the case that decent backgammon players make mistakes not because they can't properly choose the best move between their options but because they fail to even consider the best move at all. The problem being that they find two moves that look good, work out which one is the better between the two, and go with it. The problem, obviously, is that there's way more than 2 different moves in almost every backgammon position and you're giving up little edges every time you don't consider all the good moves.

This flaw exists in more than just backgammon. It's human nature to fall for this logical fallacy, and I believe I've fallen for it in terms of how I gear my Death Knight for tanking. Some background:

There's currently a stat in World of Warcraft called expertise. This stat reduces the chance a mob has of dodging or parrying your attacks. Having my autoattacks get dodged or parried costs me damage and threat. Having my specials get dodges or parried costs me time and possible damage, threat, and healing. I don't like any of those things, so it seems like I want expertise.

Unfortunately right now there is exactly one tanking piece at top level for me with expertise on it. On top of that I get to use a 2-handed weapon, which can also have a lot of expertise on it. That item is pants, and there is another great set of pants with a lot of armor on it.

When I first looked into this situation a couple months ago it looked like I had two options. Either I could wear the pants with expertise and give up on the armor pants or I could wear the armor pants and give up on the expertise. I won't say I had threat problems persay but I was definitely skirting the line so I decided to stick with my expertise pants.

The problem, of course, is that those aren't my only two options at all. Another option (the reason I'm making this post) is I could just gem expertise instead of stamina in a couple sockets to make up the difference. I need to look into exactly what I'd be giving up to do it, but it is certainly something I should have looked at in the first place. A fourth option which I hadn't even thought of until right now is switching in an expertise trinket, or enchants. In all of these cases I need to decide between threat and survivability but I should look at what I'm giving up in each case instead of just ignoring their existance.

Which takes us back to the false dichotomy. I didn't say to myself "Hey, socketing expertise isn't a good choice for reasons X and Y." No, I just didn't think of it at all. I got sucked into choosing between A and B and forgot to even look for C.

Right now the only fight that matters to me is a fight where I am dying a lot and where my threat is largely irrelevant. My gut feeling now is to just switch pants and screw expertise entirely but I want to see what the costs are this time around to make that decision!

So, my options are:

A) Status Quo: Wear Sanctified Scourgelord Legguards. 2310 ac, 123 str, 273 sta, 118 def, 92 dodge, 82 expertise
B) Just Pillars of Might: 3500 ac, 162 str, 279 sta, 10 def, 79 dodge, 77 parry

Switching to B from A gains 1190 ac, 39 str, 6 sta, 77 parry at the cost of -108 def, -13 dodge, -82 expertise.

I remain uncrittable without the defense, so I’m fine there.

C) I currently have 6 red sockets filled with purple gems at the moment. As such, it’s pretty easy to make up 60 of the expertise by switching those from dodge/sta gems to expertise/sta gems. Beyond that, I can give up 30 stamina to get the remaining 20 expertise I’m losing.
D) Enchant gloves: Expertise gives me 15 expertise at the cost of 10 parry and 2% threat. The expertise won’t make up that threat difference, so this is terrible.
E) Enchant bracers: Expertise gives 15 expertise at the cost of 40 stamina. This is a worse ratio than changing my gems, so this is also terrible.
F) Put on Victor’s Call. It is 83 expertise and costs me 228 stamina.

Now, my slightly outdated stat valuation for ac puts 18 bonus ac as worth the same as 1 stamina. So picking up 1190 extra ac is worth 66 stamina, so giving up 30 stamina to get there is good. Giving up 228 stamina to get there is not. Option F is therefore terrible.

I also don’t need to get all the way to the first expertise cap. It’s certainly good to get there (worth trading dodge for, for example) but may not be worth giving up stamina for. Especially given my current role on Lich King hard I think giving up dodge is fine but giving up stamina is not. So, assuming I switch purple gems all around my overall change by switching legs is:
1190 ac, 39 str, 6 sta, 77 parry at the cost of -108 def, -73 dodge, -22 expertise.

The parry and dodge about cancel each other out so I’m giving up 108 defense for 1190 ac and 39 strength. This is actually pretty close, since the 108 defense has very real value in terms of avoidance. In fact, my old values slot 108 defense as being worth 1166 ac. So making the swap is a marginal gain there, but it`s not the end of the world if I stick with my current setup.

I think I’m going to make the switch. It’s not a big gain but it is a gain. I also don`t have the T10.5 pants yet, just the T10 ones, so it is a big upgrade until I would get a 5th token.