1 (Not too hard). On every day of a given season, player A has a better batting average than player B. Does it follow that player A has a better batting average than player B over the whole season? Prove or give a counterexample.

2. (I don’t know the answer to this one.) At some point during the season, a player has a batting average of less than 80%. At some later point, that player has a batting average of greater than 80%. At some point, his average must have been exactly 80%. (If you don’t believe me, try it.) Why is this the case? What other percentages is this true for?
P.S. I don’t know why his batting average is so high. Maybe this is some weird hitter-friendly version of baseball.

The redoubtable J. Vogel likes RPGs in much the same way I like board games. He’s started a new and interesting blog which features, among other things, one of the greatest name/URL pairs I’ve seen: Kill the Wizard First.

I played some online Bridge today. Good fun; I forgot how much fun it is. Perhaps no other game benefits as much as Bridge from online play, especially for casual players.

Here’s why: Finding a group for Bridge is a big, fat pain. The learning curve is steep and long, and even otherwise hardcore gamers often don’t want to play. (Although for some reason, some otherwise reluctant gamers are fine with Bridge.) You need exactly four players. Fewer and there’s no game at all, more and someone will be sitting out. So overinviting to a Bridge session is troublesome, unless the players all know each other pretty darn well and can do some other activity.

Going online, you can take one partner (way easier than finding three others) and find someone else Out There to play against; you can even go alone for a pickup game.  Bidding systems are standard enough that most partnerships can work something out without too much pain.

Of the online sites that are out there, Yahoo has the advantages of being easy and download-free. Unfortunately, you’ll need to bring your own partner, because a great number of the players there are unfamiliar with any bidding conventions, wildly reckless, or flat-out insane. And that’s in the Advanced rooms. I can only imagine what the intermediate and novice rooms must be like. For those willing to invest the time in a download, Bridge Base Online offers a more sober environment with stronger opponents or pickup partners. I hear good things about OKBridge, but the service is pay, and I’m nowhere near strong enough to benefit from the services they offer.

Anyway, the online advantage is much greater than for other games:

• Chess and Go are great online, and it’s much easier to find a wide variety of opponents online. But these games don’t suffer from the difficulty of finding the exact correct number of opponents. One partner for a couple hours can make a satisfying session.
• Euchre also suffers from the exact-people problem. But Euchre is much easier to learn, so it can be more easily played as a pickup game with whoever has around; and if you have an extra person, that person can watch TV or otherwise kill 15 minutes while the game plays out. Unlike Bridge, you don’t need to get deeply into a Euchre game to play it.

It’s a long one today, but don’t give up, because it’s interesting, and I want to know what people think about the economic issues involved.

From one pirate topic to a wholly different one. The game Puzzle Pirates has been around for a while, but I finally got around to trying it a couple nights ago. I certainly have to give the designers credit for unique ideas.

The idea is this: Puzzle Pirates is a pirate-based MMORPG: ships, booty, swords, parrots, etc. You can crew a ship, but let’s face it: most of the jobs on a ship sound uninteresting to represent in a game, right? Pumping bilge water, repairing leaks, fiddling with the sails, who cares? Well, each one of these jobs is represented in a very abstract way by a puzzle. There’s a pentominoes-related puzzle, a Bejeweled playalike, a Snood playalike, a Dr. Mario playalike, and a couple others. The games are all very well-done, so they’re fun to play.

If, for instance, your ship starts taking cannon fire, it will start taking extra water in and it will slow down and lose speed, making it all the more vulnerable to whoever’s shooting at you or even in danger of being sunk. However, carpentry can repair the hull damage and bilge pumping can remove the speed penalty. If World of Warcraft were like Puzzle Pirates, then for (say) a mage, instead of trying to manage your aggro, mana, AoE, and so on, you needed to play Tetris and score points as fast as possible to deal damage to the monsters, while the tanks are trying to solve Sudoku problems to keep the monsters off of you, or whatever. PvP combat is also represented by a puzzle.

All of the other elements of an MMORPG that you would expect are there, too. Except that instead of a level, you have experience and skill ratings based on your performance in various jobs. So if a crew is looking to pickup extra people for a mission or raid, they can find someone who has good skill in the particular role that they’re lacking. There’s an economy where shops convert goods into other goods, players make money transporting resources around, etc.

Did I say economy? That brings up an interesting point, and the second interesting innovation. Clearly the designers of the game would like to make money. There are two kinds of servers. One is a fairly ordinary subscription-based server. The fee is about $10 a month and you get full access to pretty much everything. The other kind of server works on “doubloons,” which you buy for about $0.25 each. Call this model “pay-to-play.” To do anything but the most basic jobs on a ship requires a number of doubloons. For instance, a pirate’s badge that lets you hold most shipboard jobs requires 1 doubloon and lasts a months, while a captain’s badge that lets you captain your own ship costs, I think, 20. Obtaining certain pieces of equipment and certain other tasks in the world also require doubloons. You can also buy and sell doubloons with other players for the in-game currency that’s used to buy goods and such.

Although I find this scheme interesting and don’t fault them for giving it a shot, I’m also deeply skeptical. As a contrast: in their game World of Warcraft, Blizzard has gone with the pure-subscription model and, furthermore, done everything possible to combat even the perception that spending extra money might confer an in-game advantage. Buying or selling gold is punishable by a permanent account ban.

It seems to me that this stance is very important because it removes the suspicion that the company that runs the game is trying to gouge or squeeze money from its players. (Whether or not it’s true!) On a subscription server, if I add (say) a new high-level ship that goes 10% faster than the previous fastest ship for 10 times the cost, few will complain. The people who don’t have enough in-game money just won’t buy it. On the other hand, what if I put the same upgrade on a pay-to-play server, and set the price at about $25? This isn’t a killing expense, but it certainly seems like a lot of money to spend on a small advantage in an online game. What if a rival team, equipped with such a ship, hunts your ship down and takes your stuff? Will you be angry? Clearly the authors’ hope is that you’ll just be driven to buy one of your own. Or will you quit in frustration, refusing to let them extort that money from you? What if you’ve already invested doubloons in the ship you have? Will you throw good money after bad?

I’m reasonably sure I’m not compelled enough to actually take up Puzzle Pirates as a regular game, although I will admit that the mini-games are fun and all very well done. If I wanted to play often, though, I think I’d definitely go for a subscription server. Trying to justify any in-game decision in terms of a concrete dollar amount would be really jarring.

P.S. Yes, I know that on a subscription server, the currency is not U.S. dollars, but hours and minutes. The hope is that the use of the hours and minutes is entertaining enough by itself that one doesn’t mind.

P.P.S. Usually, my politics tend to the libertarian, not the socialist. But part of what I want in a game is moving away from harsh reality. Is it so wrong?

Thanks for the enthusiastic response to the pirate riddle. If you haven’t seen it, reader Alatar has posted a great analysis of the problem given a couple of slightly different assumptions about what the pirates do when a choice is made.

In other news, you may remember the version of the riddle with red and blue hats where there is a traitor. Reader John pointed out that we’ve been massively, massively overthinking it.

Full-length post to come later tonight.

Another riddle for today. I may have already shared this with some of my readers. This is one of my favorites and was supposed to go up earlier today, but something with my computer or WordPress ended up flaking out…

Five pirates have to divide a treasure of 1000 coins. The pirates have a ranking from highest to lowest. This is the system the pirates use to see who will get how much treasure:

1. The lowest ranked pirate proposes a way to divide up the treasure.
2. The pirates vote on the distribution scheme.
3. If more than half of the pirates vote yes, the treasure is divided as planned.
4. Otherwise, the pirate who proposed the scheme is forced to walk the plank, and the next lowest ranked pirate proposes a scheme.

The pirates have the following traits:

• They are perfectly logical.
• They are greedy and will vote to maximize their own treasure regardless of all other considerations.
• They are bloodthirsty and, all other things being equal, will vote against a proposal.

What should the lowest ranked pirate suggest?

Hint 1: Usually I would say something like “Now generalize the solution to any number of pirates.” But for this riddle, the solution is trivially generalizable once you have it.

Hint 2: The solution makes sense, but the distribution that it gives is pretty startling.

Other Notes: Someone asked me how the pirates came up with such a system. I wasn’t there, of course, but here are some speculations:

• There was grog involved?
• Or maybe an ancient curse?
• This is the way all pirates distribute treasure; it works better for non-perfectly-logical pirates?

Previous entries in the series here.

Name: Partnership

Problem: You have a game concept that benefits from coordination between two people being part of the challenge. Alternatively, you have a two-player game that doesn’t translate well into a larger number of players, but you’d like it to be playable as a group activity. (In this case, playing with six can sometimes work as well.)

Discussion: The two different kinds of games listed in the Problem description are actually pretty easy to distinguish. Call it Type 1 if the game would be significantly different if one player played both parts, and Type 2 if this is not the case. As the shining example of Epic Duels demonstrates, Type 2 is not worse!

Examples:

• Bridge, a Type 1 partnership game. Clearly there is a large amount of coordination required to make the bidding work well. Even once the play starts, the defenders benefit by playing as a partnership; sending each other signals by card choice, playing to maximize the use of the cards partner likely has based on his bidding and play, etc.
• Sequence, a Type 2 partnership game.  The game works fine with two people. With four separate players, the board would be far too cluttered to make any progress. In two partnerships, the game is basically the same as two-handed; you don’t know what half your cards are, but that information wouldn’t really help anyway.
• Pinochle, a Type 1 partnership game. Although the bidding isn’t as intricate as Bridge, having the cards separated is very important. First, it keeps you from knowing exactly what melds are available to your team. Second, during the play, you have to try to picture what your partner has or might have and adjust your own play accordingly to make the best use of partner’s cards.
• Epic Duels, a Type 2 partnership game. Reader Alatar reminded me about this game recently, and I should probably devote a full article to it at some point; because as a Star Wars licensed game, it’s way more fun than it has any right to be. For the purposes of this discussion: The game is reasonably light, and you’re able to table-talk enough with your partners that being separate people really doesn’t harm your ability to coordinate strategy at all. But the ability to trash-talk people, play in a social environment, and share the inevitable stories (Jango Fett killed three Jedi!) with more people makes the game much better with four people than two..

Related Patterns: There’s a pattern that might be called Cutthroat that turns a four-player game into a three-player game. The idea is that one player becomes the “declarer” (maybe by choosing a trump suit, getting to pick up extra cards, or whatever) and the other two players are temporarily united in a partnership against that player. Success is mixed; it works just fine for Pinochle but falls flat for Euchre.

I was thinking about the speed-based games Set and MyWord today. Both of these games are very difficult to play with players of differing abilities, because the difference in ability makes play frustrating for weaker players. They lose their focus as the cards they’re looking at are simply snatched away. Compare this to a similar game like Boggle. Just as often the stronger player will score a devastating win; but because both players get uninterrupted time to find words, the game doesn’t feel as frustrating.

At work today I was thinking of a simulation I was going to run to prove how much this sucks for the bad player. Here’s the game I was going to simulate:

A game takes 40 seconds and has 25 words to be found. Every second, player A finds a random word with .8 probability, and player B finds a random word with .4 probability. The catch is that only the player who finds a word first gets to score it.

I thought the score would lean heavily towards player A, but in fact, player A gets only about twice B’s score. It turns out that if each player gets to keep all the words they find, not just the ones that they find first, A gets about 160% of B’s score. The difference is noticeable, but not as shocking as I thought it would be.

I thought about fiddling with the simulation some more, but I’m tired and it would kind of be cheating anyway. I’ll ponder the question some more tomorrow, because I feel I’m noticing a real effect, even though the simulation I’m running doesn’t show it.

For the interested, the source (Python) is available upon request.

At a party this weekend, I played several games by publisher Out of the Box. Evidently there was some kind of sale. They were all card games that come in these attractive little tins, which I thought was pretty cool.

Blink: A variant of the reflexes game Speed. (scroll down to the “five cards in hand” version, although the hand is only three.) The cards have three attributes instead of two: Number, color, and shape. You have to match one of them. You will enjoy Blink if-and-only-if you like Speed, although I don’t know if I could put together a compelling case to buy the game.

Qwitch: Another variant of Speed, but this one goes a little further. The cards have two attributes (a letter A-H and number 1-8) and there’s another deck that determines whether you should try to match, increment, or decrement the letter or number. For instance, if E7 is on top of the pile and we are in Down mode, I can play any D or any 6 card. You don’t redraw immediately when cards are used up; if you run out of cards, the round stops and everybody refereshes at once. The other interesting points of note are that there is only one pile and you can play with more than two people. The single pile is a little bit of an issue in terms of people racing to it, but the game still has less potential for destruction than, say, ERS.

Qwitch, interestingly, is by the designer of Abridged.

MyWord: Basically a cross between Boggle and Set. Cards with letters are laid out on the table and as you see words, you call them out and grab them. If this sounds good to you, you’ll like the game. One hitch is that, like Set, playing with a stronger player can be frustrating because the cards go away before you’re done thinking with them. The chance of getting shut out completely is much higher than, say, Boggle. And if you’re the stronger player, you sense this happening, but what are you supposed to do? Throw the game? There’s also a tendency for consonants (or, occasionally, vowels) to pile up while you wait for any of the other kind to show up. When one does, it gets snatched away immediately. Set doesn’t have this problem because if one color keeps piling up, it’s as easy to remove as if there were a mix.

Pepper: A little harder to describe, and it’s probably best if you read the rules off the link. Unlike the other games listed today, it’s turn-based. The strategy is interesting and I’m still trying to work it out. What I’m not sure about, unfrortunately, is if the game ever will end if all players play with good strategy. Since you can control who gets to play and you can remember what cards everyone picks up, after you’ve seen everyone’s hand, you can make sure never to put them into a position where they could go out. Of course, this relies on everyone else thinking the same way.

A three part riddle today.

Part 1: You have 100 teammates, all standing in a row. They are facing forward so each can see the ones in front of him. Each one has either a red hat or a blue hat. Starting with the one in back and proceeding forward, each is asked the color of the hat on his head. If the guesser is right, the team gains a point.

The team gets a chance to decide on a strategy beforehand. However, the gamemaster is aware of their strategy, so they need to optimize the worst-case scenario. What strategy should they choose?

Part 2: Just like part 1, but work out the solution for any number of colors of hats.

Part 3: Like part 1 (only two colors of hats), but one of the teammates is a traitor. He is in league with the gamemaster and will do anything possible to mess up the team’s score. To make matters worse, the gamemaster, being aware of the team’s strategy, can place the traitor where he will do the most damage.

The players are aware of the traitor, though. What strategy can they use to optimize the worst case?