Gaming Newcomb’s Paradox II: Mechanics
Newcomb’s Paradox was created by William Newcomb of the University of California’s Lawrence Livermore Laboratory. The dread philosopher Robert Nozick published a paper on it in 1969 and it was popularized in Martin Gardner’s 1972 Scientific American column.
As a philosopher, a game master (a person who runs a tabletop role playing game) and an author of game adventures, I am rather fond of puzzles and paradoxes. As a philosopher, I can (like other philosophers) engage in the practice known as “just making stuff up.” As an adventure author, I can do the same—but I need to present the actual mechanics of each problem, puzzle and paradox. For example, a trap description has to specific exactly how the trap works, how it may be overcome and what happens if it is set off. I thought it would be interesting to look at Newcomb’s Paradox from a game master perspective and lay out the possible mechanics for it. But first, I will present the paradox and two stock attempts to solve it.
The paradox involves a game controlled by the Predictor, a being that is supposed to be masterful at predictions. Like many entities with but one ominous name, the Predictor’s predictive capabilities vary with each telling of the tale. The specific range is from having an exceptional chance of success to being infallible. The basis of the Predictor’s power also vary. In the science-fiction variants, it can be a psychic, a super alien, or a brain scanning machine. In the fantasy versions, the Predictor is a supernatural entity, such as a deity. In Nozick’s telling of the tale, the predictions are “almost certainly” correct and he stipulates that “what you actually decide to do is not part of the explanation of why he made the prediction he made”.
Once the player confronts the Predictor, the game is played as follows. The Predictor points to two boxes. Box A is clear and contains $1,000. Box B is opaque. The player has two options: just take box B or take both boxes. The Predictor then explains to the player the rules of its game: the Predictor has already predicted what the player will do. If the Predictor has predicted that the player will take just B, B will contain $1,000,000. Of course, this should probably be adjusted for inflation from the original paper. If the Predictor has predicted that the player will take both boxes, box B will be empty, so the player only gets $1,000. In Nozick’s version, if the player chooses randomly, then box B will be empty. The Predictor does not inform the player of its prediction, but box B is either empty or stuffed with cash before the players actually picks. The game begins and ends when the player makers her choice.
This paradox is regarded as a paradox because the two stock solutions are in conflict. The first stock solution is that the best choice is to take both boxes. If the Predictor has predicted the player will take both boxes, the player gets $1,000. If the Predicator has predicted (wrongly) that the player will take B, she gets $1,001,000. If the player takes just B, then she risks getting $0 (assuming the Predicator predicted wrong).
The second stock solution is that the best choice is to take B. Given the assumption that the Predictor is either infallible or almost certainly right, then if the player decides to take both boxes, she will get $1,000. If the player elects to take just B, then she will get $1,000,000. Since $1,000,000 is more than $1,000, the rational choice is to take B. Now that the paradox has been presented, I can turn to laying out some possible mechanics in gamer terms.
One obvious advantage of crafting the mechanics for a game is that the author and the game master know exactly how the mechanic works. That is, she knows the truth of the matter. While the players in role-playing games know the basic rules, they often do not know the full mechanics of a specific challenge, trap or puzzle. Instead, they need to figure out how it works—which often involves falling into spiked pits or being ground up into wizard burger. Fortunately, Newcomb’s Paradox has very simple game mechanics, but many variants.
In game mechanics, the infallible Predictor is easy to model. The game master’s description would be as follows: “have the player character (PC) playing the Predictor’s game make her choice. The Predictor is infallible, so if the player takes box B, she gets the million. If the player takes both, she gets $1,000.” In this case, the right decision is to take B. After all, the Predictor is infallible. So, the solution is easy.
Predicted choice | Actual choice | Payout |
A and B | A and B | $1,000 |
A and B | B only | $0 |
B only | A and B | $1,001,000 |
B only | B only | $1,000,000 |
A less-than infallible Predictor is also easy to model with dice. The description of the Predictor simply specifies the accuracy of its predictions. So, for example: “The Predictor is correct 99% of the time. After the player character makes her choice, roll D100 (generating a number from 1-100). If you roll 100, the Predictor was wrong. If the PC picked just box B, it is empty and she gets nothing because the Predictor predicted she would take both. If she picked both, B is full and she gets $1,001,000 because the Predictor predicted she would just take one. If you roll 1-99, the Predictor was right. If the PC picked box B, she gets $1,000,000. If she takes both, she gets $1,000 since box B is empty.” In this case, the decision is a gambling matter and the right choice can be calculated by considering the chance the Predictor is right and the relative payoffs. Assuming the Predictor is “almost always right” would make choosing only B the rational choice (unless the player absolutely and desperately needs only $1,000), since the player who picks just B will “almost always” get the $1,000,000 rather than nothing while the player who picks both will “almost always” get just $1,000. But, if the Predictor is “almost always wrong” (or even just usually wrong), then taking both would be the better choice. And so on for all the fine nuances of probability. The solution is relatively easy—it just requires doing some math based on the chance the Predictor is correct in its predictions. As such, if the mechanism of the Predicator is specified, there is no paradox and no problem at all. But, of course, in a role-playing game puzzle, the players should not know the mechanism.
If the game master is doing her job, when the players are confronted by the Predictor, they will not know the predictor’s predictive powers (and clever players will suspect some sort of trick or trap). The game master will say something like “after explaining the rules, the strange being says ‘my predictions are nearly always right/always right’ and sets two boxes down in front of you.” Really clever players will, of course, make use of spells, items, psionics or technology (depending on the game) to try to determine what is in the box and the capabilities of the Predictor. Most players will also consider just attacking the Predictor and seeing what sort of loot it has. So, for the game to be played in accord with the original version, the game master will need to provide plausible ways to counter all these efforts so that the players have no idea about the abilities of the Predictor or what is in box B. In some ways, this sort of choice would be similar to Pascal’s famous Wager: one knows that the Predictor will get it right or it won’t. But, in this case, the player has no idea about the odds of the Predictor being right. In this case, from the perspective of the player who is acting in ignorance, taking both boxes yields a 100% chance of getting $1,000 and somewhere between 0 and 100% chance of getting the extra $1,000,000. Taking the B box alone yields a 100% chance of not getting the $1,000 and some chance between 0% and 100% of getting $1,000,000. When acting in ignorance, the safe bet is to take both: the player walks away with at least $1,000. Taking just B is a gamble that might or might not pay off. The player might walk away with nothing or $1,000,000.
But, which choice is rational can depend on many possible factors. For example, suppose the players need $1,000 to buy a weapon they need to defeat the big boss monster in the dungeon, then picking the safe choice would be the smart choice: they can get the weapon for sure. If they need $1,001,000 to buy the weapon, then picking both would also be a smart choice, since that is the only way to get that sum in this game. If they need $1,000,000 to buy the weapon, then there is no rational way to pick between taking one or both, since they have no idea what gives them the best chance of getting at least $1,000,000. Picking both will get them $1,000 but only gets them the $1,000,000 if the Predictor predicted wrong. And they have no idea if it will get it wrong. Picking just B only gets them $1,000,000 if the Predictor predicted correctly. And they have no idea if it will get it right.
In the actual world, a person playing the game with the Predictor would be in the position of the players in the role-playing game: she does not know how likely it is that the Predictor will get it right. If she believes that the Predictor will probably get it wrong, then she would take both. If she thinks it will get it right, she would take just B. Since she cannot pick randomly (in Nozick’s scenario B is empty if the players decides by chance), that option is not available. As such, Newcomb’s Paradox is an epistemic problem: the player does not know the accuracy of the predictions but if she did, she would know how to pick. But, if it is known (or just assumed) the Predictor is infallible or almost always right, then taking B is the smart choice (in general, unless the person absolutely must have $1,000). To the degree that the Predictor can be wrong, taking both becomes the smarter choice (if the Predictor is always wrong, taking both is the best choice). So, there seems to be no paradox here. Unless I have it wrong, which I certainly do.
The Return of the Fourth King’s Game
Like most people, I accumulate stuff that I no longer want or need and I like to get rid of it. I also like Christmas gift giving. As an experienced game master, I also really enjoy tormenting others (in the context of the game, of course). Back in 2010 I combined all of these into the much dreaded King Bob’s Game-an event my gaming group has learned to fear and loath.
The theological basis for the game was inspired by the Three King’s Day celebration in Puerto Rico. This is a very pleasant, but very hot, place to visit and I certainly recommend going there. The Spanish fortifications in San Juan alone are worth the trip.
As the story goes, three wise men or kings (not the same thing at all, of course) brought the baby Jesus some gifts. While this served as the theological foundation for the massive commercialization of Christmas, it also gave rise to Three Kings Day, which is celebrated in Puerto Rico. The gist of the holiday is that children put out grass and water for the Kings’ camels and they get small gifts in return. This holiday is on January 6th.
Fortunately, a little research revealed that there was a 4th king, King Bob. Unlike the Three Kings, Bob was not great with directions and ended up arriving at the wrong city, albeit a few days before the other kings arrived in the proper destination.
Since King Bob could not find the baby Jesus, he decided to give away the gifts via a game, which is now known as King Bob’s game. Alternatively, it can be called The Game of the Fourth King.
Here is how the game is played.
What You Will Need
Gifts: At least 1 wrapped gift per player, preferably more. Cheap gifts are best.
Dice: Ideally you should have a D20 and some D6s, but for non gamers six sided dice will do.
The Roles
There are two roles in the game: King Bob’s stand in and player. King Bob supervises the game but does not play. He also does not get any gifts. Optionally, King Bob can also play and get gifts, but that is bad theology.
Everyone other than King Bob’s stand in is a player.
Setting Up the Game
King Bob sets up the game by creating a pile of the wrapped gifts and defending them from the greasy hands of the players until the game starts. Each player should have a die (or dice) and a board or piece of paper is needed to keep track of the order of play.
Initiative
Gamers will be familiar with this, but non-gamers will not. For the non-gamers, this is how you determine the order in which the players take their turns. To determine this, each player rolls a die (preferably the standard D20). The player with the highest roll goes first, the player with the second highest goes second and so on. In the case of a tie, reroll until it is settled.
Starting the Game
The game starts with the player who has the highest initiative. S/he selects one gift from the pile and DOES NOTopen it. Shaking and such is allowed. The second player then has his/her turn and so on for each player until it is back to the first player. After the first player has selected his gift, the other players will have more options and the first player will also have these options on his/her second turn.
Playing the Game
After the first player has a gift, the second player has his turn and so on until everyone has had a turn. The first player then has his second turn and so on. During play, a player has options. Only ONE option may be taken each turn. A player can take a different option each turn, but is not required to do so.
- Pick a Gift: the player selects a gift from the pile but DOES NOT open it. The next player then takes his/her turn.
- Open a Gift: the player opens one gift that s/he has in his/her possession and opens it. The next player then takes his/her turn.
- Steal a Gift: the player attempts to take a gift from another player. The player who is trying to steal the gift is the thief and the player who has the gift is the defender. The defender has the option of allowing the theft or resisting. If the defender allows the theft, the thief gets the gift and adds it to his/her collection. If the defender decides to resist, then the thief and the defender each roll a six sided die. If the defender matches or exceeds the thief’s roll, then s/he keeps the gift. If not, the thief adds the gift to his/her collection. The next player then takes his/her turn. Defender does not count as the defending player’s turn and s/he can defend as often as needed.
- Inflict a Gift: the player attempts to give a gift to another player. The player who is trying to give the gift is the giver and the player who has the gift is the defender. The defender has the option of allowing the giving or resisting. If the defender allows the giving, the defender gets the gift and adds it to his/her collection. If the defender decides to resist, then the giver and the defender each roll a six sided die. If the defender matches or exceeds the giver’s roll, then the gift remains with the giver. If not, the defender adds the gift to his/her collection. The next player then takes his/her turn. Defender does not count as the defending player’s turn and s/he can defend as often as needed.
Ending the Game
The game ends as soon as no more gifts remain in the gift pile (that is, the players possess all the gifts). Players must take their gifts with them when the game ends, mainly because the game is often played with the intention of getting rid of bad gifts or items that King Bob no longer wants.
Drinking Variant
Some people enjoy adding a drinking element to all games. In this case, a player who loses a roll has to take a drink.
A New Monopoly
Like most people, I have many fond (and some repressed) memories of playing Monopoly. I always tried to be the battleship, because, well, it had guns. I generally lost, but this was obviously an omen that I was intended for something other than a life of commerce.
While much of my gaming today is computer or console based (I even have a copy of Monopoly for the Xbox 360 in my house), I still have a boxed set of the classic game in my closet, along with my copy of Axis & Allies, Clue and other such games. After all, there is a lot to be said for gathering around a table with friends, snacks and some dice.
I had heard that Hasbro was updating Monopoly for the video game age, but did not think much about it until I saw a video of the new game. It struck me as some sort of horrible science fiction scenario: the beloved land of Monopoly had apparently been conquered and the black and red tower of the new master surveys the vanquished land.
This new monarch purports to be a benign overlord: it replaces the dice, money and rules of the game, thus freeing the players of the strain of rolling dice and the burden of basic math. Players play at its behest and obey its commands (or, presumably, Daleks are summoned to exterminate the transgressors).
The folks at Hasbro see this as bring a video game like experience to the game. However, I think they are fundamentally misguided.
First, as I mentioned above, there already is a Monopoly video game. So, people who want to play Monopoly as a video game can do that. There seems to be no real need to make a board game that tries to be a video game as well.
Second, the board game experience is fundamentally different from the video game experience and this difference provides something valuable. In video games, you are at the mercy of the rules set in the game (aside from using mods or hacking). With a board game, part of the game is agreeing on and applying the rules as a social group rather than having the rules inflicted by a small plastic tower. True, players sometimes try to be little dictators about the rules-but those are the game sessions that tend to really suck. As such, the new game seems to capture one of the worst aspects of live games while not providing the compensation that good video games provide in return for their lordship over the rules, such as impressive graphics. Thus, the democracy of the live game is replaced by the tyranny of the computer, without any of the awesomeness of actual video games.
Third, that tower set up looks stupid. Vaguely menacing, too. Red and black? Seriously?
The Fourth King’s Game
- Image via Wikipedia
My gaming group normally has a Christmas party, but this year events conspired to prevent us from gathering. However, we will be gaming again tomorrow, thus creating a need for a post Christmas event.
One of the reasons we did not get together earlier was due the fact that some of us travelled for the holidays. In my case, I went to Puerto Rico. While there I learned of a post Christmas holiday that inspired me to a solution to the post-Christmas problem.
As the story goes, three wise men or kings (not the same thing at all, of course) brought the baby Jesus some gifts. While this served as the theological foundation for the massive commercialization of Christmas, it also gave rise to Three Kings Day, which is celebrated in Puerto Rico. The gist of the holiday is that children put out grass and water for the Kings’ camels and they get small gifts in return. This holiday is on January 6th, which is too late for the post-Christmas event.
Fortunately, a little research revealed that there was a 4th king, King Bob. Unlike the Three Kings, Bob was not great with directions and ended up arriving at the wrong city, albeit a few days before the other kings arrived in the proper destination.
Since King Bob could not find the baby Jesus, he decided to give away the gifts via a game, which is now known as King Bob’s game. Alternatively, it can be called The Game of the 4th King.
Here is how the game is played.
What You Will Need
Gifts: At least 1 wrapped gift per player, preferably more. Cheap gifts are best.
Dice: Ideally you should have a D20 and some D6s, but for non gamers six sided dice will do.
The Roles
There are two roles in the game: King Bob’s stand in and player. King Bob supervises the game but does not play. He also does not get any gifts. Optionally, King Bob can also play and get gifts, but that is bad theology.
Everyone other than King Bob’s stand in is a player.
Setting Up the Game
King Bob sets up the game by creating a pile of the wrapped gifts and defending them from the greasy hands of the players until the game starts. Each player should have a die (or dice) and a board or piece of paper is needed to keep track of the order of play.
Initiative
Gamers will be familiar with this, but non-gamers will not. For the non-gamers, this is how you determine the order in which the players take their turns. To determine this, each player rolls a die (preferably the standard D20). The player with the highest roll goes first, the player with the second highest goes second and so on. In the case of a tie, reroll until it is settled.
Starting the Game
The game starts with the player who has the highest initiative. S/he selects one gift from the pile and DOES NOT open it. Shaking and such is allowed. The second player then has his/her turn and so on for each player until it is back to the first player. After the first player has selected his gift, the other players will have more options and the first player will also have these options on his/her second turn.
Playing the Game
After the first player has a gift, the second player has his turn and so on until everyone has had a turn. The first player then has his second turn and so on. During play, a player has options. Only ONE option may be taken each turn. A player can take a different option each turn, but is not required to do so.
- Pick a Gift: the player selects a gift from the pile but DOES NOT open it. The next player then takes his/her turn.
- Open a Gift: the player opens one gift that s/he has in his/her possession and opens it. The next player then takes his/her turn.
- Steal a Gift: the player attempts to take a gift from another player. The player who is trying to steal the gift is the thief and the player who has the gift is the defender. The defender has the option of allowing the theft or resisting. If the defender allows the theft, the thief gets the gift and adds it to his/her collection. If the defender decides to resist, then the thief and the defender each roll a six sided die. If the defender matches or exceeds the thief’s roll, then s/he keeps the gift. If not, the thief adds the gift to his/her collection. The next player then takes his/her turn. Defender does not count as the defending player’s turn and s/he can defend as often as needed.
- Inflict a Gift: the player attempts to give a gift to another player. The player who is trying to give the gift is the giver and the player who has the gift is the defender. The defender has the option of allowing the giving or resisting. If the defender allows the giving, the defender gets the gift and adds it to his/her collection. If the defender decides to resist, then the giver and the defender each roll a six sided die. If the defender matches or exceeds the giver’s roll, then the gift remains with the giver. If not, the defender adds the gift to his/her collection. The next player then takes his/her turn. Defender does not count as the defending player’s turn and s/he can defend as often as needed.
Ending the Game
The game ends as soon as no more gifts remain in the gift pile (that is, the players possess all the gifts). Players must take their gifts with them when the game ends, mainly because the game is often played with the intention of getting rid of bad gifts or items that King Bob no longer wants.
Pathfinder is the Real D&D
- Image via Wikipedia
I first started playing Dungeons & Dragons when I was 15. My mother got me started with the D&D Basic Set and I soon progressed to Advanced D&D.
While I thought the D&D system was rather awful when compared to the elegant and realistic system of games like Runequest and Call of Cthulhu, AD&D had two main selling points. First, it was so simple that even a high school stoner could roll up a character in the same time it would take them to roll a joint. Second, it had a level system that people loved. The idea of getting more an more powerful while playing has a tremendous appeal and the level progression system has become an essential aspect of almost all RPGs (computer and traditional).
I did try the 2nd Edition of D&D, but did not like it very much. To be rather vague, it did not have that “D&D feel.” I did try to run a game or two, but the magic was just not there.
When D&D 3.0 came out, I ended up giving it a shot. While it was a different sort of game (that is, it had fairly coherent and rather playable rules) from AD&D, it had the D&D feel. When 3.5 came out, I upgraded to that. When I heard that 4.0 was coming out, I looked forward to it. However, when I read the books and heard stories of people playing, I decided that it was not really D&D. I’m not going to go into the details, but the gist was that D&D 4.0 seemed more like a video game made into a traditional RPG. Crudely put, it was a bit like trying to play WoW as a tabletop RPG. While some folks like that, 4.0 lacks that D&D feel that is important to me. Some folks love the system, and I have no more to say against them than I have to say against folks who like Windows Vista.
I had looked at the Pathfinder beta (put out by Paizo) when it first came out, and had mixed feelings about it. However, when I actually played a campaign based on the rules, I realized that I rather liked it. The folks at Paizo took the 3.5 rules and revised them to address various weak points in the game. For example, they retooled the grapple rules from a mess to a workable system. They also revised the core classes in a way that gave players reasons to stick with one class from level one to level twenty. Best of all, they kept the D&D feel alive.
Of course, Pathfinder is not legally D&D, but rather a D20 system released in accord with the Open Gaming License. D&D was first owned by TSR, then it was bought up and it now belongs to Wizards of the Coast. WoC is, of course, owned by Hasbro.
This, as I see it, shows once more the downside of corporate ownership of such iconic entities. Since D&D is owned by a company, they can do pretty much anything they please with it and it will still legally be the D&D game. Of course, the fact that a company owns D&D does not entail that they own the “essence” of what it is to be D&D or that they are even fit to keep that essential nature going. The same sort of thing happens with movies. For example, Alien and Predator started off as cool and awesome movies. But, the corporate masters degraded the franchises into horrific parodies of their original awesomeness.
Naturally, I am not claiming that 4th Edition D&D is a horrible degradation on par with the Aliens vs. Predator movie. However, I am saying that it is unfortunate that the 4th edition D&D is the legally official D&D simply because the company making it legally owns D&D.
While Pathfinder is not legally D&D, to me it is D&D. It is, as I see it, the true spiritual successor to the Basic Edition I played all those years ago. So, I still play D&D, only the book sitting on the table in front of me says “Pathfinder.”
Dice Age
Having recently lost a player, my gaming group is trying to recruit. Ron, our most obsessive gamer, has been scouring the net for gamers and ways to find gamers. By the way, this is D&D style gaming-as opposed to gambling or players in the sense of those hated by player haters.
One new resource is Dice Age which is sort of a Myspace for gamers-only without (so far) the porn spammers. So far the site is very small (14 members) but it just started and might become a powerhouse of geek networking. Then again, it might fade away into oblivion.
Another resource is Gleemax, which is owned by Wizards of the Coast. It is in the Alpha stage, although it has been operating for a while.
If you are a gamer, check it out. If you are a gamer in Tallahassee and want to join a group, then contact me.
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