This is deeply related to the idea of utility functions; It's utility, not any measured resource, that you're optimizing for. The relationship between points/scores/money/friends/etc. and utility is non-linear.
Of course, games are a specific case where the utility (winning) is well-defined, and usually relative to other players, where life is much more complicated, ambiguous, and a mix of competitive and cooperative games.
games are a specific case where the utility (winning) is well-defined
Lots of board games have badly specified utility functions. The one that springs to mind is Diplomacy; if a stalemate is negotiated then the remaining players "share equally in a draw". I'd take this to mean that each player gets utility 1/n (where there are n players, and 0 is a loss and 1 is a win). But it could also be argued that they each get 1/(2n), sharing a draw (1/2) between them (to get 1/n each wouldn't they have to be "sharing equally in a win"?).
Another example is Castle Panic. It's allegedly a cooperative game. The players all "win" or "lose" together. But in the case of a win one of the players is declared a "Master Slayer". It's never stated how much the players should value being the Master Slayer over a mere win.
Interesting situations occur in these games when the players have different opinions about the value of different outcomes. One player cares more about being the Master Slayer than everyone else, so everyone else lets them be the Master Slayer. They think that they're doing much better that everyone else, but everyone else is happy so long as they all keep winning.
In Diplomacy I've never heard the 1/(2n) argument from that sentence. All it's saying is that if you are part of the draw, the person who survived with 1 supply center gets the same result as the one with all 17 on the other side of the line. Whether players actually treat it that way is up to them, of course.
But of course, my natural instinct is that winning alone is a special thing, and that winning outright is more than twice as good as a 2-way draw. When thinking about whether a 2-way draw is more or less than twice as good as a 4-way draw, I'm not sure.
In Castle Panic I think part of the fun is deciding how much you care about the title versus winning the battle, where the right answer is not zero but not enough to *seriously* risk losing the battle over that...
Multiplayer prisoner's dilemma games (with information) are absolutely a situation where I endorse Modesty. Don't play as if you are a special snowflake who is going to bamboozle your opponents. Sometimes you're increasing variance, so that you're more often first place even without raising your average score, but more often you're just shooting yourself in the foot.
If you just play a solid, modest, try-to-cooperate-against-people-who-are-also-trying-to-cooperate-against-this-strategy strategy, what I feel like tends to happen is that some opponents will profit slightly off of playing against you, and then defect a lot against each other because they're trying to be fancy, and then you win.
Previously (Putanumonit): Player of Games
Original Words of Wisdom:
Quite right, sir. Quite right.
By far the most important house rule I have for playing games is exactly that: You Play to Win the Game.
That doesn’t mean you always have to take exactly the path that maximizes your probability of winning. Style points can be a thing. Experimentation can be a thing. But in the end, you play to win the game. If you don’t think it matters, do as Herm Edwards implores us: Retire.
It’s easy to forget, sometimes, what ‘the game’ actually is, in context.
The most common and important mistake is to maximize expected points or point differential, at the cost of win probability. Alpha Go brought us many innovations, but perhaps its most impressive is its willingness to sacrifice territory it doesn’t need to minimize the chances that something will go wrong. Thus it often wins by the narrowest of point margins, but in ways that are very secure.
The larger-context version of this error is to maximize winning or points in the round rather than chance of winning the event.
In any context where points are added up over the course of an event, the game that matters is the entire event. You do play to win each round, to win each point, but strategically. You’re there to hoist the trophy.
Thus, when we face a game theory experiment like Jacob faced in Player of Games, we have to understand that we’ll face a variety of opponents with a variety of goals and methods. We’ll play a prisoner’s dilemma with them, or an iterated prisoner’s dilemma, or a guess-the-average game.
To win, one must outscore every other player. Our goal is to win the game.
Unless or until it isn’t. Jacob explicitly wasn’t trying to win at least one of the games by scoring the most points, instead choosing to win the greater game of life itself, or at least a larger subgame. This became especially clear once winning was beyond his reach. At that point, the game becomes something odd – you’re scoring points that don’t matter. It’s not much of a contest, and it doesn’t teach you much about game theory or decision theory.
It teaches you other things about human nature, instead.
A key insight is what happens when a prize is offered for the most successful player of one-shot prisoner’s dilemmas, or a series of iterated prisoner’s dilemmas.
If you cooperate, you cannot win. Period. Someone else will defect while their opponents cooperate. Maybe they’ll collude with their significant other. Maybe they’ll lie convincingly. Maybe they’ll bribe with out-of-game currency. Maybe they’ll just get lucky and face several variations on ‘cooperate bot’. Regardless of how legitimate you think those tactics are, with enough opponents, one of them will happen.
That means the only way to win is to defect and convince opponents to cooperate. Playing any other way means playing a different game.
When scoring points, make sure the points matter.
These issues will also be key to the next post as well, where we will analyze a trading board game proposed by Robin Hanson.