A Theory of Game: Psychology & The Prisoner's Dilemma
by Shannon Appelcline
It seems that much game design is done by the seat of our pants. Something strikes us, we implement it, and then we see if it works or not. There's no denying the fact that this sort of spur-of-the-moment implementation allows for a wide swath of creativity and originality which might be smothered by more exacting work. However, on the other hand, it can also deny us the ability to learn from a century of careful work done on game theory.
There has been a lot of research done on game theory. We have the von Neumann-Morgenstern theories of zero-sum games, and the later vonNeumann-Morgenstern n-person game theories. The most famous work is perhaps that of John Nash, and his idea of a "Nash Equilibrium".
Many of these ideas are insightful, but also tedious.
Less so is the idea of the Prisoner's Dilemma, and many of its variants, which predicts selfish behavior in games. This week I'd thus like to consider some of the underpinnings of game theory which most of us should at least have in our back pocket, beginning with the Prisoner's Dilemma.
The Prisoner's Dilemma
Albert W. Tucker created the Prisoner's Dilemma in 1950. It suggests a scenario where two criminals have been taken into police custody, but with only enough evidence to convict them of a minor crime, not a major crime--unless one or more of them confesses. This creates a matrix of possibilities.
If neither criminal confesses, each receives a small sentence (6 months) and then goes free. However, if just one criminal confesses, he receives no sentence and his accomplice receives a large sentence (10 years). Finally, if both criminals confess, they each receive a medium sentence (2 years).
Now, the optimal solution for the society as a whole is for neither criminal to confess, because their joint small sentences are lesser than any of the other possibilities. However, for any individual criminal the optimal solution is to confess, because this gives the possibility of no punishment and also ensures that they can't receive the maximum punishment that would occur if their accomplice confessed and they did not.
This game theory scenario explains how players might take actions in the community's worst interest, and in fact in their average worst interest, solely due to selfish self-interest. And, as we'll see, it's just one of many theories exploring this same area.
The Dollar Auction
Martin Shubik's Dollar Auction (taught to me by JC Lawrence) is another interesting variant of some of the same ideas.
Here you have an auctioneer selling a dollar bill for whatever the market will bear. But, there's a catch: both the winner and the second-place bidder must pay their entire bid. The auction starts off simply, with someone bidding a penny, but then someone else puts in their two cents, and the first bidder realizes that he's out his penny with no return. When the bidding gets up to a dollar, the 99 cent bidder realizes that he can either be out 99 cents or else bid $1.01, and only be out a penny (because he receives the dollar in exchange) ... and so the bidding continues. Marvin Minsky, a CS professor at MIT, says that the dollar usually goes for between $3 and $5.
Though artificial in its game theory construct, the idea of the dollar auction does have real world parallels. Consider a war, where each of two forces are expending resources. The side that loses the war will have spent everything fo now return, while the side that "wins" might just be reducing the total cost of their investment through that "victory".
Again, it's a classic situation of greed overcoming community good.
The Free Rider Paradox
Social Scientist Mancur Olson is the author of a theory of collective action that is called a generalization of the prisoner's dilemma: the Free Rider Paradox. Bruno Faidutti directly integrated this idea into his game, Terra.
The basic principle is this: there are certain collective actions (forming a labor union, cutting down on ozone emissions, etc.) that will collectively benefit a number of people. But, each individual participating in the action suffers some deficit (union fees, less efficient vehicles) while everyone in a broader group of people appreciates a benefit (better wages, cleaner air) whether they participate in the action or not.
In other words, due to refusing to sign on to the Kyoto Accords, George W. Bush can expect his Americans to maintain their current level of energy usage, and at the same time appreciate the benefits of less skin cancer and a better ecosystem that accrue from other countries who do sign on. America is thus a free rider in the battle against the greenhouse effect.
Of course, if everyone adopted this same policy of not taking this action, there would be no benefit.
To put it back into the earlier terms, individual players decide upon an individual greedy answer, even if the community good might be better if everyone cooperated.
The Ultimatum Game
The Ultimatum Game, which comes out of economic theory, is an interesting game theory experiment because it shows the opposite response as the Prisoner's Dilemma: players reject greed for community good.
Again, the experiment is simple. There are two players, and one of them is given a set amount of money to split (say, $100). The decision maker makes one decision as to how to split that money (say, $60 and $40), and the other player then accepts it or rejects it; if he accepts the split each player gets the set amount of money, while if he rejects neither player gets anything.
As with the previous two theories, there is a community good here, which I'd measure as whether "fair" splits usually occur. There's also greed, with each player being given the opportunity to get free money. However, in most cases the respondant will reject any offer of less than 20% of the total sum. In other words, he'll set aside his personal gain for the hope of improving the community in the future.
Or maybe it's just spite.
This probably goes back to an idea called the "iterative prisoner's dilemma", where a number of rounds of the prisoner's dilemma are played, giving players the opportunity for reprisal for previous actions. In this case the pendulum starts to shift, and people start to move toward the community good, of neither player confessing, rather than individual good--because there are consequences.
Greed, community good, and now reprisal--they're all tied up into psychological (and game theory) knots, and these four theories just start to explain what reactions we might expect.
Making the Abstract Concrete
So, what does all this abstract game theory mean in the world of real game design?
I think that the Prisoner's Dilemma, and its kin, point toward interesting decisions that you could incorporate into any game. What is the individual good? What is the community good? What happens when you force players to make a decision between community and individual benefit? What happens when you also introduce ideas of reprisal?
To be even more specific, consider a problem of online game design--for example, players clearing out spawn regions--and then look toward the behaviors predicted by these game theories, and how they might resolve the problem.
In most online games, there's some sort of spawning mechanism, whereby players have the option to enter hunting grounds and kill what's there, to gain experience and equipment. There is a very strong individual good here (XP + warez) and a weak community good (increased enjoyment for all players).
As predicted, most players go for the strong individual good, clearing out spawn regions with no concern for other players' enjoyment.
What if we now increased that community good, so that spawn points spawned more, the more often they're left alone? Suddenly we've started to shift a simplistic Prisoner's Dilemma to a more complex Free Rider's Dilemma, where we give some players the option to benefit from the good will of others. And, in the mechanism, we've probably created the basis for some interesting social and community structures.
What if we now took an additional step and said that all the players who share a particular spawn point have the option to "veto" that spawn point, removing all rewards for that spawn point for the last day. Now, if a player feels like he's not getting his fair share, because he's holding back while others are overhunting, he can offer reprisal. Enter the Ultimatum Game.
And enter the possibilty for these abstract game theory ideas to have a real effect on game design.
Food for thought.
As I've written this piece, I've come to realize that it shares some ideas in common with Christopher Allen's recent blog entry, Dunbar, Altruistic Punishment, and Meta-Moderation, which also talks about how altruistic punishment (like disagreeing on that $100 split, at a cost to yourself) can improve a community.
If you're interested in more on this topic, I suggest reading that next, then the various Wikipedia entries I've linked to in this article.