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解析游戏博弈论与游戏设计之间的关系

发布时间:2013-05-06 14:40:28 Tags:,,,,

作者:Frank Lantz

游戏博弈论是一种数学性研究,能够应用于多种领域,包括哲学,军事,经济和策略等等。游戏设计师并不是在使用特有的游戏博弈论。

游戏博弈论是对于情境的数学分析,即参与者需要在此做出选择。让我们以服装设计为例吧,即所有人需要想出最独特的服饰。这一决定是基于其他人的决定,并且能够进行数学建模。

游戏博弈论的关键术语分别是玩家(参与者),策略(选择)以及收益(结果)。基于所有情境,你可以创造一个收益矩阵。

最初的游戏博弈论是由伟大的数学家John von Neumann所提出。他也喜欢聚会并玩扑克。他所撰写的《Theory of Parlour Games》开启了我们对于游戏博弈论的研究。

“轮盘赌”是一款估算可能性的游戏;“象棋”是基于估算决策树的游戏。而扑克则完全不同;它是关于欺骗,出价,并基于敌人的选择做决定的游戏。

“象棋并不算是游戏,只能算是一种基于计算的谜题。真正的游戏应该是带有欺骗性。”

游戏例子1——《Cutting the Cake》

Cutting-Cake(from santafty.com)

Cutting-Cake(from santafty.com)

在这款游戏中,一个玩家负责切蛋糕,而另一个玩家将挑选他的第一块蛋糕。

第二个玩家将面对最佳选择。即一般情况下他都会选择较大的一块。

而对于第一个玩家而言,他所面对的选择总是极大极小。他将把手伸向可能带来最小损失的选择(即尽可能均匀地切蛋糕)。

研究对手的眼神是游戏博弈论的核心。

在零和游戏中,所有的选择加在一起便为零。也就是永远都是一方获胜另一方失败。

游戏例子2——硬币匹配游戏

在这种游戏中,两名玩家将分别选择硬币的一面。如果两个硬币相匹配,那么第一个玩家将获得硬币。如果两个硬币并不同,那就是第二个玩家获胜并获得硬币。

在此的最佳策略便是添加一些随机性去挣脱对手。

如果游戏是非零和游戏会怎样?例如“与比尔盖茨玩硬币匹配游戏。”仍旧基于相同的规则,除非两个硬币都正面朝上,否则比尔盖茨就需要给你100万美元。如此对你来说最佳选择便是时不时扔出正面朝上的硬币,如此比尔盖茨便会希望硬币是反面朝上。在棒球中亦是如此,如果扔出好球的话别人便能更好地猜测你接下来的方向,所以你最好时不时地扔出一些坏球。

游戏例子3——《Chicken》

在这款游戏中,玩家双方都将驾驶着汽车朝悬崖前进。第一个转弯的玩家便是输家。但是如果两个玩家都转弯的话,双方的结果便都是死亡。这是一种非零和游戏。

科学家们发现这种收益矩阵也可以用于自然界中动物之间的打斗。

John Maynard Smith关于“老鹰vs.鸽子”的理论便映射出这种行为。他将游戏博弈论与生物学原理联系在一起,并发现动物们遵循着一种稳定进化策略。如果出现过多鸽子,那便会激起老鹰内心的斗志。但是如果存在太多老鹰,它们便会一直相互抗击去争抢仅有的鸽子。最终将达到双方的进化平衡。

而在《Chicken》中,删除选择将推动着其他玩家做出决定。例如一个玩家删除了他的方向盘,他便不可能进行转弯,这便意味着另外一个玩家将被迫转弯。有时候,非理性也会战胜合理性。

游戏例子4——《Prisoner’s Dilemna》

在这款游戏中,两名犯人将被关在不同的牢房里。他们将选择背叛同伴或者保持沉默。如果双方都选择沉默,他们便会被关在牢中一年,但是如果都招供了,他们就需要被关在牢中3年。而如果他们做出的不同选择,那么背叛者将被无罪释放而另外一个人则需要被囚禁10年。

这真的是一种进退两难的选择。并不存在任何优势策略。

最佳策略便是招供。这可能带来更好的结果并避免10年的牢狱之灾。如果两名犯人都是聪明的游戏博弈论专家,他们便会都选择这一策略并共同在牢里待3年。而另外一种情况是两名犯人同时被囚禁在牢中,但却不能说话并且未获得任何信息。如同他们同时选择沉默的话便能获得更好的结果。“最佳”方法将导致次优结果,但是沉默的方法却能导致更好的结果。这是什么情况?

RAND cooperation便是由一群军事理论家所组成。Von Neumann(游戏邦注:“现代电子计算机之父”)和RAND都认为冷战的最佳解决方法便是先发制人。但事实上却是非理性决定拯救了世界。

游戏博弈论将我们置于核战争理论的边缘,但是非理性的悖论也是来自RAND。也有一种情况便是游戏博弈论和扑克(即创造出赌徒)拯救了世界。

Robert Axelrod问了一个问题:“我们将如何实现合作结果?”他创造了一个能够模拟囚犯反复困境的计算机程序,并要求bot在这种模拟中正视彼此。最成功的bot便是Anatol Rapoport的以牙还牙策略,这是具有合作性的方法,但是如果在之前的回合展开了攻击,那也会带有攻击性。在今后十年里这种策略将继续制胜,并且所有人也将知道这是打败对手的有效策略。我们也能在生物学中找到以牙还牙的情况。

在此之后William Press和Freeman Dyson也将以牙还牙归到零确定性策略中。

游戏例子5——最后通牒游戏

在这种游戏中,一名玩家将划分100美元并为另一名玩家提供交易。另一名玩家可以选择接受或拒绝该交易,而最终导致没有一名玩家能够赚到钱。

Thomas Schelling写下了相关游戏理念。这不只是关于冲突和竞争,还涉及了合作,竞争与协商。

如何将游戏博弈论应用到游戏设计中?

1.继承。游戏博弈论是源自像扑克这样的现实游戏。此外,历史上许多游戏博弈论专家也设计过游戏。就像提出了纳什均衡理论的John Nash便设计过一款有关网络和关系的游戏。

2.正式分析。游戏博弈论为游戏平衡和游戏经济提供了一个很棒的基础。

3.灵感。游戏博弈论决策很简单,但却并非微不足道。我们可以从桌面游戏,现实游戏以及许多社交游戏中看到真相。Frank Lantz的游戏(如《Parking Wars》, 《Spore Islands》,《Power Plant》和《The Friend Game》)都是关于玩家基于彼此间去做决策。

Journey(from forbes)

Journey(from forbes)

4.和谐。最近我们发现像《Journey》,《Proteus》,《亲爱的艾斯特》,《The Graveyard》以及《骤雨》等游戏都删除了可能影响平衡的机制,并专注于情感体验。

我们需要注意的是,“理性不一定是最佳方法。”解释并不能详尽讨论一个主题。就像知道人们是由原子组成也不能去压缩他们。

游戏博弈论并不只是关于理性。人文与科学是相互区分的,而我们是否能不压缩其中的一方而将其整合在一起?

数学,策略和规则是与故事,情感和爱共存的。游戏设计是数学与美学的整合。

即使游戏博弈论不具备实际用途的话也没关系。我们并不需要找寻实用性,我们真正要寻找的是真相。

本文为游戏邦/gamerboom.com编译,拒绝任何不保留版权的转载,如需转载请联系:游戏邦

GDC2013 – Strange Love: The Relationship Between Game Theory and Game Design

This talk was given by Frank Lantz.

Game theory is a mathematical study that applies to many fields including philosophy, military, economy, and strategy. Game designers don’t use proper game theory much at all.

Game theory is the mathematical analysis of situations where multiple parties are making choices. Take for example a costume party where everyone is trying to come up with an awesome unique costume. This decision will depend on other people’s decisions and can be modeled mathematically.

The key terms of game theory are players (the parties involved), strategies (the choices), and payoffs (the results). With all situations, you can make a payoff matrix.

The origins of game theory come from John von Neumann, who was a great mathematician. He also liked to party and play Poker. He wrote a book called the “Theory of Parlour Games” which started game theory.

Roulette is a game that can be calculated with possibilities; chess is a game of calculated decision trees. Poker, however, was completely different. Poker was about bluffing and bidding, about making choices based on opponent’s choices.

“Chess is not a game, it’s a computational puzzle. Real games, like life, are about bluffing…”

Game example #1 – Cutting the Cake

In this game, one player cuts the cake and the other player gets first pick of the piece she’ll take.

For the second player, there is a dominated strategy, the choice that is obvious and best. She’ll always pick the bigger piece.

For the first player, the decision is minimax. He’ll go for the choice with the minimal losses (cut the cake as evenly as possible).

The process of looking through the opponent’s eyes is the core of game theory.

In a zero-sum game, all the choices add up to zero. One wins and the other loses.

Game example #2 – Matching pennies

?In this game, two players choose a side on a penny respectively. If both pennies match, the first player takes the pennies. If both pennies are different, the second player wins and takes them.

Often the best strategy here is to add some randomness to throw off the opponent.

What if the game was non-zero sum? Consider “matching pennies with Bill Gates.” The same rules apply as before, except if both pennies match on heads, Bill gives you a million dollars. The obvious choice for you is to play heads, but Bill will always play tails. There is a dominant strategy, but it’s good to throw in the other choice to throw off your opponent. This is likewise in baseball, where throwing your best pitch will make you become predictable, so it’s good to throw in a bad pitch every now and then.

Game example #3 – Chicken

In this game, both players drive cars towards a cliff. The first player to swerve is the loser. However, if neither players swerve, then the outcome is very negative for both, resulting in death. This is a non-zero sum game.

Scientists have found that this payoff matrix applies to animal fights in nature.

John Maynard Smith’s theory of Hawks vs. Doves maps out this behavior. He looked at game theory in relationship to biology and found that animals follow an evolutionary stable strategy. If there are too many doves, then it pays off to be the hawk with the aggressive strength. If there are too many hawks, then they end up fighting each other all the time and it pays off to be a dove. Ultimately, the two species strike an evolutionary balance.

In the game of Chicken, eliminating choices will force a decision for the other player. If for example, one player removed his steering wheel altogether, he can’t swerve because he has no means to, which means the other player is forced to swerve. Sometimes, the irrational beats rationality.

Game example #4 – Prisoner’s Dilemna

In this game, two criminals are apprehended and placed in different cells. They are asked to betray their partner (defect) or stay silent (cooperate). If both cooperate, they get one year in prison, but if they defect against each other, they’ll both get 3 years in prison. If they choose different options, the defector will get off jail scotfree while the other will get 10 years in prison.

This is a real dilemna because the decision is difficult. There is no dominant strategy.

The best optimal decision is to defect. It yields the better deals and avoids the nasty 10 year sentence. If the two prisoners are smart game theorists, they will choose this strategy and get 3 year sentences. However, another pair of prisoners can come into this situation, and they’re dumb and uninformed. They’ll both choose to cooperate, resulting in a much better payoff. The “optimal” solution lead to suboptimal results, but the dumb solution lead to higher payoffs. What?

The RAND cooperation was a group of military game theorists. Von Neumann and RAND both thought the optimal solution during the Cold War was to pre-emptively bomb Russia first. But it turns out that the irrational decision (not to bomb) saved the world.

Game theory put us in the brink of nuclear war, but the paradox of irrationality also came from RAND. There’s a chance that game theory and Poker, a game that was meant for degenerate gamblers, saved the world.

Robert Axelrod asks the question, “How do we get to cooperative payoffs?” He created a computer program that simulated the iterated prisoner’s dilemna and asked for open submissions for bots to face each other in this simultation. The most successful bot was Anatol Rapoport’s Tit for Tot strategy, which always cooperated but will attack if attacked the turn before. This strategy continued to win in future decades even when everyone knew it was the strategy to beat. Tit for Tot is also discovered and observed in biology.

William Press and Freeman Dyson later categorized Tit for Tot as a subset of zero-deterministic strategies.

Game example #5 – Ultimatum game

In this game, one player must divide $100 and offer the deal to another player. The other player can accept the deal or reject it, in which case neither player gets any money.

Thomas Schelling wrote about game theory. It’s not just about conflict and competition, but about coordination, competition, and negotiation.

What are the applications of game theory to game design?

1.??Legacy. Game theory came from real world games like Poker. In addition, many historical game theorists have designed games. John Nash, who developed the Nash’s equilibrium, designed a game about networks and relationships.

2.Formal Analysis. Game theory provides a great basis for game balance, yomi, and game economies.

3.Inspiration. Game theory decisions are simple, but not trivial. You can see examples of it in board games, reality game shows, and many social games. Frank Lantz’s games (Parking Wars, Spore Islands, Power Plant, The Friend Game) are all also related to decisions players make in regards to each other.

4.Reconciliation. Recently, we’ve seen games like Journey, Proteus, Dear Esther, The Graveyard, and Heavy Rain. These are games that removes min-maxing and focus heavily on emotional experience.

?The key takeaway is that “the rational is not incompatible with the sublime.” Explanations do not exhaust a topic. Knowing that a person is made up of atoms doesn’t diminish the person in any way.

Game theory is not just about rationality. Humanities and sciences have separated, but can we bring them both together without diminishing either of them?

Math, strategy, and rules can co-exist with stories, emotions, and love. Game design is where math and beauty meet.

?It’s okay if game theory has no practical use. We’re not necessarily looking for utility, we’re looking for truth.(source:blogspot)


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