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。
游戏例子1——《Cutting the Cake》
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。
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。
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。
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。
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)