Game Theory: A Very Short Introduction
Ken Binmore, 2007
Holmes decides at which station to get off, and Moriarty decides where to wait for him. Ken Binmore, the scientist who led the design of Britain's mobile licence auction, shows that games like this surround us everywhere: on the road, in bargaining, at an auction, and even in a lake where fish live. You will learn what a Nash equilibrium is, why sensible people do not cooperate in a single meeting, and what makes them cooperate when the meetings are repeated.
Who it is for
For people who want to understand how we choose when the result depends on others, and why cooperation arises even among selfish people. There will be no formulas.
We do not need an outside coercive force, real or imagined, in order to cooperate successfully.
Holmes, Moriarty and a station where you cannot make a mistake
Picture Sherlock Holmes on a train. Professor Moriarty is chasing him. Holmes has to decide at which station to get off, and Moriarty has to decide at which station to wait for him. If they choose the same station, Moriarty wins. If they choose different ones, Holmes escapes. Ken Binmore opens his book with exactly this scene from Conan Doyle's story. In his words, the detective and the criminal played a real game on the way to their last clash at the Reichenbach Falls.
What is the sensible thing for Holmes to do? The first thing that comes to mind is a chain of thought: “he thinks that I think that he thinks…”. Edgar Allan Poe proposed the same chain in “The Purloined Letter”. Binmore shows where it breaks: if the opponent reasons just as cleverly, the chain never ends, and that method cannot let both sides win at the same time. The answer of game theory is far more surprising, and we will come to it.
Binmore trained as a mathematician, but he is known as an economist. He led the team that designed Britain's mobile licence auction, so he writes about strategy as a practitioner. His book is very short and almost free of formulas. The main idea is simple: games are everywhere. A driver in heavy traffic is playing a game, and so is a buyer bidding on eBay. Game theory studies how to play such games sensibly.
Here you will learn what a game and an equilibrium are, why it is sometimes better to toss a coin, why sensible people do not cooperate in one short meeting, and why cooperation still arises when meetings are repeated. You will also see how the same laws work among fish and bees, at auctions and in bargaining.
Equilibrium: a point no one gains by leaving alone
Binmore explains game theory with small toy games. There are usually two players, Alice and Bob. A game begins where your result depends on someone else's choice: it is not enough for Alice to decide what to do, she has to guess what Bob will do.
The prize in a game is not necessarily money. Theory measures it in conventional units called utility. John von Neumann showed how to measure it: by how much risk a person is willing to take for a result. This is where the idea of risk aversion comes in. It means that each next thousand pleases you less than the one before. Such a person gladly buys insurance. Binmore stresses that the strength of risk aversion depends only on personal taste, like a choice between chocolate and vanilla ice cream.
What matters too is what game theory calls sensible. It does not judge whether your wants are good. It is interested in revealed preferences, that is, what you actually choose, not what you say about yourself. A sensible player simply does not contradict themselves. A common mistake is to think that sensible means cold and greedy for money.
The central concept of the book comes from John Nash, the hero of the film “A Beautiful Mind”. Binmore reminds us that Nash was still a student when he laid the foundations of the theory of sensible bargaining. A Nash equilibrium is a set of moves in which each player's move is the best answer to the others' moves. Nobody gains by changing their behaviour alone.
Binmore's simplest example is the game of driving on the road. If everyone drives on the right, it pays you to drive on the right too. If everyone drives on the left, it pays to drive on the left. Both orders are equilibria, and theory does not say which one is correct. It says why everyone sticks to it. Binmore recalls how Sweden switched from left-hand to right-hand traffic overnight in 1967. Switching one by one would have been deadly, so the whole country moved at once.
This gives a practical question for any stuck situation: does it pay anyone alone to act differently? If not, you are looking at an equilibrium, and persuading one person will not shift it. You have to move together.
About the author
Ken Binmore is a British mathematician and economist, emeritus professor at University College London and one of the leading researchers in game theory. He led the design of Britain's 2000 mobile licence auction.
All books by this author: Ken Binmore
Terms from the book
- Rubinstein
- Hamilton
- Utility
- Galileo
- Holmes
- Kant
- Edgar
- Nash
- Prisoner's dilemma
- Axelrod
- Evolutionarily stable strategy
- Solomon
- Moriarty: Professor Moriarty, Holmes's main opponent in Conan Doyle's stories.
- Doyle: Arthur Conan Doyle, British writer and author of the Sherlock Holmes stories. The chase Binmore analyses is described in the story “The Final Problem”.
- Reichenbach: The Reichenbach Falls in Switzerland, where according to Conan Doyle's story Holmes and Moriarty fought their final battle.
- eBay: American online marketplace where, among other ways, items are sold at auction.
- von Neumann: John von Neumann, American mathematician of Hungarian origin. With Oskar Morgenstern he laid the foundations of game theory, proved the minimax theorem and proposed a way to measure utility.
- Risk aversion: The trait of a person for whom each next unit of money is worth less than the one before, so he is willing to pay to avoid risk.
- Revealed preferences: An approach that judges a person's wants by what they actually choose, not by what they say.
- Nash equilibrium: A set of moves in which each player's move is the best answer to the others' moves, so that nobody gains by changing their behaviour alone.
- Mixed strategy: Choosing a move at random with chosen odds, so that the opponent cannot guess it.
- Minimax: Von Neumann's minimax theorem: in a game of pure conflict you can choose a move that gives the biggest gain even if the opponent replies in the worst way for you.
- Folk theorem: A claim of game theory that in an infinitely repeated game patient players can sustain cooperation on their own, without outside force. The name is a joke: for a long time the result circulated among scientists without an author.
- TIT-FOR-TAT: A “tit for tat” strategy for a repeated game: start by cooperating, then repeat the opponent's last move.
- Fitness: In biology, a measure of how many descendants a carrier of a gene or a trait leaves.
- Bluegill: The bluegill sunfish, a small freshwater fish from the lakes of North America, popular with anglers.
- Vickrey: William Vickrey, Canadian-American economist and Nobel Prize laureate in economics in 1996. He described an auction in which the winner pays the second-highest bid.
All terms
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