Mathematical Ideas that Shaped the World Game theory Plan for this class What is the idea of Game Theory? What kind of situations is it useful in? Find out how nice the person next to you is! How can Game Theory explain human altruism? How (and why!) does eBay work? What is Game Theory? “Game Theory attempts to mathematically capture behaviour in games where an individual’s success in making choices depends on the choices of others.” Examples of where to use game theory A shop owner deciding the price of biscuits Competing with your mates for a girl/boyfriend Khrushchev and Kennedy in the Cuban Missile Crisis Combating climate change Deciding whether to help someone Auctions Voting systems Animal behaviour and evolution Nim There are n objects 2 players On your turn, you may take either 1, 2 or 3 of the objects The person who takes the last object wins the game. Can you find a winning strategy? Types of game Cooperative vs. non-cooperative Zero sum Perfect vs. non-perfect information John von Neumann published the first real paper on Game Theory, in 1928. He focused on 2-person zero-sum games. John von Neumann (1903 – 1957) Born in Budapest to a Jewish family. By the age of 6 could speak classical Greek and recite telephone directories. Got PhD at the age of 22 and worked in Berlin before emigrating to the US in 1930. John von Neumann (1903 – 1957) Worked out the key steps in the nuclear physics needed to create the hydrogen bomb. Was on the committee which decided where to drop the first atomic bomb. Died of cancer under military security. I know that you know that I know... The mischievous leprechaun You and a friend get to the end of a rainbow and find a leprechaun with a pot of gold. It has 1 billion pounds. The leprechaun offers you £1. If you refuse, he offers £10 to your friend. If they refuse, he offers £100 to you, etc. When should you take the money? The Ultimatum Game Two players. Player 1 has £100. They can choose how to split this between themselves and Player 2. If Player 2 accepts the split, they get the money. Otherwise, they both get nothing. How much should Player 1 offer to Player 2? Assumptions In Game Theory we make the following assumptions: That the players are rational That players do not trust each other The behaviour predicted by Game Theory for the Ultimatum Game is rarely seen. Does this mean that humans are irrational? John Nash (1928 – now) Born in West Virginia. The letter of recommendation for his PhD was a single line: “This man is a genius.” Completed his thesis at Princeton – it consisted of only 28 pages. Nash’s thesis Nash studied the theory of non-cooperative games and the idea of equilibria. In his thesis, Nash attacked the longaccepted theories of Adam Smith, whose maxim was “every man for himself”. This is highlighted in his biographical film A Beautiful Mind, in which he explains why going for your first choice is not always a good idea… Strategic voting Bill, Phil and Gill are thinking of painting the living room. They are going to vote on whether to keep the walls white or paint them blue. Phil proposes instead to paint them pink. They decide to vote first between blue and pink. They then vote between the winning colour and white. Strategic voting Bill Phil Gill 1. Blue 1. Pink 1. White 2. White 2. Blue 2. Blue 3. Pink 3. White 3. Pink Why is it in Gill’s best interest to vote for pink? Payoff matrices A payoff matrix records the winnings of each player in every possible outcome of the game. Player 2 odd even 3 odd Player 1 3 6 0 0 even 6 1 2 equilibri um Chicken Player 2 stay run 0 run 0 Player 1 stay 5 -1 -1 5 -4 -4 Nash equilibria In Chicken, there is no dominant strategy. But there are two places where a Nash equilibrium occurs. A Nash equilibrium is where, if you think your opponent will keep the same strategy, you will do best if you keep using the strategy that you currently have. Nash equilibria Player 2 run stay 0 run 0 Player 1 5 -1 -1 stay 5 Nash equilibrium -4 -4 Nash equilibrium Golden Balls This was a game show where the finale involved two people and a jackpot of money. Each person could choose whether to share or steal the money. If they both steal, they get nothing If they both share, they get 50% each. If one steals and one shares, the stealer gets everything. Prisoners Dilemma Two people are on trial for a crime. Each of them has a chance to grass on the other one. If both stay quiet, there is little evidence to convict and they only get a 2 year jail term. If they both grass, they get 5 years in jail. If one grasses and the other doesn’t, he gets released while the other does 6 years. Prisoners payoff Player 2 Stay quiet Stay quiet Defect -2 0 -2 -6 Player 1 -6 Defect 0 -5 -5 equilibrium Prisoners analysis The game-theoretic analysis of the Prisoners Dilemma seems counterintuitive – clearly both players would be better off if they cooperated with each other. In experiments, 40% of subjects cooperated. What if the game is played a number of times and the scores added? The iterated Prisoners Dilemma Did you have a different strategy in the extended version of the game? What kinds of strategies do you think are likely to be effective? Can this explain anything about human behaviour? Do people act morally because it is in their selfinterest to do so? Applications The Prisoners Dilemma applies in any situation where it is in the collective interest to work together, but where there is always an incentive to defect. Stealing Combating climate change Price fixing in supermarkets Animals sharing a resource Athletes taking drugs John Nash - illness Nash was admitted to hospital in 1959 for paranoid schizophrenia. Upon release he sought asylum in France but was deported back to the US. Spent another 9 years on medication and visiting hospitals. John Nash - recognition After 1970 he refused to take any more medication, and began doing mathematics again. In 1994 he finally received a Nobel Prize for his work on Game Theory. In total 8 game theorists have won Nobel prizes in economics. Lessons to take home Game Theory can be applied to almost any situation: in politics, sociology, economics, biology, philosophy, law. Standard game theory does not always accurately predict human behaviour... But on the other hand it can go some way towards explaining deep philosophical concepts such as morality.