1 / 24

Game Theory

Game Theory. Developed to explain the optimal strategy in two-person interactions. Initially, von Neumann and Morganstern Zero-sum games John Nash Nonzero-sum games Harsanyi, Selten Incomplete information. An example: Big Monkey and Little Monkey. Monkeys usually eat ground-level fruit

lamya
Download Presentation

Game Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Game Theory • Developed to explain the optimal strategy in two-person interactions. • Initially, von Neumann and Morganstern • Zero-sum games • John Nash • Nonzero-sum games • Harsanyi, Selten • Incomplete information

  2. An example:Big Monkey and Little Monkey • Monkeys usually eat ground-level fruit • Occasionally climb a tree to get a coconut (1 per tree) • A Coconut yields 10 Calories • Big Monkey expends 2 Calories climbing the tree. • Little Monkey expends 0 Calories climbing the tree.

  3. An example:Big Monkey and Little Monkey • If BM climbs the tree • BM gets 6 C, LM gets 4 C • LM eats some before BM gets down • If LM climbs the tree • BM gets 9 C, LM gets 1 C • BM eats almost all before LM gets down • If both climb the tree • BM gets 7 C, LM gets 3 C • BM hogs coconut • How should the monkeys each act so as to maximize their own calorie gain?

  4. An example:Big Monkey and Little Monkey • Assume BM decides first • Two choices: wait or climb • LM has four choices: • Always wait, always climb, same as BM, opposite of BM. • These choices are called actions • A sequence of actions is called a strategy

  5. An example:Big Monkey and Little Monkey c w Big monkey c w c Little monkey w 0,0 9,1 6-2,4 7-2,3 • What should Big Monkey do? • If BM waits, LM will climb – BM gets 9 • If BM climbs, LM will wait – BM gets 4 • BM should wait. • What about LM? • Opposite of BM (even though we’ll never get to the right side • of the tree)

  6. An example:Big Monkey and Little Monkey • These strategies (w and cw) are called best responses. • Given what the other guy is doing, this is the best thing to do. • A solution where everyone is playing a best response is called a Nash equilibrium. • No one can unilaterally change and improve things. • This representation of a game is called extensive form.

  7. An example:Big Monkey and Little Monkey • What if the monkeys have to decide simultaneously? c w Big monkey c w c Little monkey w 0,0 9,1 6-2,4 7-2,3 Now Little Monkey has to choose before he sees Big Monkey move Two Nash equilibria (c,w), (w,c) Also a third Nash equilibrium: Big Monkey chooses between c & w with probability 0.5 (mixed strategy)

  8. An example:Big Monkey and Little Monkey • It can often be easier to analyze a game through a different representation, called normal form Little Monkey c v Big Monkey 5,3 4,4 c v 9,1 0,0

  9. Choosing Strategies • In the simultaneous game, it’s harder to see what each monkey should do • Mixed strategy is optimal. • Trick: How can a monkey maximize its payoff, given that it knows the other monkeys will play a Nash strategy? • Oftentimes, other techniques can be used to prune the number of possible actions.

  10. Eliminating Dominated Strategies • The first step is to eliminate actions that are worse than another action, no matter what. c w Big monkey c w c w c 9,1 4,4 w Little monkey We can see that Big Monkey will always choose w. So the tree reduces to: 9,1 0,0 9,1 6-2,4 7-2,3 Little Monkey will Never choose this path. Or this one

  11. Eliminating Dominated Strategies • We can also use this technique in normal-form games: Column a b 9,1 4,4 a Row b 0,0 5,3

  12. Eliminating Dominated Strategies • We can also use this technique in normal-form games: a b 9,1 4,4 a b 0,0 5,3 For any column action, row will prefer a.

  13. Eliminating Dominated Strategies • We can also use this technique in normal-form games: a b 9,1 4,4 a b 0,0 5,3 Given that row will pick a, column will pick b. (a,b) is the unique Nash equilibrium.

  14. Prisoner’s Dilemma • Each player can cooperate or defect Column cooperate defect cooperate -1,-1 -10,0 Row defect -8,-8 0,-10

  15. Prisoner’s Dilemma • Each player can cooperate or defect Column cooperate defect cooperate -1,-1 -10,0 Row defect -8,-8 0,-10 Defecting is a dominant strategy for row

  16. Prisoner’s Dilemma • Each player can cooperate or defect Column cooperate defect cooperate -1,-1 -10,0 Row defect -8,-8 0,-10 Defecting is also a dominant strategy for column

  17. Prisoner’s Dilemma • Even though both players would be better off cooperating, mutual defection is the dominant strategy. • What drives this? • One-shot game • Inability to trust your opponent • Perfect rationality

  18. Prisoner’s Dilemma • Relevant to: • Arms negotiations • Online Payment • Product descriptions • Workplace relations • How do players escape this dilemma? • Play repeatedly • Find a way to ‘guarantee’ cooperation • Change payment structure

  19. Tragedy of the Commons • Game theory can be used to explain overuse of shared resources. • Extend the Prisoner’s Dilemma to more than two players. • A cow costs a dollars and can be grazed on common land. • The value of milk produced (f(c) ) depends on the number of cows on the common land. • Per cow: f(c) / c

  20. Tragedy of the Commons • To maximize total wealth of the entire village: max f(c) – ac. • Maximized when marginal product = a • Adding another cow is exactly equal to the cost of the cow. • What if each villager gets to decide whether to add a cow? • Each villager will add a cow as long as the cost of adding that cow to that villager is outweighed by the gain in milk.

  21. Tragedy of the Commons • When a villager adds a cow: • Output goes from f(c) /c to f(c+1) / (c+1) • Cost is a • Notice: change in output to each farmer is less than global change in output. • Each villager will add cows until output- cost = 0. • Problem: each villager is making a local decision (will I gain by adding cows), but creating a net global effect (everyone suffers)

  22. Tragedy of the Commons • Problem: cost of maintenance is externalized • Farmers don’t adequately pay for their impact. • Resources are overused due to inaccurate estimates of cost. • Relevant to: • IT budgeting • Bandwidth and resource usage, spam • Shared communication channels • Environmental laws, overfishing, whaling, pollution, etc.

  23. Avoiding Tragedy of the Commons • Private ownership • Prevents TOC, but may have other negative effects. • Social rules/norms, external control • Nice if they can be enforced. • Taxation • Try to internalize costs; accounting system needed. • Solutions require changing the rules of the game • Change individual payoffs • Mechanism design

  24. Coming next time • How to select an optimal strategy • How to deal with incomplete information • How to handle multi-stage games

More Related