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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

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Game Theory

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  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

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