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 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.

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?

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

An example: Big Monkey and Little Monkey c Big monkey w 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)

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.

An example: Big Monkey and Little Monkey What if the monkeys have to decide simultaneously? c Big monkey w 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)

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

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.

Eliminating Dominated Strategies The first step is to eliminate actions that are worse than another action, no matter what. w c Big monkey w c 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

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

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.

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.

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

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

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

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

Prisoner’s Dilemma Relevant to: How do players escape this dilemma? 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

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

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.

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)

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.

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

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