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lecture2 notes - 6.254 Game Theory with Engineering...

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6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games Asu Ozdaglar MIT February 4, 2009 1
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Game Theory: Lecture 2 Introduction Outline Decisions, utility maximization Strategic form games Best responses and dominant strategies Dominated strategies and iterative elimination of strictly dominated strategies Nash Equilibrium Examples Reading: Fudenberg and Tirole, Chapter 1. 2
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Game Theory: Lecture 2 Introduction Motivation In many social and engineered systems, agents make a variety of choices. For example: How to map your drive in the morning (or equivalently how to route your network traffic). How to invest in new technologies. Which products to buy. How to evaluate information obtained from friends, neighbors, coworkers and media. In all of these cases, interactions with other agents affect your payoff, well-being, utility. How to make decisions in such situations? “multiagent decision theory” or game theory . 3
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Game Theory: Lecture 2 Introduction “Rational Decision-Making” Powerful working hypothesis in economics: individuals act rationally in the sense of choosing the option that gives them higher “payoff”. Payoff here need not be monetary payoff. Social and psychological factors influence payoffs and decisions. Nevertheless, the rational decision-making paradigm is useful because it provides us with a (testable) theory of economic and social decisions. We often need only ordinal information; i.e., two options a and b , and we imagine a preference relation that represents the ranking of different options, and we simply check whether a b or a b . But in game theory we often need cardinal information because decisions are made under natural or strategic uncertainty. The theory of decision-making under uncertainty was originally developed by John von Neumann and Oskar Morgenstern. 4
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Game Theory: Lecture 2 Introduction Decision-Making under Uncertainty von Neumann and Morgenstern posited a number of “reasonable” axioms that rational decision-making under uncertainty should satisfy. From these, they derived the expected utility theory . Under uncertainty, every choice induces a lottery , that is, a probability distribution over different outcomes. E.g., one choice would be whether to accept a gamble which pays $10 with probability 1/2 and makes you lose $10 with probability 1/2. von Neumann and Morgenstern’s expected utility theory shows that (under their axioms) there exists a utility function (also referred to as Bernoulli utility function) u ( c ) , which gives the utility of consequence (outcome) c . Then imagine that choice a induces a probability distribution F a ( c ) over consequences. 5
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Game Theory: Lecture 2 Introduction Decision-Making under Uncertainty (continued) Then the utility of this choice is given by the expected utility according to the probability distribution F a ( c ) : U ( a ) = u ( c ) dF a ( c ) .
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