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Unformatted text preview: Copyright David Scoones 2010 Page 1 Economics 112 Lecture Notes Lecture 4. Mixed Strategies Pure strategies where players choose a single strategy to play with certainty are special cases of mixed strategies where players let the strategy they choose be determined by a probability distribution . For example, rather than choosing to turn left for sure (a pure strategy) or to turn right for sure (another pure strategy), a player may decide to toss a coin and turn left if it comes up heads and right if it comes up tails (a mixed strategy). The coin serves as a randomization device , which (if its fair) generates a probability distribution with equal likelihood of each pure strategy being chosen. Mixed strategies are strategies that involve players choosing lotteries over pure strategies, the kind of strategy we have been considering until now. Mixed strategies are technically useful, in that sometimes a mixed strategy equilibrium exists when no pure strategy equilibrium does. Mixed strategies are also realistic in some cases: that is, sometimes players really do make choices with randomization. And even when that is not true, mixed strategies can sometimes be interpreted in other, interesting ways. In some ways, this is the most difficult part of the course . These techniques will arise at various points, but if you understand their use here, it should be no problem to follow the later applications, even when the context is quite different. The difficulties are in part that we will need to do a tiny bit of algebra, of the kind familiar (or not) from high school math class. The larger challenge is conceptual: the math is much easier if you understand what we are trying to find, and for most people, these ideas seem quite peculiar when first encountered. I urge you to think carefully about matching pennies, especially about why you are willing to randomize your choice. Would you do so always? How does your choice depend on the choice of your opponent? Key terms: pure strategy, mixed strategy, mixed strategy Nash Equilibrium 1. Assigning probabilities Mixed strategies assign probabilities to a set of pure strategies. At one extreme, all the probability is assigned to a single strategy. In this (special) case, the mixed strategy is really just a pure strategy. More typically when game theorists refer to mixed strategies they mean that more than one pure strategy might be chosen with positive probability. Recall that probabilities must add to one. It is possible to have some pure strategies played with probability zero while others are played with positive probabilities. A completely mixed strategy places positive probability on all pure strategies available to a player. Copyright David Scoones 2010 Page 2 2. Why would a player be willing to randomize?...
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