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The Art of Strat RULE 4: Having exhausted the simple avenues of looking for dom nant strategies or ruling out dominated ones, next search all the cells of the game table for a pair of mutual best responses in the same cell, which is a Nash equilibrium of the game. 124 GAMES WITH INFINITELY MANY STRATEGIES In each of the versions of the pricing game we discussed so far we allowed each firm only a small number of price points: only $80 and $70 in chapter 3, and only between $42 and $38 in $1 steps in this chapter. Our purpose was only to convey the concepts of the prisoners' dilemma and Nash equilibrium in the simplest possible context. In reality, prices can be any number of dollars and cent and for all intents and purposes it is as if they can be chosen over continuous range of numbers. Our theory can cope with this further extension quite easily using nothing more than basic high-school algebra and geometry We can show the prices of the two firms in a two-dimensional graph, measuring RE's price along the horizontal or X axis and BB's price along the vertical or Y axis. We can show the best B. B. LEAN'S 42 PRICE 40 39 38 38 Nash equilibrium 39 RE's best response 40 41 BB's best response 42 RAINBOW'S END'S PRICE
A Beautiful Equilibrium responses in this graph instead of showing bold italic profit num- bers in a game table of discrete price points. We do this for the original example where the cost of each shirt merely tell you the result.7 The formula for BB's best response in to each store was $20. We omit the details of the mathematics and terms of RE's price (or BB's belief about the price RE is setting) is BB's best response price=24+0.4 x RE's price (or BB's belief about it). 125 This is shown as the flatter of the two lines in the above graph. We see that for each $1 cut in RE's price, BB's best response should be to cut its own price but by less, namely 40 cents. This is the result of BB's calculation, striking the best balance between losing cus- tomers to RE and accepting a lower profit margin. The steeper of the two curves in the figure is RE's best response to its belief about BB's price. Where the two curves intersect, the best response of each is consistent with the other's beliefs; we have Nash equilibrium. The figure shows that this occurs when each firm charges $40. Moreover, it shows that this particular game has exactly one Nash equilibrium. Our finding a unique Nash equilib- rium in the table where prices had to be multiples of $1 was not an artificial consequence of that restriction. Such graphs or tables that allow much more detail than we could in the simple examples are a standard method for comput ing Nash equilibria. The calculation or graphing can quickly get too complicated for paper-and-pencil methods, and too boring besides, but that's what computers are for. The simple examples give us a basic understanding of the concept, and we should reserve our human thinking skills for the higher-level activity of assessing its usefulness. Indeed, that is our very next topic. A BEAUTIFUL EQUILIBRIUM? John Nash's equilibrium has a lot of conceptual claim to be the solution of a game where each player has the freedom of choice. Perhaps the strongest argument in its favor takes the form of a