Relationship to Rationality Is Common Knowledge Relationship to Rationality Is Common Knowledge To explore the relationship between those strategies which survive the itera- tive deletion of strictly dominated strategies (IDSDS) and Nash equilibria, let’s start with an example. Consider a Nash equilibrium for a three-player game in which player 1 uses strategy x, player 2 […]
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Foundations of Nash Equilibrium Foundations of Nash Equilibrium THUS FAR, WE’VE OFFERED two approaches to solving a game: iterative deletion of strictly dominated strategies and Nash equilibrium. It is natural to wonder how they are related, so we’ll address this issue next. Then there is the matter of how a strategy is interpreted in the
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Whoever has the highest performance is promoted Whoever has the highest performance is promoted. Each contestant has one unit of effort that she can allocate in three possible ways: she can use it to enhance her own performance (which we’ll refer to as a “positive” effort) or to denigrate one of the two competing players
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SITUATION: PROMOTION AND SABOTAGE Suppose you are engaged in a contest in which the person with the highest performance wins a prize. Currently, you’re in second place. What can you do to improve your chances of winning? One thought is to work hard to improve your performance. But what might prove more effective is engaging
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SITUATION: VOTING, SINCERE OR DEVIOUS? A company has three shareholders. Shareholder 1 controls 25% of the shares, shareholder 2 controls 35%, and shareholder 3 controls 40%. The company has offers from two other companies, denoted A and B, to purchase it. There is also a third option, which is to decline both offers. Shareholder 1
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SITUATION: AMERICAN IDOL FANDOM Alicia, Kaitlyn, and Lauren are ecstatic. They’ve just landed tickets to attend this week’s segment of American Idol. The three teens have the same favorite among the nine contestants that remain: Ace Young. They’re determined to take this opportunity to make a statement. While IMing, they come up with a plan
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CONFLICT AND MUTUAL INTEREST IN GAMES Rock–Paper–Scissors is a game of pure conflict. What do I mean by that? Well, take note of an interesting property of the payoff matrix in Figure 4.8: Players’ payoffs always sum to the same number (which happens to be zero). For example, if both Bart and Lisa choose rock,
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SITUATION: A GAME OF COORDINATION AND CONFLICT—TELEPHONE In the driving conventions game, there were two Nash equilibria and the play- ers were indifferent between them: driving on the right was just as good as driving on the left. Now let us consider a setting that also has two equilibria, but the players rank them differently.
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SITUATION: A COORDINATION GAME—DRIVING CONVENTIONS We next look at an example of a coordination game, which has the property that players have a common interest in coordinating their actions. A coordi- nation game that most adults engage in every day is the choice of the side of the road upon which to drive. It’s not
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Defining Nash Equilibrium 91 which ensures that a player’s strategy is optimal, given the other players’ strate- gies. Thus, all players are simultaneously doing their best. A violation of one or more of those conditions means that a strategy profile is not a Nash equilib- rium. Unlike the game of horseshoes, you don’t come “close
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