Abstract
Strategic decisions are an integral part of our daily lives. Game theory provides a foundation for analyzing strategy, and as such, is a vitally important subject. At work, you may need to decide whether it is worth it to put in extra hours to earn a bigger bonus than your colleagues. Strategic decisions are especially common in competitive games, such as chess, tennis, football, and the TV show Survivor. These are considered games because strategic interaction is important to success. That is, to decide your best course of action, you must consider how others are likely to behave. Two examples make this point clear.
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Notes
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There are also games of uncertainty, where nature plays a role in the game. For example, nature may decide whether a player (firm or worker) has a high or low level of productivity. For further discussion, see Rasmusen (2007).
- 4.
This is similar to the concept of Pareto optimality, which is reached when there is no way to make one person better off without making someone else worse off. Generally, Pareto optimality applies to everyone in society, while a Pareto outcome in a coordination game applies only to the players in the game. For a review of the concept of Pareto optimality, see Frank and Bernanke (2008), Bernheim and Whinston (2008), Pindyck and Rubenfield (2009), or Varian (2010).
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Of course, you might think that this enables Chris and Pat to avoid being disappointed with each other and actually leads to the highest payoff. This possibility is ruled out by their payoffs, which leaves them both with 0, as compared to a payoff of (3, 1) or (1, 3). In other words, they each really dislike the meat–white and fish–red outcome.
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It is weakly dominated if there exists a feasible strategy that earns a payoff that is at least as high.
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The game theory rules listed in this chapter borrow from Dixit and Nalebuff (1991).
- 9.
Of course, real criminals know this and may agree beforehand never to squeal when caught by the police. With organized crime, for example, a crime syndicate may threaten to harm the family members of squealers and make this a dynamic game. These possibilities are ruled out here, and this game is simply meant to serve as a thought-provoking example.
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The dominant pig earns 2 by choosing Push and earns 0 by choosing Not.
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The lag is probably due to the fact that pigs have incomplete information, cannot talk, and it takes time to learn the payoffs from each possible course of action. These are called evolutionary games, which are described by Samuelson (1997).
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Where the first term, Push, represents player 1’s (the big pig’s) strategy and the second, Not, represents player 2’s (the small pig’s) strategy.
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Nash won the prize with two other economists: Reinhard Selten (1965) who applied Nash’s concept to dynamic games and John Harsanyi (1967, 1968a, 1968b) who incorporated imperfect information into game theory. You can learn more about Nash’s contributions to game theory and his life struggles with mental illness by reading Nasar’s (1998) book, A Beautiful Mind. The movie with the same title is based on the book but is a Hollywood rendition that is not very accurate. In true Hollywood fashion, in the movie Nash gains inspiration for his game theory ideas from a bar scene where he and his friends discuss strategies for meeting women. In reality, his ideas came to him in an economics class in international trade.
- 14.
A set is closed if it contains all of its boundary points, and it is bounded if the distance between any two points in the set is less than infinity.
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The greater than or equal to sign is important to Nash’s proof that at least one Nash equilibrium will exist in static games with a finite number of players and the strategy space that meets certain regularity conditions (i.e., it is nonempty, convex, closed, and bounded). For further discussion and a more advanced treatment, see Mas-Colell et al. (1995) and Rasmusen (2007).
- 16.
It turns out that for most people, this is difficult to do. To get around this problem, you could decide to use the second hand on your watch to determine your action. If the second hand is between 1 and 30, choose heads; if it is between 31 and 60, choose tails.
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This is called the expected value of a risky event, where the expected value is the weighted average of the return associated with each possible outcome and the weights are the probabilities of each respective outcome. In this case, there are two possible returns: 1 (associated with the H–H outcome that has a probability of pr2) and –1 (associated with the H–T outcome that has a probability of 1 – pr2).
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This does not appear to be correct, because the smallest change in price is 1¢. Nevertheless, the owner could make infinitesimally small changes in quantity, giving the owner an infinite number of possible prices per ounce. We also assume product differentiation, an issue we discuss in later chapters.
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A maximum is reached because the profit equation is concave. If it were convex, this procedure would produce the price that minimizes profit. See the Mathematics and Econometrics Appendix at the end of the book for further discussion.
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This is comparative static analysis, which we discuss in Chap. 5 and in the Mathematics and Econometrics Appendix.
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The first issue is addressed in perfect Bayesian models, which deal with how players form beliefs and behave when faced with imperfect information. The second problem where there is a chance that one or more players make errors is addressed in trembling hand models (Selten 1975).
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Problems in this section have a unique SPNE. Unfortunately, some dynamic games will have more than one SPNE. See the Mathematics and Econometrics Appendix for further discussion of dynamic programming techniques.
- 23.
The first two are called proper subgames to distinguish them from the entire game itself.
- 24.
Actually, most game theorists assume it to be 100¢ − 0¢. This makes player 2 indifferent between accepting and rejecting. When indifferent, participants are assumed to play the game or accept the offer. In more complex games, this can simplify the mathematical analysis.
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For a survey of the evidence, see Levitt and List (2007).
- 26.
But, you might ask why player 2 does not make the following offer to player 1: “give me the $1 and I promise to give back $1.01 and make us both better off.” Because a legal contract is not enforceable in this game, player 2’s promise is not credible and would not be believed by player 1. Thus, the optimal strategy is for player 1 to give nothing in the first stage of the game.
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This will earn Ford a total payoff of 15 = (5 + 10 + 0) instead of 10 = (5 + 5 + 0).
- 29.
You might ask why we would make this assumption when no game can be played forever. Many companies plan to survive well into the future, regardless of the management team running the company. We can get dramatically different results when we assume an infinitely repeated game. You will also find that solving such games will be easier than you may think.
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In other words, \( \mathop{\sum }\limits_{{t = 1}}^{\infty } {D^t}\cdot 1 = 0.{95} + 0.{9}0{25} + 0.{8574} + 0.{8145} + \cdots = {2}0. \)
- 31.
This is sometimes called a “grim strategy,” because the players can never get back to the cooperative equilibrium once one player fails to cooperate.
- 32.
C’s decision to reject may take so long that the ice cream has melted away, or C may toss the ice cream on the ground in disgust with the offer.
- 33.
It would be more precise to assume a split of (100 – ε) and ε percent, where ε is infinitesimally small, but this notation is rather cumbersome.
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This is not always true in reality. For example, negotiations between firms and unions frequently break down and lead to strikes. Delayed settlements can also result if we relax the assumption of perfect and complete information (Sobel and Takahashi 1983).
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In other words, C faces just one big decision node or information set that contains C1 and C2, which means that the player does not know what has happened in the past.
- 37.
Another situation where the teller gives up the money right away, even if the robber is rational, is called a trembling-hand equilibrium (Selten 1975). In this case, there is a possibility that the robber will make a mistake and blow up the bank due to a trembling hand that is holding the ignition button to the bomb. If that probability of blowing up the bank is high enough, then the SPNE is for the teller to give up the money immediately.
- 38.
The original centipede game developed by Rosenthal (1991) had 100 periods.
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Tremblay, V.J., Tremblay, C.H. (2012). Introductory Game Theory and Economic Information. In: New Perspectives on Industrial Organization. Springer Texts in Business and Economics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3241-8_3
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