Discusión
5. Diferencias en grado de dolor según experiencia de la enfermera
Many real-world situations of strategic behavior are repeated over an extended period of time. Sometimes, this can be modeled by an appropriate static model. For example, in the previous section, we saw how a two-stage game can be used to model competition in long-term and short-term variables.
Consider, however, the strategic phenomenon of retaliation, that is, the situation whereby a player changes its strategic variable in response to a rival’s action. Clearly, this cannot be achieved in a static, simultaneous-move game, for in such game there is no time for a player to react to another player’s actions.
Figure 4.10 Stage Game.
A useful way to model the situation whereby players react to each other’s strategic moves is to consider a repeated game. Consider a simultaneous-choice game like the one in figure 4.1. Because in this game each player chooses one action only once, we refer to it as a one-shot game. A repeated game is defined by a one-shot game—also referred to as stage game—which is repeated a number of times. If repetition takes place a finite number of times, then we have a finitely repeated game; otherwise, we have an infinitely repeated game.
In one-shot games, strategies are easy to define. In fact, strategies are identified with actions. In repeated games, however, it is useful to distinguish between actions and strategies. Consider the one-shot game in figure 4.10. In this game, each player has three actions/strategies to choose from: T , M , B for Player 1; and L, C, R for Player 2.
Now suppose that this one-shot game is repeated twice. In each period, Player 1 still has three actions to choose from. However, the set of possible strategies for Player 1 is now much more complex. A strategy for Player 1 has to indicate what to choose in period 1 and what to choose in period 2 as a function of the actions that were taken in period 1. Generally, a strategy is defined as a player’s complete contingent plan of action for all possible occurrences in the game. Because there are nine possible outcomes in the first period, three possible actions in the second period, and three possible actions in the first period, Player 1 has 3 times 3 to the power of 9, or 59,049, possible strategies!
Does this proliferation of strategies add anything of interest that was not present in the one-shot version of the game? In many cases, the answer is “yes.” Let us start by looking at the equilibria of the one-shot game. Direct inspection reveals that this game has two Nash equilibria:(M, C) and (B, R).fNotice that the best payoff for both players would
fTechnical note: We are referring only to equilibria in pure strategies.
be(T, L), yielding each a payoff of 5, but such an outcome is not a Nash equilibrium.
The best Nash equilibrium yields each player a payoff of 4.g
gThis game is identical to that in figure 4.1 except that we add a third strategy to each player. Although this third strategy leads to an extra Nash equilibrium, the main feature of the game in figure 4.1 is still valid—namely, the conflict between individual and joint incentives that characterizes the
“prisoner’s dilemma.”
Let us now derive the equilibria of the repeated game. One first observation is that the repeated play of the equilibrium strategies of the one-shot game forms an equilibrium
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of the repeated game. So, for example,(M, C) in both periods is an equilibrium. The implicit strategies that lead to such equilibrium of the repeated game are, for Player 1, “choose M in period 1 and choose M in period 2 regardless of what happened in period 1”; and likewise for Player 2. That is, players choose history-independent strategies.
The interesting question is whether there are equilibria of the repeated game that do not correspond to equilibria of the one-shot game. Consider the following strategy for Player 1: play T in period 1. In period 2, play M if period 1 actions were(T, L); otherwise, play B. As for Player 2, take the following strategy: play L in period 1. In period 2, play C if period 1 actions were(T, L); otherwise, play R.
Let us now check that these strategies constitute an equilibrium of the repeated game. In period 2, assuming that the period 1 outcome was (T, L), the designated strategies call for players to choose(M, C). Because these actions form a Nash equilibrium of the one-shot game, it must be in the players’ best interest to choose them in the second period of a two-period repeated game. That is, no player would be able to improve its payoff by choosing something different. Likewise, in period 2 and assuming that the period 1 outcome was different from(T, L), the designated strategies call for players to choose(B, R). Because the latter also constitute an equilibrium of the one-shot game, the same reasoning applies.
Finally, we have to check that period 1 actions are also part of a Nash equilibrium strategy. Take Player 1: Choosing the action T , as indicated by the designated strategy, yields a payoff of 5 in the first period. Because, by assumption, Player 2 is playing the designated strategy (L in period 1), Player 1’s period 1 choice will lead to(M, C) in period 2, yielding Player 1 an additional payoff of 4. Total payoff is therefore 9.
Now suppose that, in period 1, Player 1 chooses M instead. Period 1 payoff would then be 6, because Player 2 chooses L. However, choosing M in period 1 would lead to the play of(B, R) in period 2, yielding Player 1 an additional payoff of only 1. Total payoff would therefore be 7, which is less than 9. A similar comparison is obtained if we consider other deviations from the designated strategies by either Player 1 or Player 2. We conclude that the designated strategies constitute a Nash equilibrium.
In words, the previously designated strategies may be described in the following way. The players agree to choose the payoff-maximizing actions in the first period:
(T, L). Although this cannot be sustained in a one-shot game—both players would have an incentive to deviate—an arrangement can be made whereby (T, L) is part of an equilibrium in the two-period game. The idea is that period 2 actions are used to “punish”
players in case they deviate from the designated period 1 actions. Because of this period 2 “punishment,” a period 1 deviation that would be profitable in the short run (that is, in the one-shot game) is not profitable once the two periods are taken into consideration. In fact, the period 1 gain from deviation (6 minus 5) is less than the loss in period 2 payoff that results from Player 2’s “retaliation” (4 minus 1).
We conclude:
Because players can react to other players’ past actions, repeated games allow for equilibrium outcomes that would not be an equilibrium in the corresponding one-shot game.
As we see in chapter8, this idea of “agreements” between players that are enforced by mutual retaliation plays an important role in explaining the working of cartels and, more generally, the nature of collusive behavior.
Summary
. A game is a model that depicts a situation of strategic behavior. A game consists of a set of players, rules, and a set of payoff functions.
. Games may be represented in normal form (matrix) or in extensive form (game tree).
Normally, games with simultaneous choices are represented in the normal form, whereas games with sequential choices are represented in the extensive form.
. Simultaneous strategy choices should not be interpreted literally: When observation lags are significant, it is as if players were simultaneously choosing strategies.
. The equilibrium of a game indicates the strategies that one would expect players to choose. The most common equilibrium concept is that of Nash equilibrium—a situation such that no player would unilaterally find it optimal to change its strategy.
. In analyzing games, it is important not only whether players are rational. It is also important whether players believe the other players are rational.
. Sequential games should be solved backward. Such procedure excludes strategies that are not credible.
. Committing to take a future action which is ex-post suboptimal may have an ex-ante strategic value.
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. Repeated games are a way of modeling repeated interaction between players. Because players can react to other players’ past actions, repeated games allow for equilibrium outcomes that would not be an equilibrium in the corresponding one-shot game.
Key Concepts
. game
. normal form and extensive form
. dominant and dominated strategies
. Nash equilibrium
. backward induction
. credible commitment
. repeated game
Review and Practice Exercises
4.1 What are the assumptions regarding player rationality implicit in solving a game by elimination of dominated strategies? Contrast this with the case of dominant strategies.
4.2 The U.K. Office of Fair Trading has recently unveiled a plan that will offer immunity from prosecution to firms who blow the whistle on their co-cartel conspirators.
In the United States, this tactic has proven extremely successful: Since its introduction in 1993, the total amount of fines for anticompetitive behavior has increased twentyfold.
Show how the tactic initiated by the U.S. Department of Justice, soon to be followed by the U.K. Office of Fair Trading, changes the rules of the game played between firms in a secret cartel.
4.3 Figure4.11 represents a series of two-player games that illustrate the rivalry between Time magazine and Newsweek. Each magazine’s strategy consists of choosing a cover story: “Impeachment” and “Financial Crisis” are the two choices.h
hIn each cell, the first number is the payoff for the row player (Time).
The first version of the game corresponds to the case when the game is symmetric (Time and Newsweek are equally well positioned). As the payoff matrix suggests, “Im-peachment” is a better story but payoffs are lower when both magazines choose the same story. The second version of the game corresponds to the assumption that Time is a more
Figure 4.11 The Cover-Story Game.
popular magazine (Time’s payoff is greater than Newsweek’s when both magazines cover the same story). Finally, the third version of the game illustrates the case in which the magazines are sufficiently different that some readers will buy both magazines even if they cover the same story.
For each of the three versions of the game,
a. Determine whether the game can be solved by dominant strategies.
b. Determine all Nash equilibria.
c. Indicate clearly which assumptions regarding rationality are required in order to reach the solutions in (a) and (b).
4.4∗ In the movie E.T., a trail of Reese’s Pieces, one of Hershey’s chocolate brands, is used to lure the little alien out of the woods. As a result of the publicity created by this scene, sales of Reese’s Pieces trebled, allowing Hershey to catch up with rival Mars.
Universal Studio’s original plan was to use a trail of Mars’ M&Ms. However, Mars turned down the offer, presumably because it thought $1 million, the price demanded by the producer of E.T., was very high. The makers of E.T. then turned to Hershey, who accepted the deal.
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Suppose that the publicity generated by having M&Ms included in the movie would increase Mars’ profits by $800,000. Suppose moreover that Hershey’s increase in market share cost Mars a loss of $500,000. Finally, let b be the benefit for Hershey from having its brand be the chosen one.
Describe the preceding events as a game in extensive form. Determine the equilib-rium as a function of b. If the equilibequilib-rium differs from the actual events, how do you think they can be reconciled?
4.5 Hernan Cort´ez, the Spanish navigator and explorer, is said to have burnt his ships upon arrival to Mexico. By so doing, he effectively eliminated the option of him and his soldiers returning to their homeland. Discuss the strategic value of this action, knowing the Spanish colonists were faced with potential resistance from the Mexican natives.
4.6 Consider the following game depicting the process of standard setting in high-definition television (HDTV).47The United States and Japan must simultaneously decide whether to invest a high or a low value into HDTV research. Each country’s payoffs are summarized in figure4.12.
a. Are there any dominant strategies in this game? What is the Nash equilibrium of the game? What are the rationality assumptions implicit in this equilibrium?
b. Suppose now that the United States has the option of committing to a strategy before Japan’s decision is reached. How would you model this new situation? What are the Nash equilibria of this new game?
c. Comparing the answers to (a) and (b), what can you say about the value of commitment for the United States?
d. “When precommitment has a strategic value, the player that makes that commitment ends up ‘regretting’ its actions, in the sense that, given the rival’s choices, it could achieve a higher payoff by choosing a different action.” In light of your answer to (b), how would you comment on this statement?
Figure 4.12 The HDTV Game: Each Country Chooses a High or a Low Level of R&D on HDTV.
4.7 Consider a one-shot game with two equilibria and suppose that this game is repeated twice. Explain in words why there may be equilibria in the two-period game that are different from the equilibria of the one-shot game.
Extension Exercise
4.8∗∗ Consider the game in figure4.13.48Show, by backward induction, that rational players choose d at every node of the game, yielding a payoff of 2 for Player 1 and zero for Player 2. Is this equilibrium reasonable? What are the rationality assumptions implicit in it?
Figure 4.13 The Centipede Game.
In the payoff vectors, the top number is player 1’s payoff, the bottom one player 2’s.