This section presents the basic tools used to analyze simultaneous-move, one-shot games.
Recall that in a simultaneous-move game, players must make decisions without knowledge of the decisions made by other players. The fact that a game is “one-shot” simply means that the players will play the game only once.
Knowledge of simultaneous-move, one-shot games is important to managers making decisions in an environment of interdependence. For example, it can be used to analyze situ- ations where the profits of a firm depend not only on the firm’s action but on the actions of rival firms as well. Before we look at specific applications of simultaneous-move, one-shot games, let us examine the general theory used to analyze such decisions.
Theory
We begin with two key definitions. First, a strategy is a decision rule that describes the actions a player will take at each decision point. Second, the normal-form game indicates the players in the game, the possible strategies of the players, and the payoffs to the players that will result from alternative strategies.
Perhaps the best way to understand precisely what is meant by strategy and normal-form game is to examine a simple example. The normal form of a simultaneous-move game is pre- sented in Table 10–1. There are two players, whom we will call A and B to emphasize that the theory is completely general; that is, the players can be any two entities that are engaged in a situation of strategic interaction. If you wish, you may think of the players as the managers of two firms competing in a duopoly.
Player A has two possible strategies: He can choose up or down. Similarly, the feasible strategies for player B are left or right. This illustrates that the players may have different strategic options. Again, by calling the strategies up, down, and so on, we emphasize that these actions can represent virtually any decisions. For instance, up might represent raising strategy
In game theory, a decision rule that describes the actions a player will take at each decision point.
normal-form game A representation of a game indicating the players, their possible strategies, and the payoffs resulting from alternative strategies.
Table 10–1
A Normal-Form Game
Player B
Strategy Left Right
Player A Up 10, 20 15, 8
Down −10, 7 10, 10
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the price and down lowering price, or up a high level of advertising and down a low level of advertising.
Finally, the payoffs to the two players are given by the entries in each cell of the matrix.
The first entry refers to the payoff to player A, and the second entry denotes the payoff to player B. An important thing to notice about the description of the game is that the payoff to player A crucially depends on the strategy player B chooses. For example, if A chooses up and B chooses left, the resulting payoffs are 10 for A and 20 for B. Similarly, if player A’s strategy is up while B’s strategy is right, A’s payoff is 15 while B’s payoff is 8.
Since the game in Table 10–1 is a simultaneous-move, one-shot game, the players get to make one, and only one, decision and must make their decisions at the same time. For player A, the decision is simply up or down. Moreover, the players cannot make conditional deci- sions; for example, A can’t choose up if B chooses right or down if B chooses left. The fact that the players make decisions at the same time precludes each player from basing his or her decisions on what the other player does.
What is the optimal strategy for a player in a simultaneous-move, one-shot game? As it turns out, this is a very complex question and depends on the nature of the game being played.
There is one instance, however, in which it is easy to characterize the optimal decision—a situation that involves a dominant strategy. A strategy is a dominant strategy if it results in the highest payoff regardless of the action of the opponent.
In Table 10–1, the dominant strategy for player A is up. To see this, note that if player B chooses left, the best choice by player A is up since 10 units of profits are better than the –10 he would earn by choosing down. If B chose right, the best choice by A would be up since 15 units of profits are better than the 10 he would earn by choosing down. In short, regardless of whether player B’s strategy is left or right, the best choice by player A is up. Up is a dominant strategy for player A.
dominant strategy A strategy that results in the highest payoff to a player regardless of the opponent’s action.
P R I N C I P L E
Play Your Dominant Strategy
Check to see if you have a dominant strategy. If you have one, play it.
In simultaneous-move, one-shot games where a player has a dominant strategy, the optimal decision is to choose the dominant strategy. By doing so, you will maximize your payoff regard- less of what your opponent does. In some games a player may not have a dominant strategy.
DEMONSTRATION PROBLEM 10–1
In the game presented in Table 10–1, does player B have a dominant strategy?
ANSWER:
Player B does not have a dominant strategy. To see this, note that if player A chose up, the best choice by player B would be left since 20 is better than the payoff of 8 she would earn by choosing right.
But if A chose down, the best choice by B would be right since 10 is better than the payoff of 7 she would realize by choosing left. Thus, there is no dominant strategy for player B; the best choice by B depends on what A does.
What should a player do in the absence of a dominant strategy? One possibility would be to play a secure strategy—a strategy that guarantees the highest payoff given the worst possible scenario. As we will see in a moment, this approach is not generally the optimal way to play a game, but it is useful to explain the reasoning that underlies this strategy. By using a secure strategy, a player maximizes the payoff that would result in the “worst-case scenario.”
In other words, to find a secure strategy, a player examines the worst payoff that could arise for each of his or her actions and chooses the action that has the highest of these worst payoffs.
secure strategy A strategy that guarantees the highest payoff given the worst possible scenario.
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A very natural way of formalizing the “end result” of such a thought process is captured in the definition of Nash equilibrium. A set of strategies constitute a Nash equilibrium if, given the strategies of the other players, no player can improve her payoff by unilaterally changing her own strategy. The concept of Nash equilibrium is very important because it represents a situation where every player is doing the best he or she can given what other players are doing.
Nash equilibrium A condition describing a set of strategies in which no player can improve her payoff by unilaterally changing her own strategy, given the other players’ strategies.
DEMONSTRATION PROBLEM 10–2
What is the secure strategy for player B in the game presented in Table 10–1?
ANSWER:
The secure strategy for player B is right. By choosing left B can guarantee a payoff of only 7, but by choosing right she can guarantee a payoff of 8. Thus, the secure strategy by player B is right.
While useful, the notion of a secure strategy suffers from two shortcomings. First, it is a very conservative strategy and should be considered only if you have a good reason to be extremely averse to risk. Second, it does not take into account the optimal decisions of your rival and thus may prevent you from earning a significantly higher payoff. In particular, player B in Table 10–1 should recognize that a dominant strategy for player A is to play up.
Thus, player B should reason as follows: “Player A will surely choose up since up is a dom- inant strategy. Therefore, I should not choose my secure strategy (right) but instead choose left.” Assuming player A indeed chooses the dominant strategy (up), player B will earn 20 by choosing left, but only 8 by choosing the secure strategy (right).
P R I N C I P L E Put Yourself in Your Rival’s Shoes
If you do not have a dominant strategy, look at the game from your rival’s perspective. If your rival has a dominant strategy, anticipate that he or she will play it.
DEMONSTRATION PROBLEM 10–3
In the game presented in Table 10–1, what are the Nash equilibrium strategies for players A and B?
ANSWER:
The Nash equilibrium strategy for player A is up, and for player B it is left. To see this, suppose A chooses up and B chooses left. Would either player have an incentive to change his or her strategy?
No. Given that player A’s strategy is up, the best player B can do is choose left. Given that B’s strat- egy is left, the best A can do is choose up. Hence, given the strategies (up, left), each player is doing the best he or she can given the other player’s decision.
Why aren’t any of the other strategy combinations—(up, right), (down, right), and (down, left)—a Nash equilibrium? This is because, for each combination, at least one player would like to change his or her strategy given the strategy of the other player. Consider each in turn. The strategies (up, right) are not a Nash equilibrium because, given Player A is playing up, Player B would do better by playing left instead of right. The strategies (down, right) are not a Nash equilibrium because, given Player B is playing right, Player A would do better by playing up instead of down. The strategies (down, left) are not a Nash equilibrium because both players could do better: given Player A is play- ing down, Player B would do better by playing right instead of left; and given Player B is playing left, Player A would do better by playing up instead of down.
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Director Ron Howard scored a home run by strategically releasing A Beautiful Mind just in time to win four Golden Globe Awards in 2002. The film—based loosely on the life of Nobel Laureate John Forbes Nash Jr., whose “Nash equi- librium” revolutionized economics and game theory—won best dramatic picture and best screenplay. Actor Russell Crowe also won a Golden Globe for his portrayal of the brilliant man whose battle with delusions, mental illness, and paranoid schizophrenia almost kept him from win- ning the 1994 Nobel Prize in Economics. While some know Ron Howard for his accomplishments as a director, he is best known as the kid who played Opie Taylor and Richie Cunningham in the popular Andy Griffith and Happy Days TV shows. For this reason, Eddie Murphy once dubbed him
“Little Opie Cunningham” in a Saturday Night Live skit.
While A Beautiful Mind is an enjoyable film, its portrait of Nash’s life is at odds with Sylvia Nasar’s carefully documented and best-selling book with the same title. More relevant to students of game theory, the film does not accurately illus- trate the concept for which Nash is renowned. Translation:
Don’t rent the movie as a substitute for learning how to use Nash’s equilibrium concept to make business decisions.
Hollywood attempts to illustrate Nash’s insight into game theory in a bar scene in which Nash and his buddies are eyeing one absolutely stunning blonde and several of her brunette friends. All of the men prefer the blonde. Nash pon- ders the situation and says, “If we all go for the blonde, we block each other. Not a single one of us is going to get her.
So then we go for her friends. But they will all give us the cold shoulder because nobody likes to be second choice.
But what if no one goes for the blonde? We don’t get in each other’s way, and we don’t insult the other girls. That’s the only way we win.” The camera shows a shot of the blonde sitting all alone at the bar while the men dance happily with
the brunettes. The scene concludes with Nash rushing off to write a paper on his new concept of equilibrium.
What’s wrong with this scene? Recall that a Nash equi- librium is a situation where no player can gain by changing his decision, given the decisions of the other players. In Hollywood’s game, the men are players and their decisions are which of the women to pursue. If the other men opt for the brunettes, the blonde is all alone just waiting to dance.
This means that the remaining man’s best response, given the decisions of the others, is to pursue the lonely blonde!
Hollywood’s dance scene does not illustrate a Nash equilib- rium, but the exact opposite: a situation where any one of the men could unilaterally gain by switching to the blonde, given that the other men are dancing with brunettes! What is the correct term for Hollywood’s dance scene in which the blonde is left all alone? Personally, we like the term
“Opie equilibrium” because it honors the director of the film and sounds much more upbeat than “disequilibrium.”
Hollywood also uses the dance scene to spin its view that “Adam Smith was wrong.” In particular, since the men are better off dancing with the brunettes than all pursuing the blonde, viewers are to conclude that it is never socially efficient for individuals to pursue their own selfish desires.
While Chapter 14 of this book shows a number of situations where markets may fail, Hollywood’s illustration is not one of them. Its “Opie equilibrium” outcome is actually socially inefficient because none of the men get to enjoy the com- pany of the stunning blonde. In contrast, a real Nash equilib- rium to the game entails one man dancing with the blonde and the others dancing with brunettes. Any Nash equilibrium to Hollywood’s game not only has the property that each man is selfishly maximizing his own satisfaction, given the strategies of the others, but the outcome is also socially effi- cient because it doesn’t squander a dance with the blonde.