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In Chapter 11, we had to expand the concept of equilibrium to apply it to market outcomes in oligopolistic industries. Because each oligopolistic firm’s price and quantity decisions influence those of its competitor, equilibrium in oligopolies requires more than just an equilibrium price and quantity for the industry as a whole. The stable market equilibrium also requires that no firm in the industry wants to change its decision given the choices its competitors are making.

A Nash equilibrium is a natural concept for determining the likely outcome of a game. In a Nash equilibrium, no player wants to unilaterally change her strategy given whatever strategies the other players are choosing. That is, the players are all doing the best they can, given the others’ actions.

Lots of dominated strategies in a game, as in the Warner Brothers–Disney example above, makes finding a Nash equilibrium easy. However, in most games eliminating dominated strategies alone won’t completely pin down Nash equilibria. That’s because the conditions that must hold in a Nash equilibrium are not as stringent as those that define dominant and dominated strategies. A dominant strategy is always the best thing for a player to do, no matter what an opponent does. But a Nash equilibrium only requires that an action is the best thing a player can do given the action the opponent happens to be taking. Therefore, a game can have a Nash equilibrium even if it has no dominated or dominant strategies. Finding the equilibrium in these more general cases can still be straightforward if there aren’t too many players or possible actions.

The first step to finding a Nash equilibrium in any game is to lay out the game’s players, strategies, and the payoffs that result from each combination of strategies. Table 12.1 does this for the Warner Brothers–Disney game. This particular way of organizing the economic content of a game—that is, by putting each player’s payoffs into a matrix based on the strategies chosen by the players—is called the normal form. Putting a game into normal form makes it easier to determine what its equilibria are.

Let’s do an example with a new game. Two competing magazines, I’m Famous Weekly and Look at Me! Magazine, are each considering what to use as the cover story for its next issue. They each have the same two choices, which we’ll label reality show (RS) and celebrity interview (CI). We assume that neither magazine can observe its competitor’s cover choice until it’s on the newsstands, so both magazines must make their choices simultaneously. In addition, we assume some readers only like stories about reality shows, others only like celebrity interviews, and the readers who prefer celebrity interviews outnumber those who like reality show articles. If a magazine is the only one offering a particular type of story, it will get all of those readers. If both offer the same story, they will split those readers between them, but Famous will receive a larger share than Me! because a majority of readers prefer Famous, all else equal.

The profits (in thousands of dollars) that each magazine earns depend on its own choice for a cover story as well as its competitor’s choice. These choices and their payoffs are shown in Table 12.2. As before, the red number before the comma in each cell is the payoff to I’m Famous Weekly, and the blue number after the comma is the payoff to Look at Me! Magazine.

To zero in on the game’s Nash equilibrium, let’s first consider Famous’ best responses to Me!’s possible actions. If Me! chooses an RS cover story (left column), then Famous’ best action is to choose a CI cover. This is because it will earn a $500,000 payoff by choosing CI but only $300,000 by choosing RS. To keep track of this best response, we’ll put a check next to Famous’ payoff in the bottom left cell. If Me! runs a CI cover, Famous’ best response is still to choose CI because it earns $450,000 instead of $400,000. Because this is Famous’ best response in this situation, we put another check next to the $450. We can see that choosing a celebrity interview as its cover story is a dominant strategy for Famous; no matter what Me! does, Famous makes more money by going with the CI cover.

*Payoffs are measured in thousands of dollars of profit.

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Now let’s do the same exercise for Me!. If Famous chooses a reality show cover, then Me! should choose a celebrity interview cover, because it will make $400,000 of profit rather than $0. We put a check next to the $400 payoff in the upper-right cell to indicate that this is a best response for Me!. However, if Famous chooses a CI cover, Me! is better off choosing an RS cover. It will make $300,000 instead of $200,000 by doing so, because it will have the reality show readers all to itself rather than splitting the set of celebrity interview fans. Another check goes next to the $300 payoff for Me! in the lower-left corner.

Looking at the pattern of checks (i.e., the pattern of best-response strategies), we can see a couple of things right away. First, Famous will choose a celebrity interview for its cover because this is a dominant strategy. Why? Remember that a dominant strategy is the best strategy for a player to follow no matter what the other player does. Because choosing a celebrity interview cover results in higher payoffs (profits) for Famous no matter what Me! does (look at the check marks for Famous’ payoffs in Table 12.2), the celebrity interview is a dominant strategy for Famous. And, because a reality show cover is never the best option for Famous, RS is a dominated strategy.

Me! Magazine, on the other hand, has no dominant strategy: If Famous chooses an RS cover, then Me! is better off choosing CI. If Famous instead chooses a CI cover, Me! will earn a higher profit by choosing a reality show cover. Thus, Me!’s best strategy changes with the choice made by Famous; no one strategy is best under all circumstances, so Me! has no dominant strategy. It also has no dominated strategy because both strategies (RS and CI) can each be a best strategy for Me!, depending on Famous’ action.

Second, even though we can’t eliminate many possible outcomes by getting rid of dominated strategies, there is still a Nash equilibrium to the game. In fact, we can see that Famous choosing a CI cover story and Me! choosing an RS cover are a mutual best response—both players have checks by their payoffs in this cell. Therefore, this is a Nash equilibrium. As is true of Nash equilibria in general, even if you gave either magazine the ability to unilaterally change its strategy choice, once one magazine knew what the other was doing, it wouldn’t want to switch because doing so would only reduce its profit. That stability is a primary reason why the Nash equilibrium is useful for predicting the outcomes of games.

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Predicting likely outcomes of a game can be fairly easy when there’s only one Nash equilibrium, as there is with prisoners’ dilemmas or with the magazine cover story game we just analyzed. Because of the mutual-best-response logic of the Nash equilibrium, it’s likely that the players will end up following the Nash equilibrium strategies. But things get tougher to figure out if there is more than one Nash equilibrium, and that outcome is fairly common.

Let’s revisit Warner Brothers and Disney, but this time we look at their choice of when to release their next movies. Suppose that both companies currently have animated movies ready for distribution, so now they must decide when these movies should open in theaters around the country.

Both companies want to take advantage of certain periods of the year when people have especially high demand for watching movies. Let’s simplify things by saying the companies basically have three periods from which to choose their opening date. One is the Memorial Day weekend (which we’ll call May). This is traditionally a big movie-going time of the year and, as such, is an appealing option for an opening date. The second possibility is the time around Christmas and New Year’s (which we’ll call December). This is also typically a high-demand period of the year, though not quite as high as Memorial Day. Finally, there is mid-March, usually a low-demand period.

Warner Brothers and Disney both understand these demand patterns. And if either were a monopolist, it could easily rank its opening date choices: May is best, then December, then March. But they’re not monopolists. Their profits from choosing a particular opening date will depend on the other company’s choice as well. This is, then, a game theory problem. The two companies realize that if they both choose the same opening date, it will be a disaster. They’ll split the animated feature market down the middle rather than having the whole market to themselves (though at different times in the year).

So how will Warner Brothers and Disney balance these opposing considerations? Table 12.3 shows the specifics of the game they are playing, including the payouts that both companies would receive for any possible set of opening date strategies.

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*Payoffs are measured in millions of dollars of profit.

If either company chooses a May opening when the other does not, the May opener makes $300 million. If either company opens by itself in December, its movie will make $200 million. Finally, if either company has a March opener to itself, it makes $100 million.

If both companies open their movies in the same month, however, neither fares so well. In fact, we assume they’ll earn lower profit regardless of the opening date (although they’ll lose more if they open during worse times of the year). If they both choose to open in May, they make only $50 million each. If both movies open in December, each company earns zero profit. And if both firms opt to open in March, they will each incur a $50 million loss.

What is the Nash equilibrium for this game? To find out, we apply our box-check method to mark each player’s best-response strategies. First, let’s do it for Warner Brothers. If Disney chooses a May opening date, Warner Brothers’ best response is to choose a December opening ($200 million > $100 million > $50 million). This outcome is shown in the middle cell of the left column. If Disney chooses December, then May is Warner Brothers’ best response (top cell of middle column). May is also Warner Brothers’ best response if Disney chooses a March opening date (top cell of right column). We’ve put checks next to the payoffs for all of Warner Brothers’ best responses. (Remember that Warner Brothers’ numbers are in red and to the left of the comma.)

Everything for Disney’s optimal responses is just a mirror image, so we can check off its best responses in the same manner. If Warner Brothers chooses a May opening date, Disney should choose December ($200 million is its largest payoff possible; top cell of middle column). If Warner Brothers chooses either a December or March opening date, Disney earns more profit by choosing May ($300 million is its largest payoff possible; middle and bottom cells of the left column).

Now we’ve identified each firm’s best responses to any possible strategy its competitor might pursue. Notice that a March opening is a dominated strategy for both players. That is, it is never optimal to open in March regardless of what the competitor might do. So we can ignore any outcomes that involve either firm opening in March. That simplifies the game to the four squares in the upper left of the game table. These are shown by themselves in Table 12.4.

Look at the mutual best responses, where there are two checks in the same box. There are two of them. Either Warner Brothers picks December and Disney chooses May, or the other way around. Both outcomes are Nash equilibria of the game. This game shows the difficulty of predicting the outcome of games with multiple equilibria. We can narrow down the possibilities—no rational company would choose a March date, for example, and both companies would avoid opening their movies in the same period—but we cannot say precisely what the one outcome will be. We know that one company would likely pick May and the other December, but given the information in the table, we have no basis to determine what company chooses which date.

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*Payoffs are measured in millions of dollars of profit.

Later in the chapter, we see that changes in the structure of the game, such as changes in the sequence of moves or modest shifts in payouts, can lead to different outcomes than the multiple Nash equilibria shown here. For the time being, however, we’ve gone as far as game theory can take us with this game.

We have been focusing on games in which a player makes a choice between different actions, choosing the specific strategy that maximizes her payoff. This type of strategy is known as a pure strategy. However, it may not always be in the player’s best interest to follow a pure strategy. In some situations, it may be best for the player to choose her actions randomly from the set of pure strategies available to her. This type of strategy is called a mixed strategy.

As an example of a game that can be played using a mixed strategy, think of a soccer game in which a player is taking a penalty kick. You’ve probably seen this on TV, or perhaps done it yourself on the field. The kicker faces the goalie, alone, and kicks the ball from close range. The distance is so close, in fact, that the goalie just dives to one side or the other and prays that it’s the right way. It turns out that penalty kicks can be analyzed using game theory. There are two players (the kicker and the goalie), with each having a set of strategy choices (“kick left” or “kick right” for the kicker, “dive left” or “dive right” for the goalie), and payoffs that depend on the chosen strategies of both players. This game is shown in Table 12.5.

Let’s assume we’re considering a pivotal penalty kick: If the goalie picks the same side as the kicker, the goalie’s team wins, and if the players choose opposite sides, the kicker’s team wins. We can assign payoffs of 1 to the winner and 0 to the loser. That’s what is shown in the table. Note that, because the kicker and goalie face each other, a kicker choosing to kick to the right will hope that the goalie dives to the left—that is, the goalie’s own right. To avoid confusion, we set up the table so a dive by the goalie toward where the kicker kicks the ball is considered to be the same direction choice—that is, both choose right or both choose left.

Let’s use our check method again. The kicker always wants to do the opposite of what the goalie does. If the goalie chooses left, the kicker’s best response is to choose right. If the goalie dives right, the kicker wants to kick left. However, the goalie always wants to match the kicker’s choice: If the kicker chooses left, the goalie’s best response is to go left and go right if the kicker chooses right. Putting checks by all these responses gives us the situation in the table.

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Notice what has happened: There is no box with two checks in it. That’s because no set of strategies exists such that both players simultaneously choose the best response to the other’s choice. Given what one player chooses, the other player will want to switch. Our check method therefore doesn’t indicate any pure-strategy Nash equilibrium to the game, at least in the way we have been defining a Nash equilibrium. In pure strategies, a Nash equilibrium always needs two checks in a box.

There actually is a Nash equilibrium to this game, but it is not one in pure strategies in which each player chooses a single action. Up to now, the games we’ve analyzed have had at least one (and sometimes more) pure-strategy equilibria, such as both firms choosing to advertise, magazines selecting different cover stories, or film studios ensuring their films are not released at the same time.

The Nash equilibrium that exists in the penalty-kicks game involves randomizing across different strategies, that is, sometimes choosing one option (kick right) and at other times choosing another (kick left). Suppose the kicker, rather than simply choosing left or right as a strategy to follow all of the time, formed a strategy like “I will kick right 80% of the time and kick left 20% of the time.” The goalie could have a similar strategy of diving one way or the other with a set probability. These randomized actions are mixed strategies. In mixed strategies, the randomization pattern (whether the probability mix is 80/20, 50/50, or anything else) is itself a strategy, just as kicking left for certain is a strategy.

Although the 80/20 right/left strategy example above is a mixed strategy, it is not a Nash equilibrium strategy. Think about it: If the kicker is going right 80% of the time, then the goalie’s best response is to always go right. But the kicker’s best response to a goalie diving right every time is to kick left all of the time. But if the kicker always goes left, the goalie also wants to go left all the time. If they want to switch once they know what the other player is doing, it’s not a Nash equilibrium.

The only mixed-strategy Nash equilibrium in this game is for both the kicker and goalie to go left and right exactly half of the time. If the kicker randomly kicks to the right half of the time and to the left the other half, the goalie gets the same payoff whether she jumps left or right, so she is happy to randomize between the two. And if the goalie splits 50/50, the same is true of the kicker. Therefore, this is a mutual best response.

The strange thing about a mixed-strategy Nash equilibrium is that, given that they are using the optimal probabilities, both players are indifferent between the actions they randomize over. If they weren’t indifferent, they would prefer to play a pure strategy, where they would kick (dive) one direction all of the time. To arrive at a mixed-strategy equilibrium, each player needs to pick the probability of taking each action so that the player’s opponent is indifferent between her own actions. By choosing a 50/50 split between kicking right and left, the kicker leaves the goalie equally well off whether she chooses to dive left or dive right. As a result, there is no way for the goalie herself to do better than dive right 50% of the time and dive left 50% of the time. This is one of the most confusing and counterintuitive results in all of economics, so if it seems bizarre, you are not alone in thinking so.

In game theory, the idea of Nash equilibrium is based on players rationally considering all the possible payoffs and coldly calculating their optimal best responses to their opponent’s choices. You can probably imagine, then, that the notion of players making systematic errors (errors that are nonrandom and occur over and over again in the same direction) poses a big problem for predicting the outcomes of games by looking for Nash equilibria. If a player cannot trust her opponents to at least do what’s in the opponents’ best interests, how should she act?

In Chapter 18, we look at some of the new economic research on behavioral economics, the branch of economics that argues people don’t always behave rationally and often make various systematic errors in judgment.

We have a lot to say about irrationality in Chapter 18, but now is a good time to bring up a type of strategy known in game theory as the maximin strategy (short for “maximize the minimum”). Maximin is a conservative strategy because the player is not going for the highest payoff, but is instead choosing a strategy to minimize losses. Therefore, maximin can be useful in games in which one or more players might be irrational. The idea of the maximin strategy (and the origin of its name) is that a player takes actions that minimize the damage to her in the worst-case scenario—maximizing her minimum payoff. In other words, if an opponent chooses the exact strategy that would punish a player as much as possible (even if it hurts the opponent, too), in what way can the player respond that will minimize the damage? A maximin strategy limits the game’s downside; instead of looking for the best outcome given an opponent’s actions, the player just tries to minimize her exposure to losses.

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*Payoffs are measured in thousands of dollars of profit.

Let’s go back to our earlier example in which two magazines are choosing a cover story for their next issue. We saw that a celebrity interview (CI) cover story was a dominant strategy for I’m Famous Weekly, and that the Nash equilibrium had Look at Me! Magazine going with a reality show (RS) cover as a best response. The payoffs are shown in Table 12.6 (which is the same as Table 12.2).

But suppose for a minute that Me!’s management believes its competitor isn’t just Famous but also dumb—so dumb that it can’t even tell that CI is its dominant strategy. If the Famous editorial board goes with the reality show cover (because they’re irrational, confused, or in a hurry), and Me! Magazine proceeded with an RS cover, too, it would be disastrous for Me! because its profit would be zero. (Remember that both magazines are making these cover story decisions simultaneously.) If Me! goes with a CI cover instead, the worst it can do is make $200,000 if Famous opts for a CI cover (as it rationally should), and the best it can do is $400,000 if Famous acts irrationally and does an RS cover.

If Me! is risk-averse enough, it will choose to go with a celebrity interview cover even though the Nash equilibrium suggests a reality show. This choice (choosing a CI cover) is a maximin strategy for Me!: It chooses the strategy that maximizes its minimum possible outcome, by raising it from $0 under RS to $200,000 under CI. The outcome of the game in this case is not a Nash equilibrium or a profit-maximizing equilibrium, however. Me! could do better (earning $300,000) by unilaterally changing its cover from CI to RS if it knew for sure that Famous was going to choose CI. Me! is giving up that best response in order to avoid what it considers to be a potential disaster should Famous be too dumb to do the rational thing.

Something that’s interesting to note about this sort of situation is that a firm might realize it can influence its competitor’s behavior by seeming to be crazy. The mere threat of irrational behavior—if it’s credible enough—can be used in some cases to manipulate an opponent’s actions in a player’s favor.