Chapter 10: Game Theory Inside Oligopoly Flashcards
Game Theory
Payoff depends on the actions taken by other players.
In a game, the players are firm managers who make decisions
The planned decisions of the players are called strategies.
The payoffs to the players are the profits or losses that result from the strategies.
Due to interdependence, the payoff to a player depends not only on that player’s strategy but also on the strategies employed by other players.
The order in which players make decisions is important
Oligopolistic Market
In an oligopolistic market consisting of two firms, each player must make a pricing decision.
Simultaneous-Move Game
Game in which each player makes decisions without knowledge of the other players’ decisions.
Example: Rock-Paper-Scissors
Sequential-Move Game
Game in which one player makes a move after observing the other player’s move.
Example: chess (players alternate moves).
Oligopoly Games: Setting Prices
If two firms must set prices without knowledge of each other’s decisions, it is a simultaneous-move game; if one firm sets its price after observing its rival’s price, it is a sequential-move game.
One-Shot Game
Game in which the underlying game is played only once.
Repeated Game
Game in which the underlying game is played more than once.
Normal-Form Game
A representation of a game indicating the players, their possible strategies, and the payoffs resulting from alternative strategies.
Strategy
In game theory, a decision rule that describes the actions a player will take at each decision point.
Simultaneous-Move, One-Shot Games
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.
Hint: 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.
Since the game 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.
The players cannot make conditional decisions; 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.
Payoffs in a Normal Form Game: Example
The payoff to player A crucially depends on the strategy player B chooses.
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.
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.
Optimal Strategy for a Player in a Simultaneous-Move, One-Shot Game
In simultaneous-move, one-shot games where a player has a dominant strategy, the optimal decision is to choose the dominant strategy. Doing so will maximize your payoff regardless of what your opponent does.
Note: A player may not have a dominant strategy in some games.
Dominant Strategy
A strategy that results in the highest payoff to a player regardless of the opponent’s action.
Secure Strategy (In the Absence of a Dominant Strategy)
A strategy that guarantees the highest payoff given the worst possible scenario.
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.
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 consider your rival’s optimal decisions and thus may prevent you from earning a significantly higher payoff.
Secure Strategy: Example
Example:
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 dominant 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).
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.
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.
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.
it represents a situa- tion where every player is doing the best he or she can, given what other players are doing.
Nash Equilibrium: Example
The Nash equilibrium strategy for player A is up, and for player B it is left.
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 strategy 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 other player’s strategy.
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 playing 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.
Applications of Simultatenous Move, One-Shot Games: Pricing Decisions in a Bertrand Duopoly
Two firms face a situation where they must decide whether to charge low or high prices.
The first number in each cell represents firm A’s profits and the second number represents firm B’s profits. If firm A charges a high price while firm B charges a low price, A loses 10 units of profits while B earns 50 units of profits.
Note: While the numbers in Table 10–2 are arbitrary, their relative magnitude is consistent with the nature of Bertrand competition.
The profits of both firms are higher when both charge high prices than when they both charge low prices because, in each instance, consumers have no incentive to switch to the other firm. On the other hand, if one firm charges a high price and the other firm undercuts that price, the lower-priced firm will gain all the other firm’s customers and thus earn higher profits at the expense of the competitor.
The firms meet once, and only once, in the market (one-shot game). Moreover, the game is a simultaneous-move game in that each firm makes a pricing decision without knowing the other firm’s decision.
Nash Equilibrium Strategy in a Simultatenous Move, One-Shot Bertrand Duopoly Game
In a one-shot play of the game, the Nash equilibrium strategies are for each firm to charge the low price.
If firm B charges a high price, firm A’s best choice is to charge a low price since 50 units of profits are better than the 10 units it would earn if A charged the high price. Similarly, if firm B charges the low price, firm A’s best choice is to charge the low price since 0 units of profits are preferred to the 10 units of losses that would result if A charged the high price.
Similar arguments hold from firm B’s perspective. Firm A is always better off charging the low price regardless of what firm B does, and B is always better off charging the low price regardless of what A does. Hence, charging a low price is a dominant strategy for both firms.
In the one-shot version of this game, each firm’s best strategy is to charge a low price regardless of the other firm’s action. The game’s outcome is that both firms charge low prices and earn zero profits.
Collusion in a Game: A Dilemma
Profits are less than the firms would earn if they colluded and “agreed” to both charge high prices.
For example, in Table 10–2 we see that each firm would earn profits of 10 units if both charged high prices. This is a classic result in economics and is called a dilemma because the Nash equilibrium outcome is inferior (from the viewpoint of the firms) to the situation where they both “agree” to charge high prices.
Why Firm Collusion Fails: Illegal & Cheating
Why can’t firms collude and agree to charge high prices? One answer is that collusion is illegal; firms are not allowed to meet and “conspire” to set high prices. There are other reasons, however.
Suppose the managers did secretly meet and agree to charge high prices. Would they have an incentive to live up to their promises? Consider firm A’s point of view.
If it “cheated” on the collusive agreement by lowering its price, it would increase its profits from 10 to 50. Thus, firm A has an incentive to induce firm B to charge a high price so that it can “cheat” to earn higher profits. Of course, firm B recognizes this incentive, which precludes the agreement from being reached in the first place.
However, suppose the manager of firm A is “honest” and would never cheat on a promise to charge a high price. (She is “honest” enough to keep her word to the other manager but not so honest as to obey the law against collusion.) What happens to firm A if the manager of firm B cheats on the collusive agreement? If B cheats, A experiences losses of $10. When firm A’s stockholders ask the manager why they lost $10 when the rival firm earned profits of $50, how can the manager answer? She cannot admit she was cheated on in a collusive agreement, for doing so might send her to jail for having violated the law. Whatever her answer, she risks being fired or sent to prison.
Applications of Simultatenous Move, One-Shot Games: Advertising and Quality Decisions
In oligopolistic markets, firms advertise and/or increase their product quality in an attempt to increase the demand for their products.
Note: While both quality and advertising can be used to increase the demand for a product, our discussion will use advertising as a placeholder for both quality and advertising.
Oligopolistic Markets: Advertising
In most oligopolistic markets, advertising increases the demand for a firm’s product by taking customers away from other firms in the industry.
An increase in one firm’s advertising increases its profits at the expense of other firms in the market; there is interdependency among the advertising decisions of firms.
Example: Breakfast Cereal Industry, By advertising its brand of cereal, a particular firm does not induce many consumers to eat cereal for lunch and dinner; instead, it induces customers to switch to its brand from another brand. This can lead to a situation where each firm advertises just to “cancel out” the effects of other firms’ advertising, resulting in high levels of advertising, no change in industry or firm demand, and low profits.
Coordination Decisions
There are situations where firms have competing objectives: One firm can gain only at the expense of other firms. But, not all games have this structure.
Example: Producers of electrical appliances have a choice of which type of electrical outlets to put on appliances: 90-volt, four-prong outlets or 120-volt, two-prong outlets. In an environment where different appliances require different outlets, a consumer who desires several appliances would have to spend a considerable sum wiring the house to accommodate all the appliances. This would reduce the amount the consumer has available for buying appliances and therefore would adversely affect the profits of appliance manufacturers. In contrast, if the appliance manufacturers can “coordinate” their decisions (i.e., produce appliances that require the same types of wiring), they will earn higher profits.
Coordination Game: Example
Two firms must decide whether to produce appliances requiring 120-volt or 90-volt outlets. If each firm produces appliances requiring 120-volt outlets, each firm will earn profits of $100.
If each firm produces appliances requiring 90-volt outlets, each firm will earn $100. However, if the two firms produce appliances requiring different types of outlets, each firm will earn zero profits due to the lower demand that will result from consumers’ need to spend more money wiring their houses.
What would you do if you were the manager of firm A in this example? If you do not know what firm B is going to do, you have a very tough decision. All you can do is “guess” what B will do. If you think B will produce 120-volt appliances, you should produce 120-volt appliances as well. If you think B will produce 90-volt appliances, you should do likewise. You will thus maximize profits by doing what firm B does. Effectively, both you and firm B will do better by “coordinating” your decisions.
Coordinate Game: Nash Equilibrium
The game in Table 10–4 has two Nash equilibria.
One Nash equilibrium is for each firm to produce 120-volt appliances; the other is for each firm to produce 90-volt appliances. The question is how the firms will get to one of these equilibria. If the firms could “talk” to each other, they could agree to produce 120-volt systems. Alternatively, the government could set a standard that electrical outlets be required to operate on 120-volt, two-prong outlets.
In effect, this would allow the firms to “coordinate” their decisions. Notice that once they agree to produce 120-volt appliances, there is no incentive to cheat on this agreement. The game in Table 10–4 is not analogous to the pricing or advertising games analyzed earlier; it is a game of coordination rather than a game of conflicting interests.
Game Theory: Monitoring Employees
Game theory can also be used to analyze interactions between workers and the manager.
Managers desire workers to work hard, while workers enjoy leisure.
One way a manager can reduce workers’ incentives to shirk is to engage in “random” spot checks of the workplace. Game theory provides a way of seeing why this can work. Consider a game between a worker and a manager. The manager has two possible actions: (1) monitor the worker or (2) don’t monitor the worker. The worker has two choices: (1) work or (2) shirk. These possible actions and resulting payoffs are depicted in Table 10–5.
Monitoring Employees Payoffs
The interpretation of this normal-form game is as follows: If the manager monitors while the worker works, the worker “wins” and the manager “loses.” The manager has spent time monitoring a worker who was already working. In this case, suppose the manager’s payoff is −1, and the worker’s payoff is 1. The payoffs are the same if the manager does not monitor the worker and the worker shirks; the worker wins because she gets away with shirking.
In contrast, if the manager monitors while the worker shirks, the manager wins 1, and the worker who gets caught loses 1. Similarly, if the worker works and the manager does not monitor, the manager wins 1, and the worker loses 1. The numbers in Table 10–5 are purely hypothetical, but they are consistent with the relative payoffs that arise in such situations.
Monitoring Employees: No Nash Equilibrium
Notice that the game in Table 10–5 does not have a Nash equilibrium, at least in the usual sense of the term. To see this, suppose the manager’s strategy is to monitor the worker. Then the best choice of the worker is to work. But if the worker works, the manager does better by changing his strategy: choosing not to monitor. Thus, “monitoring” is not part of a Nash equilibrium strategy. The paradox, however, is that “not monitoring” isn’t part of a Nash equilibrium either.
Suppose the manager’s strategy is “don’t monitor.” Then the worker will maximize her payoff by shirking. Given that the worker shirks, the manager does better by changing the strategy to “monitor” to increase his payoff from −1 to 1. Thus, we see that “don’t monitor” is not part of a Nash equilibrium strategy either.
Mixed (Randomized) Strategy
A strategy whereby a player randomizes over two or more available actions in order to keep rivals from being able to predict his or her action.
Mixed (Randomized) Strategy: Example
In Table 5-1, the worker and the manager want to keep their actions “secret”; if the manager knows what the worker is doing, it will be curtains for the worker, and vice versa. In such situations, players find it in their interest to engage in a mixed (randomized) strategy.
What this means is that players “randomize” over their available strategies; for instance, the manager flips a coin to determine whether or not to monitor. By doing so, the worker cannot predict whether the manager will be present to monitor her and, consequently, cannot outguess the manager.
Simultaneous-Move, One-Shot Game Nash Bargaining Game
In a Nash bargaining game, two players “bargain” over some object of value. In a simultaneous-move, one-shot bargaining game, the players have only one chance to reach an agreement, and the offers made in bargaining are made simultaneously.
Simultaneous-Move, One-Shot Game Nash Bargaining Game: Example
Suppose management and a labor union are bargaining over how much of a $100 surplus to give to the union. Suppose, for simplicity, that the $100 can be split only into $50 increments.
The players have one shot to reach an agreement. The parties simultaneously write the amount they desire on a piece of paper (0, 50, or 100). If the sum of the amounts each party asks for does not exceed $100, the players get the specified amounts. But if the sum of the amounts requested exceeds $100, bargaining ends in a stalemate. Suppose that the delays caused by this stalemate cost both the union and management $1.
Simultaneous-Move, One-Shot Game Nash Bargaining Game: Example
Table 10–6 presents the normal form of this hypothetical bargaining game. If you were management, what amount would you ask for? Suppose you wrote down $100. Then, you would only get any money if the union asked for zero. Notice that if management asked for $100 and the union asked for $0, neither party would have an incentive to change its amounts; we would be in Nash equilibrium.
Before concluding that you should ask for $100, think again. Suppose the union wrote down $50. Management’s best response to this move would be to ask for $50. And given that management asked for $50, the union would have no incentive to change its amount. Thus, a 50–50 split of the $100 also would be a Nash equilibrium.
Finally, suppose management asked for $0, and the union asked for the entire $100. This, too, would constitute a Nash equilibrium. Neither party could improve its payoff by changing its strategy given the strategy of the other.
Thus, there are three Nash equilibrium outcomes to this bargaining game. One outcome splits the money evenly among the parties, while the other two outcomes give all the money to either the union or management.
This example illustrates that the outcomes of simultaneous-move bargaining games are difficult to predict because there are generally multiple Nash equilibria. This multiplicity of equilibria leads to inefficiencies when the parties fail to “coordinate” on an equilibrium. In Table 10–6, for instance, six of the nine potential outcomes are inefficient in that they result in total payoffs that are less than the amount to be divided. Three of these outcomes entail negative payoffs due to stalemate. Unfortunately, stalemate is common in labor disputes: Agreements often fail or are delayed because the two sides ask for more (in total) than there is to split.
Experimental evidence suggests that bargainers often perceive a 50–50 split to be “fair.” Consequently, many players in real-world settings tend to choose strategies that result in such a split even though there are other Nash equilibria. Clearly, for the game in Table 10–6, if you expect the union to ask for $50, you, as management, should ask for $50.
Infinitely Repeated Game
A game that is played over and over again forever and in which players receive payoffs during each play of the game.
Infinitely Repeated Game: Payoffs
When a game is played, again and again, players receive payoffs during each repetition of the game.
Due to the time value of money, a dollar earned during the first repetition of the game is worth more than a dollar earned in later repetitions; players must appropriately discount future payoffs when they make current decisions.
Present Value of Future Profits
The value of a firm is the present value of all future profits earned by the firm.
If the interest rate is i, π0 represents profits today, π1 profits one year from today, π2 profits two years from today, and so on, the value of a firm that will be in business for T years.
Present Value of Future Profits: Formula
If the profits earned by the firm are the same in each period (πt = π for each period t) and the horizon is infinite (T = ∞), this formula simplifies to
Supporting Collusion with Trigger Strategies: Simultaneous-Move Bertrand Pricing Game
The Nash equilibrium in a one-shot play of this game is for each firm to charge low prices and earn zero profits.
Suppose the firms play the game in Table 10–7 day after day, week after week, for all eternity. Thus, we are considering an infinitely repeated Bertrand pricing game, not a one-shot game. In this section, we will examine the impact of repeated play on the equilibrium outcome of the game.
Trigger Strategy
A strategy that is contingent on the past play of a game and in which some particular past action “triggers” a different action by the first player.
When firms repeatedly face a payoff matrix such as that in Table 10–7, it is possible for them to “collude” without fear of being cheated on. They do this by using trigger strategies.
