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.