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Game Theory: The Prisoner's Dilemma

This presentation try to explain Oligopoly Market using game theory (the prisoner's dilemma) and Nash equilibrium theory
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Stephen Gunawan

on 19 September 2011

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Transcript of Game Theory: The Prisoner's Dilemma

OLIGOPOLY Is a market structure in which: A small number of firms compete Natural or legal barriers prevent the entry of new firms. Formed by economies of scale and demand These factors can create a "Natural Oligopoly" Example of Natural Oligopoly: D (demand curve), shows the demand for taxi rides in a town. If the average total cost curve of a taxi company is ATC1 in part (a), the market is a natural duopoly - an oligopoly market with two firms The lowest price at which the firm would remain in business is $10 a ride. At that price, the quantity of rides demanded is 60 a day. It's the quantity that can be provided by just two firms. There is no room in this market for three firms. But if there were only one firm, it would make an economic profit and a second firm would enter to take some of the business and economic profit If the average total cost curve of a taxi company is ATC2 in part (b), the efficient scale of one firm is 20 rides a day. This market large enough for three firms A legal oligopoly arises when a legal barrier to entry protects the small number of firms in a market. For example, a city might license two taxi firms or two bus companies (even though the combination of demand and economies of scale leaves room for more than two firms) Because barriers to entry exist, oligopoly consists of a small number of firms, each of which has a large share of the market. Such firms are:

Interdependent

They face a temptation to cooperate to increase their joint economic profit With a small number of firms in the market, each firm's actions influence the profits of all the other firms. When a small number of firms share a market, they can increase their profits by forming a cartel and acting like monopoly is a group of firms acting together - colluding - to limit output, raise price, and increase economic profit Suppose you run one of three gas stations in a small town. You're trying to decide whether to cut your price. To make your decision, you must predict how the other firms will react and calculate the effects of those reactions on your profit.
If you cut your price and your competitors don't cut theirs, you sell more and the other two firms sell less.
But won't the other firms cut their prices too and make your profit falls.

SO WHAT WILL YOU DO? The Kinked Demand Curve Model The kinked demand curve model of oligopoly is based on the assumption that each firm believes that if it raises its price, others will not follow, but if it cuts its price, other firms will cut theirs (so the marginal revenue will diminished) Figure above shows the demand curve (D) that a firm believes it faces. The demand curve has a kink at the current price, P, and quantity, Q.
At prices above P, a small price rise brings a big decrease in the quantity sold. The other firm will hold their current price and the firm has the highest price for the good, so it loses market share.
At price below P, even a large price cut brings only a small increase in the quantity sold. In this case, other firms match the price cut, so the firm gets no price advantage over its competitors.
The kink in the demand curve creates a break in the marginal revenue curve (MR). To maximize proft, the firm produces the quantity at which marginal cost equals marginal revenue. That quantity (Q) is where the marginal cost curve passes through the gap AB in the marginal revenue curve.
If marginal cost fluctuates between A and B, like the marginal cost curve MC0 and MC1, the firm does not change its price or its output.
Only if marginal cost fluctuates outside the range AB does the firm change its price and output.
So the kinked demand curve model predicts that price and quantity are insensitive to small cost change. But this model has a problem. If marginal cost increases by enough to cause the firm to increase its price and if all firms experience the same increase in marginal cost, they all increase their price together. The firm's believe and basic assumption (that others will not join it in a price rise) is incorrect.
A firm that bases its actions on wrong assumptions will not maximize profit and might even end up incurring an economic loss. Several models have been developed to explain the prices and quantities in oligopoly markets. The models fall into two broad groups:
Traditional models, and
Game theory models Dominant Firm Oligopoly Dominant form oligopoly arises when one firm (the dominant firm) has a big cost advantage over the other firms and produce a large part of the industry output. The dominant firm sets the market price and the other firms are price takers. To see how a dominant firm oligopoly works, suppose that eleven (11) firms operate gas stations in a city. Big-G is a dominant firm. In part (a), the demand curve D tells us the total quantity of gas demanded in the city at each price. The supply curve S10 is the supply curve of the 10 small firms. Part (b) shows the situation facing Big-G.
Big-G marginal cost curve is MC and its marginal revenue curve is MR. Big-G faces the demand curve XD that shows the excess demand not met by the 10 small firms (for example: at a price of $1 a gallon, the quantity demanded is 20,000 gallons, the quantity supplied by the 10 small firms is 10,000 gallons, and the excess quantity demanded is 10,000 gallons, measured by the distance AB in both parts of the figure.
To maximize profit, Big-G operates like monopoly. Its sells 10,000 gallons a week, where its marginal revenue equals its marginal cost, for a price of $1 a gallon.
The 10 small firms take the price of $1 a gallon. They behave just like firms in perfect competition.
This way Big-G sells 10,000 gallons and the 10 small firms each sell 1,000 gallons. The traditional models don't enable us to understand all oligopoly markets and we're now going to study some newer models based on game theory. Economist think about oligopoly as a game, and to study oligopoly markets they use a set of tools called game theory.
Game theory is a tool for studying strategic behaviour (behaviour that takes into account the expected behaviour of others and the recognition of mutual interdependence.
Game theory was invented by John von Neumann in 1937 and extended by von Neumann and Oskar Morgenstern in 1944. What is a game? There are many different games. There are ball games, board games, game of chance, game of skill, etc. But what is it about all these different activities that make them games? What do all these games have in common? -THE PRISONER'S DILEMMA- Art and Bob have been caught red-handed, stealing a car. Facing airtight cases, they will receive a sentence of two years each for their crime. During his interviews with the two prisoners, the district attorney begins to suspect that he has stumbled on the two people who were responsible for a multi-million dollar bank robbery some months earlier. But this is just a suspicion. He has no evidence on which he can convict them of the greater crime unless he can get them to confess. But how can he extract a confession? The answer is by making the prisoners play a game. So, the district attorney makes the prisoners play the game that we will now describe: All games share four common features:
Rules
Strategies
Payoffs
Outcome RULES Each prisoners (player) is placed in a separate room and cannot communicate with the other prisoner. Each is told that he is suspected of having carried out the bank robbery and that:

If both of them confess to the larger crime, each will receive a sentence of 3 years for both crimes.

If he alone confess and his accomplice does not, he will receive only a 1-year sentence while his accomplice will receive a 10-year sentence. STRATEGIES In game theory, strategies are all the possible actions of each player. Art and Bob each have two possible actions:

Confess to the bank robbery.

Deny having commited the bank robbery.

Because there are two players, each with two strategies, there are four possible outcomes:

Both confess

Both deny

Art confess and Bob denies

Bob confess and Art denies. PAYOFFS Each prisoners can work out his payoffs in each of these situations, and we can tabulate the four possible payoffs for each of the prisoners in what is called a payoff matrix for the game. Payoff matrix is a table that shown the payoffs for every possible action by each player for every possible action by each other player OUTCOME The choice of both players determine the outcome of the game. To predict that outcome, we can use equilibrium idea proposed by John Nash of Princeton University (who receive Nobel Prize for Economic Science in 1994 and was the subject of the 2001 movie "A Beautiful Mind").

In Nash Equilibrium, player A takes the best possible action given the action of player B and player B takes the best possible action given the action of player A.

Art's point of view: If Bob confesses, Art's best action is to confess because in that case, he is sentenced to 3 years rather than 10 years. If Bob denies, Art's best action is still to confess because in that case he receives 1 year rather than 2 years. So Art's best action is to confess.

Bob's point of view: If Art confesses, Bob's best action is to confess because in that case, he is sentenced to 3 years rather than 10 years. If Art denies, Bob's best action is still to confess because in that case he receives 1 year rather than 2 years. So Bob's best action is to confess.

Because each player's best action is to confess, each does confess, each goes to jail for 3 years, and the district attorney has solved the bank robbery. This is the Nash Equilibrium of the game. The Dilemma - Now that you have found the outcome to the prisoner's dilemma, you can better see the dilemma. The dilemma arises as each prisoner contemplates the consequences of denying. Each prisoners knows that if both of them deny, they will receive only a 2-year sentence for stealing the car. But neither has any way of knowing that his accomplice will deny. The dilemma leads to the equilibrium of the game. A Bad Outcome:

For the prisoners, the equilibrium of the game, with each confessing, is not the best outcome. If neither of them confess, each gets only 2 years for the lesser crime. Isn't there someway in which this better outcome can be achieved? It seems that there is not, because the player can't communicate with each other. Each player can put himself in the other player's place, and so each player can figure out that there is a best strategy for each of them. Each prisoners knows that it is not in the best interest of the other to deny. So each prisoner knows that he must confess, thereby delivering a bad outcome for both.

The firm in oligopoly are in a similar situation to Art and Bob in the prisoner's dilemma game. Let's see how we can use this game to understand oligopoly. We can use game theory and a game like the prisoner's dilemma to understand price fixing, price wars, and other aspects of the behaviour of firms in oligopoly. We'll begin with a price-fixing game.
To understand price fixing, we're going to study the special case of Duopoly. Duopoly is easier to study than oligopoly with three or more firms, and it captures the essence of all oligopoly situations. Somehow, the firms must share the market. And how they share it depends on the actions of each. COST AND DEMAND CONDITION Two firms, Trick and Gear, produce switchgears. They have identical cost and also produce identical swithcgears (co one firm's swithgear is a perfect substitute for the other's). The market price for each firm's product is identical. COLLUSION We'll suppose that Trick and Gear enter into a collusive agreement. A collusive agreement is an agreement between two (or more) producers to form a cartel to restrict output, raise the price, and increase profits.

The strategies that firms in cartel can pursue are to:
Comply = A firm that complies carries out the agreement.
Cheat = A firm that cheats break the agreement to its own benefit and to the cost of other firm.

Four possible combinations of actions for the firms:
Both firms comply
Both firms cheat
Trick complies and Gear cheats.
Gear complies and Trick cheats. COLLUDING TO MAXIMIZE PROFIT The only thing that the firms in duopoly must do beyond what a monopoly does is to agree on how much of the total output each of them will produce.

Part (a) shows the situation for each firm, and part (b) shows the situation for the industry as a whole. Conclusion: each firms produce 2,000 items and sell the product at $9,000/unit. Making profit for $2 million for each firm. We have just describe one possible outcome for a duopoly game: the two firms collude to produce the monopoly profit maximizing output and divide that output equally between themselves. From the industry point of view, this solution is identical to a monopoly. The economic profit that is made by monopoly is the maximum total profit that can be made by the duopoly when the firms collude. But with price greater than marginal cost, either firm might think of trying to increase profit by cheating on the agreement and producing more than the agreed amount ONE FIRM CHEATS ON A COLLUSIVE AGREEMENT To set the stage for cheating on their agreement, Trick convinces Gear that demand has decreased and that it cannot sell 2,000 units a week. Trick tells Gear that it plans to cut its price so that it can sell the agreed 2,000 units each week. Because the two firms produce an identical product, Gear matches Tricks's price cut but still produces only 2,000 units a week.
In fact, there is no decrease in demand. Trick plans to increase output, which he knows will lower the price, and Trick wants to ensure that Gear's output remains at the agreed level. Conclusion: Trick (the cheat) makes an economic profit of $4.5 million, and Gear (the complier) incurs an economic loss of $1 million. The industry makes an economic profit of $3.5 million. This industry profit is $0.5 million less than the economic profit that a monopoly would make. But the profit is distributed unevenly. Trick makes a bigger economic profit than it would under the collusive agreement, while Gear incurs an economic loss.

Let's next see what happens if both firms cheat. BOTH FIRMS CHEAT Suppose that both firms cheat and that each firm behaves like the cheating firm that we have just analyzed. Each tells the other that it is unable to sell its output at the going price and that it plans to cut its price. But because both firms cheat, each will propose a succesively lower price. As long as price exceeds marginal cost, each firm has an incentive to increase its production - to cheat. Only when price equals marginal cost is there no further incentive to cheat. This situation arise when the price has reached $6000. At a price less than $6,000, each firm incurs an economic loss. At a price $6,000, each firm covers all its costs and makes zero economic profit. We have now describe a third possible outcome of this duopoly game: Both firms cheat. If both firms cheat on the collusive agreement, the output of each firm is 3,000 unit a week and the price is $6,000 a unit. Each firm makes zero economic profit PAYOFF MATRIX AND NASH EQUILIBRIUM IN THE DUOPOLISTS' DILEMMA Table 2 sets out the payoff matrix for this game. It is constructed in the same way as the payoff matrix for the prisoners' dilemma in Table 1. In this case, the payoffs are profits. Do they comply or cheat?

Gear's point of view: Suppose that Tricks cheats. If i comply, I will incur an economic loss of $1 million. If I also cheat, I will make zero economic profit. Zero is better than minus $1 million, so I'm better of if I cheat. Now suppose Trick complies. If I cheat, I will make an economic profit of $4.5 million, and if I comply, I will make an economic profit of $2 million. A $4.5 million profit is better than a $2 million, so I'm better of if I cheat. So regardless of whether Trick cheats or complies, it pays Gear to cheat. Cheating is Gear's best strategy.

Trick comes to the same conclusion as Gear because the two firms face an identical situation. So both firms cheat.

The Nash Equilibrium of the duopoly game is that both firms cheat. And although the industry has only two firms, they act like in a perfect competition and makes zero economic profit. IT'S NOT ALWAYS LIKE THAT... This conclusion is not general and will not always arise. There are a lot other oligopoly games that are like the prisoner's dilemma.

The worst possible outcome for each player arises from cooperating when the other player cheats.
The best possible outcome, for each player to cooperate, is not Nash equilibrium because it is in neither player's self-interest to cooperate if the other one cooperates.

It is this failure to achieve the best outcome for both players - the best social outcome if the two players are the entire economy - that led John Nash to claim (as he was portrayed as doing in the movie A Beautiful Mind) that he had challenged Adam Smith's idea that we are always guided, as if by an invinsible hand, to promote the social interest when we are pursuing our self interest. Case application 10-3: The Airlines' Fare War and the Prisoners' Dilemma Condition: during 1990 until 1991 domestic airlines lost more than $6 billion. Pan Am & Eastern Airlines went out of business. Continental, TWA, America West filed for bankruptcy protection.

Rather than establishing price dicipline, however, American's new fare structure started a process of competitive price cuts that led to another disastrous price war during the summer of 1992.

Even though deep price cuts increased summer travel sharply, all airlines incurred losses (the low fares failed to cover the industry average cost). Three attempts to increase air fares by 30% above presale levels in the fall of 1992 failed when one or more of the carriers did not go along.

In short, U.S. Airlines seemed to be in a prisoners' dilemma and, UNABLE TO COOPERATE, faced heavy losses.

Only with the strong rebound in air travel in 1995 did airlines refrain from engaging in another price war and thus earned profits.

But tranquility and profits did not last long. With very low marginal cost of adding passengers to a flight after it has been scheduled, there is strong incentive for all airlines to cut fares to fill all seats on a flight. Syndicate 5:

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