Oligopoly and games

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Under perfect competition or monopolistic competition, there are so many firms in the industry that each one can ignore the immediate effect of its own actions on particular rivals. However, in an oligopolistic industry each firm must consider how its actions affect the decisions of its relatively few competitors. Each firm must guess how its rivals will react. Before discussing what constitutes an intelligent guess, we investigate whether they are likely to collude or compete. Collusion is a means of reducing competition with a view to increasing profit.

$\mid$ Collusion is an explicit or implicit agreement to avoid competition with a view to increasing profit.

A particular form of collusion occurs when firms co-operate to form a cartel, as we saw in the last chapter. Collusion is more difficult if there are many firms in the industry, if the product is not standardized, or if demand and cost conditions are changing rapidly. In the absence of collusion, each firm’s demand curve depends upon how competitors react: If Air Canada contemplates offering customers a seat sale on a particular route, how will West Jet react? Will it, too, make the same offer to buyers? If Air Canada thinks about West Jet’s likely reaction, will it go ahead with the contemplated offer? A conjecture is a belief that one firm forms about the strategic reaction of another competing firm.

$\mid$ A conjecture is a belief that one firm forms about the strategic reaction of another competing firm.

Good poker players will attempt to anticipate their opponents’ moves or reactions. Oligopolists are like poker players, in that they try to anticipate their rivals’ moves. To study interdependent decision making, we use game theory. A game is a situation in which contestants plan strategically to maximize their profits, taking account of rivals’ behaviour.

$\mid$ A game is a situation in which contestants plan strategically to maximize their profits, taking account of rivals’ behaviour.

The players in the game try to maximize their own payoffs. In an oligopoly, the firms are the players and their payoffs are their profits. Each player must choose a strategy, which is a plan describing how a player moves or acts in different situations

$\mid$ A strategy is a game plan describing how a player acts, or moves, in each possible situation.

How do we arrive at an equilibrium in these games? Let us begin by defining a commonly used concept of equilibrium. A Nash equilibrium is one in which each player chooses the best strategy, given the strategies chosen by the other players and there is no incentive to move or change choice.

$\mid$ A Nash equilibrium is one in which each player chooses the best strategy, given the strategies chosen by the other player, and there is no incentive for any player to move.

In such an equilibrium, no player wants to change strategy, since the other players’ strategies were already figured into determining each player’s own best strategy. This concept and theory are attributable to the Princeton mathematician John Nash, who was popularized by the Hollywood movie version of his life A Beautiful Mind.

In most games, each player’s best strategy depends on the strategies chosen by their opponents. Sometimes, though not always, a player’s best strategy is independent of those chosen by rivals. Such a strategy is called a dominant strategy

$\mid$ A dominant strategy is a player’s best strategy, whatever the strategies adopted by rivals.

We now illustrate these concepts with the help of two different games. These games differ in their outcomes and strategies. Figure 11.2 contains the domestic happiness game 1. Will and Kate are attempting to live in harmony, and their happiness depends upon each of them carrying out domestic chores such as shopping, cleaning and cooking. If both contribute to domestic life they each receive a happiness or utility level of 5 units. If one contributes and the other does not the happiness levels are 2 for the contributor and 6 for the free-rider. If neither contributes happiness levels are 3 each. When each follows the same strategy the payoffs are on the diagonal, when they follow different strategies the payoffs are on the off-diagonal. Since the elements of the table define the payoffs resulting from various choices, this type of matrix is called a payoff matrix.

$\mid$ A payoff matrix defines the rewards to each player resulting from particular choices.

So how is the game likely to unfold? In response to Will’s choice of a contribute strategy, Kate’s utility maximizing choice involves lazing: she gets 6 units by not contributing as opposed to 5 by contributing. Instead, if Will decides to be lazy what is in Kate’s best interest? Clearly it is to be lazy also because that strategy yields 3 units of happiness compared to 2 units if she contributes. In sum, Kate’s best strategy is to be lazy, regardless of Will’s behaviour. So the strategy of not contributing is a dominant strategy.

Will also has a dominant strategy – identical to Kate’s. This is not surprising since the payoffs are symmetric in the table. Hence, since each has a dominant strategy of not contributing the Nash equilibrium is in the bottom right cell, where each receives a payoff of 3 units. Interestingly this equilibrium is not the one that yields maximum combined happiness.

Table 11.2 A game with dominant strategies
		Kate’s choice
		Contribute	Laze
Will’s choice	Contribute	5,5	2,6
Will’s choice	Laze	6,2	3,3

The first element in each cell denotes the payoff or utility to Will; the second element the utility to Kate.

The reason that the equilibrium yields less utility for each player in this game is that the game is competitive: each player tends to their own interest and seeks the best outcome conditional on the choice of the other player. This is evident from the (5, 5) combination. From this position Kate would do better to defect to the Laze strategy, because her utility would increase 2 .

To summarize: this game has a unique equilibrium and each player has a dominant strategy. But let us change the payoffs just slightly to the values in Table 11.4. The off-diagonal elements have changed. The contributor now gets no utility as a result of his or her contributions: even though the household is a better place, he or she may be so annoyed with the other person that no utility flows to the contributor.

Table 11.3 A game without dominant strategies
		Kate’s choice
		Contribute	Laze
Will’s choice	Contribute	5,5	0,4
Will’s choice	Laze	4,0	3,3

The first element in each cell denotes the payoff or utility toWill; the second element the utility to Kate.

What are the optimal choices here? Starting again fromWill choosing to contribute, what is Kate’s best strategy? Clearly it is to contribute: she gets 5 units from contributing and 4 from lazing, hence she is better contributing. But what is her best strategy if Will decides to laze? Clearly it is to laze, because that yields her 3 units as opposed to 0 by contributing. This set of payoffs therefore contains no dominant strategy for either player.

As a result of there being no dominant strategy, there arises the possibility of more than one equilibrium outcome. In fact there are two equilibria in this game now: if the players find themselves both contributing and obtaining a utility level of (5, 5) it would not be sensible for either one to defect to a laze option. By the same reasoning, if they find themselves at the (laze, laze) combination there is no incentive to move to a contribute strategy.

Once again, it is to be emphasized that the twin equilibria emerge in a competitive environment. If this game involved cooperation or collusion the players should be able to reach the (5, 5) equilibrium rather than the (3, 3) equilibrium. But in the competitive environment we cannot say ex ante which equilibrium will be attained.

This game illustrates the tension between collusion and competition. While we have developed the game in the context of the household, it can equally be interpreted in the context of a profit maximizing game between two market competitors. Suppose the numbers define profit levels rather than utility as in Table 11.4. The contribute option can be interpreted as ‘cooperate’ or ‘collude’ as we described for a cartel in the previous chapter. They collude by agreeing to restrict output, set a price with profit maximizing monopoly behaviour as a goal and split the profits. The combined best profit outcome (5, 5) arises when each firm restricts its output.

Table 11.4 Collusion possibilities
		Kate’s profit
		Contribute	Laze
Firm W’s profit	Low output	5,5	2,6
Firm W’s profit	High output	6,2	3,3

The first element in each cell denotes the profit to Firm W; the second element the profit to Firm K.

But again there arises an incentive to defect: if Firm W agrees to maintain a high price and restrict output, then Firm K has an incentive to renege and increase output, hoping to improve its profit through the willingness of Firm W to restrict output. Since the game is symmetric, each firm has an incentive to renege.

Obviously there arises the question of whether these firms can find an operating mechanism that would ensure they each generate a profit of 5 units rather than 3 units, while remaining purely self-interested. This question brings us to the realm of games that are repeated many times. Such games are beyond our scope, but here is a flavour of how they work.

One way is that a binding commitment be put in place, which restricts future actions. If both players in this game were to sign an enforceable agreement to produce a low output, then each could earn a profit of five units rather than the three units earned in the non-collusive or ‘competitive’ equilibrium. A binding commitment gets both players to produce a low output and gain thereby. Although this commitment concept is powerful, its application to oligopoly requires care: cartels within a country are usually illegal.

$\mid$ A binding commitment is an agreement, entered into voluntarily, that restricts future actions.

Is there a less formal way to avoid “cheating” in the cartel solution? In some situations there may be. The key ingredient for such a possibility is that the punishment that will be inflicted upon the firm violating the agreement must be in the interest of the firm that inflicts the punishment. Such a situation gives credibility to the punishment.

$\mid$ A credible threat is one that, after the fact, is still optimal to carry out.

We now examine a particular type of game involving two firms – a duopoly.

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Oligopoly and games