I have nearly finished the Open Yale course on game theory. Just the final exam is still to be done. Although my exam won’t be graded, I’ll use the opportunity to tackle those problems with the new tools and mind-set and see, what I have learned and understood.
The last topics in the course were about incomplete information and signaling. In game theory incomplete information refers to games, where at least one player is uncertain of the other players’ so called types. A type describes a player’s preferences, available strategies and payoffs. For example, in a real life negotiation we often have incomplete information, being uncertain how tough the opponent is and how he values the potential outcomes. A related concept, imperfect information, we have already encountered when a player does not know, which strategy his opponent has played, although he knows all the potential ways the game could be played and the related preferences and payoffs.
Incomplete information is a realistic condition and thus often encountered when modeling real world phenomena. Often incomplete information is also seen as an asset from the informed party’s point of view: for example, a company knowing both its own and its competitor’s cost structure is intuitively thought to have an advantage, since it knows more, and information is power, as the cliché goes. But what exactly is the benefit of knowing your competitors’ costs and how valuable is it? A further question, with an even less intuitive answer, is, whether such a better-informed company would actually profit from making its own cost structure public. In the following, I will answer these questions from the perspective of the Cournot duopoly model, but taking it a bit further from the standard treatment.
The following is based on the Open Yale course Econ 159a Game Theory and on Strategies and Games: Theory and Practice, 1999 by Prajit K. Dutta.
Incomplete information in a Cournot duopoly
In the standard Cournot duopoly two profit-maximizing companies manufacture substitute products with identical constant marginal cost and serve the same market. The outcome of the model is that in equilibrium both companies produce the same amount of goods, in total more than the monopoly quantity but less than the quantity in free-market competition. Consequently, the companies also get higher than free-market prices and profits, but not the monopoly prices or profits due to the competition between the two rivals.
In an incomplete information Cournot duopoly at least one company (I will only consider this case in the following) has its marginal cost different from those of its competitor’s and this exact cost is unknown to the competitor, while the competitors marginal cost is known by both parties. On average, the company with non-public marginal cost has the same marginal cost as its competitor and the competitor knows this. In equilibrium, the company with the unpublicized marginal cost produces more, reducing the market price, and gets higher profits when its marginal cost is lower than the average. If its marginal cost is above the average, its produced quantity, the price and its profit move to the opposite directions. Somewhat surprisingly, if the company can without costs and credibly reveal its lower marginal cost to the competitor, it will benefit even more, while a company with above average marginal cost would suffer even more.
In table 1 I have summarized the produced quantities, prices and profits to show that revealing its lower than average cost structure in a Cournot duopoly is indeed beneficial for a company. The intuition is the following. When the low-cost producer (company 2) does not reveal its marginal costs, the competitor (company 1) reacts based on the average marginal cost, producing the standard Cournot-quantity. This is the logical reaction, since on average company 2 has the same marginal cost as company 1. However, company knows its own cost structure and produces more than the standard Cournot-quantity, since this is profitable due to the lower than average marginal cost. Likewise, a high-cost company 2 would produce less than the standard Cournot-quantity, but would not suffer from its competitor’s producing more than its standard Cournot-quantity, since again company 1 is reacting as if company 2 had the average marginal cost.
If company 2 could credibly make its lower than average marginal cost public, company 1 would know that company 2 can and will produce more due to its lower marginal cost, driving the price down. In this case company 1 would react to the actual lower than average marginal cost of company 2, not on company 2’s expected average marginal cost. Consequently, company 1 will produce less to counter the price erosion and maximize its margins in this situation. Also, knowing the reaction of company 1, company will produce more, making also higher profits than in the case of incomplete information. Correspondingly, a high-cost producer will also suffer more if its cost structure becomes public, so it will try to keep this information secret.
Table 1 summarizes the Cournot-quantities, prices and profits in different cases of a Cournot duopoly, with companies 1 and 2, company 2 being the low- / high-cost producer who always knows company 1’s constant marginal cost. In the table I have used the following notation:
- P = a – bQ > 0, where P is the unit price a and b are non-negative constants
- Q = Q1 + Q2, where Q1 and Q2 are the quantities produced
- c is the average marginal cost
- ε is the difference of the low-/high-cost company’s from the average marginal cost
- c + ε > 0
- a low-cost company 2 has ε < 0, a high-cost company 2 has ε > 0
I have also used P’, P’’ and similar expressions for the quantities to separate the cases of standard Cournot duopoly from those of incomplete and complete information with company 2 having lower or higher than average marginal cost.
From table 1 it becomes clear that profits for company 2 increase (decrease) for negative (positive) values of ε, when we move from the standard Cournot duopoly to a duopoly with different marginal costs between companies with incomplete and finally complete information. Thus, a cost advantage is strictly profitable and making it public increases the profit. The profits of company 1 decrease (increase) for negative (positive) values of ε, when moving from the standard Cournot duopoly to a duopoly with different marginal costs between companies with incomplete information. Thus, a cost disadvantage of one company is strictly profitable for the other companies and their profits increase with this cost disadvantage. With some algebra, it can be shown that company 1’s profits in the case of complete information are lower than in the case of incomplete information and that the difference is (ε/6)*(a + c + 2ε), which is negative (positive) for negative (positive) values of ε.
Information cascading and revelation
As argued previously, a low-cost producer in a Cournot duopoly has the incentive to make its cost structure public to maximize its benefits, i.e. profits. More precisely, the producer with the lowest marginal cost has this incentive, since non-disclosure would lead the competition treating the company as an average-cost producer. Therefore, the producer with the lowest marginal cost will reveal its costs. Now, if there are more competitors in the market, the producer with the second lowest marginal cost also has the incentive to reveal its costs, although they are higher than those of the cost leader. If the producer with the second lowest marginal cost did not reveal its costs, it would now be treated as an average producer in the remaining group of companies: the lowest cost producer has now been excluded from the average, since its cost structure is known. But being treated as an average-cost producer is clearly sub-optimal, if a company’s marginal cost is below the average. Thus, the producer with the second lowest marginal cost will also reveal its costs. This logic can be followed right to the last company on the market, to the one with the highest marginal cost.
When one company, the one with the lowest marginal cost, reveals its costs, this leads to information cascading, since the companies with the next lowest costs now want to differentiate from competitors with higher marginal cost, even if they have costs above the average over all companies at the market. This makes the marginal costs of all companies public, or at least their relation to one another. The last company is evidently going to be the one with the highest costs, since it wishes to stay hidden among the masses and has thus not yet revealed its costs. It follows that the last company does not have to reveal its costs; the competition will be able to infer, that the last company has the highest marginal cost and will react accordingly, even if not knowing the exact costs.
The dog didn’t bark – the value of undisclosed information
The previous exercise shows an important, broader real-world application of revealing information. The absence of evidence or the absence of information can be a substantial piece of information. If a company does not want to reveal its cost structure, it is likely due to its high (marginal) costs, at least in the context of the Cournot duopoly model. But not all markets are like the Cournot model, so the implications are not completely generalizable. But still, even in a free market where all companies are price takers, revealing your costs might help you; not in gaining market share, since your produced quantities do not affect the prices, but in keeping competitors from entering price wars. By revealing your costs, you can potentially indicate that you have the largest margin and can thus come out on top, should a competitor start a price war. But by revealing your costs, you may be able to keep your opponents at bay, since they know in advance that their chances of winning a price war are slim.
The difficulty in revealing the costs and gaining the associated advantage is that it is not always evident, which company has the lowest costs. Therefore, companies may restrain from revealing their costs, even if they would benefit from their low costs more when if became public. But without knowing that it is a low-cost producer a company risks becoming disadvantaged, should it in fact be a high-cost producer.
In the case of very similar consumables and bulk goods (e.g. oil, agriculture products) good estimates on the relative costs among competitors may be made and thus own cost advantages can be made public to keep the competition from starting a price war. For highly specialized and small-series products with small markets, revealing a cost advantage would give the power to affect the demand, prices and quantities produced, but finding out who is the low-cost producer might be more difficult. Specialty products often enjoy higher margins, so that product prices are poor indicators of true costs.
In conclusion, revealing information to your rivals may give you an advantage, and not revealing information already conveys information about your situation. Furthermore, even if you are not in the best position in a group of competitors, being able to separate yourself from the even weaker ones may give you an advantage, and to do this you must credibly signal that you have an advantage over some of the competitors. This will then lead to all with a relative advantage to revealing this information.
Signaling is yet another topic in game theory and discusses how different types of players, e.g. good and bad workers, can credibly be identified: an employer has an incentive to get the good and, presumably, more productive good workers, and the good workers have the incentive to reveal themselves in hopes of a higher salary. But the bad workers also have an incentive to be taken for good workers, due to the potential higher salary. Therefore, if the higher workers are to earn more and employers are to pay “correct” wages to the two types of workers, we need a credible signal that the good worker can and will give, but the bad worker cannot or does not want to give. I will return to this topic in my next post.